Subgroups — D₄×#14555510 #31794277

Order 17536.

1648 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555510	C₂	1	2	yes	maximal, complemented
2	#22578074	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578062	C₂	1	2	yes	maximal, complemented
5	#22578058	C₂	1	1	yes	maximal, characteristic, complemented
6	#22578054	C₂	1	2	yes	maximal, complemented
7	#22578049	C₂	1	2	yes	maximal, complemented
8	#22578043	C₂	1	4	yes	maximal, complemented
9	D₄×D₅₄₈	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₂²⋊D₁₃₇	C₂	1	2	yes	maximal, complemented
11	#22577953	C₂	1	1	yes	maximal, characteristic, complemented
12	#22577950	C₂	1	2	yes	maximal, complemented
13	#22577914	C₂	1	2	yes	maximal, complemented
14	#22577912	C₂	1	4	yes	maximal, complemented
15	C₄×#14555510	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×D₄×D₁₃₇	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#14555510	—	2	16	no	—
19	C₂×#14555510	C₂²	1	1	yes	characteristic
20	C₂×#14555510	C₂²	1	4	yes	complemented
21	C₂×#14555510	C₂²	1	2	yes	complemented
22	C₂×#14555510	C₂²	1	4	yes	complemented
23	C₂×#14555510	C₂²	1	2	yes	complemented
24	C₂×#14555510	C₂²	1	2	yes	complemented
25	C₂²×Dic₂₇₄	C₂²	1	2	yes	complemented
26	C₂²×D₅₄₈	C₂²	1	2	yes	complemented
27	C₂²×C₂²⋊D₁₃₇	C₂²	1	4	yes	complemented
28	C₂×D₄×D₁₃₇	C₂²	1	2	yes	—
29	C₂²×C₄×D₁₃₇	C₂²	1	4	yes	complemented
30	#17984088	C₂²	1	1	yes	characteristic, complemented
31	#17984087	C₂²	1	2	yes	complemented
32	#17984085	C₂²	1	1	yes	characteristic, complemented
33	#17984084	C₂²	1	2	yes	complemented
34	#17984076	C₂²	1	2	yes	complemented
35	#17984074	C₂²	1	4	yes	complemented
36	#17984073	C₂²	1	4	yes	complemented
37	#17984071	C₂²	1	4	yes	complemented
38	#17984070	C₂²	1	4	yes	complemented
39	#17984067	C₂²	1	2	yes	complemented
40	#17984067	C₂²	1	2	yes	complemented
41	#17984066	C₂²	1	4	yes	complemented
42	#17984065	C₂²	1	4	yes	complemented
43	#17984063	C₂²	1	2	yes	complemented
44	#17984062	C₂²	1	2	yes	complemented
45	#17984062	C₂²	1	2	yes	complemented
46	#17984062	C₂²	1	2	yes	complemented
47	#17984062	C₂²	1	2	yes	complemented
48	#17984061	C₂²	1	4	yes	complemented
49	#17984061	C₂²	1	4	yes	complemented
50	#17984060	C₂²	1	4	yes	complemented
51	#17984059	C₂²	1	2	yes	complemented
52	#17984058	C₂²	1	4	yes	complemented
53	#17984057	C₂²	1	4	yes	complemented
54	#17984056	C₂²	1	4	yes	complemented
55	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic, complemented
56	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic, complemented
57	C₄×D₅₄₈	C₂²	1	2	yes	complemented
58	C₄×D₅₄₈	C₂²	1	2	yes	complemented
59	C₄×D₅₄₈	C₂²	1	1	yes	characteristic, complemented
60	C₄×D₅₄₈	C₂²	1	1	yes	characteristic, complemented
61	D₄×Dic₁₃₇	C₂²	1	2	yes	—
62	#17984050	C₂²	1	2	yes	complemented
63	C₄×C₂²⋊D₁₃₇	C₂²	1	4	yes	complemented
64	C₄×C₂²⋊D₁₃₇	C₂²	1	2	yes	complemented
65	C₄×C₂²⋊D₁₃₇	C₂²	1	2	yes	complemented
66	C₄×C₂²⋊D₁₃₇	C₂²	1	4	yes	complemented
67	#17984048	C₂²	1	4	yes	complemented
