Subgroups — C₁₃₇×#25868 #46875254

Order 35072.

1230 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#536×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#491×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#488×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2629×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2043×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2042×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1959×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1956×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1788×C₁₃₇	C₂	1	2	yes	maximal
10	#1787×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1750×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	#536×C₁₃₇	—	2	4	no	—
15	#536×C₁₃₇	—	2	4	no	—
16	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×#124×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×#123×C₁₃₇	—	2	2	no	—
22	C₂×C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
23	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
24	#491×C₁₃₇	—	2	4	no	—
25	#488×C₁₃₇	—	2	4	no	—
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×#103×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
31	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
32	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
33	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
35	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
37	#398×C₁₃₇	—	2	2	no	—
38	#396×C₁₃₇	—	2	2	no	—
39	#389×C₁₃₇	C₂²	1	2	yes	—
40	#384×C₁₃₇	C₂²	1	1	yes	characteristic
41	#380×C₁₃₇	C₂²	1	2	yes	—
42	#336×C₁₃₇	C₂²	1	2	yes	—
43	C₂×#41×C₁₃₇	—	2	2	no	—
44	C₂×#41×C₁₃₇	—	2	2	no	—
45	C₂×#41×C₁₃₇	—	2	4	no	—
46	C₂×#41×C₁₃₇	—	4	2	no	—
47	C₂²×Q₈×C₁₃₇	—	2	2	no	—
48	C₂²×Q₈×C₁₃₇	—	2	2	no	—
49	C₂²×Q₈×C₁₃₇	—	2	1	no	—
50	C₂²×Q₈×C₁₃₇	—	2	4	no	—
51	C₂²×Q₈×C₁₃₇	—	4	2	no	—
52	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₁₃₇	—	2	1	no	—
56	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₁₃₇	—	2	1	no	—
58	C₂³×C₄×C₁₃₇	—	2	1	no	—
59	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
63	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
64	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
65	C₄⋊Q₈×C₁₃₇	D₄	1	4	yes	—
66	#125×C₁₃₇	—	2	2	no	—
67	#125×C₁₃₇	—	2	2	no	—
68	#124×C₁₃₇	—	2	2	no	—
69	#124×C₁₃₇	—	2	2	no	—
70	#124×C₁₃₇	—	2	2	no	—
71	#124×C₁₃₇	D₄	1	4	yes	—
72	#123×C₁₃₇	—	2	16	no	—
73	C₂²⋊Q₈×C₁₃₇	—	2	16	no	—
74	#117×C₁₃₇	—	2	2	no	—
75	#117×C₁₃₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
78	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
81	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
82	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
88	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
90	#103×C₁₃₇	—	2	4	no	—
91	#103×C₁₃₇	—	2	4	no	—
92	#103×C₁₃₇	—	2	4	no	—
93	#103×C₁₃₇	—	4	2	no	—
94	#103×C₁₃₇	—	4	2	no	—
95	#103×C₁₃₇	—	4	2	no	—
96	#102×C₁₃₇	—	2	4	no	—
