Subgroups — C₁₃₇×#25312 #46874698

Order 35072.

764 conjugacy classes of subgroups.

The subgroup lattice has 518 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, complemented
2	#3077×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#459×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	#2045×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1978×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1965×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1950×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1931×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	C₄×#100×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×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×D₄×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	#478×C₁₃₇	—	2	2	no	—
20	#476×C₁₃₇	—	2	1	no	—
21	#474×C₁₃₇	—	2	1	no	—
22	#459×C₁₃₇	—	2	1	no	—
23	#459×C₁₃₇	—	2	2	no	—
24	#459×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	#418×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×#100×C₁₃₇	C₂²	1	1	yes	characteristic
38	C₂×#100×C₁₃₇	C₂²	1	2	yes	—
39	C₂×#100×C₁₃₇	C₂²	1	1	yes	characteristic
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×D₄×C₁₃₇	—	2	1	no	—
48	C₂²×D₄×C₁₃₇	—	2	1	no	—
49	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
50	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₁₃₇	—	2	1	no	—
52	C₂×D₈×C₁₃₇	—	2	4	no	—
53	C₂×D₈×C₁₃₇	—	2	2	no	—
54	C₂×D₈×C₁₃₇	—	2	2	no	—
55	C₂×D₈×C₁₃₇	—	4	2	no	—
56	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
60	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
64	#127×C₁₃₇	—	2	1	no	—
65	#127×C₁₃₇	—	2	1	no	—
66	#127×C₁₃₇	—	2	1	no	—
67	#127×C₁₃₇	—	2	1	no	—
68	#125×C₁₃₇	D₄	1	2	yes	—
69	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
70	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
71	C₄×Q₈×C₁₃₇	—	2	1	no	—
72	C₄×Q₈×C₁₃₇	—	2	1	no	—
73	C₄×D₄×C₁₃₇	—	2	1	no	—
74	C₄×D₄×C₁₃₇	—	2	1	no	—
75	C₄×D₄×C₁₃₇	—	2	1	no	—
76	C₄×D₄×C₁₃₇	—	2	1	no	—
77	#117×C₁₃₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×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	#102×C₁₃₇	—	2	2	no	—
92	#102×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	#100×C₁₃₇	—	2	2	no	—
100	#100×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	1	no	—
110	C₂⁴×C₁₃₇	—	2	2	no	—
111	C₂×Q₈×C₁₃₇	—	2	1	no	—
112	C₂×Q₈×C₁₃₇	—	2	1	no	—
113	C₂×D₄×C₁₃₇	—	2	1	no	—
114	C₂×D₄×C₁₃₇	—	2	1	no	—
115	C₂×D₄×C₁₃₇	—	2	2	no	—
116	C₂×D₄×C₁₃₇	—	2	2	no	—
117	C₂×D₄×C₁₃₇	—	2	1	no	—
118	C₂×D₄×C₁₃₇	—	2	2	no	—
119	C₂×D₄×C₁₃₇	—	2	2	no	—
120	C₂×D₄×C₁₃₇	—	2	1	no	—
121	C₂×D₄×C₁₃₇	—	2	2	no	—
122	C₂×D₄×C₁₃₇	—	2	1	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	C₂×D₄×C₁₃₇	#41	1	2	yes	—
131	C₂²×C₄×C₁₃₇	—	2	1	no	—
132	C₂²×C₄×C₁₃₇	—	4	1	no	—
133	C₂²×C₄×C₁₃₇	—	4	1	no	—
134	C₂²×C₄×C₁₃₇	—	8	2	no	—
135	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