68	#17984047	C₂²	1	1	yes	characteristic, complemented
69	#17984045	C₂²	1	2	yes	complemented
70	C₄⋊C₄×D₁₃₇	C₂²	1	2	yes	complemented
71	C₂²⋊C₄×D₁₃₇	C₂²	1	4	yes	complemented
72	C₄²×D₁₃₇	C₂²	1	2	yes	complemented
73	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
74	C₂³×C₄×C₁₃₇	C₂²	1	2	yes	complemented
75	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
76	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
77	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
78	C₄×D₄×C₁₃₇	C₂²	1	2	yes	complemented
79	C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
80	C₄×D₄×C₁₃₇	C₂²	1	2	yes	complemented
81	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
82	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	complemented
83	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
84	C₂³×D₁₃₇	C₂³	1	4	yes	—
85	#14555510	—	2	8	no	—
86	#14555510	—	2	8	no	—
87	#14555510	—	2	16	no	—
88	#14555510	—	2	16	no	—
89	#14555510	—	2	8	no	—
90	#14555510	D₄	1	8	yes	direct product factor
91	C₂×Dic₂₇₄	—	2	8	no	—
92	C₂×Dic₂₇₄	C₂³	1	4	yes	—
93	C₂×Dic₂₇₄	C₂³	1	2	yes	—
94	C₂×Dic₂₇₄	C₂³	1	1	yes	characteristic
95	C₂×D₅₄₈	—	2	8	no	—
96	C₂×D₅₄₈	C₂³	1	2	yes	—
97	C₂×D₅₄₈	C₂³	1	4	yes	—
98	C₂×D₅₄₈	C₂³	1	2	yes	—
99	C₂×D₅₄₈	C₂³	1	2	yes	—
100	C₂×D₅₄₈	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊D₁₃₇	—	2	16	no	—
102	C₂×C₂²⋊D₁₃₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊D₁₃₇	C₂³	1	4	yes	—
104	C₂×C₂²⋊D₁₃₇	C₂³	1	4	yes	—
105	C₂×C₂²⋊D₁₃₇	C₂³	1	4	yes	—
106	C₂×C₂²⋊D₁₃₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊D₁₃₇	C₂³	1	2	yes	—
108	D₄×D₁₃₇	—	2	8	no	—
109	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
110	C₂×C₄×D₁₃₇	—	2	16	no	—
111	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
112	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
113	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
114	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
115	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
116	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
117	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
118	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
119	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
120	C₄⋊Dic₁₃₇	C₂³	1	2	yes	—
121	#14555489	C₂³	1	4	yes	—
122	#14555489	C₂³	1	2	yes	—
123	#14555489	C₂³	1	2	yes	—
124	#14555489	C₂³	1	4	yes	—
125	C₂²⋊Dic₁₃₇	C₂³	1	4	yes	—
126	C₂²⋊Dic₁₃₇	C₂³	1	2	yes	—
127	C₂²⋊Dic₁₃₇	C₂³	1	4	yes	—
128	#14555487	C₂³	1	4	yes	—
129	#14555487	C₂³	1	4	yes	—
130	#14555487	C₂³	1	2	yes	—
131	#14555487	C₂³	1	2	yes	—
132	#14555487	C₂³	1	4	yes	—
133	#14555487	C₂³	1	4	yes	—
134	C₄×Dic₁₃₇	C₂³	1	2	yes	—
135	C₄×Dic₁₃₇	C₂³	1	2	yes	—
136	C₄×Dic₁₃₇	C₂³	1	2	yes	—
137	C₂⁴×C₁₃₇	C₂³	1	2	yes	—
138	C₂×D₄×C₁₃₇	—	2	4	no	—
139	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
140	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
141	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
142	C₂²×C₄×C₁₃₇	—	2	8	no	—
