Subgroups — C₁₃₇×#24727 #46874113

Order 35072.

1050 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#463×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#461×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2547×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2068×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2042×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1956×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1939×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1877×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1802×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₁₃₇	—	2	1	no	—
20	#463×C₁₃₇	—	2	8	no	—
21	#461×C₁₃₇	—	2	4	no	—
22	C₄×D₈×C₁₃₇	—	2	4	no	—
23	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
27	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
29	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
31	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
33	#397×C₁₃₇	—	2	2	no	—
34	#391×C₁₃₇	C₂²	1	2	yes	—
35	#389×C₁₃₇	C₂²	1	2	yes	—
36	#385×C₁₃₇	—	2	2	no	—
37	#384×C₁₃₇	C₂²	1	1	yes	characteristic
38	#380×C₁₃₇	—	2	1	no	—
39	#380×C₁₃₇	—	2	1	no	—
40	#380×C₁₃₇	—	2	1	no	—
41	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
42	#340×C₁₃₇	C₂²	1	2	yes	—
43	#336×C₁₃₇	C₂²	1	2	yes	—
44	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
45	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₁₃₇	—	2	1	no	—
48	C₂³×C₄×C₁₃₇	—	2	1	no	—
49	C₂³×C₄×C₁₃₇	—	2	2	no	—
50	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
51	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
52	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
53	C₂×D₈×C₁₃₇	—	2	2	no	—
54	C₂×D₈×C₁₃₇	—	2	2	no	—
55	C₂×D₈×C₁₃₇	D₄	1	4	yes	—
56	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
60	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
61	#124×C₁₃₇	—	2	4	no	—
62	#124×C₁₃₇	—	4	2	no	—
63	C₄×D₄×C₁₃₇	—	2	4	no	—
64	C₄×D₄×C₁₃₇	—	2	2	no	—
65	C₄×D₄×C₁₃₇	—	2	2	no	—
66	C₄×D₄×C₁₃₇	—	2	2	no	—
67	C₄×D₄×C₁₃₇	—	4	2	no	—
68	C₄×D₄×C₁₃₇	D₄	1	4	yes	—
69	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₁₃₇	—	4	4	no	—
75	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
85	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
86	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
87	#107×C₁₃₇	—	2	2	no	—
88	C₄⋊C₈×C₁₃₇	—	2	4	no	—
89	C₄⋊C₈×C₁₃₇	—	2	2	no	—
90	#103×C₁₃₇	—	2	4	no	—
91	#103×C₁₃₇	—	2	4	no	—
92	#103×C₁₃₇	—	4	2	no	—
93	#102×C₁₃₇	—	2	2	no	—
94	#102×C₁₃₇	—	2	4	no	—
95	#102×C₁₃₇	—	2	2	no	—
96	C₄×C₈×C₁₃₇	—	2	2	no	—
97	#95×C₁₃₇	—	2	2	no	—
98	#95×C₁₃₇	—	2	2	no	—
99	#95×C₁₃₇	—	2	2	no	—
100	#95×C₁₃₇	D₄	1	2	yes	—
101	C₂⁴×C₁₃₇	—	2	1	no	—
102	C₂⁴×C₁₃₇	—	2	1	no	—
103	C₂×Q₈×C₁₃₇	—	2	2	no	—
104	C₂×Q₈×C₁₃₇	—	2	2	no	—
105	C₂×Q₈×C₁₃₇	—	2	2	no	—
106	C₂×Q₈×C₁₃₇	—	2	2	no	—
107	C₂×Q₈×C₁₃₇	—	2	2	no	—
108	C₂×Q₈×C₁₃₇	—	2	2	no	—
109	C₂×D₄×C₁₃₇	—	2	1	no	—
110	C₂×D₄×C₁₃₇	—	2	1	no	—
111	C₂×D₄×C₁₃₇	—	2	1	no	—
112	C₂×D₄×C₁₃₇	—	2	1	no	—
113	C₂×D₄×C₁₃₇	—	2	1	no	—
114	C₂×D₄×C₁₃₇	—	2	1	no	—
115	C₂×D₄×C₁₃₇	—	2	1	no	—
116	C₂×D₄×C₁₃₇	—	2	1	no	—
117	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₁₃₇	#41	1	2	yes	—