136	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
137	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
138	D₈×C₁₃₇	—	4	4	no	—
139	D₈×C₁₃₇	—	4	4	no	—
140	M₁₆×C₁₃₇	—	2	1	no	—
141	M₁₆×C₁₃₇	—	2	1	no	—
142	M₁₆×C₁₃₇	—	2	1	no	—
143	M₁₆×C₁₃₇	—	2	1	no	—
144	M₁₆×C₁₃₇	—	2	2	no	—
145	M₁₆×C₁₃₇	—	2	1	no	—
146	M₁₆×C₁₃₇	—	2	2	no	—
147	M₁₆×C₁₃₇	—	2	1	no	—
148	M₁₆×C₁₃₇	—	2	1	no	—
149	M₁₆×C₁₃₇	#41	1	2	yes	—
150	C₂×C₈×C₁₃₇	—	2	1	no	—
151	C₂×C₈×C₁₃₇	—	2	2	no	—
152	C₂×C₈×C₁₃₇	—	2	2	no	—
153	C₂×C₈×C₁₃₇	—	2	1	no	—
154	C₂×C₈×C₁₃₇	—	2	1	no	—
155	C₂×C₈×C₁₃₇	—	2	1	no	—
156	C₂×C₈×C₁₃₇	—	2	2	no	—
157	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₁₃₇	—	2	2	no	—
160	C₄⋊C₄×C₁₃₇	—	2	2	no	—
161	C₄⋊C₄×C₁₃₇	—	2	1	no	—
162	C₄⋊C₄×C₁₃₇	—	2	1	no	—
163	C₄⋊C₄×C₁₃₇	—	2	1	no	—
164	C₄⋊C₄×C₁₃₇	—	2	1	no	—
165	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
166	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
167	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
168	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
169	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
170	C₄²×C₁₃₇	—	2	2	no	—
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	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
175	C₂³×C₁₃₇	—	2	2	no	—
176	C₂³×C₁₃₇	—	4	1	no	—
177	C₂³×C₁₃₇	—	4	1	no	—
178	C₂³×C₁₃₇	—	4	2	no	—
179	C₂³×C₁₃₇	—	4	2	no	—
180	C₂³×C₁₃₇	—	4	2	no	—
181	C₂³×C₁₃₇	—	8	1	no	—
182	C₂³×C₁₃₇	—	8	1	no	—
183	C₂³×C₁₃₇	—	8	2	no	—
184	C₂³×C₁₃₇	—	8	2	no	—
185	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
186	Q₈×C₁₃₇	—	4	2	no	—
187	Q₈×C₁₃₇	—	8	1	no	—
188	D₄×C₁₃₇	—	2	2	no	—
189	D₄×C₁₃₇	—	2	2	no	—
190	D₄×C₁₃₇	—	4	2	no	—
191	D₄×C₁₃₇	—	4	1	no	—
192	D₄×C₁₃₇	—	4	2	no	—
193	D₄×C₁₃₇	—	4	2	no	—
194	D₄×C₁₃₇	—	4	1	no	—
195	D₄×C₁₃₇	—	4	2	no	—
196	D₄×C₁₃₇	—	8	2	no	—
197	D₄×C₁₃₇	—	8	2	no	—
198	D₄×C₁₃₇	—	8	2	no	—
199	D₄×C₁₃₇	—	8	2	no	—
200	C₂×C₄×C₁₃₇	—	2	2	no	—
201	C₂×C₄×C₁₃₇	—	2	2	no	—
202	C₂×C₄×C₁₃₇	—	2	2	no	—
203	C₂×C₄×C₁₃₇	—	4	1	no	—
204	C₂×C₄×C₁₃₇	—	4	1	no	—
205	C₂×C₄×C₁₃₇	—	8	2	no	—
206	C₂×C₄×C₁₃₇	—	8	2	no	—
207	C₂×C₄×C₁₃₇	—	8	1	no	—
208	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
209	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
210	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
211	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
212	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
217	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
218	C₈×C₁₃₇	—	2	2	no	—
219	C₈×C₁₃₇	—	2	2	no	—
220	C₈×C₁₃₇	—	2	2	no	—
221	C₈×C₁₃₇	—	4	1	no	—
222	C₈×C₁₃₇	—	4	1	no	—
223	C₈×C₁₃₇	—	4	1	no	—
224	C₈×C₁₃₇	—	4	2	no	—
225	C₂²×C₁₃₇	—	2	1	no	—
226	C₂²×C₁₃₇	—	2	1	no	—
227	C₂²×C₁₃₇	—	4	2	no	—
228	C₂²×C₁₃₇	—	8	2	no	—
229	C₂²×C₁₃₇	—	8	2	no	—
230	C₂²×C₁₃₇	—	8	1	no	—
231	C₂²×C₁₃₇	—	8	1	no	—
232	C₂²×C₁₃₇	—	8	2	no	—
233	C₂²×C₁₃₇	—	8	2	no	—
234	C₂²×C₁₃₇	—	8	2	no	—
235	C₂²×C₁₃₇	—	16	2	no	—
236	C₂²×C₁₃₇	—	16	2	no	—
