Subgroups — #542×D₁₃₇ #31794159

Order 17536.

1132 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578027	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578026	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578025	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578024	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578023	C₂	1	2	yes	maximal, complemented
6	#22578022	C₂	1	1	yes	maximal, characteristic, complemented
7	#22578021	C₂	1	2	yes	maximal, complemented
8	#22578020	C₂	1	1	yes	maximal, characteristic, complemented
9	#22578019	C₂	1	2	yes	maximal, complemented
10	#22578018	C₂	1	1	yes	maximal, characteristic, complemented
11	#22578017	C₂	1	2	yes	maximal, complemented
12	C₄⋊Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#126×D₁₃₇	C₂	1	2	yes	maximal, complemented
14	#124×D₁₃₇	C₂	1	2	yes	maximal, complemented
15	#124×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
16	#123×D₁₃₇	C₂	1	2	yes	maximal, complemented
17	C₂²⋊Q₈×D₁₃₇	C₂	1	2	yes	maximal, complemented
18	C₂²⋊Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
19	#121×D₁₃₇	C₂	1	1	yes	maximal, characteristic
20	C₄×Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
21	C₄×D₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
22	#117×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
23	#542×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×Q₈×D₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×D₁₃₇	C₂²	1	2	yes	—
26	C₂×D₄×D₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₁₃₇	C₂²	1	2	yes	—
28	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₁₃₇	C₂²	1	2	yes	—
30	#17984088	C₂²	1	1	yes	characteristic, complemented
31	#17984087	C₂²	1	1	yes	characteristic, complemented
32	#17984087	C₂²	1	2	yes	complemented
33	#17984087	C₂²	1	1	yes	characteristic, complemented
34	#17984086	C₂²	1	2	yes	complemented
35	#17984083	C₂²	1	2	yes	complemented
36	#17984083	C₂²	1	2	yes	complemented
37	#17984083	C₂²	1	2	yes	complemented
38	#17984082	C₂²	1	2	yes	complemented
39	#17984081	C₂²	1	2	yes	complemented
40	#17984081	C₂²	1	2	yes	complemented
41	#17984081	C₂²	1	2	yes	complemented
42	#17984077	C₂²	1	2	yes	complemented
43	#17984077	C₂²	1	1	yes	characteristic, complemented
44	#17984076	C₂²	1	2	yes	complemented
45	#17984076	C₂²	1	1	yes	characteristic, complemented
46	#17984075	C₂²	1	1	yes	characteristic, complemented
47	#17984075	C₂²	1	2	yes	complemented
48	#17984074	C₂²	1	2	yes	complemented
49	#17984074	C₂²	1	2	yes	complemented
50	#17984074	C₂²	1	2	yes	complemented
51	#17984074	C₂²	1	2	yes	complemented
52	#17984074	C₂²	1	2	yes	complemented
53	#17984074	C₂²	1	2	yes	complemented
54	#17984073	C₂²	1	2	yes	complemented
55	#17984073	C₂²	1	2	yes	complemented
56	#17984072	C₂²	1	2	yes	complemented
57	#17984071	C₂²	1	2	yes	complemented
58	#17984070	C₂²	1	2	yes	complemented
59	#17984070	C₂²	1	2	yes	complemented
60	#17984069	C₂²	1	2	yes	complemented
61	#17984068	C₂²	1	1	yes	characteristic, complemented
62	#17984068	C₂²	1	2	yes	complemented
63	#17984067	C₂²	1	2	yes	complemented
64	#17984067	C₂²	1	1	yes	characteristic, complemented
65	#17984066	C₂²	1	2	yes	complemented
66	#17984066	C₂²	1	2	yes	complemented
67	#17984066	C₂²	1	2	yes	complemented
68	#17984065	C₂²	1	2	yes	complemented
69	#17984065	C₂²	1	2	yes	complemented