97	#102×C₁₃₇	—	4	2	no	—
98	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
99	#97×C₁₃₇	—	2	2	no	—
100	C₄×C₈×C₁₃₇	—	2	2	no	—
101	#95×C₁₃₇	—	2	2	no	—
102	#95×C₁₃₇	—	2	2	no	—
103	C₂⁴×C₁₃₇	—	2	1	no	—
104	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	#41×C₁₃₇	—	4	8	no	—
106	#41×C₁₃₇	—	4	8	no	—
107	C₂×Q₈×C₁₃₇	—	2	4	no	—
108	C₂×Q₈×C₁₃₇	—	2	1	no	—
109	C₂×Q₈×C₁₃₇	—	2	1	no	—
110	C₂×Q₈×C₁₃₇	—	2	1	no	—
111	C₂×Q₈×C₁₃₇	—	2	2	no	—
112	C₂×Q₈×C₁₃₇	—	2	2	no	—
113	C₂×Q₈×C₁₃₇	—	2	2	no	—
114	C₂×Q₈×C₁₃₇	—	2	2	no	—
115	C₂×Q₈×C₁₃₇	—	2	2	no	—
116	C₂×Q₈×C₁₃₇	—	2	2	no	—
117	C₂×Q₈×C₁₃₇	—	4	8	no	—
118	C₂×Q₈×C₁₃₇	—	4	8	no	—
119	C₂×Q₈×C₁₃₇	—	4	2	no	—
120	C₂×Q₈×C₁₃₇	—	4	2	no	—
121	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
122	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
123	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
124	C₂×D₄×C₁₃₇	—	2	4	no	—
125	C₂×D₄×C₁₃₇	—	2	1	no	—
126	C₂×D₄×C₁₃₇	—	2	2	no	—
127	C₂×D₄×C₁₃₇	—	2	2	no	—
128	C₂×D₄×C₁₃₇	—	4	2	no	—
129	C₂×D₄×C₁₃₇	—	4	2	no	—
130	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₁₃₇	—	2	1	no	—
132	C₂²×C₄×C₁₃₇	—	2	1	no	—
133	C₂²×C₄×C₁₃₇	—	2	2	no	—
134	C₂²×C₄×C₁₃₇	—	2	4	no	—
135	C₂²×C₄×C₁₃₇	—	2	4	no	—
136	C₂²×C₄×C₁₃₇	—	2	1	no	—
137	C₂²×C₄×C₁₃₇	—	2	1	no	—
138	C₂²×C₄×C₁₃₇	—	2	1	no	—
139	C₂²×C₄×C₁₃₇	—	2	2	no	—
140	C₂²×C₄×C₁₃₇	—	2	2	no	—
141	C₂²×C₄×C₁₃₇	—	4	2	no	—
142	C₂²×C₄×C₁₃₇	—	4	2	no	—
143	C₂²×C₄×C₁₃₇	—	4	2	no	—
144	C₂²×C₄×C₁₃₇	—	4	2	no	—
145	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
149	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
150	C₂×C₈×C₁₃₇	—	2	2	no	—
151	C₂×C₈×C₁₃₇	—	2	2	no	—
152	C₂×C₈×C₁₃₇	—	2	2	no	—
153	C₂×C₈×C₁₃₇	—	2	2	no	—
154	C₂×C₈×C₁₃₇	—	2	2	no	—
155	C₂×C₈×C₁₃₇	—	2	2	no	—
156	C₄⋊C₄×C₁₃₇	—	2	8	no	—
157	C₄⋊C₄×C₁₃₇	—	2	8	no	—
158	C₄⋊C₄×C₁₃₇	—	2	4	no	—
159	C₄⋊C₄×C₁₃₇	—	2	2	no	—
160	C₄⋊C₄×C₁₃₇	—	2	2	no	—
161	C₄⋊C₄×C₁₃₇	—	2	2	no	—
162	C₄⋊C₄×C₁₃₇	—	2	2	no	—
163	C₄⋊C₄×C₁₃₇	—	4	4	no	—
164	C₄⋊C₄×C₁₃₇	—	8	2	no	—
165	C₄⋊C₄×C₁₃₇	—	8	2	no	—
166	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
167	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
168	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
169	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
170	C₄²×C₁₃₇	—	2	1	no	—
171	C₄²×C₁₃₇	—	2	1	no	—
172	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
173	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
174	C₂³×C₁₃₇	—	2	1	no	—
175	C₂³×C₁₃₇	—	2	1	no	—
176	C₂³×C₁₃₇	—	2	2	no	—
177	C₂³×C₁₃₇	—	2	2	no	—
178	C₂³×C₁₃₇	—	2	2	no	—
179	C₂³×C₁₃₇	—	4	1	no	—
180	C₂³×C₁₃₇	—	4	1	no	—
181	C₂³×C₁₃₇	—	4	2	no	—
182	C₂³×C₁₃₇	—	4	2	no	—
183	C₂³×C₁₃₇	#124	1	2	yes	—
184	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
185	Q₈×C₁₃₇	—	2	4	no	—
186	Q₈×C₁₃₇	—	2	4	no	—
187	Q₈×C₁₃₇	—	2	4	no	—
188	Q₈×C₁₃₇	—	4	8	no	—
189	Q₈×C₁₃₇	—	4	2	no	—
190	Q₈×C₁₃₇	—	4	2	no	—
191	Q₈×C₁₃₇	—	4	2	no	—
192	D₄×C₁₃₇	—	2	4	no	—
193	D₄×C₁₃₇	—	4	8	no	—
194	D₄×C₁₃₇	—	4	2	no	—