143	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
144	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
145	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
146	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
147	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
148	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
149	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
150	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
151	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
152	C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
153	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
154	C₂²⋊C₄×C₁₃₇	C₂³	1	2	yes	—
155	C₂²⋊C₄×C₁₃₇	C₂³	1	2	yes	—
156	C₂²⋊C₄×C₁₃₇	C₂³	1	2	yes	—
157	C₂²⋊C₄×C₁₃₇	C₂³	1	2	yes	—
158	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
159	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
160	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
161	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
162	C₂²×D₁₃₇	—	2	8	no	—
163	C₂²×D₁₃₇	—	2	8	no	—
164	C₂²×D₁₃₇	—	4	4	no	—
165	C₂²×D₁₃₇	C₂⁴	1	4	yes	—
166	C₂²×D₁₃₇	C₂⁴	1	2	yes	—
167	Dic₂₇₄	—	2	4	no	—
168	Dic₂₇₄	—	2	8	no	—
169	Dic₂₇₄	C₂×D₄	1	4	yes	complemented
170	D₅₄₈	—	2	8	no	—
171	D₅₄₈	—	2	4	no	—
172	D₅₄₈	—	2	4	no	—
173	D₅₄₈	—	4	2	no	—
174	D₅₄₈	C₂×D₄	1	4	yes	complemented
175	C₂²⋊D₁₃₇	—	2	8	no	—
176	C₂²⋊D₁₃₇	—	2	8	no	—
177	C₂²⋊D₁₃₇	—	2	8	no	—
178	C₂²⋊D₁₃₇	—	2	8	no	—
179	C₂²⋊D₁₃₇	—	4	4	no	—
180	C₂²⋊D₁₃₇	C₂×D₄	1	8	yes	complemented
181	C₂×Dic₁₃₇	—	2	8	no	—
182	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
183	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
184	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
185	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
186	C₄×D₁₃₇	—	2	8	no	—
187	C₄×D₁₃₇	—	2	8	no	—
188	C₄×D₁₃₇	—	2	4	no	—
189	C₄×D₁₃₇	—	2	8	no	—
190	C₄×D₁₃₇	—	4	2	no	—
191	C₄×D₁₃₇	C₂×D₄	1	8	yes	complemented
192	C₂³×C₁₃₇	—	2	4	no	—
193	C₂³×C₁₃₇	—	2	4	no	—
194	C₂³×C₁₃₇	—	4	2	no	—
195	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
196	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
197	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
198	D₄×C₁₃₇	—	2	4	no	—
199	D₄×C₁₃₇	—	2	2	no	—
200	D₄×C₁₃₇	#41	1	4	yes	complemented
201	C₂×C₄×C₁₃₇	—	2	4	no	—
202	C₂×C₄×C₁₃₇	—	2	4	no	—
203	C₂×C₄×C₁₃₇	—	2	4	no	—
204	C₂×C₄×C₁₃₇	—	2	4	no	—
205	C₂×C₄×C₁₃₇	—	2	4	no	—
206	C₂×C₄×C₁₃₇	—	2	2	no	—
207	C₂×C₄×C₁₃₇	—	4	1	no	—
208	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	complemented
209	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
210	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
211	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
212	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
213	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
214	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
215	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
216	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
217	C₂²×C₁₃₇	—	2	2	no	—
