Subgroups — C₃₇×#28709 #22649962

Order 9472.

774 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₇×#436	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2562	C₂	1	2	yes	maximal
4	C₃₇×#2076	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2075	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1989	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1809	C₂	1	4	yes	maximal, complemented
8	C₄×#107×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×M₁₆×C₃₇	C₂²	1	2	yes	—
10	C₂×#125×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#499	—	2	4	no	—
13	C₃₇×#436	—	2	4	no	—
14	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×#107×C₃₇	C₂²	1	2	yes	—
17	C₂×#106×C₃₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#400	—	2	2	no	—
22	C₃₇×#394	—	2	2	no	—
23	C₃₇×#388	—	2	4	no	—
24	C₃₇×#380	C₂²	1	4	yes	—
25	C₃₇×#380	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
27	C₃₇×#340	C₂²	1	4	yes	—
28	C₃₇×#340	C₂²	1	4	yes	—
29	C₃₇×#340	C₂²	1	2	yes	—
30	C₂²×D₄×C₃₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₃₇	C₂³	1	2	yes	—
32	C₂×M₁₆×C₃₇	—	2	4	no	—
33	C₂×M₁₆×C₃₇	—	2	8	no	—
34	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	#125×C₃₇	—	2	4	no	—
38	#125×C₃₇	—	2	4	no	—
39	C₄×D₄×C₃₇	—	2	2	no	—
40	C₄×D₄×C₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
48	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
50	C₂×C₄²×C₃₇	—	2	2	no	—
51	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
52	#107×C₃₇	—	2	2	no	—
53	#107×C₃₇	—	2	2	no	—
54	#107×C₃₇	—	2	4	no	—
55	#106×C₃₇	—	2	4	no	—
56	C₄⋊C₈×C₃₇	—	2	2	no	—
57	C₂²⋊C₈×C₃₇	—	2	4	no	—
58	C₄×C₈×C₃₇	D₄	1	4	yes	—
59	#95×C₃₇	—	2	2	no	—
60	#95×C₃₇	—	2	4	no	—
61	#95×C₃₇	—	2	4	no	—
62	C₂⁴×C₃₇	C₂×D₄	1	2	yes	—
63	C₂×D₄×C₃₇	—	2	2	no	—
64	C₂×D₄×C₃₇	—	4	2	no	—
65	C₂²×C₄×C₃₇	—	2	2	no	—
66	C₂²×C₄×C₃₇	—	2	2	no	—
67	C₂²×C₄×C₃₇	—	2	4	no	—
68	C₂²×C₄×C₃₇	—	2	4	no	—
69	C₂²×C₄×C₃₇	—	2	4	no	—
70	C₂²×C₄×C₃₇	—	2	2	no	—
71	C₂²×C₄×C₃₇	—	2	2	no	—
72	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₃₇	—	4	8	no	—
77	C₂×C₈×C₃₇	—	2	2	no	—
78	C₂×C₈×C₃₇	—	2	2	no	—
79	C₂×C₈×C₃₇	—	2	4	no	—
80	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₃₇	#41	1	4	yes	—
83	C₄⋊C₄×C₃₇	—	2	2	no	—
84	C₄⋊C₄×C₃₇	—	2	2	no	—
85	C₄⋊C₄×C₃₇	—	2	2	no	—
86	C₄⋊C₄×C₃₇	—	2	4	no	—
87	C₄⋊C₄×C₃₇	—	2	2	no	—
88	C₄⋊C₄×C₃₇	—	2	2	no	—
89	C₄⋊C₄×C₃₇	—	2	4	no	—
90	C₄⋊C₄×C₃₇	—	4	4	no	—
91	C₄⋊C₄×C₃₇	—	4	4	no	—
92	C₂²⋊C₄×C₃₇	—	2	4	no	—
93	C₂²⋊C₄×C₃₇	—	4	8	no	—
94	C₂²⋊C₄×C₃₇	—	8	4	no	—
95	C₄²×C₃₇	—	8	2	no	—
96	C₄²×C₃₇	C₂×D₄	1	2	yes	—
97	C₄²×C₃₇	#41	1	2	yes	—
98	C₂³×C₃₇	—	2	2	no	—
99	C₂³×C₃₇	—	2	4	no	—
100	C₂³×C₃₇	—	2	4	no	—
101	C₂³×C₃₇	—	2	2	no	—
102	C₂³×C₃₇	—	2	2	no	—