195	C₂×C₄×C₁₃₇	—	2	1	no	—
196	C₂×C₄×C₁₃₇	—	2	1	no	—
197	C₂×C₄×C₁₃₇	—	2	2	no	—
198	C₂×C₄×C₁₃₇	—	2	2	no	—
199	C₂×C₄×C₁₃₇	—	2	4	no	—
200	C₂×C₄×C₁₃₇	—	2	4	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	—	4	8	no	—
208	C₂×C₄×C₁₃₇	—	4	8	no	—
209	C₂×C₄×C₁₃₇	—	4	1	no	—
210	C₂×C₄×C₁₃₇	—	4	1	no	—
211	C₂×C₄×C₁₃₇	—	4	1	no	—
212	C₂×C₄×C₁₃₇	—	4	1	no	—
213	C₂×C₄×C₁₃₇	—	4	1	no	—
214	C₂×C₄×C₁₃₇	—	4	1	no	—
215	C₂×C₄×C₁₃₇	—	4	2	no	—
216	C₂×C₄×C₁₃₇	—	4	2	no	—
217	C₂×C₄×C₁₃₇	—	4	2	no	—
218	C₂×C₄×C₁₃₇	—	4	2	no	—
219	C₂×C₄×C₁₃₇	—	4	2	no	—
220	C₂×C₄×C₁₃₇	—	4	2	no	—
221	C₂×C₄×C₁₃₇	#120	1	4	yes	—
222	C₂×C₄×C₁₃₇	#124	1	2	yes	—
223	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
226	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
227	C₈×C₁₃₇	—	4	2	no	—
228	C₈×C₁₃₇	—	4	2	no	—
229	C₂²×C₁₃₇	—	2	4	no	—
230	C₂²×C₁₃₇	—	2	2	no	—
231	C₂²×C₁₃₇	—	4	1	no	—
232	C₂²×C₁₃₇	—	4	1	no	—
233	C₂²×C₁₃₇	—	4	2	no	—
234	C₂²×C₁₃₇	—	4	2	no	—
235	C₂²×C₁₃₇	—	4	4	no	—
236	C₂²×C₁₃₇	—	8	2	no	—
237	C₂²×C₁₃₇	—	8	1	no	—
238	C₂²×C₁₃₇	—	8	1	no	—
239	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
240	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
241	C₂²×C₁₃₇	#535	1	1	yes	characteristic
242	C₂²×C₁₃₇	#577	1	2	yes	—
243	C₂²×C₁₃₇	#578	1	2	yes	—
244	C₄×C₁₃₇	—	2	2	no	—
245	C₄×C₁₃₇	—	2	4	no	—
246	C₄×C₁₃₇	—	4	2	no	—
247	C₄×C₁₃₇	—	4	2	no	—
248	C₄×C₁₃₇	—	4	2	no	—
249	C₄×C₁₃₇	—	4	2	no	—
250	C₄×C₁₃₇	—	8	2	no	—
251	C₄×C₁₃₇	—	8	2	no	—
252	C₄×C₁₃₇	#546	1	2	yes	—
253	C₄×C₁₃₇	#546	1	2	yes	—
254	C₂×C₁₃₇	—	4	2	no	—
255	C₂×C₁₃₇	—	8	2	no	—
256	C₂×C₁₃₇	#2435	1	1	yes	characteristic
257	C₂×C₁₃₇	#3013	1	2	yes	—
258	C₂×C₁₃₇	#3137	1	2	yes	—
259	C₂×C₁₃₇	#3221	1	1	yes	characteristic
260	C₂×C₁₃₇	#3223	1	1	yes	characteristic
261	#25868	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₁₃₇	#25868	1	1	yes	characteristic, direct product factor, Sylow
263	C₂×#536	C₂×C₁₃₇	1	1	yes	characteristic
264	C₂×#491	C₂×C₁₃₇	1	1	yes	characteristic, complemented
265	C₂×#488	C₂×C₁₃₇	1	1	yes	characteristic, complemented
266	#2629	C₂×C₁₃₇	1	1	yes	characteristic, complemented
267	#2043	C₂×C₁₃₇	1	2	yes	complemented
268	#2042	C₂×C₁₃₇	1	2	yes	complemented
269	#1959	C₂×C₁₃₇	1	1	yes	characteristic, complemented
270	#1956	C₂×C₁₃₇	1	1	yes	characteristic, complemented
271	#1788	C₂×C₁₃₇	1	2	yes	—
272	#1787	C₂×C₁₃₇	1	2	yes	complemented
273	#1750	C₂×C₁₃₇	1	1	yes	characteristic, complemented
274	C₂²×#41	C₂²×C₁₃₇	1	1	yes	characteristic
275	C₂³×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
276	#536	—	2	4	no	—
277	#536	—	2	4	no	—
278	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
279	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
280	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
281	C₂×#124	C₂²×C₁₃₇	1	1	yes	characteristic
282	C₂×#124	C₂²×C₁₃₇	1	1	yes	characteristic
283	C₂×#123	—	2	2	no	—
284	C₂×C₂²⋊Q₈	—	2	2	no	—
285	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
286	#491	—	2	4	no	—
