Subgroups — C₁₃₇×#25315 #46874701

Order 35072.

732 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3138×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#3078×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#460×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2962×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2063×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2046×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1977×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1965×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#1950×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1931×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	C₄×#101×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
13	#533×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	#490×C₁₃₇	—	2	2	no	—
17	#486×C₁₃₇	—	2	1	no	—
18	#486×C₁₃₇	—	2	1	no	—
19	#479×C₁₃₇	—	2	2	no	—
20	#477×C₁₃₇	—	2	1	no	—
21	#475×C₁₃₇	—	2	1	no	—
22	#460×C₁₃₇	—	2	1	no	—
23	#460×C₁₃₇	—	2	2	no	—
24	#460×C₁₃₇	—	2	1	no	—
25	#446×C₁₃₇	—	2	2	no	—
26	#446×C₁₃₇	—	2	1	no	—
27	C₈×Q₈×C₁₃₇	—	2	1	no	—
28	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
29	#428×C₁₃₇	C₂²	1	2	yes	—
30	#424×C₁₃₇	C₂²	1	2	yes	—
31	#423×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
33	#419×C₁₃₇	C₂²	1	2	yes	—
34	#418×C₁₃₇	C₂²	1	2	yes	—
35	#418×C₁₃₇	C₂²	1	2	yes	—
36	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
37	C₂×#101×C₁₃₇	C₂²	1	1	yes	characteristic
38	C₂×#101×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₂×#100×C₁₃₇	C₂²	1	2	yes	—
40	C₂×#100×C₁₃₇	C₂²	1	2	yes	—
41	#405×C₁₃₇	C₂²	1	2	yes	—
42	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
44	#343×C₁₃₇	C₂²	1	1	yes	characteristic
45	#343×C₁₃₇	C₂²	1	1	yes	characteristic
46	#341×C₁₃₇	C₂²	1	2	yes	—
47	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₁₃₇	—	2	1	no	—
49	C₂²×D₄×C₁₃₇	—	2	1	no	—
50	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
51	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
52	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
53	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
54	C₂×SD₁₆×C₁₃₇	—	4	2	no	—
55	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
59	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
63	#127×C₁₃₇	—	2	1	no	—
64	#127×C₁₃₇	—	2	1	no	—
65	#127×C₁₃₇	—	2	1	no	—
66	#127×C₁₃₇	—	2	1	no	—
67	#125×C₁₃₇	D₄	1	2	yes	—
68	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
69	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
70	C₄×Q₈×C₁₃₇	—	2	1	no	—
71	C₄×Q₈×C₁₃₇	—	2	1	no	—
72	C₄×Q₈×C₁₃₇	—	2	1	no	—
73	C₄×Q₈×C₁₃₇	—	2	1	no	—
74	C₄×Q₈×C₁₃₇	—	2	1	no	—
75	C₄×Q₈×C₁₃₇	—	2	1	no	—
76	#117×C₁₃₇	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₁₃₇	—	2	1	no	—
80	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
81	#108×C₁₃₇	—	4	2	no	—
82	#107×C₁₃₇	—	2	2	no	—
83	#106×C₁₃₇	—	2	2	no	—
84	C₄⋊C₈×C₁₃₇	—	2	2	no	—
85	C₄⋊C₈×C₁₃₇	—	2	2	no	—
86	C₄⋊C₈×C₁₃₇	—	2	1	no	—
87	C₄⋊C₈×C₁₃₇	—	2	1	no	—