70	#17984065	C₂²	1	2	yes	complemented
71	#17984064	C₂²	1	2	yes	complemented
72	#17984064	C₂²	1	1	yes	characteristic, complemented
73	#17984063	C₂²	1	1	yes	characteristic
74	#17984062	C₂²	1	1	yes	characteristic
75	#17984061	C₂²	1	2	yes	—
76	#17984060	C₂²	1	2	yes	—
77	#17984059	C₂²	1	1	yes	characteristic
78	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
79	C₄×Dic₂₇₄	C₂²	1	2	yes	complemented
80	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic, complemented
81	#17984053	C₂²	1	1	yes	characteristic, complemented
82	#17984053	C₂²	1	2	yes	complemented
83	C₄×D₅₄₈	C₂²	1	1	yes	characteristic, complemented
84	D₄×Dic₁₃₇	C₂²	1	1	yes	characteristic, complemented
85	#17984050	C₂²	1	1	yes	characteristic, complemented
86	C₄×C₂²⋊D₁₃₇	C₂²	1	2	yes	complemented
87	#17984048	C₂²	1	2	yes	complemented
88	#17984047	C₂²	1	1	yes	characteristic, complemented
89	#17984046	C₂²	1	1	yes	characteristic, complemented
90	#17984046	C₂²	1	1	yes	characteristic, complemented
91	#17984045	C₂²	1	1	yes	characteristic, complemented
92	#17984045	C₂²	1	1	yes	characteristic, complemented
93	#17984044	C₂²	1	1	yes	characteristic, complemented
94	#17984044	C₂²	1	1	yes	characteristic, complemented
95	C₄⋊C₄×D₁₃₇	C₂²	1	1	yes	characteristic
96	C₄⋊C₄×D₁₃₇	C₂²	1	2	yes	—
97	C₄⋊C₄×D₁₃₇	C₂²	1	2	yes	—
98	C₄⋊C₄×D₁₃₇	C₂²	1	1	yes	characteristic
99	C₄⋊C₄×D₁₃₇	C₂²	1	1	yes	characteristic
100	C₄⋊C₄×D₁₃₇	C₂²	1	1	yes	characteristic
101	C₄⋊C₄×D₁₃₇	C₂²	1	2	yes	—
102	C₂²⋊C₄×D₁₃₇	C₂²	1	2	yes	—
103	C₂²⋊C₄×D₁₃₇	C₂²	1	2	yes	—
104	C₂²⋊C₄×D₁₃₇	C₂²	1	2	yes	—
105	C₂²⋊C₄×D₁₃₇	C₂²	1	1	yes	characteristic
106	C₂²⋊C₄×D₁₃₇	C₂²	1	1	yes	characteristic
107	C₂²⋊C₄×D₁₃₇	C₂²	1	2	yes	—
108	C₂²⋊C₄×D₁₃₇	C₂²	1	2	yes	—
109	C₄²×D₁₃₇	C₂²	1	2	yes	—
110	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
111	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
112	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
113	#126×C₁₃₇	C₂²	1	2	yes	complemented
114	#124×C₁₃₇	C₂²	1	2	yes	complemented
115	#124×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
116	#123×C₁₃₇	C₂²	1	2	yes	complemented
117	C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	complemented
118	C₂²⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
119	#121×C₁₃₇	C₂²	1	1	yes	characteristic
120	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
121	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
122	#117×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₁₃₇	C₂³	1	1	yes	characteristic
124	C₂³×D₁₃₇	C₂³	1	2	yes	—
125	C₂×Dic₂₇₄	C₂³	1	2	yes	—
126	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
127	C₂×Dic₂₇₄	C₂³	1	2	yes	—
128	C₂×Dic₂₇₄	C₂³	1	2	yes	—
129	C₂×Dic₂₇₄	C₂³	1	2	yes	—
130	Q₈×D₁₃₇	—	2	4	no	—
131	Q₈×D₁₃₇	—	4	4	no	—
132	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
133	C₂×D₅₄₈	C₂³	1	2	yes	—
134	C₂×C₂²⋊D₁₃₇	C₂³	1	2	yes	—
135	C₂×C₂²⋊D₁₃₇	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₁₃₇	C₂³	1	2	yes	—
137	D₄×D₁₃₇	—	2	4	no	—
138	D₄×D₁₃₇	—	4	4	no	—
139	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
140	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
141	C₂×C₄×D₁₃₇	—	2	4	no	—
142	C₂×C₄×D₁₃₇	—	4	4	no	—
143	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
144	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
145	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