287	#488	—	2	4	no	—
288	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
289	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
290	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
291	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
292	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
293	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
294	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
295	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
296	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
297	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
298	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
299	#398	—	2	2	no	—
300	#396	—	2	2	no	—
301	#389	C₂²×C₁₃₇	1	2	yes	—
302	#384	C₂²×C₁₃₇	1	1	yes	characteristic
303	#380	C₂²×C₁₃₇	1	2	yes	—
304	#336	C₂²×C₁₃₇	1	2	yes	—
305	C₂×#41	—	2	2	no	—
306	C₂×#41	—	2	2	no	—
307	C₂×#41	—	2	4	no	—
308	C₂×#41	—	4	2	no	—
309	C₂²×Q₈	—	2	2	no	—
310	C₂²×Q₈	—	2	2	no	—
311	C₂²×Q₈	—	2	1	no	—
312	C₂²×Q₈	—	2	4	no	—
313	C₂²×Q₈	—	4	2	no	—
314	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
315	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
316	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
319	C₂³×C₄	—	2	1	no	—
320	C₂³×C₄	—	2	1	no	—
321	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
323	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
324	C₄⋊Q₈	—	2	2	no	—
325	C₄⋊Q₈	—	2	2	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	D₄×C₁₃₇	1	4	yes	—
328	#125	—	2	2	no	—
329	#125	—	2	2	no	—
330	#124	—	2	2	no	—
331	#124	—	2	2	no	—
332	#124	—	2	2	no	—
333	#124	D₄×C₁₃₇	1	4	yes	—
334	#123	—	2	16	no	—
335	C₂²⋊Q₈	—	2	16	no	—
336	#117	—	2	2	no	—
337	#117	—	2	2	no	—
338	C₂×C₄⋊C₄	—	2	2	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₄⋊C₄	—	2	2	no	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
350	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
352	#103	—	2	4	no	—
353	#103	—	2	4	no	—
354	#103	—	2	4	no	—
355	#103	—	4	2	no	—
356	#103	—	4	2	no	—
357	#103	—	4	2	no	—
358	#102	—	2	4	no	—
359	#102	—	4	2	no	—
360	C₂²⋊C₈	—	2	4	no	—
361	#97	—	2	2	no	—
362	C₄×C₈	—	2	2	no	—
363	#95	—	2	2	no	—
364	#95	—	2	2	no	—
365	C₂⁴	—	2	1	no	—
366	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
367	#41	—	4	8	no	—
368	#41	—	4	8	no	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	4	8	no	—
380	C₂×Q₈	—	4	8	no	—
381	C₂×Q₈	—	4	2	no	—
382	C₂×Q₈	—	4	2	no	—
383	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
384	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
385	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	4	2	no	—
391	C₂×D₄	—	4	2	no	—
392	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	4	2	no	—
404	C₂²×C₄	—	4	2	no	—
405	C₂²×C₄	—	4	2	no	—
406	C₂²×C₄	—	4	2	no	—
407	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
408	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
409	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