88	C₄⋊C₈×C₁₃₇	D₄	1	1	yes	characteristic
89	C₄⋊C₈×C₁₃₇	D₄	1	1	yes	characteristic
90	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
91	#103×C₁₃₇	—	2	2	no	—
92	#103×C₁₃₇	—	2	2	no	—
93	#102×C₁₃₇	—	2	2	no	—
94	#102×C₁₃₇	—	2	2	no	—
95	#102×C₁₃₇	—	2	2	no	—
96	#102×C₁₃₇	—	2	2	no	—
97	#102×C₁₃₇	—	2	2	no	—
98	#102×C₁₃₇	—	2	2	no	—
99	#101×C₁₃₇	—	2	2	no	—
100	#101×C₁₃₇	—	2	2	no	—
101	#100×C₁₃₇	—	2	4	no	—
102	#100×C₁₃₇	—	4	2	no	—
103	#97×C₁₃₇	—	2	2	no	—
104	#97×C₁₃₇	—	2	1	no	—
105	#97×C₁₃₇	—	2	1	no	—
106	C₄×C₈×C₁₃₇	—	2	2	no	—
107	C₄×C₈×C₁₃₇	—	2	1	no	—
108	C₄×C₈×C₁₃₇	—	2	1	no	—
109	C₂⁴×C₁₃₇	—	2	2	no	—
110	C₂×Q₈×C₁₃₇	—	2	1	no	—
111	C₂×Q₈×C₁₃₇	—	2	1	no	—
112	C₂×Q₈×C₁₃₇	—	2	1	no	—
113	C₂×Q₈×C₁₃₇	—	2	1	no	—
114	C₂×Q₈×C₁₃₇	—	2	1	no	—
115	C₂×Q₈×C₁₃₇	—	2	1	no	—
116	C₂×Q₈×C₁₃₇	—	2	1	no	—
117	C₂×Q₈×C₁₃₇	#41	1	2	yes	—
118	C₂×D₄×C₁₃₇	—	2	2	no	—
119	C₂×D₄×C₁₃₇	—	2	2	no	—
120	C₂×D₄×C₁₃₇	—	2	2	no	—
121	C₂×D₄×C₁₃₇	—	2	2	no	—
122	C₂×D₄×C₁₃₇	—	2	2	no	—
123	C₂×D₄×C₁₃₇	—	2	2	no	—
124	C₂×D₄×C₁₃₇	—	4	1	no	—
125	C₂×D₄×C₁₃₇	—	4	1	no	—
126	C₂×D₄×C₁₃₇	—	4	1	no	—
127	C₂×D₄×C₁₃₇	—	4	1	no	—
128	C₂×D₄×C₁₃₇	—	8	2	no	—
129	C₂×D₄×C₁₃₇	—	8	2	no	—
130	C₂²×C₄×C₁₃₇	—	2	1	no	—
131	C₂²×C₄×C₁₃₇	—	2	1	no	—
132	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
133	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
134	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
135	SD₁₆×C₁₃₇	—	4	4	no	—
136	SD₁₆×C₁₃₇	—	4	4	no	—
137	M₁₆×C₁₃₇	—	2	1	no	—
138	M₁₆×C₁₃₇	—	2	2	no	—
139	M₁₆×C₁₃₇	—	2	1	no	—
140	M₁₆×C₁₃₇	—	2	1	no	—
141	M₁₆×C₁₃₇	—	2	1	no	—
142	M₁₆×C₁₃₇	—	2	1	no	—
143	M₁₆×C₁₃₇	—	2	2	no	—
144	M₁₆×C₁₃₇	—	2	1	no	—
145	M₁₆×C₁₃₇	—	2	1	no	—
146	M₁₆×C₁₃₇	#41	1	2	yes	—
147	C₂×C₈×C₁₃₇	—	2	1	no	—
148	C₂×C₈×C₁₃₇	—	2	2	no	—
149	C₂×C₈×C₁₃₇	—	2	2	no	—
150	C₂×C₈×C₁₃₇	—	2	1	no	—
151	C₂×C₈×C₁₃₇	—	2	1	no	—
152	C₂×C₈×C₁₃₇	—	2	1	no	—
153	C₂×C₈×C₁₃₇	—	2	2	no	—
154	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₁₃₇	—	2	2	no	—
157	C₄⋊C₄×C₁₃₇	—	2	2	no	—
158	C₄⋊C₄×C₁₃₇	—	2	1	no	—
159	C₄⋊C₄×C₁₃₇	—	2	1	no	—
160	C₄⋊C₄×C₁₃₇	—	2	1	no	—
161	C₄⋊C₄×C₁₃₇	—	2	1	no	—
162	C₄⋊C₄×C₁₃₇	—	4	1	no	—
163	C₄⋊C₄×C₁₃₇	—	4	1	no	—
164	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
166	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
167	C₄²×C₁₃₇	—	2	2	no	—
168	C₄²×C₁₃₇	—	4	1	no	—
169	C₄²×C₁₃₇	—	4	1	no	—
170	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
174	C₂³×C₁₃₇	—	4	2	no	—
175	C₂³×C₁₃₇	—	4	2	no	—
176	C₂³×C₁₃₇	—	4	2	no	—
177	C₂³×C₁₃₇	—	8	2	no	—
178	C₂³×C₁₃₇	—	8	2	no	—
179	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
180	Q₈×C₁₃₇	—	2	2	no	—
181	Q₈×C₁₃₇	—	2	2	no	—
182	Q₈×C₁₃₇	—	4	1	no	—
183	Q₈×C₁₃₇	—	4	1	no	—
184	Q₈×C₁₃₇	—	4	2	no	—
185	Q₈×C₁₃₇	—	8	1	no	—
186	D₄×C₁₃₇	—	4	2	no	—
187	D₄×C₁₃₇	—	4	2	no	—
188	D₄×C₁₃₇	—	4	2	no	—
189	D₄×C₁₃₇	—	4	2	no	—
190	D₄×C₁₃₇	—	8	2	no	—
191	D₄×C₁₃₇	—	8	2	no	—
192	D₄×C₁₃₇	—	8	2	no	—
193	D₄×C₁₃₇	—	8	2	no	—
194	C₂×C₄×C₁₃₇	—	2	2	no	—
195	C₂×C₄×C₁₃₇	—	2	2	no	—
196	C₂×C₄×C₁₃₇	—	2	2	no	—
197	C₂×C₄×C₁₃₇	—	2	2	no	—
198	C₂×C₄×C₁₃₇	—	4	1	no	—