146	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
147	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
148	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
149	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
150	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
151	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
152	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
153	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
154	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
155	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
156	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
157	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
158	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
159	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
160	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
161	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
162	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
163	#14555489	C₂³	1	1	yes	characteristic
164	#14555489	C₂³	1	1	yes	characteristic
165	#14555489	C₂³	1	1	yes	characteristic
166	#14555489	C₂³	1	1	yes	characteristic
167	#14555489	C₂³	1	2	yes	—
168	#14555489	C₂³	1	2	yes	—
169	#14555489	C₂³	1	2	yes	—
170	#14555489	C₂³	1	2	yes	—
171	#14555489	C₂³	1	2	yes	—
172	#14555489	C₂³	1	2	yes	—
173	#14555489	C₂³	1	2	yes	—
174	#14555489	C₂³	1	2	yes	—
175	C₂²⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
176	C₂²⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
177	C₂²⋊Dic₁₃₇	C₂³	1	2	yes	—
178	C₂²⋊Dic₁₃₇	C₂³	1	2	yes	—
179	C₂²⋊Dic₁₃₇	C₂³	1	2	yes	—
180	C₂²⋊Dic₁₃₇	C₂³	1	2	yes	—
181	C₂²⋊Dic₁₃₇	C₂³	1	2	yes	—
182	#14555487	C₂³	1	1	yes	characteristic
183	#14555487	C₂³	1	1	yes	characteristic
184	#14555487	C₂³	1	1	yes	characteristic
185	#14555487	C₂³	1	1	yes	characteristic
186	#14555487	C₂³	1	2	yes	—
187	#14555487	C₂³	1	2	yes	—
188	#14555487	C₂³	1	2	yes	—
189	#14555487	C₂³	1	2	yes	—
190	#14555487	C₂³	1	2	yes	—
191	#14555487	C₂³	1	2	yes	—
192	#14555487	C₂³	1	2	yes	—
193	#14555487	C₂³	1	2	yes	—
194	#14555487	C₂³	1	2	yes	—
195	#14555487	C₂³	1	2	yes	—
196	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
197	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
198	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
199	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
200	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
201	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
202	C₄×Dic₁₃₇	C₂³	1	2	yes	—
203	C₄×Dic₁₃₇	C₂³	1	2	yes	—
204	C₄×Dic₁₃₇	C₂³	1	2	yes	—
205	C₂×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
206	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
207	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
208	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
209	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
210	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
211	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
212	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
213	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
214	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
215	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
216	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
217	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