237	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
238	C₂²×C₁₃₇	#529	1	1	yes	characteristic
239	C₂²×C₁₃₇	#542	1	1	yes	characteristic
240	C₄×C₁₃₇	—	2	1	no	—
241	C₄×C₁₃₇	—	2	1	no	—
242	C₄×C₁₃₇	—	2	1	no	—
243	C₄×C₁₃₇	—	4	2	no	—
244	C₄×C₁₃₇	—	4	2	no	—
245	C₄×C₁₃₇	—	8	1	no	—
246	C₄×C₁₃₇	#472	1	2	yes	—
247	C₄×C₁₃₇	#546	1	1	yes	characteristic
248	C₄×C₁₃₇	#548	1	1	yes	characteristic
249	C₄×C₁₃₇	#548	1	1	yes	characteristic
250	C₄×C₁₃₇	#550	1	1	yes	characteristic
251	C₂×C₁₃₇	—	2	1	no	—
252	C₂×C₁₃₇	—	2	1	no	—
253	C₂×C₁₃₇	—	8	2	no	—
254	C₂×C₁₃₇	—	16	2	no	—
255	C₂×C₁₃₇	#2426	1	1	yes	characteristic
256	C₂×C₁₃₇	C₂×#472	1	1	yes	characteristic
257	C₂×C₁₃₇	#3068	1	1	yes	characteristic
258	#25312	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₁₃₇	#25312	1	1	yes	characteristic, direct product factor, Sylow
260	#3138	C₂×C₁₃₇	1	1	yes	characteristic, complemented
261	#3077	C₂×C₁₃₇	1	1	yes	characteristic, complemented
262	C₂×#459	C₂×C₁₃₇	1	1	yes	characteristic
263	#2962	C₂×C₁₃₇	1	1	yes	characteristic, complemented
264	#2063	C₂×C₁₃₇	1	2	yes	complemented
265	#2045	C₂×C₁₃₇	1	2	yes	complemented
266	#1978	C₂×C₁₃₇	1	2	yes	complemented
267	#1965	C₂×C₁₃₇	1	1	yes	characteristic, complemented
268	#1950	C₂×C₁₃₇	1	1	yes	characteristic, complemented
269	#1931	C₂×C₁₃₇	1	2	yes	complemented
270	C₄×#100	C₂×C₁₃₇	1	1	yes	characteristic, complemented
271	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
272	#533	C₂²×C₁₃₇	1	1	yes	characteristic
273	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
274	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
275	#490	—	2	2	no	—
276	#486	—	2	1	no	—
277	#486	—	2	1	no	—
278	#478	—	2	2	no	—
279	#476	—	2	1	no	—
280	#474	—	2	1	no	—
281	#459	—	2	1	no	—
282	#459	—	2	2	no	—
283	#459	—	2	1	no	—
284	#446	—	2	2	no	—
285	#446	—	2	1	no	—
286	C₈×Q₈	—	2	1	no	—
287	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
288	#428	C₂²×C₁₃₇	1	2	yes	—
289	#424	C₂²×C₁₃₇	1	2	yes	—
290	#423	C₂²×C₁₃₇	1	1	yes	characteristic
291	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
292	#418	C₂²×C₁₃₇	1	2	yes	—
293	#418	C₂²×C₁₃₇	1	2	yes	—
294	#418	C₂²×C₁₃₇	1	2	yes	—
295	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
296	C₂×#100	C₂²×C₁₃₇	1	2	yes	—
297	C₂×#100	C₂²×C₁₃₇	1	1	yes	characteristic
298	C₂×#100	C₂²×C₁₃₇	1	1	yes	characteristic
299	C₂×#100	C₂²×C₁₃₇	1	2	yes	—
300	#405	C₂²×C₁₃₇	1	2	yes	—
301	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
302	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
303	#343	C₂²×C₁₃₇	1	1	yes	characteristic
304	#343	C₂²×C₁₃₇	1	1	yes	characteristic
305	#341	C₂²×C₁₃₇	1	2	yes	—
306	C₂²×D₄	—	2	1	no	—
307	C₂²×D₄	—	2	1	no	—
308	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
309	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂×D₈	—	2	4	no	—
312	C₂×D₈	—	2	2	no	—
313	C₂×D₈	—	2	2	no	—
314	C₂×D₈	—	4	2	no	—
315	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