199	C₂×C₄×C₁₃₇	—	4	1	no	—
200	C₂×C₄×C₁₃₇	—	4	1	no	—
201	C₂×C₄×C₁₃₇	—	4	1	no	—
202	C₂×C₄×C₁₃₇	—	8	1	no	—
203	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
204	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
205	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
206	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
207	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
213	C₈×C₁₃₇	—	2	2	no	—
214	C₈×C₁₃₇	—	2	2	no	—
215	C₈×C₁₃₇	—	2	2	no	—
216	C₈×C₁₃₇	—	4	1	no	—
217	C₈×C₁₃₇	—	4	1	no	—
218	C₈×C₁₃₇	—	4	1	no	—
219	C₈×C₁₃₇	—	4	2	no	—
220	C₂²×C₁₃₇	—	2	1	no	—
221	C₂²×C₁₃₇	—	2	1	no	—
222	C₂²×C₁₃₇	—	8	2	no	—
223	C₂²×C₁₃₇	—	8	2	no	—
224	C₂²×C₁₃₇	—	8	2	no	—
225	C₂²×C₁₃₇	—	16	2	no	—
226	C₂²×C₁₃₇	—	16	2	no	—
227	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
228	C₂²×C₁₃₇	#529	1	1	yes	characteristic
229	C₂²×C₁₃₇	#542	1	1	yes	characteristic
230	C₄×C₁₃₇	—	2	1	no	—
231	C₄×C₁₃₇	—	2	1	no	—
232	C₄×C₁₃₇	—	2	1	no	—
233	C₄×C₁₃₇	—	4	2	no	—
234	C₄×C₁₃₇	—	4	2	no	—
235	C₄×C₁₃₇	—	4	2	no	—
236	C₄×C₁₃₇	—	8	1	no	—
237	C₄×C₁₃₇	#471	1	2	yes	—
238	C₄×C₁₃₇	#546	1	1	yes	characteristic
239	C₄×C₁₃₇	#548	1	1	yes	characteristic
240	C₄×C₁₃₇	#548	1	1	yes	characteristic
241	C₄×C₁₃₇	#550	1	1	yes	characteristic
242	C₂×C₁₃₇	—	2	1	no	—
243	C₂×C₁₃₇	—	2	1	no	—
244	C₂×C₁₃₇	—	16	2	no	—
245	C₂×C₁₃₇	#2426	1	1	yes	characteristic
246	C₂×C₁₃₇	C₂×#471	1	1	yes	characteristic
247	C₂×C₁₃₇	#3068	1	1	yes	characteristic
248	#25315	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₁₃₇	#25315	1	1	yes	characteristic, direct product factor, Sylow
250	#3138	C₂×C₁₃₇	1	1	yes	characteristic
251	#3078	C₂×C₁₃₇	1	1	yes	characteristic, complemented
252	C₂×#460	C₂×C₁₃₇	1	1	yes	characteristic
253	#2962	C₂×C₁₃₇	1	1	yes	characteristic, complemented
254	#2063	C₂×C₁₃₇	1	2	yes	complemented
255	#2046	C₂×C₁₃₇	1	2	yes	complemented
256	#1977	C₂×C₁₃₇	1	2	yes	complemented
257	#1965	C₂×C₁₃₇	1	1	yes	characteristic
258	#1950	C₂×C₁₃₇	1	1	yes	characteristic, complemented
259	#1931	C₂×C₁₃₇	1	2	yes	complemented
260	C₄×#101	C₂×C₁₃₇	1	1	yes	characteristic, complemented
261	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
262	#533	C₂²×C₁₃₇	1	1	yes	characteristic
263	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
264	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
265	#490	—	2	2	no	—
266	#486	—	2	1	no	—
267	#486	—	2	1	no	—
268	#479	—	2	2	no	—
269	#477	—	2	1	no	—
270	#475	—	2	1	no	—
271	#460	—	2	1	no	—
272	#460	—	2	2	no	—
273	#460	—	2	1	no	—
274	#446	—	2	2	no	—
275	#446	—	2	1	no	—
276	C₈×Q₈	—	2	1	no	—
277	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
278	#428	C₂²×C₁₃₇	1	2	yes	—
279	#424	C₂²×C₁₃₇	1	2	yes	—
280	#423	C₂²×C₁₃₇	1	1	yes	characteristic
281	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
282	#419	C₂²×C₁₃₇	1	2	yes	—
283	#418	C₂²×C₁₃₇	1	2	yes	—
284	#418	C₂²×C₁₃₇	1	2	yes	—
285	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
286	C₂×#101	C₂²×C₁₃₇	1	1	yes	characteristic
287	C₂×#101	C₂²×C₁₃₇	1	1	yes	characteristic