218	C₂²⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
219	C₂²⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
220	C₂²⋊C₄×C₁₃₇	C₂³	1	2	yes	—
221	C₂²⋊C₄×C₁₃₇	C₂³	1	2	yes	—
222	C₂²⋊C₄×C₁₃₇	C₂³	1	2	yes	—
223	C₂²⋊C₄×C₁₃₇	C₂³	1	2	yes	—
224	C₂²⋊C₄×C₁₃₇	C₂³	1	2	yes	—
225	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
226	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
227	C₄²×C₁₃₇	C₂³	1	2	yes	—
228	C₂²×D₁₃₇	—	2	2	no	—
229	C₂²×D₁₃₇	—	2	4	no	—
230	C₂²×D₁₃₇	—	2	4	no	—
231	C₂²×D₁₃₇	—	2	2	no	—
232	C₂²×D₁₃₇	—	2	2	no	—
233	C₂²×D₁₃₇	—	2	4	no	—
234	C₂²×D₁₃₇	C₂⁴	1	1	yes	characteristic
235	C₂²×D₁₃₇	C₂⁴	1	2	yes	—
236	C₂²×D₁₃₇	C₂⁴	1	1	yes	characteristic
237	Dic₂₇₄	—	2	2	no	—
238	Dic₂₇₄	—	2	4	no	—
239	Dic₂₇₄	—	4	2	no	—
240	Dic₂₇₄	—	4	2	no	—
241	Dic₂₇₄	—	4	2	no	—
242	D₅₄₈	—	2	2	no	—
243	D₅₄₈	—	4	2	no	—
244	C₂²⋊D₁₃₇	—	2	4	no	—
245	C₂²⋊D₁₃₇	—	4	2	no	—
246	C₂²⋊D₁₃₇	—	4	2	no	—
247	C₂×Dic₁₃₇	—	2	2	no	—
248	C₂×Dic₁₃₇	—	4	2	no	—
249	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
250	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
251	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
252	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
253	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
254	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
255	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
256	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
257	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
258	C₄×D₁₃₇	—	2	2	no	—
259	C₄×D₁₃₇	—	2	4	no	—
260	C₄×D₁₃₇	—	2	2	no	—
261	C₄×D₁₃₇	—	2	4	no	—
262	C₄×D₁₃₇	—	2	4	no	—
263	C₄×D₁₃₇	—	2	2	no	—
264	C₄×D₁₃₇	—	2	2	no	—
265	C₄×D₁₃₇	—	2	2	no	—
266	C₄×D₁₃₇	—	2	2	no	—
267	C₄×D₁₃₇	—	2	2	no	—
268	C₄×D₁₃₇	—	4	2	no	—
269	C₄×D₁₃₇	—	4	2	no	—
270	C₄×D₁₃₇	#41	1	4	yes	—
271	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
272	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
273	Q₈×C₁₃₇	—	2	2	no	—
274	Q₈×C₁₃₇	—	4	2	no	—
275	D₄×C₁₃₇	—	2	2	no	—
276	D₄×C₁₃₇	—	4	2	no	—
277	C₂×C₄×C₁₃₇	—	2	2	no	—
278	C₂×C₄×C₁₃₇	—	4	2	no	—
279	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
280	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
281	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
282	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
283	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
284	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
285	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
286	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
287	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
288	C₂²×C₁₃₇	—	2	1	no	—
289	C₂²×C₁₃₇	—	2	2	no	—
290	C₂²×C₁₃₇	—	2	2	no	—
291	C₂²×C₁₃₇	—	2	1	no	—
292	C₂²×C₁₃₇	—	2	1	no	—
293	C₂²×C₁₃₇	—	2	2	no	—
294	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
295	C₂×D₁₃₇	—	2	1	no	—
296	C₂×D₁₃₇	—	2	2	no	—
297	C₂×D₁₃₇	—	2	2	no	—
298	C₂×D₁₃₇	—	2	1	no	—
299	C₂×D₁₃₇	—	2	1	no	—
300	C₂×D₁₃₇	—	2	2	no	—
301	C₂×D₁₃₇	—	4	4	no	—
302	C₂×D₁₃₇	—	4	8	no	—
303	C₂×D₁₃₇	C₂×#41	1	2	yes	—
304	C₂×D₁₃₇	H(5,𝔽₂)	1	2	yes	—
305	C₂×D₁₃₇	2₋¹⁺⁴	1	2	yes	—
306	C₄×C₁₃₇	—	2	1	no	—
307	C₄×C₁₃₇	—	2	2	no	—
308	C₄×C₁₃₇	—	2	1	no	—
309	C₄×C₁₃₇	—	2	2	no	—
310	C₄×C₁₃₇	—	2	2	no	—
311	C₄×C₁₃₇	—	2	1	no	—
312	C₄×C₁₃₇	—	2	1	no	—
313	C₄×C₁₃₇	—	2	1	no	—
314	C₄×C₁₃₇	C₂×#41	1	2	yes	—
315	Dic₁₃₇	—	2	1	no	—
316	Dic₁₃₇	—	2	2	no	—