218	C₂²×C₁₃₇	—	2	1	no	—
219	C₂²×C₁₃₇	—	2	1	no	—
220	C₂²×C₁₃₇	—	2	2	no	—
221	C₂²×C₁₃₇	—	2	2	no	—
222	C₂²×C₁₃₇	—	2	2	no	—
223	C₂²×C₁₃₇	—	2	2	no	—
224	C₂²×C₁₃₇	—	4	8	no	—
225	C₂²×C₁₃₇	C₂²×D₄	1	2	yes	—
226	C₂²×C₁₃₇	C₂×#41	1	4	yes	complemented
227	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
228	C₂×D₁₃₇	—	2	2	no	—
229	C₂×D₁₃₇	—	2	2	no	—
230	C₂×D₁₃₇	—	2	4	no	—
231	C₂×D₁₃₇	—	2	4	no	—
232	C₂×D₁₃₇	—	2	4	no	—
233	C₂×D₁₃₇	—	4	16	no	—
234	C₂×D₁₃₇	C₂²×D₄	1	4	yes	—
235	C₄×C₁₃₇	—	2	2	no	—
236	C₄×C₁₃₇	—	2	2	no	—
237	C₄×C₁₃₇	—	2	1	no	—
238	C₄×C₁₃₇	—	2	1	no	—
239	C₄×C₁₃₇	—	2	1	no	—
240	C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
241	C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
242	C₄×C₁₃₇	C₂×#41	1	2	yes	complemented
243	Dic₁₃₇	—	2	4	no	—
244	Dic₁₃₇	—	2	2	no	—
245	Dic₁₃₇	—	2	2	no	—
246	Dic₁₃₇	C₂²×D₄	1	4	yes	—
247	C₂×C₁₃₇	—	2	4	no	—
248	C₂×C₁₃₇	—	2	2	no	—
249	C₂×C₁₃₇	—	4	2	no	—
250	C₂×C₁₃₇	C₂³×D₄	1	1	yes	characteristic
251	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
252	C₂×C₁₃₇	#585	1	1	yes	characteristic
253	D₁₃₇	—	2	4	no	—
254	D₁₃₇	—	4	4	no	—
255	C₁₃₇	D₄×#41	1	1	yes	characteristic, complemented, Sylow
256	D₄×#41	—	137	1	no	maximal, Sylow
257	C₂²×#41	—	137	2	no	—
258	#550	—	137	1	no	—
259	D₄×Q₈	—	137	1	no	—
260	#548	—	137	2	no	—
261	#547	—	137	1	no	—
262	#547	—	137	2	no	—
263	#546	—	137	2	no	—
264	#546	—	137	4	no	—
265	D₄²	—	137	2	no	—
266	D₄²	—	137	1	no	—
267	#532	—	137	2	no	—
268	#532	—	137	1	no	—
269	#525	—	137	2	no	—
270	#525	—	137	4	no	—
271	C₄×#41	—	137	1	no	—
272	C₂×C₄×D₄	—	137	2	no	—
273	C₂×C₄×D₄	D₁₃₇	1	1	yes	characteristic, complemented
274	C₂×#41	—	274	16	no	—
275	C₂×#41	—	137	2	no	—
276	C₂×#41	—	137	4	no	—
277	C₂×#41	—	137	2	no	—
278	C₂×#41	—	137	4	no	—
279	C₂×#41	—	137	2	no	—
280	C₂×#41	—	137	1	no	—
281	C₂²×Q₈	—	137	2	no	—
282	C₂²×D₄	—	137	4	no	—
283	C₂²×D₄	—	137	2	no	—
284	C₂²×D₄	—	137	2	no	—
285	C₂²×D₄	C₂×D₁₃₇	1	1	yes	characteristic
286	C₂³×C₄	—	137	4	no	—
287	C₂³×C₄	C₂×D₁₃₇	1	2	yes	complemented
288	C₄⋊Q₈	—	137	2	no	—
289	C₄⋊Q₈	—	137	1	no	—
290	#127	—	137	2	no	—
291	#127	—	137	1	no	—
292	#124	—	137	4	no	—
293	#124	—	137	2	no	—
294	#123	—	137	4	no	—
295	#123	—	137	4	no	—
296	#123	—	137	4	no	—
297	C₂²⋊Q₈	—	137	2	no	—
298	C₂²⋊Q₈	—	137	2	no	—
299	C₂²⋊Q₈	—	137	4	no	—
300	C₂²⋊Q₈	—	137	4	no	—
301	#121	—	137	4	no	—
302	#121	—	137	2	no	—
303	#121	—	137	2	no	—
304	#121	—	137	2	no	—
305	#121	—	137	2	no	—
306	#121	—	137	2	no	—
307	#121	—	137	2	no	—
308	#121	—	137	4	no	—
309	#121	—	137	4	no	—
310	#120	—	137	4	no	—
311	#120	—	137	4	no	—
312	#120	—	137	4	no	—
313	C₄×Q₈	—	137	1	no	—
314	C₄×Q₈	—	137	1	no	—
315	C₄×D₄	—	137	2	no	—
316	C₄×D₄	—	137	4	no	—
317	C₄×D₄	—	137	2	no	—
318	C₄×D₄	—	137	2	no	—
319	C₄×D₄	—	137	4	no	—
320	C₄×D₄	—	137	2	no	—
321	C₄×D₄	—	137	4	no	—
322	C₄×D₄	—	137	2	no	—
323	C₄×D₄	—	137	1	no	—
324	C₄×D₄	—	137	1	no	—
325	C₄×D₄	—	137	2	no	—
326	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic
327	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic
328	C₄×D₄	C₂×D₁₃₇	1	2	yes	complemented
329	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic, complemented
330	C₄×D₄	C₂×D₁₃₇	1	2	yes	complemented
331	C₄×D₄	C₂×D₁₃₇	1	1	yes	characteristic, complemented
332	#117	—	137	1	no	—
333	#117	—	137	2	no	—
334	C₂×C₄⋊C₄	—	137	2	no	—
335	C₂×C₄⋊C₄	C₂×D₁₃₇	1	1	yes	characteristic, complemented
336	C₂×C₂²⋊C₄	—	137	4	no	—
337	C₂×C₂²⋊C₄	C₂×D₁₃₇	1	2	yes	complemented
338	C₂×C₄²	—	137	2	no	—
339	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic, complemented
340	C₂⁴	—	137	4	no	—
341	C₂⁴	C₂²×D₁₃₇	1	2	yes	—
342	#41	—	274	8	no	—
343	#41	—	274	16	no	—
344	#41	—	274	8	no	—
345	#41	—	274	16	no	—
346	#41	—	274	8	no	—
347	#41	—	137	8	no	—
348	C₂×Q₈	—	274	8	no	—
349	C₂×Q₈	—	137	2	no	—
350	C₂×Q₈	—	137	4	no	—
351	C₂×Q₈	—	137	1	no	—
352	C₂×D₄	—	2	4	no	—
353	C₂×D₄	—	274	8	no	—
354	C₂×D₄	—	274	16	no	—
355	C₂×D₄	—	274	8	no	—
356	C₂×D₄	—	137	4	no	—
357	C₂×D₄	—	137	4	no	—
358	C₂×D₄	—	137	4	no	—
359	C₂×D₄	—	137	4	no	—
360	C₂×D₄	—	137	4	no	—
361	C₂×D₄	—	137	2	no	—
362	C₂×D₄	—	137	2	no	—
363	C₂×D₄	—	137	4	no	—
364	C₂×D₄	—	137	2	no	—
365	C₂×D₄	—	137	2	no	—
366	C₂×D₄	—	137	1	no	—
367	C₂×D₄	C₂²×D₁₃₇	1	2	yes	—
368	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
369	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
370	C₂²×C₄	—	2	8	no	—
371	C₂²×C₄	—	274	16	no	—
372	C₂²×C₄	—	137	4	no	—
373	C₂²×C₄	—	137	2	no	—
374	C₂²×C₄	—	137	4	no	—
375	C₂²×C₄	—	137	4	no	—
376	C₂²×C₄	—	137	4	no	—
377	C₂²×C₄	—	137	2	no	—
378	C₂²×C₄	—	137	2	no	—
379	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
380	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
381	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
382	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
383	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
384	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
385	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
386	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
387	C₄⋊C₄	—	137	4	no	—
388	C₄⋊C₄	—	137	2	no	—
389	C₄⋊C₄	—	137	2	no	—
390	C₄⋊C₄	—	137	4	no	—
391	C₄⋊C₄	—	137	2	no	—
392	C₄⋊C₄	—	137	1	no	—
393	C₄⋊C₄	—	137	1	no	—
394	C₄⋊C₄	—	137	2	no	—
395	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
396	C₄⋊C₄	C₂²×D₁₃₇	1	1	yes	characteristic
397	C₄⋊C₄	C₂²×D₁₃₇	1	2	yes	—
398	C₂²⋊C₄	—	137	4	no	—
399	C₂²⋊C₄	—	137	2	no	—
400	C₂²⋊C₄	—	137	4	no	—
401	C₂²⋊C₄	—	137	4	no	—
402	C₂²⋊C₄	—	137	4	no	—
403	C₂²⋊C₄	—	137	2	no	—
404	C₂²⋊C₄	—	137	2	no	—
405	C₂²⋊C₄	—	137	4	no	—
406	C₂²⋊C₄	—	137	4	no	—
407	C₂²⋊C₄	C₂²×D₁₃₇	1	2	yes	—
408	C₂²⋊C₄	C₂²×D₁₃₇	1	2	yes	—
409	C₂²⋊C₄	C₂²×D₁₃₇	1	2	yes	—
410	C₂²⋊C₄	C₂²×D₁₃₇	1	2	yes	—
411	C₄²	—	137	2	no	—
412	C₄²	—	137	2	no	—
413	C₄²	—	137	2	no	—
414	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
415	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
416	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
417	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