288	C₂×#100	C₂²×C₁₃₇	1	2	yes	—
289	C₂×#100	C₂²×C₁₃₇	1	2	yes	—
290	#405	C₂²×C₁₃₇	1	2	yes	—
291	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
292	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
293	#343	C₂²×C₁₃₇	1	1	yes	characteristic
294	#343	C₂²×C₁₃₇	1	1	yes	characteristic
295	#341	C₂²×C₁₃₇	1	2	yes	—
296	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
297	C₂²×D₄	—	2	1	no	—
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
300	C₂×SD₁₆	—	2	4	no	—
301	C₂×SD₁₆	—	2	2	no	—
302	C₂×SD₁₆	—	2	2	no	—
303	C₂×SD₁₆	—	4	2	no	—
304	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
305	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
306	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
307	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
308	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
310	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
311	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
312	#127	—	2	1	no	—
313	#127	—	2	1	no	—
314	#127	—	2	1	no	—
315	#127	—	2	1	no	—
316	#125	D₄×C₁₃₇	1	2	yes	—
317	C₂²⋊Q₈	—	2	1	no	—
318	C₂²⋊Q₈	—	2	1	no	—
319	C₄×Q₈	—	2	1	no	—
320	C₄×Q₈	—	2	1	no	—
321	C₄×Q₈	—	2	1	no	—
322	C₄×Q₈	—	2	1	no	—
323	C₄×Q₈	—	2	1	no	—
324	C₄×Q₈	—	2	1	no	—
325	#117	C₂³×C₁₃₇	1	2	yes	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
330	#108	—	4	2	no	—
331	#107	—	2	2	no	—
332	#106	—	2	2	no	—
333	C₄⋊C₈	—	2	2	no	—
334	C₄⋊C₈	—	2	2	no	—
335	C₄⋊C₈	—	2	1	no	—
336	C₄⋊C₈	—	2	1	no	—
337	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
338	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
339	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
340	#103	—	2	2	no	—
341	#103	—	2	2	no	—
342	#102	—	2	2	no	—
343	#102	—	2	2	no	—
344	#102	—	2	2	no	—
345	#102	—	2	2	no	—
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	#101	—	2	2	no	—
349	#101	—	2	2	no	—
350	#100	—	2	4	no	—
351	#100	—	4	2	no	—
352	#97	—	2	2	no	—
353	#97	—	2	1	no	—
354	#97	—	2	1	no	—
355	C₄×C₈	—	2	2	no	—
356	C₄×C₈	—	2	1	no	—
357	C₄×C₈	—	2	1	no	—
358	C₂⁴	—	2	2	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	#41×C₁₃₇	1	2	yes	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	4	1	no	—
374	C₂×D₄	—	4	1	no	—
375	C₂×D₄	—	4	1	no	—
376	C₂×D₄	—	4	1	no	—
377	C₂×D₄	—	8	2	no	—
378	C₂×D₄	—	8	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
382	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
383	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
384	SD₁₆	—	4	4	no	—
385	SD₁₆	—	4	4	no	—
386	M₁₆	—	2	1	no	—
387	M₁₆	—	2	2	no	—
388	M₁₆	—	2	1	no	—
389	M₁₆	—	2	1	no	—
390	M₁₆	—	2	1	no	—
391	M₁₆	—	2	1	no	—
392	M₁₆	—	2	2	no	—
393	M₁₆	—	2	1	no	—
394	M₁₆	—	2	1	no	—
395	M₁₆	#41×C₁₃₇	1	2	yes	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
404	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	4	1	no	—
412	C₄⋊C₄	—	4	1	no	—
413	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
414	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₄²	—	2	2	no	—
417	C₄²	—	4	1	no	—