317	Dic₁₃₇	—	2	1	no	—
318	Dic₁₃₇	—	2	2	no	—
319	Dic₁₃₇	—	2	2	no	—
320	Dic₁₃₇	—	2	1	no	—
321	Dic₁₃₇	—	2	1	no	—
322	Dic₁₃₇	—	2	1	no	—
323	Dic₁₃₇	C₂×#41	1	2	yes	—
324	C₂×C₁₃₇	—	4	1	no	—
325	C₂×C₁₃₇	—	4	2	no	—
326	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
327	C₂×C₁₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
328	C₂×C₁₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
329	D₁₃₇	—	4	1	no	—
330	D₁₃₇	—	4	2	no	—
331	D₁₃₇	#542	1	4	yes	direct product factor
332	C₁₃₇	C₂×#542	1	1	yes	characteristic, complemented, Sylow
333	C₂×#542	—	137	1	no	maximal, Sylow
334	#542	—	137	1	no	—
335	#542	—	137	2	no	—
336	#542	—	137	2	no	—
337	#542	—	137	1	no	—
338	#542	—	137	2	no	—
339	#542	—	137	2	no	—
340	#542	—	137	1	no	—
341	#542	—	137	1	no	—
342	#542	—	137	1	no	—
343	#542	—	137	1	no	—
344	#542	—	137	1	no	—
345	#542	D₁₃₇	1	1	yes	characteristic, direct product factor
346	C₂×C₄⋊Q₈	—	137	1	no	—
347	C₂×#126	—	137	2	no	—
348	C₂×#124	—	137	1	no	—
349	C₂×#124	—	137	2	no	—
350	C₂×#123	—	137	2	no	—
351	C₂×C₂²⋊Q₈	—	137	2	no	—
352	C₂×C₂²⋊Q₈	—	137	1	no	—
353	C₂×#121	—	137	1	no	—
354	C₂×C₄×Q₈	—	137	1	no	—
355	C₂×C₄×D₄	—	137	1	no	—
356	C₂×#117	—	137	1	no	—
357	C₂²×Q₈	—	137	1	no	—
358	C₂²×Q₈	—	137	2	no	—
359	C₂²×D₄	—	137	1	no	—
360	C₂²×D₄	—	137	2	no	—
361	C₂³×C₄	—	137	1	no	—
362	C₂³×C₄	—	137	2	no	—
363	C₄⋊Q₈	—	137	2	no	—
364	C₄⋊Q₈	—	137	2	no	—
365	C₄⋊Q₈	—	137	1	no	—
366	C₄⋊Q₈	—	137	1	no	—
367	C₄⋊Q₈	—	137	1	no	—
368	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
369	#126	—	137	2	no	—
370	#126	—	137	2	no	—
371	#126	—	137	2	no	—
372	#126	—	137	2	no	—
373	#126	—	137	2	no	—
374	#126	—	137	2	no	—
375	#126	—	137	2	no	—
376	#126	C₂×D₁₃₇	1	2	yes	complemented
377	#124	—	137	1	no	—
378	#124	—	137	2	no	—
379	#124	—	137	1	no	—
380	#124	—	137	2	no	—
381	#124	—	137	2	no	—
382	#124	—	137	2	no	—
383	#124	—	137	2	no	—
384	#124	—	137	2	no	—
385	#124	—	137	2	no	—
386	#124	—	137	2	no	—
387	#124	—	137	1	no	—
388	#124	—	137	2	no	—
389	#124	C₂×D₁₃₇	1	1	yes	characteristic, complemented
390	#124	C₂×D₁₃₇	1	2	yes	complemented
391	#123	—	137	2	no	—
392	#123	—	137	2	no	—
393	#123	—	137	2	no	—
394	#123	—	137	2	no	—
395	#123	—	137	2	no	—
396	#123	—	137	2	no	—
397	#123	—	137	2	no	—
398	#123	C₂×D₁₃₇	1	2	yes	complemented
399	C₂²⋊Q₈	—	137	2	no	—
400	C₂²⋊Q₈	—	137	1	no	—
401	C₂²⋊Q₈	—	137	2	no	—
402	C₂²⋊Q₈	—	137	1	no	—
403	C₂²⋊Q₈	—	137	2	no	—
404	C₂²⋊Q₈	—	137	2	no	—
405	C₂²⋊Q₈	—	137	2	no	—
406	C₂²⋊Q₈	—	137	2	no	—
407	C₂²⋊Q₈	—	137	1	no	—
408	C₂²⋊Q₈	—	137	2	no	—
409	C₂²⋊Q₈	—	137	2	no	—
410	C₂²⋊Q₈	—	137	2	no	—
411	C₂²⋊Q₈	C₂×D₁₃₇	1	2	yes	complemented
412	C₂²⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
413	#121	—	137	1	no	—
414	#121	—	137	2	no	—
415	#121	—	137	1	no	—
416	#121	—	137	1	no	—
417	#121	—	137	2	no	—
418	#121	C₂×D₁₃₇	1	1	yes	characteristic
419	C₄×Q₈	—	137	1	no	—
420	C₄×Q₈	—	137	2	no	—
421	C₄×Q₈	—	137	1	no	—
422	C₄×Q₈	—	137	2	no	—
423	C₄×Q₈	—	137	1	no	—
424	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic, complemented
425	C₄×D₄	—	137	1	no	—
426	C₄×D₄	—	137	2	no	—
427	C₄×D₄	—	137	1	no	—
428	C₄×D₄	—	137	2	no	—
429	C₄×D₄	—	137	1	no	—
430	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic, complemented
431	#117	—	137	1	no	—
432	#117	—	137	1	no	—
433	#117	—	137	1	no	—
434	#117	—	137	1	no	—
435	#117	—	137	1	no	—
436	#117	—	137	1	no	—