121	C₂²×C₄×C₁₃₇	—	2	1	no	—
122	C₂²×C₄×C₁₃₇	—	2	4	no	—
123	C₂²×C₄×C₁₃₇	—	2	1	no	—
124	C₂²×C₄×C₁₃₇	—	2	2	no	—
125	C₂²×C₄×C₁₃₇	—	2	2	no	—
126	C₂²×C₄×C₁₃₇	—	4	4	no	—
127	C₂²×C₄×C₁₃₇	—	4	4	no	—
128	C₂²×C₄×C₁₃₇	—	4	2	no	—
129	C₂²×C₄×C₁₃₇	—	4	2	no	—
130	C₂²×C₄×C₁₃₇	—	4	2	no	—
131	C₂²×C₄×C₁₃₇	—	4	2	no	—
132	C₂²×C₄×C₁₃₇	—	4	2	no	—
133	C₂²×C₄×C₁₃₇	—	8	2	no	—
134	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
139	SD₁₆×C₁₃₇	—	4	8	no	—
140	D₈×C₁₃₇	—	2	8	no	—
141	C₂×C₈×C₁₃₇	—	2	1	no	—
142	C₂×C₈×C₁₃₇	—	2	1	no	—
143	C₂×C₈×C₁₃₇	—	2	1	no	—
144	C₂×C₈×C₁₃₇	—	2	2	no	—
145	C₂×C₈×C₁₃₇	—	2	2	no	—
146	C₂×C₈×C₁₃₇	—	2	2	no	—
147	C₂×C₈×C₁₃₇	—	2	1	no	—
148	C₂×C₈×C₁₃₇	—	2	1	no	—
149	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₁₃₇	—	2	2	no	—
151	C₄⋊C₄×C₁₃₇	—	2	2	no	—
152	C₄⋊C₄×C₁₃₇	—	2	1	no	—
153	C₄⋊C₄×C₁₃₇	—	2	2	no	—
154	C₄⋊C₄×C₁₃₇	—	2	1	no	—
155	C₄⋊C₄×C₁₃₇	—	2	1	no	—
156	C₄⋊C₄×C₁₃₇	—	2	2	no	—
157	C₄⋊C₄×C₁₃₇	—	4	4	no	—
158	C₄⋊C₄×C₁₃₇	—	4	2	no	—
159	C₄⋊C₄×C₁₃₇	—	4	2	no	—
160	C₄⋊C₄×C₁₃₇	—	4	2	no	—
161	C₄⋊C₄×C₁₃₇	—	8	1	no	—
162	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
163	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
164	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
165	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
166	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
167	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
168	C₄²×C₁₃₇	—	2	2	no	—
169	C₄²×C₁₃₇	—	2	2	no	—
170	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
171	C₄²×C₁₃₇	#41	1	2	yes	—
172	C₂³×C₁₃₇	—	2	2	no	—
173	C₂³×C₁₃₇	—	2	2	no	—
174	C₂³×C₁₃₇	—	2	2	no	—
175	C₂³×C₁₃₇	—	2	2	no	—
176	C₂³×C₁₃₇	—	4	1	no	—
177	C₂³×C₁₃₇	—	4	1	no	—
178	C₂³×C₁₃₇	—	4	1	no	—
179	C₂³×C₁₃₇	—	4	1	no	—
180	C₂³×C₁₃₇	—	4	1	no	—
181	C₂³×C₁₃₇	—	4	1	no	—
182	C₂³×C₁₃₇	—	4	1	no	—
183	C₂³×C₁₃₇	—	4	1	no	—
184	C₂³×C₁₃₇	—	4	1	no	—
185	C₂³×C₁₃₇	—	4	1	no	—
186	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₁₃₇	—	4	2	no	—
188	Q₈×C₁₃₇	—	4	2	no	—
189	Q₈×C₁₃₇	—	4	2	no	—
190	Q₈×C₁₃₇	—	4	2	no	—
191	D₄×C₁₃₇	—	2	2	no	—
192	D₄×C₁₃₇	—	2	2	no	—
193	D₄×C₁₃₇	—	2	2	no	—
194	D₄×C₁₃₇	—	2	2	no	—
195	D₄×C₁₃₇	—	2	2	no	—
196	D₄×C₁₃₇	—	4	1	no	—
197	D₄×C₁₃₇	—	4	1	no	—
198	D₄×C₁₃₇	#121	1	4	yes	—
199	C₂×C₄×C₁₃₇	—	2	1	no	—
200	C₂×C₄×C₁₃₇	—	2	1	no	—
201	C₂×C₄×C₁₃₇	—	2	2	no	—
202	C₂×C₄×C₁₃₇	—	2	2	no	—
203	C₂×C₄×C₁₃₇	—	2	2	no	—
204	C₂×C₄×C₁₃₇	—	2	2	no	—
205	C₂×C₄×C₁₃₇	—	2	2	no	—
206	C₂×C₄×C₁₃₇	—	2	2	no	—
207	C₂×C₄×C₁₃₇	—	2	2	no	—
208	C₂×C₄×C₁₃₇	—	4	1	no	—
209	C₂×C₄×C₁₃₇	—	4	1	no	—
210	C₂×C₄×C₁₃₇	—	4	8	no	—
211	C₂×C₄×C₁₃₇	—	4	1	no	—
212	C₂×C₄×C₁₃₇	—	4	2	no	—
213	C₂×C₄×C₁₃₇	—	4	2	no	—
214	C₂×C₄×C₁₃₇	—	4	2	no	—
215	C₂×C₄×C₁₃₇	—	4	2	no	—
216	C₂×C₄×C₁₃₇	—	4	1	no	—
217	C₂×C₄×C₁₃₇	—	4	2	no	—
218	C₂×C₄×C₁₃₇	—	4	1	no	—
219	C₂×C₄×C₁₃₇	—	4	2	no	—
220	C₂×C₄×C₁₃₇	—	8	8	no	—
221	C₂×C₄×C₁₃₇	—	8	4	no	—
222	C₂×C₄×C₁₃₇	#121	1	2	yes	—
223	C₂×C₄×C₁₃₇	#121	1	2	yes	—
224	C₂×C₄×C₁₃₇	#135	1	2	yes	—
225	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
226	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
231	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
232	C₈×C₁₃₇	—	2	2	no	—