418	C₂³	—	2	4	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	274	8	no	—
422	C₂³	—	274	8	no	—
423	C₂³	—	548	4	no	—
424	C₂³	—	137	4	no	—
425	C₂³	—	137	2	no	—
426	C₂³	C₂³×D₁₃₇	1	2	yes	—
427	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
428	C₂³	C₂³×D₁₃₇	1	2	yes	—
429	Q₈	—	274	8	no	—
430	Q₈	—	274	4	no	—
431	Q₈	—	137	4	no	—
432	D₄	—	2	4	no	—
433	D₄	—	2	2	no	—
434	D₄	—	274	8	no	—
435	D₄	—	274	8	no	—
436	D₄	—	274	8	no	—
437	D₄	—	274	8	no	—
438	D₄	—	274	8	no	—
439	D₄	—	274	4	no	—
440	D₄	—	274	4	no	—
441	D₄	—	548	2	no	—
442	D₄	—	548	4	no	—
443	D₄	—	137	8	no	—
444	D₄	—	137	4	no	—
445	D₄	#14555510	1	4	yes	direct product factor
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	274	8	no	—
454	C₂×C₄	—	274	8	no	—
455	C₂×C₄	—	274	8	no	—
456	C₂×C₄	—	274	8	no	—
457	C₂×C₄	—	274	4	no	—
458	C₂×C₄	—	548	2	no	—
459	C₂×C₄	—	137	2	no	—
460	C₂×C₄	—	137	2	no	—
461	C₂×C₄	—	137	4	no	—
462	C₂×C₄	—	137	2	no	—
463	C₂×C₄	—	137	8	no	—
464	C₂×C₄	D₄×D₁₃₇	1	4	yes	complemented
465	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
466	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
467	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
468	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
469	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
470	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
471	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
472	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
473	C₂²	—	2	2	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	2	2	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	4	8	no	—
481	C₂²	—	274	2	no	—
482	C₂²	—	274	2	no	—
483	C₂²	—	274	4	no	—
484	C₂²	—	274	4	no	—
485	C₂²	—	274	4	no	—
486	C₂²	—	548	16	no	—
487	C₂²	—	137	4	no	—
488	C₂²	C₂×D₄×D₁₃₇	1	2	yes	—
489	C₂²	C₂×#14555510	1	4	yes	complemented
490	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	2	1	no	—
496	C₄	—	274	4	no	—
497	C₄	—	274	2	no	—
498	C₄	—	274	2	no	—
499	C₄	—	137	4	no	—
500	C₄	C₂×D₄×D₁₃₇	1	2	yes	—
501	C₄	C₂×D₄×D₁₃₇	1	2	yes	—
502	C₄	C₂×#14555510	1	2	yes	complemented
503	C₂	—	2	2	no	—
504	C₂	—	2	4	no	—
505	C₂	—	4	2	no	—
506	C₂	—	274	4	no	—
507	C₂	—	548	4	no	—
508	C₂	C₂²×D₄×D₁₃₇	1	1	yes	characteristic
509	C₂	C₂²×#14555510	1	1	yes	characteristic
510	C₂	#22578239	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃₇
commutator, socle, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂×C₄×D₄×C₁₃₇	C₂
hypercenter	C₂×C₄×D₄	D₁₃₇
hypocenter	C₁₃₇	D₄×#41
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555510	C₂	semidirect
2. #22578074	C₂	semidirect
3. D₄×Dic₂₇₄	C₂	semidirect
4. #22578062	C₂	semidirect
5. #22578058	C₂	semidirect
6. #22578054	C₂	semidirect
7. #22578049	C₂	semidirect
8. #22578043	C₂	semidirect
9. D₄×D₅₄₈	C₂	semidirect
10. D₄×C₂²⋊D₁₃₇	C₂	semidirect
11. #22577953	C₂	semidirect
12. #22577950	C₂	semidirect
13. #22577914	C₂	semidirect
14. #22577912	C₂	semidirect
15. C₄×#14555510	C₂	semidirect
16. C₄×D₄×D₁₃₇	C₂	semidirect
17. C₂×C₄×D₄×C₁₃₇	C₂	semidirect