437	#117	—	137	1	no	—
438	#117	C₂×D₁₃₇	1	1	yes	characteristic, complemented
439	C₂×C₄⋊C₄	—	137	1	no	—
440	C₂×C₄⋊C₄	—	137	1	no	—
441	C₂×C₄⋊C₄	—	137	1	no	—
442	C₂×C₄⋊C₄	—	137	1	no	—
443	C₂×C₄⋊C₄	—	137	2	no	—
444	C₂×C₄⋊C₄	—	137	2	no	—
445	C₂×C₄⋊C₄	—	137	2	no	—
446	C₂×C₂²⋊C₄	—	137	1	no	—
447	C₂×C₂²⋊C₄	—	137	1	no	—
448	C₂×C₂²⋊C₄	—	137	2	no	—
449	C₂×C₂²⋊C₄	—	137	2	no	—
450	C₂×C₂²⋊C₄	—	137	2	no	—
451	C₂×C₂²⋊C₄	—	137	2	no	—
452	C₂×C₂²⋊C₄	—	137	2	no	—
453	C₂×C₄²	—	137	1	no	—
454	C₂×C₄²	—	137	1	no	—
455	C₂×C₄²	—	137	2	no	—
456	C₂⁴	—	137	1	no	—
457	C₂⁴	—	137	2	no	—
458	C₂×Q₈	—	274	4	no	—
459	C₂×Q₈	—	548	4	no	—
460	C₂×Q₈	—	137	1	no	—
461	C₂×Q₈	—	137	2	no	—
462	C₂×Q₈	—	137	2	no	—
463	C₂×Q₈	—	137	2	no	—
464	C₂×Q₈	—	137	2	no	—
465	C₂×Q₈	C₂²×D₁₃₇	1	1	yes	characteristic
466	C₂×Q₈	C₂²×D₁₃₇	1	2	yes	—
467	C₂×D₄	—	274	4	no	—
468	C₂×D₄	—	548	4	no	—
469	C₂×D₄	—	137	2	no	—
470	C₂×D₄	—	137	2	no	—
471	C₂×D₄	—	137	2	no	—
472	C₂×D₄	—	137	1	no	—
473	C₂×D₄	—	137	2	no	—
474	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
475	C₂×D₄	C₂²×D₁₃₇	1	2	yes	—
476	C₂²×C₄	—	274	4	no	—
477	C₂²×C₄	—	548	4	no	—
478	C₂²×C₄	—	137	1	no	—
479	C₂²×C₄	—	137	1	no	—
480	C₂²×C₄	—	137	1	no	—
481	C₂²×C₄	—	137	2	no	—
482	C₂²×C₄	—	137	1	no	—
483	C₂²×C₄	—	137	1	no	—
484	C₂²×C₄	—	137	2	no	—
485	C₂²×C₄	—	137	1	no	—
486	C₂²×C₄	—	137	2	no	—
487	C₂²×C₄	—	137	1	no	—
488	C₂²×C₄	—	137	1	no	—
489	C₂²×C₄	—	137	2	no	—
490	C₂²×C₄	—	137	2	no	—
491	C₂²×C₄	—	137	2	no	—
492	C₂²×C₄	—	137	1	no	—
493	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
494	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
495	C₄⋊C₄	—	137	1	no	—
496	C₄⋊C₄	—	137	1	no	—
497	C₄⋊C₄	—	137	1	no	—
498	C₄⋊C₄	—	137	1	no	—
499	C₄⋊C₄	—	137	2	no	—
500	C₄⋊C₄	—	137	2	no	—
501	C₄⋊C₄	—	137	2	no	—
502	C₄⋊C₄	—	137	2	no	—
503	C₄⋊C₄	—	137	2	no	—
504	C₄⋊C₄	—	137	2	no	—
505	C₄⋊C₄	—	137	2	no	—
506	C₄⋊C₄	—	137	2	no	—
507	C₄⋊C₄	—	137	1	no	—
508	C₄⋊C₄	—	137	1	no	—
509	C₄⋊C₄	—	137	1	no	—
510	C₄⋊C₄	—	137	1	no	—
511	C₄⋊C₄	—	137	2	no	—
512	C₄⋊C₄	—	137	2	no	—
513	C₄⋊C₄	—	137	2	no	—
514	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
516	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
517	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
518	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
519	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
520	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
521	C₂²⋊C₄	—	137	1	no	—
522	C₂²⋊C₄	—	137	1	no	—
523	C₂²⋊C₄	—	137	2	no	—
524	C₂²⋊C₄	—	137	2	no	—
525	C₂²⋊C₄	—	137	2	no	—
526	C₂²⋊C₄	—	137	2	no	—
527	C₂²⋊C₄	—	137	2	no	—
528	C₂²⋊C₄	—	137	1	no	—
529	C₂²⋊C₄	—	137	1	no	—
530	C₂²⋊C₄	—	137	1	no	—
531	C₂²⋊C₄	—	137	1	no	—
532	C₂²⋊C₄	—	137	2	no	—
533	C₂²⋊C₄	—	137	2	no	—
534	C₂²⋊C₄	—	137	2	no	—
535	C₂²⋊C₄	—	137	2	no	—
536	C₂²⋊C₄	—	137	2	no	—
537	C₂²⋊C₄	—	137	2	no	—
538	C₂²⋊C₄	—	137	2	no	—
539	C₂²⋊C₄	—	137	2	no	—
540	C₂²⋊C₄	—	137	2	no	—
541	C₂²⋊C₄	—	137	2	no	—
542	C₂²⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
543	C₂²⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
544	C₂²⋊C₄	C₂²×D₁₃₇	1	2	yes	—
545	C₂²⋊C₄	C₂²×D₁₃₇	1	2	yes	—
546	C₂²⋊C₄	C₂²×D₁₃₇	1	2	yes	—
547	C₂²⋊C₄	C₂²×D₁₃₇	1	2	yes	—
548	C₂²⋊C₄	C₂²×D₁₃₇	1	2	yes	—
549	C₄²	—	137	1	no	—
550	C₄²	—	137	1	no	—
551	C₄²	—	137	1	no	—
552	C₄²	—	137	1	no	—
553	C₄²	—	137	1	no	—
554	C₄²	—	137	1	no	—