418	C₄²	—	4	1	no	—
419	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
420	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
421	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
422	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	8	2	no	—
427	C₂³	—	8	2	no	—
428	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
429	Q₈	—	2	2	no	—
430	Q₈	—	2	2	no	—
431	Q₈	—	4	2	no	—
432	Q₈	—	4	1	no	—
433	Q₈	—	4	1	no	—
434	Q₈	—	8	1	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	4	2	no	—
439	D₄	—	8	2	no	—
440	D₄	—	8	2	no	—
441	D₄	—	8	2	no	—
442	D₄	—	8	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	8	1	no	—
452	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
453	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
454	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
455	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
456	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
457	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
458	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
459	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
460	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
461	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
462	C₈	—	2	2	no	—
463	C₈	—	2	2	no	—
464	C₈	—	2	2	no	—
465	C₈	—	4	1	no	—
466	C₈	—	4	1	no	—
467	C₈	—	4	1	no	—
468	C₈	—	4	2	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	8	2	no	—
472	C₂²	—	8	2	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	16	2	no	—
475	C₂²	—	16	2	no	—
476	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
477	C₂²	#529×C₁₃₇	1	1	yes	characteristic
478	C₂²	#542×C₁₃₇	1	1	yes	characteristic
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	2	no	—
485	C₄	—	8	1	no	—
486	C₄	#471×C₁₃₇	1	2	yes	—
487	C₄	#546×C₁₃₇	1	1	yes	characteristic
488	C₄	#548×C₁₃₇	1	1	yes	characteristic
489	C₄	#548×C₁₃₇	1	1	yes	characteristic
490	C₄	#550×C₁₃₇	1	1	yes	characteristic
491	C₂	—	2	1	no	—
492	C₂	—	2	1	no	—
493	C₂	—	16	2	no	—
494	C₂	#2426×C₁₃₇	1	1	yes	characteristic
495	C₂	C₂×#471×C₁₃₇	1	1	yes	characteristic
496	C₂	#3068×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #3078×C₁₃₇	C₂	semidirect
4. #2962×C₁₃₇	C₂	semidirect
5. #2063×C₁₃₇	C₂	semidirect
6. #2046×C₁₃₇	C₂	semidirect
7. #1977×C₁₃₇	C₂	semidirect
9. #1950×C₁₃₇	C₂	semidirect
10. #1931×C₁₃₇	C₂	semidirect
11. C₄×#101×C₁₃₇	C₂	semidirect
248. #25315	C₁₃₇	direct (N × H)
249. C₁₃₇	#25315	direct (N × H)
251. #3078	C₂×C₁₃₇	semidirect
253. #2962	C₂×C₁₃₇	semidirect
254. #2063	C₂×C₁₃₇	semidirect
255. #2046	C₂×C₁₃₇	semidirect
256. #1977	C₂×C₁₃₇	semidirect
258. #1950	C₂×C₁₃₇	semidirect
259. #1931	C₂×C₁₃₇	semidirect
260. C₄×#101	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3138×C₁₃₇	2
#3078×C₁₃₇	2
C₂×#460×C₁₃₇	2
#2962×C₁₃₇	2
#2063×C₁₃₇	2
#2046×C₁₃₇	2
#1977×C₁₃₇	2
#1965×C₁₃₇	2
#1950×C₁₃₇	2
#1931×C₁₃₇	2
C₄×#101×C₁₃₇	2
#25315	137

Subgroups — C137×#25315 #46874701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25315 #46874701