20. C₂×#14555510	C₂²	semidirect
21. C₂×#14555510	C₂²	semidirect
22. C₂×#14555510	C₂²	semidirect
23. C₂×#14555510	C₂²	semidirect
24. C₂×#14555510	C₂²	semidirect
25. C₂²×Dic₂₇₄	C₂²	semidirect
26. C₂²×D₅₄₈	C₂²	semidirect
27. C₂²×C₂²⋊D₁₃₇	C₂²	semidirect
29. C₂²×C₄×D₁₃₇	C₂²	semidirect
30. #17984088	C₂²	semidirect
31. #17984087	C₂²	semidirect
32. #17984085	C₂²	semidirect
33. #17984084	C₂²	semidirect
34. #17984076	C₂²	semidirect
35. #17984074	C₂²	semidirect
36. #17984073	C₂²	semidirect
37. #17984071	C₂²	semidirect
38. #17984070	C₂²	semidirect
39. #17984067	C₂²	semidirect
40. #17984067	C₂²	semidirect
41. #17984066	C₂²	semidirect
42. #17984065	C₂²	semidirect
43. #17984063	C₂²	semidirect
44. #17984062	C₂²	semidirect
45. #17984062	C₂²	semidirect
46. #17984062	C₂²	semidirect
47. #17984062	C₂²	semidirect
48. #17984061	C₂²	semidirect
49. #17984061	C₂²	semidirect
50. #17984060	C₂²	semidirect
51. #17984059	C₂²	semidirect
52. #17984058	C₂²	semidirect
53. #17984057	C₂²	semidirect
54. #17984056	C₂²	semidirect
55. C₄×Dic₂₇₄	C₂²	semidirect
56. C₄×Dic₂₇₄	C₂²	semidirect
57. C₄×D₅₄₈	C₂²	semidirect
58. C₄×D₅₄₈	C₂²	semidirect
59. C₄×D₅₄₈	C₂²	semidirect
60. C₄×D₅₄₈	C₂²	semidirect
62. #17984050	C₂²	semidirect
63. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
64. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
65. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
66. C₄×C₂²⋊D₁₃₇	C₂²	semidirect
67. #17984048	C₂²	semidirect
68. #17984047	C₂²	semidirect
69. #17984045	C₂²	semidirect
70. C₄⋊C₄×D₁₃₇	C₂²	semidirect
71. C₂²⋊C₄×D₁₃₇	C₂²	semidirect
72. C₄²×D₁₃₇	C₂²	semidirect
74. C₂³×C₄×C₁₃₇	C₂²	semidirect
77. C₄×D₄×C₁₃₇	C₂²	semidirect
78. C₄×D₄×C₁₃₇	C₂²	semidirect
79. C₄×D₄×C₁₃₇	C₂²	semidirect
80. C₄×D₄×C₁₃₇	C₂²	semidirect
81. C₂×C₄⋊C₄×C₁₃₇	C₂²	semidirect
82. C₂×C₂²⋊C₄×C₁₃₇	C₂²	semidirect
83. C₂×C₄²×C₁₃₇	C₂²	semidirect
90. #14555510	D₄	direct (N × H)
169. Dic₂₇₄	C₂×D₄	semidirect
174. D₅₄₈	C₂×D₄	semidirect
180. C₂²⋊D₁₃₇	C₂×D₄	semidirect
191. C₄×D₁₃₇	C₂×D₄	semidirect
200. D₄×C₁₃₇	#41	semidirect
208. C₂×C₄×C₁₃₇	C₂×D₄	semidirect
226. C₂²×C₁₃₇	C₂×#41	semidirect
242. C₄×C₁₃₇	C₂×#41	semidirect
255. C₁₃₇	D₄×#41	semidirect
273. C₂×C₄×D₄	D₁₃₇	semidirect
287. C₂³×C₄	C₂×D₁₃₇	semidirect
328. C₄×D₄	C₂×D₁₃₇	semidirect
329. C₄×D₄	C₂×D₁₃₇	semidirect
330. C₄×D₄	C₂×D₁₃₇	semidirect
331. C₄×D₄	C₂×D₁₃₇	semidirect
335. C₂×C₄⋊C₄	C₂×D₁₃₇	semidirect
337. C₂×C₂²⋊C₄	C₂×D₁₃₇	semidirect
339. C₂×C₄²	C₂×D₁₃₇	semidirect
445. D₄	#14555510	direct (N × H)
464. C₂×C₄	D₄×D₁₃₇	semidirect
489. C₂²	C₂×#14555510	semidirect
502. C₄	C₂×#14555510	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14555510	2
#22578074	2
D₄×Dic₂₇₄	2
#22578062	2
#22578058	2
#22578054	2
#22578049	2
#22578043	2
D₄×D₅₄₈	2
D₄×C₂²⋊D₁₃₇	2
#22577953	2
#22577950	2
#22577914	2
#22577912	2
C₄×#14555510	2
C₄×D₄×D₁₃₇	2
C₂×C₄×D₄×C₁₃₇	2
D₄×#41	137

Subgroups — D4×#14555510 #31794277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#14555510 #31794277