103	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₃₇	—	4	4	no	—
105	C₂×C₄×C₃₇	—	2	2	no	—
106	C₂×C₄×C₃₇	—	2	1	no	—
107	C₂×C₄×C₃₇	—	2	1	no	—
108	C₂×C₄×C₃₇	—	2	2	no	—
109	C₂×C₄×C₃₇	—	2	4	no	—
110	C₂×C₄×C₃₇	—	4	2	no	—
111	C₂×C₄×C₃₇	—	4	2	no	—
112	C₂×C₄×C₃₇	—	4	4	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	2	no	—
115	C₂×C₄×C₃₇	—	4	4	no	—
116	C₂×C₄×C₃₇	—	4	4	no	—
117	C₂×C₄×C₃₇	—	4	4	no	—
118	C₂×C₄×C₃₇	—	4	4	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	#121	1	2	yes	—
122	C₂×C₄×C₃₇	#121	1	2	yes	—
123	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
125	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
127	C₈×C₃₇	—	2	2	no	—
128	C₈×C₃₇	—	4	2	no	—
129	C₈×C₃₇	#124	1	4	yes	—
130	#28709	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₃₇	—	4	4	no	—
132	C₂²×C₃₇	—	4	4	no	—
133	C₂²×C₃₇	—	4	2	no	—
134	C₂²×C₃₇	—	4	2	no	—
135	C₂²×C₃₇	—	4	2	no	—
136	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
137	C₂²×C₃₇	#547	1	2	yes	—
138	C₂²×C₃₇	#575	1	2	yes	—
139	C₂²×C₃₇	#576	1	1	yes	characteristic
140	C₂²×C₃₇	#578	1	1	yes	characteristic
141	C₄×C₃₇	—	2	2	no	—
142	C₄×C₃₇	—	2	2	no	—
143	C₄×C₃₇	—	4	2	no	—
144	C₄×C₃₇	—	8	4	no	—
145	C₄×C₃₇	—	8	2	no	—
146	C₄×C₃₇	—	8	2	no	—
147	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
148	C₄×C₃₇	#536	1	1	yes	characteristic
149	C₄×C₃₇	#556	1	2	yes	—
150	C₂×#499	C₂×C₃₇	1	1	yes	characteristic, complemented
151	C₂×#436	C₂×C₃₇	1	1	yes	characteristic
152	#2562	C₂×C₃₇	1	2	yes	—
153	#2076	C₂×C₃₇	1	2	yes	complemented
154	#2075	C₂×C₃₇	1	2	yes	complemented
155	#1989	C₂×C₃₇	1	2	yes	complemented
156	#1809	C₂×C₃₇	1	4	yes	complemented
157	C₄×#107	C₂×C₃₇	1	1	yes	characteristic, complemented
158	C₂×C₃₇	—	8	2	no	—
159	C₂×C₃₇	#2547	1	1	yes	characteristic
160	C₂×C₃₇	#3054	1	1	yes	characteristic
161	C₂×C₃₇	#3061	1	1	yes	characteristic
162	C₂×C₃₇	#3352	1	2	yes	—
163	C₂×C₃₇	#3355	1	2	yes	—
164	C₂²×M₁₆	C₂²×C₃₇	1	2	yes	—
165	C₂×#125	C₂²×C₃₇	1	2	yes	—
166	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
167	#499	—	2	4	no	—
168	#436	—	2	4	no	—
169	C₂×#107	C₂²×C₃₇	1	2	yes	—
170	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
171	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
172	C₂×#106	C₂²×C₃₇	1	2	yes	—
173	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
174	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
176	#400	—	2	2	no	—
177	#394	—	2	2	no	—
178	#388	—	2	4	no	—
179	#380	C₂²×C₃₇	1	4	yes	—
180	#380	C₂²×C₃₇	1	4	yes	—
181	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
182	#340	C₂²×C₃₇	1	4	yes	—
183	#340	C₂²×C₃₇	1	4	yes	—
184	#340	C₂²×C₃₇	1	2	yes	—
185	C₃₇	#28709	1	1	yes	characteristic, direct product factor, Sylow