316	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
317	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
318	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
319	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
321	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
322	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
323	#127	—	2	1	no	—
324	#127	—	2	1	no	—
325	#127	—	2	1	no	—
326	#127	—	2	1	no	—
327	#125	D₄×C₁₃₇	1	2	yes	—
328	C₂²⋊Q₈	—	2	1	no	—
329	C₂²⋊Q₈	—	2	1	no	—
330	C₄×Q₈	—	2	1	no	—
331	C₄×Q₈	—	2	1	no	—
332	C₄×D₄	—	2	1	no	—
333	C₄×D₄	—	2	1	no	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	—	2	1	no	—
336	#117	C₂³×C₁₃₇	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
340	#108	—	4	2	no	—
341	#107	—	2	2	no	—
342	#106	—	2	2	no	—
343	C₄⋊C₈	—	2	2	no	—
344	C₄⋊C₈	—	2	2	no	—
345	C₄⋊C₈	—	2	1	no	—
346	C₄⋊C₈	—	2	1	no	—
347	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
348	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
349	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
350	#102	—	2	2	no	—
351	#102	—	2	2	no	—
352	#102	—	2	2	no	—
353	#102	—	2	2	no	—
354	#102	—	2	2	no	—
355	#102	—	2	2	no	—
356	#102	—	2	2	no	—
357	#102	—	2	2	no	—
358	#100	—	2	2	no	—
359	#100	—	2	2	no	—
360	#100	—	2	4	no	—
361	#100	—	4	2	no	—
362	#97	—	2	2	no	—
363	#97	—	2	1	no	—
364	#97	—	2	1	no	—
365	C₄×C₈	—	2	2	no	—
366	C₄×C₈	—	2	1	no	—
367	C₄×C₈	—	2	1	no	—
368	C₂⁴	—	2	1	no	—
369	C₂⁴	—	2	2	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	4	1	no	—
384	C₂×D₄	—	4	1	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	4	1	no	—
387	C₂×D₄	—	8	2	no	—
388	C₂×D₄	—	8	2	no	—
389	C₂×D₄	#41×C₁₃₇	1	2	yes	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	4	1	no	—
392	C₂²×C₄	—	4	1	no	—
393	C₂²×C₄	—	8	2	no	—
394	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
395	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
397	D₈	—	4	4	no	—
398	D₈	—	4	4	no	—
399	M₁₆	—	2	1	no	—
400	M₁₆	—	2	1	no	—
401	M₁₆	—	2	1	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	1	no	—
404	M₁₆	—	2	1	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	—	2	1	no	—
407	M₁₆	—	2	1	no	—
408	M₁₆	#41×C₁₃₇	1	2	yes	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
417	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
425	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	4	1	no	—
428	C₂²⋊C₄	—	4	1	no	—
429	C₄²	—	2	2	no	—
430	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
431	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
433	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
434	C₂³	—	2	2	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	2	no	—
440	C₂³	—	8	1	no	—
441	C₂³	—	8	1	no	—
442	C₂³	—	8	2	no	—
443	C₂³	—	8	2	no	—
444	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
445	Q₈	—	4	2	no	—
446	Q₈	—	8	1	no	—
447	D₄	—	2	2	no	—