555	C₄²	—	137	2	no	—
556	C₄²	—	137	2	no	—
557	C₄²	—	137	2	no	—
558	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
559	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
560	C₄²	C₂²×D₁₃₇	1	2	yes	—
561	C₂³	—	274	2	no	—
562	C₂³	—	274	4	no	—
563	C₂³	—	274	4	no	—
564	C₂³	—	274	2	no	—
565	C₂³	—	274	2	no	—
566	C₂³	—	274	4	no	—
567	C₂³	—	137	1	no	—
568	C₂³	—	137	2	no	—
569	C₂³	—	137	1	no	—
570	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
571	C₂³	C₂³×D₁₃₇	1	2	yes	—
572	Q₈	—	2	2	no	—
573	Q₈	—	4	2	no	—
574	Q₈	—	274	4	no	—
575	Q₈	—	274	2	no	—
576	Q₈	—	548	2	no	—
577	Q₈	—	548	2	no	—
578	Q₈	—	548	2	no	—
579	D₄	—	2	2	no	—
580	D₄	—	4	2	no	—
581	D₄	—	274	4	no	—
582	D₄	—	274	2	no	—
583	D₄	—	548	2	no	—
584	D₄	—	548	2	no	—
585	D₄	—	548	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	274	2	no	—
589	C₂×C₄	—	274	2	no	—
590	C₂×C₄	—	274	4	no	—
591	C₂×C₄	—	274	4	no	—
592	C₂×C₄	—	274	4	no	—
593	C₂×C₄	—	274	2	no	—
594	C₂×C₄	—	274	2	no	—
595	C₂×C₄	—	274	2	no	—
596	C₂×C₄	—	274	2	no	—
597	C₂×C₄	—	274	2	no	—
598	C₂×C₄	—	274	2	no	—
599	C₂×C₄	—	548	2	no	—
600	C₂×C₄	—	548	2	no	—
601	C₂×C₄	—	548	2	no	—
602	C₂×C₄	—	137	1	no	—
603	C₂×C₄	—	137	1	no	—
604	C₂×C₄	—	137	2	no	—
605	C₂×C₄	—	137	1	no	—
606	C₂×C₄	—	137	1	no	—
607	C₂×C₄	—	137	2	no	—
608	C₂×C₄	—	137	1	no	—
609	C₂×C₄	—	137	2	no	—
610	C₂×C₄	—	137	1	no	—
611	C₂×C₄	—	137	4	no	—
612	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
613	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
614	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
615	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
616	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
617	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
618	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
619	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
620	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
621	C₂²	—	2	1	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	274	1	no	—
628	C₂²	—	274	2	no	—
629	C₂²	—	274	2	no	—
630	C₂²	—	274	1	no	—
631	C₂²	—	274	1	no	—
632	C₂²	—	274	2	no	—
633	C₂²	—	548	4	no	—
634	C₂²	—	548	8	no	—
635	C₂²	—	137	2	no	—
636	C₂²	—	137	2	no	—
637	C₂²	—	137	2	no	—
638	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
639	C₄	—	2	1	no	—
640	C₄	—	2	2	no	—
641	C₄	—	2	1	no	—
642	C₄	—	2	2	no	—
643	C₄	—	2	2	no	—
644	C₄	—	2	1	no	—
645	C₄	—	2	1	no	—
646	C₄	—	2	1	no	—
647	C₄	—	274	1	no	—
648	C₄	—	274	2	no	—
649	C₄	—	274	1	no	—
650	C₄	—	274	2	no	—
651	C₄	—	274	2	no	—
652	C₄	—	274	1	no	—
653	C₄	—	274	1	no	—
654	C₄	—	274	1	no	—
655	C₄	—	137	2	no	—
656	C₄	#41×D₁₃₇	1	2	yes	—
657	C₂	—	4	1	no	—
658	C₂	—	4	2	no	—
659	C₂	—	548	1	no	—
660	C₂	—	548	2	no	—
661	C₂	—	137	4	no	—
662	C₂	C₂×#41×D₁₃₇	1	1	yes	characteristic
663	C₂	H(5,𝔽₂)×D₁₃₇	1	1	yes	characteristic
664	C₂	2₋¹⁺⁴×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₇
commutator, socle, cosocle	C₂²×C₁₃₇	C₂⁵
Fitting	#542×C₁₃₇	C₂
prime radical	#121×D₁₃₇	C₂
hypercenter	#542	D₁₃₇
hypocenter	C₁₃₇	C₂×#542
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #542

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22578027	C₂	semidirect
2. #22578026	C₂	semidirect
3. #22578025	C₂	semidirect
4. #22578024	C₂	semidirect
5. #22578023	C₂	semidirect
6. #22578022	C₂	semidirect
7. #22578021	C₂	semidirect
8. #22578020	C₂	semidirect