186	C₂²×D₄	D₄×C₃₇	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₃₇	1	2	yes	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	2	8	no	—
190	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
191	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
193	#125	—	2	4	no	—
194	#125	—	2	4	no	—
195	C₄×D₄	—	2	2	no	—
196	C₄×D₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₂²⋊C₄	D₄×C₃₇	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
208	#107	—	2	2	no	—
209	#107	—	2	2	no	—
210	#107	—	2	4	no	—
211	#106	—	2	4	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	C₄×C₈	D₄×C₃₇	1	4	yes	—
215	#95	—	2	2	no	—
216	#95	—	2	4	no	—
217	#95	—	2	4	no	—
218	C₂⁴	C₂×D₄×C₃₇	1	2	yes	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
232	M₁₆	—	4	8	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
238	C₂×C₈	#41×C₃₇	1	4	yes	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	4	8	no	—
250	C₂²⋊C₄	—	8	4	no	—
251	C₄²	—	8	2	no	—
252	C₄²	C₂×D₄×C₃₇	1	2	yes	—
253	C₄²	#41×C₃₇	1	2	yes	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
260	D₄	—	4	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	#121×C₃₇	1	2	yes	—
278	C₂×C₄	#121×C₃₇	1	2	yes	—
279	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
281	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
282	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	#124×C₃₇	1	4	yes	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
292	C₂²	C₃₇×#547	1	2	yes	—
293	C₂²	C₃₇×#575	1	2	yes	—
294	C₂²	C₃₇×#576	1	1	yes	characteristic
295	C₂²	C₃₇×#578	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
303	C₄	C₃₇×#536	1	1	yes	characteristic
304	C₄	C₃₇×#556	1	2	yes	—
305	C₂	—	8	2	no	—
306	C₂	C₃₇×#2547	1	1	yes	characteristic
307	C₂	C₃₇×#3054	1	1	yes	characteristic
308	C₂	C₃₇×#3061	1	1	yes	characteristic
309	C₂	C₃₇×#3352	1	2	yes	—
310	C₂	C₃₇×#3355	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₄×C₃₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄×D₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#499	C₂	semidirect
4. C₃₇×#2076	C₂	semidirect
5. C₃₇×#2075	C₂	semidirect
6. C₃₇×#1989	C₂	semidirect
7. C₃₇×#1809	C₂	semidirect
8. C₄×#107×C₃₇	C₂	semidirect
130. #28709	C₃₇	direct (N × H)
150. C₂×#499	C₂×C₃₇	semidirect
153. #2076	C₂×C₃₇	semidirect
154. #2075	C₂×C₃₇	semidirect
155. #1989	C₂×C₃₇	semidirect
156. #1809	C₂×C₃₇	semidirect
157. C₄×#107	C₂×C₃₇	semidirect
185. C₃₇	#28709	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#499	2
C₂×C₃₇×#436	2
C₃₇×#2562	2
C₃₇×#2076	2
C₃₇×#2075	2
C₃₇×#1989	2
C₃₇×#1809	2
C₄×#107×C₃₇	2
#28709	37

Subgroups — C37×#28709 #22649962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28709 #22649962