410	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	2	8	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	4	4	no	—
426	C₄⋊C₄	—	8	2	no	—
427	C₄⋊C₄	—	8	2	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	8	2	no	—
432	C₄²	—	2	1	no	—
433	C₄²	—	2	1	no	—
434	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
435	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	1	no	—
438	C₂³	—	2	2	no	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	4	1	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	2	no	—
445	C₂³	#124×C₁₃₇	1	2	yes	—
446	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
447	Q₈	—	2	4	no	—
448	Q₈	—	2	4	no	—
449	Q₈	—	2	4	no	—
450	Q₈	—	4	8	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	D₄	—	2	4	no	—
455	D₄	—	4	8	no	—
456	D₄	—	4	2	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	—	4	8	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	#120×C₁₃₇	1	4	yes	—
484	C₂×C₄	#124×C₁₃₇	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
487	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
488	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
489	C₈	—	4	2	no	—
490	C₈	—	4	2	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	4	no	—
498	C₂²	—	8	2	no	—
499	C₂²	—	8	1	no	—
500	C₂²	—	8	1	no	—
501	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
502	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
503	C₂²	#535×C₁₃₇	1	1	yes	characteristic
504	C₂²	#577×C₁₃₇	1	2	yes	—
505	C₂²	#578×C₁₃₇	1	2	yes	—
506	C₄	—	2	2	no	—
507	C₄	—	2	4	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	8	2	no	—
513	C₄	—	8	2	no	—
514	C₄	#546×C₁₃₇	1	2	yes	—
515	C₄	#546×C₁₃₇	1	2	yes	—
516	C₂	—	4	2	no	—
517	C₂	—	8	2	no	—
518	C₂	#2435×C₁₃₇	1	1	yes	characteristic
519	C₂	#3013×C₁₃₇	1	2	yes	—
520	C₂	#3137×C₁₃₇	1	2	yes	—
521	C₂	#3221×C₁₃₇	1	1	yes	characteristic
522	C₂	#3223×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×#41×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×#117×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#491×C₁₃₇	C₂	semidirect
3. C₂×#488×C₁₃₇	C₂	semidirect
4. #2629×C₁₃₇	C₂	semidirect
5. #2043×C₁₃₇	C₂	semidirect
6. #2042×C₁₃₇	C₂	semidirect
7. #1959×C₁₃₇	C₂	semidirect
8. #1956×C₁₃₇	C₂	semidirect
10. #1787×C₁₃₇	C₂	semidirect
11. #1750×C₁₃₇	C₂	semidirect
261. #25868	C₁₃₇	direct (N × H)
262. C₁₃₇	#25868	direct (N × H)
264. C₂×#491	C₂×C₁₃₇	semidirect
265. C₂×#488	C₂×C₁₃₇	semidirect
266. #2629	C₂×C₁₃₇	semidirect
267. #2043	C₂×C₁₃₇	semidirect
268. #2042	C₂×C₁₃₇	semidirect
269. #1959	C₂×C₁₃₇	semidirect
270. #1956	C₂×C₁₃₇	semidirect
272. #1787	C₂×C₁₃₇	semidirect
273. #1750	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#536×C₁₃₇	2
C₂×#491×C₁₃₇	2
C₂×#488×C₁₃₇	2
#2629×C₁₃₇	2
#2043×C₁₃₇	2
#2042×C₁₃₇	2
#1959×C₁₃₇	2
#1956×C₁₃₇	2
#1788×C₁₃₇	2
#1787×C₁₃₇	2
#1750×C₁₃₇	2
#25868	137

Subgroups — C137×#25868 #46875254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25868 #46875254