9. #22578019	C₂	semidirect
10. #22578018	C₂	semidirect
11. #22578017	C₂	semidirect
12. C₄⋊Q₈×D₁₃₇	C₂	semidirect
13. #126×D₁₃₇	C₂	semidirect
14. #124×D₁₃₇	C₂	semidirect
15. #124×D₁₃₇	C₂	semidirect
16. #123×D₁₃₇	C₂	semidirect
17. C₂²⋊Q₈×D₁₃₇	C₂	semidirect
18. C₂²⋊Q₈×D₁₃₇	C₂	semidirect
20. C₄×Q₈×D₁₃₇	C₂	semidirect
21. C₄×D₄×D₁₃₇	C₂	semidirect
22. #117×D₁₃₇	C₂	semidirect
23. #542×C₁₃₇	C₂	semidirect
30. #17984088	C₂²	semidirect
31. #17984087	C₂²	semidirect
32. #17984087	C₂²	semidirect
33. #17984087	C₂²	semidirect
34. #17984086	C₂²	semidirect
35. #17984083	C₂²	semidirect
36. #17984083	C₂²	semidirect
37. #17984083	C₂²	semidirect
38. #17984082	C₂²	semidirect
39. #17984081	C₂²	semidirect
40. #17984081	C₂²	semidirect
41. #17984081	C₂²	semidirect
42. #17984077	C₂²	semidirect
43. #17984077	C₂²	semidirect
44. #17984076	C₂²	semidirect
45. #17984076	C₂²	semidirect
46. #17984075	C₂²	semidirect
47. #17984075	C₂²	semidirect
48. #17984074	C₂²	semidirect
49. #17984074	C₂²	semidirect
50. #17984074	C₂²	semidirect
51. #17984074	C₂²	semidirect
52. #17984074	C₂²	semidirect
53. #17984074	C₂²	semidirect
54. #17984073	C₂²	semidirect
55. #17984073	C₂²	semidirect
56. #17984072	C₂²	semidirect
57. #17984071	C₂²	semidirect
58. #17984070	C₂²	semidirect
59. #17984070	C₂²	semidirect
60. #17984069	C₂²	semidirect
61. #17984068	C₂²	semidirect
62. #17984068	C₂²	semidirect
63. #17984067	C₂²	semidirect
64. #17984067	C₂²	semidirect
65. #17984066	C₂²	semidirect
66. #17984066	C₂²	semidirect
67. #17984066	C₂²	semidirect
68. #17984065	C₂²	semidirect
69. #17984065	C₂²	semidirect
70. #17984065	C₂²	semidirect
71. #17984064	C₂²	semidirect
72. #17984064	C₂²	semidirect
78. Q₈×Dic₁₃₇	C₂²	semidirect
79. C₄×Dic₂₇₄	C₂²	semidirect
80. C₄×Dic₂₇₄	C₂²	semidirect
81. #17984053	C₂²	semidirect
82. #17984053	C₂²	semidirect
83. C₄×D₅₄₈	C₂²	semidirect
84. D₄×Dic₁₃₇	C₂²	semidirect
85. #17984050	C₂²	semidirect
86. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
87. #17984048	C₂²	semidirect
88. #17984047	C₂²	semidirect
89. #17984046	C₂²	semidirect
90. #17984046	C₂²	semidirect
91. #17984045	C₂²	semidirect
92. #17984045	C₂²	semidirect
93. #17984044	C₂²	semidirect
94. #17984044	C₂²	semidirect
112. C₄⋊Q₈×C₁₃₇	C₂²	semidirect
113. #126×C₁₃₇	C₂²	semidirect
114. #124×C₁₃₇	C₂²	semidirect
115. #124×C₁₃₇	C₂²	semidirect
116. #123×C₁₃₇	C₂²	semidirect
117. C₂²⋊Q₈×C₁₃₇	C₂²	semidirect
118. C₂²⋊Q₈×C₁₃₇	C₂²	semidirect
120. C₄×Q₈×C₁₃₇	C₂²	semidirect
121. C₄×D₄×C₁₃₇	C₂²	semidirect
122. #117×C₁₃₇	C₂²	semidirect
331. D₁₃₇	#542	direct (N × H)
332. C₁₃₇	C₂×#542	semidirect
345. #542	D₁₃₇	direct (N × H)
368. C₄⋊Q₈	C₂×D₁₃₇	semidirect
376. #126	C₂×D₁₃₇	semidirect
389. #124	C₂×D₁₃₇	semidirect
390. #124	C₂×D₁₃₇	semidirect
398. #123	C₂×D₁₃₇	semidirect
411. C₂²⋊Q₈	C₂×D₁₃₇	semidirect
412. C₂²⋊Q₈	C₂×D₁₃₇	semidirect
424. C₄×Q₈	C₂×D₁₃₇	semidirect
430. C₄×D₄	C₂×D₁₃₇	semidirect
438. #117	C₂×D₁₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22578027	2
#22578026	2
#22578025	2
#22578024	2
#22578023	2
#22578022	2
#22578021	2
#22578020	2
#22578019	2
#22578018	2
#22578017	2
C₄⋊Q₈×D₁₃₇	2
#126×D₁₃₇	2
#124×D₁₃₇	2
#123×D₁₃₇	2
C₂²⋊Q₈×D₁₃₇	2
#121×D₁₃₇	2
C₄×Q₈×D₁₃₇	2
C₄×D₄×D₁₃₇	2
#117×D₁₃₇	2
#542×C₁₃₇	2
C₂×#542	137

Subgroups — #542×D137 #31794159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #542×D₁₃₇ #31794159