233	C₈×C₁₃₇	—	4	2	no	—
234	C₈×C₁₃₇	—	4	1	no	—
235	C₂²×C₁₃₇	—	2	4	no	—
236	C₂²×C₁₃₇	—	4	2	no	—
237	C₂²×C₁₃₇	—	4	2	no	—
238	C₂²×C₁₃₇	—	4	2	no	—
239	C₂²×C₁₃₇	—	4	2	no	—
240	C₂²×C₁₃₇	—	4	2	no	—
241	C₂²×C₁₃₇	—	4	2	no	—
242	C₂²×C₁₃₇	—	4	2	no	—
243	C₂²×C₁₃₇	—	4	2	no	—
244	C₂²×C₁₃₇	—	4	2	no	—
245	C₂²×C₁₃₇	—	8	1	no	—
246	C₂²×C₁₃₇	—	8	1	no	—
247	C₂²×C₁₃₇	—	8	1	no	—
248	C₂²×C₁₃₇	—	8	1	no	—
249	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
250	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
251	C₂²×C₁₃₇	#548	1	1	yes	characteristic
252	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
253	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₂²×C₁₃₇	#577	1	1	yes	characteristic
255	C₂²×C₁₃₇	#578	1	1	yes	characteristic
256	C₄×C₁₃₇	—	2	2	no	—
257	C₄×C₁₃₇	—	4	2	no	—
258	C₄×C₁₃₇	—	4	2	no	—
259	C₄×C₁₃₇	—	4	2	no	—
260	C₄×C₁₃₇	—	8	2	no	—
261	C₄×C₁₃₇	—	8	2	no	—
262	C₄×C₁₃₇	—	8	1	no	—
263	C₄×C₁₃₇	#463	1	4	yes	complemented
264	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
265	C₄×C₁₃₇	#537	1	1	yes	characteristic
266	C₄×C₁₃₇	#547	1	1	yes	characteristic
267	C₄×C₁₃₇	#553	1	1	yes	characteristic
268	C₂×C₁₃₇	—	4	4	no	—
269	C₂×C₁₃₇	—	8	2	no	—
270	C₂×C₁₃₇	#2416	1	1	yes	characteristic
271	C₂×C₁₃₇	C₂×#463	1	1	yes	characteristic
272	C₂×C₁₃₇	#3046	1	1	yes	characteristic
273	C₂×C₁₃₇	#3272	1	1	yes	characteristic
274	C₂×C₁₃₇	#3283	1	1	yes	characteristic
275	C₂×C₁₃₇	#3360	1	1	yes	characteristic
276	C₂×C₁₃₇	#3367	1	1	yes	characteristic
277	#24727	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₁₃₇	#24727	1	1	yes	characteristic, direct product factor, Sylow
279	C₂×#463	C₂×C₁₃₇	1	2	yes	—
280	C₂×#461	C₂×C₁₃₇	1	1	yes	characteristic, complemented
281	C₂×C₄×D₈	C₂×C₁₃₇	1	1	yes	characteristic
282	#2547	C₂×C₁₃₇	1	1	yes	characteristic, complemented
283	D₄×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
284	#2068	C₂×C₁₃₇	1	2	yes	complemented
285	#2042	C₂×C₁₃₇	1	2	yes	complemented
286	#1956	C₂×C₁₃₇	1	1	yes	characteristic, complemented
287	#1939	C₂×C₁₃₇	1	1	yes	characteristic, complemented
288	#1877	C₂×C₁₃₇	1	2	yes	complemented
289	#1802	C₂×C₁₃₇	1	1	yes	characteristic, complemented
290	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
291	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
292	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
293	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
294	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
295	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
296	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
297	C₂²×C₄⋊C₄	—	2	1	no	—
298	#463	—	2	8	no	—
299	#461	—	2	4	no	—
300	C₄×D₈	—	2	4	no	—
301	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
302	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
303	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
304	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
305	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
306	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
307	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
308	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