448	D₄	—	2	2	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	1	no	—
451	D₄	—	4	2	no	—
452	D₄	—	4	2	no	—
453	D₄	—	4	1	no	—
454	D₄	—	4	2	no	—
455	D₄	—	8	2	no	—
456	D₄	—	8	2	no	—
457	D₄	—	8	2	no	—
458	D₄	—	8	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	8	2	no	—
465	C₂×C₄	—	8	2	no	—
466	C₂×C₄	—	8	1	no	—
467	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
468	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
469	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
470	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
471	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
473	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
474	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
475	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
476	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
477	C₈	—	2	2	no	—
478	C₈	—	2	2	no	—
479	C₈	—	2	2	no	—
480	C₈	—	4	1	no	—
481	C₈	—	4	1	no	—
482	C₈	—	4	1	no	—
483	C₈	—	4	2	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	8	2	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	8	1	no	—
490	C₂²	—	8	1	no	—
491	C₂²	—	8	2	no	—
492	C₂²	—	8	2	no	—
493	C₂²	—	8	2	no	—
494	C₂²	—	16	2	no	—
495	C₂²	—	16	2	no	—
496	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
497	C₂²	#529×C₁₃₇	1	1	yes	characteristic
498	C₂²	#542×C₁₃₇	1	1	yes	characteristic
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	2	1	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	8	1	no	—
505	C₄	#472×C₁₃₇	1	2	yes	—
506	C₄	#546×C₁₃₇	1	1	yes	characteristic
507	C₄	#548×C₁₃₇	1	1	yes	characteristic
508	C₄	#548×C₁₃₇	1	1	yes	characteristic
509	C₄	#550×C₁₃₇	1	1	yes	characteristic
510	C₂	—	2	1	no	—
511	C₂	—	2	1	no	—
512	C₂	—	8	2	no	—
513	C₂	—	16	2	no	—
514	C₂	#2426×C₁₃₇	1	1	yes	characteristic
515	C₂	C₂×#472×C₁₃₇	1	1	yes	characteristic
516	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×#459×C₁₃₇	C₂
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
1. #3138×C₁₃₇	C₂	semidirect
2. #3077×C₁₃₇	C₂	semidirect
4. #2962×C₁₃₇	C₂	semidirect
5. #2063×C₁₃₇	C₂	semidirect
6. #2045×C₁₃₇	C₂	semidirect
7. #1978×C₁₃₇	C₂	semidirect
8. #1965×C₁₃₇	C₂	semidirect
9. #1950×C₁₃₇	C₂	semidirect
10. #1931×C₁₃₇	C₂	semidirect
11. C₄×#100×C₁₃₇	C₂	semidirect
258. #25312	C₁₃₇	direct (N × H)
259. C₁₃₇	#25312	direct (N × H)
260. #3138	C₂×C₁₃₇	semidirect
261. #3077	C₂×C₁₃₇	semidirect
263. #2962	C₂×C₁₃₇	semidirect
264. #2063	C₂×C₁₃₇	semidirect
265. #2045	C₂×C₁₃₇	semidirect
266. #1978	C₂×C₁₃₇	semidirect
267. #1965	C₂×C₁₃₇	semidirect
268. #1950	C₂×C₁₃₇	semidirect
269. #1931	C₂×C₁₃₇	semidirect
270. C₄×#100	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
#3077×C₁₃₇	2
C₂×#459×C₁₃₇	2
#2962×C₁₃₇	2
#2063×C₁₃₇	2
#2045×C₁₃₇	2
#1978×C₁₃₇	2
#1965×C₁₃₇	2
#1950×C₁₃₇	2
#1931×C₁₃₇	2
C₄×#100×C₁₃₇	2
#25312	137

Subgroups — C137×#25312 #46874698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25312 #46874698