309	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
310	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
311	#397	—	2	2	no	—
312	#391	C₂²×C₁₃₇	1	2	yes	—
313	#389	C₂²×C₁₃₇	1	2	yes	—
314	#385	—	2	2	no	—
315	#384	C₂²×C₁₃₇	1	1	yes	characteristic
316	#380	—	2	1	no	—
317	#380	—	2	1	no	—
318	#380	—	2	1	no	—
319	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
320	#340	C₂²×C₁₃₇	1	2	yes	—
321	#336	C₂²×C₁₃₇	1	2	yes	—
322	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
323	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
324	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
325	C₂³×C₄	—	2	1	no	—
326	C₂³×C₄	—	2	1	no	—
327	C₂³×C₄	—	2	2	no	—
328	C₂×SD₁₆	—	2	4	no	—
329	C₂×SD₁₆	—	2	4	no	—
330	C₂×SD₁₆	—	2	4	no	—
331	C₂×D₈	—	2	2	no	—
332	C₂×D₈	—	2	2	no	—
333	C₂×D₈	D₄×C₁₃₇	1	4	yes	—
334	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
335	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
337	C₄⋊Q₈	—	2	2	no	—
338	C₄⋊Q₈	—	2	2	no	—
339	#124	—	2	4	no	—
340	#124	—	4	2	no	—
341	C₄×D₄	—	2	4	no	—
342	C₄×D₄	—	2	2	no	—
343	C₄×D₄	—	2	2	no	—
344	C₄×D₄	—	2	2	no	—
345	C₄×D₄	—	4	2	no	—
346	C₄×D₄	D₄×C₁₃₇	1	4	yes	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	4	4	no	—
353	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
354	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₂²⋊C₄	—	2	1	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
364	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
365	#107	—	2	2	no	—
366	C₄⋊C₈	—	2	4	no	—
367	C₄⋊C₈	—	2	2	no	—
368	#103	—	2	4	no	—
369	#103	—	2	4	no	—
370	#103	—	4	2	no	—
371	#102	—	2	2	no	—
372	#102	—	2	4	no	—
373	#102	—	2	2	no	—
374	C₄×C₈	—	2	2	no	—
375	#95	—	2	2	no	—
376	#95	—	2	2	no	—
377	#95	—	2	2	no	—
378	#95	D₄×C₁₃₇	1	2	yes	—
379	C₂⁴	—	2	1	no	—
380	C₂⁴	—	2	1	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
396	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
397	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
398	C₂×D₄	#41×C₁₃₇	1	2	yes	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	4	4	no	—
405	C₂²×C₄	—	4	4	no	—
406	C₂²×C₄	—	4	2	no	—
407	C₂²×C₄	—	4	2	no	—
408	C₂²×C₄	—	4	2	no	—
409	C₂²×C₄	—	4	2	no	—
410	C₂²×C₄	—	4	2	no	—
411	C₂²×C₄	—	8	2	no	—
412	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
413	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
415	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
417	SD₁₆	—	4	8	no	—
418	D₈	—	2	8	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	4	4	no	—
436	C₄⋊C₄	—	4	2	no	—
437	C₄⋊C₄	—	4	2	no	—
438	C₄⋊C₄	—	4	2	no	—
439	C₄⋊C₄	—	8	1	no	—
440	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	—	4	4	no	—
443	C₂²⋊C₄	—	4	4	no	—
444	C₂²⋊C₄	—	4	4	no	—
445	C₂²⋊C₄	—	4	2	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
449	C₄²	#41×C₁₃₇	1	2	yes	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	4	1	no	—
455	C₂³	—	4	1	no	—
456	C₂³	—	4	1	no	—
457	C₂³	—	4	1	no	—
458	C₂³	—	4	1	no	—
459	C₂³	—	4	1	no	—
460	C₂³	—	4	1	no	—
461	C₂³	—	4	1	no	—
462	C₂³	—	4	1	no	—
463	C₂³	—	4	1	no	—
464	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
465	Q₈	—	4	2	no	—
466	Q₈	—	4	2	no	—
467	Q₈	—	4	2	no	—
468	Q₈	—	4	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	2	no	—
471	D₄	—	2	2	no	—
472	D₄	—	2	2	no	—
473	D₄	—	2	2	no	—
474	D₄	—	4	1	no	—
475	D₄	—	4	1	no	—
476	D₄	#121×C₁₃₇	1	4	yes	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	8	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	8	8	no	—
499	C₂×C₄	—	8	4	no	—
500	C₂×C₄	#121×C₁₃₇	1	2	yes	—
501	C₂×C₄	#121×C₁₃₇	1	2	yes	—
502	C₂×C₄	#135×C₁₃₇	1	2	yes	—
503	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
504	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
506	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
507	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
508	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
509	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
510	C₈	—	2	2	no	—
511	C₈	—	4	2	no	—
512	C₈	—	4	1	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	8	1	no	—
524	C₂²	—	8	1	no	—
525	C₂²	—	8	1	no	—
526	C₂²	—	8	1	no	—
527	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
528	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
529	C₂²	#548×C₁₃₇	1	1	yes	characteristic
530	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
531	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
532	C₂²	#577×C₁₃₇	1	1	yes	characteristic
533	C₂²	#578×C₁₃₇	1	1	yes	characteristic
534	C₄	—	2	2	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	2	no	—
537	C₄	—	4	2	no	—
538	C₄	—	8	2	no	—
539	C₄	—	8	2	no	—
540	C₄	—	8	1	no	—
541	C₄	#463×C₁₃₇	1	4	yes	complemented
542	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
543	C₄	#537×C₁₃₇	1	1	yes	characteristic
544	C₄	#547×C₁₃₇	1	1	yes	characteristic
545	C₄	#553×C₁₃₇	1	1	yes	characteristic
546	C₂	—	4	4	no	—
547	C₂	—	8	2	no	—
548	C₂	#2416×C₁₃₇	1	1	yes	characteristic
549	C₂	C₂×#463×C₁₃₇	1	1	yes	characteristic
550	C₂	#3046×C₁₃₇	1	1	yes	characteristic
551	C₂	#3272×C₁₃₇	1	1	yes	characteristic
552	C₂	#3283×C₁₃₇	1	1	yes	characteristic
553	C₂	#3360×C₁₃₇	1	1	yes	characteristic
554	C₂	#3367×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₈×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄×C₄⋊C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#461×C₁₃₇	C₂	semidirect
4. #2547×C₁₃₇	C₂	semidirect
5. D₄×C₄⋊C₄×C₁₃₇	C₂	semidirect
6. #2068×C₁₃₇	C₂	semidirect
7. #2042×C₁₃₇	C₂	semidirect
8. #1956×C₁₃₇	C₂	semidirect
9. #1939×C₁₃₇	C₂	semidirect
10. #1877×C₁₃₇	C₂	semidirect
11. #1802×C₁₃₇	C₂	semidirect
263. C₄×C₁₃₇	#463	semidirect
277. #24727	C₁₃₇	direct (N × H)
278. C₁₃₇	#24727	direct (N × H)
280. C₂×#461	C₂×C₁₃₇	semidirect
282. #2547	C₂×C₁₃₇	semidirect
283. D₄×C₄⋊C₄	C₂×C₁₃₇	semidirect
284. #2068	C₂×C₁₃₇	semidirect
285. #2042	C₂×C₁₃₇	semidirect
286. #1956	C₂×C₁₃₇	semidirect
287. #1939	C₂×C₁₃₇	semidirect
288. #1877	C₂×C₁₃₇	semidirect
289. #1802	C₂×C₁₃₇	semidirect
541. C₄	#463×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#463×C₁₃₇	2
C₂×#461×C₁₃₇	2
C₂×C₄×D₈×C₁₃₇	2
#2547×C₁₃₇	2
D₄×C₄⋊C₄×C₁₃₇	2
#2068×C₁₃₇	2
#2042×C₁₃₇	2
#1956×C₁₃₇	2
#1939×C₁₃₇	2
#1877×C₁₃₇	2
#1802×C₁₃₇	2
#24727	137

Subgroups — C137×#24727 #46874113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24727 #46874113