Subgroups — C₃₇×#28707 #22649960

Order 9472.

806 conjugacy classes of subgroups.

The subgroup lattice has 306 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×C₈×D₄×C₃₇	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×#1807	C₂	1	4	yes	maximal, complemented
8	C₄×#106×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×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	C₈×D₄×C₃₇	—	2	4	no	—
14	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×#106×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	C₂²×C₈×C₃₇	—	2	4	no	—
33	C₂²×C₈×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	#106×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	4	no	—
64	C₂×D₄×C₃₇	—	2	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	C₂×C₈×C₃₇	—	2	2	no	—
77	C₂×C₈×C₃₇	—	2	2	no	—
78	C₂×C₈×C₃₇	—	2	4	no	—
79	C₂×C₈×C₃₇	—	4	8	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	2	no	—
101	C₂³×C₃₇	—	2	4	no	—
102	C₂³×C₃₇	#135	1	4	yes	—
103	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₃₇	—	2	8	no	—
105	C₂×C₄×C₃₇	—	2	2	no	—
106	C₂×C₄×C₃₇	—	2	2	no	—
107	C₂×C₄×C₃₇	—	2	2	no	—
108	C₂×C₄×C₃₇	—	2	4	no	—
109	C₂×C₄×C₃₇	—	4	2	no	—
110	C₂×C₄×C₃₇	—	4	2	no	—
111	C₂×C₄×C₃₇	—	4	4	no	—
112	C₂×C₄×C₃₇	—	4	2	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	4	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	2	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	#121	1	2	yes	—
121	C₂×C₄×C₃₇	#121	1	2	yes	—
122	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
126	C₈×C₃₇	—	2	2	no	—
127	C₈×C₃₇	—	4	2	no	—
128	C₈×C₃₇	#124	1	4	yes	—
129	#28707	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₃₇	—	2	8	no	—
131	C₂²×C₃₇	—	2	4	no	—
132	C₂²×C₃₇	—	4	4	no	—
133	C₂²×C₃₇	—	4	2	no	—
134	C₂²×C₃₇	—	4	2	no	—
135	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
136	C₂²×C₃₇	#547	1	2	yes	—
137	C₂²×C₃₇	C₂×#135	1	2	yes	—
138	C₂²×C₃₇	#577	1	2	yes	—
139	C₄×C₃₇	—	2	2	no	—
140	C₄×C₃₇	—	2	2	no	—
141	C₄×C₃₇	—	4	2	no	—
142	C₄×C₃₇	—	8	4	no	—
143	C₄×C₃₇	—	8	2	no	—
144	C₄×C₃₇	—	8	2	no	—
145	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
146	C₄×C₃₇	#536	1	1	yes	characteristic
147	C₄×C₃₇	#556	1	2	yes	—
148	C₂×#499	C₂×C₃₇	1	1	yes	characteristic, complemented
149	C₂×C₈×D₄	C₂×C₃₇	1	1	yes	characteristic
150	#2562	C₂×C₃₇	1	2	yes	—
151	#2076	C₂×C₃₇	1	2	yes	complemented
152	#2075	C₂×C₃₇	1	2	yes	complemented
153	#1989	C₂×C₃₇	1	2	yes	complemented
154	#1807	C₂×C₃₇	1	4	yes	complemented
155	C₄×#106	C₂×C₃₇	1	1	yes	characteristic, complemented
156	C₂×C₃₇	—	4	4	no	—
157	C₂×C₃₇	#2547	1	1	yes	characteristic
158	C₂×C₃₇	#3050	1	1	yes	characteristic
159	C₂×C₃₇	#3060	1	1	yes	characteristic
160	C₂×C₃₇	#3337	1	4	yes	—
161	C₂³×C₈	C₂²×C₃₇	1	2	yes	—
162	C₂×#125	C₂²×C₃₇	1	2	yes	—
163	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
164	#499	—	2	4	no	—
165	C₈×D₄	—	2	4	no	—
166	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
167	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
168	C₂×#106	C₂²×C₃₇	1	2	yes	—
169	C₂×#106	C₂²×C₃₇	1	2	yes	—
170	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
171	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
173	#400	—	2	2	no	—
174	#394	—	2	2	no	—
175	#388	—	2	4	no	—
176	#380	C₂²×C₃₇	1	4	yes	—
177	#380	C₂²×C₃₇	1	4	yes	—
178	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
179	#340	C₂²×C₃₇	1	4	yes	—
180	#340	C₂²×C₃₇	1	4	yes	—
181	#340	C₂²×C₃₇	1	2	yes	—
182	C₃₇	#28707	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×D₄	D₄×C₃₇	1	1	yes	characteristic
184	C₂³×C₄	C₂³×C₃₇	1	2	yes	—
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	—	2	8	no	—
187	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
188	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
190	#125	—	2	4	no	—
191	#125	—	2	4	no	—
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₂²⋊C₄	D₄×C₃₇	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
205	#107	—	2	2	no	—
206	#107	—	2	2	no	—
207	#106	—	2	4	no	—
208	#106	—	2	4	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₄×C₈	D₄×C₃₇	1	4	yes	—
212	#95	—	2	2	no	—
213	#95	—	2	4	no	—
214	#95	—	2	4	no	—
215	C₂⁴	C₂×D₄×C₃₇	1	2	yes	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	8	no	—
233	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
235	C₂×C₈	#41×C₃₇	1	4	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	4	8	no	—
247	C₂²⋊C₄	—	8	4	no	—
248	C₄²	—	8	2	no	—
249	C₄²	C₂×D₄×C₃₇	1	2	yes	—
250	C₄²	#41×C₃₇	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	4	no	—
255	C₂³	#135×C₃₇	1	4	yes	—
256	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
257	D₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	#121×C₃₇	1	2	yes	—
274	C₂×C₄	#121×C₃₇	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
276	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
277	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
278	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	4	2	no	—
281	C₈	#124×C₃₇	1	4	yes	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	4	no	—
287	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
288	C₂²	C₃₇×#547	1	2	yes	—
289	C₂²	C₂×#135×C₃₇	1	2	yes	—
290	C₂²	C₃₇×#577	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	4	no	—
297	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
298	C₄	C₃₇×#536	1	1	yes	characteristic
299	C₄	C₃₇×#556	1	2	yes	—
300	C₂	—	4	4	no	—
301	C₂	C₃₇×#2547	1	1	yes	characteristic
302	C₂	C₃₇×#3050	1	1	yes	characteristic
303	C₂	C₃₇×#3060	1	1	yes	characteristic
304	C₂	C₃₇×#3337	1	4	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×#1807	C₂	semidirect
8. C₄×#106×C₃₇	C₂	semidirect
129. #28707	C₃₇	direct (N × H)
148. C₂×#499	C₂×C₃₇	semidirect
151. #2076	C₂×C₃₇	semidirect
152. #2075	C₂×C₃₇	semidirect
153. #1989	C₂×C₃₇	semidirect
154. #1807	C₂×C₃₇	semidirect
155. C₄×#106	C₂×C₃₇	semidirect
182. C₃₇	#28707	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×C₈×D₄×C₃₇	2
C₃₇×#2562	2
C₃₇×#2076	2
C₃₇×#2075	2
C₃₇×#1989	2
C₃₇×#1807	2
C₄×#106×C₃₇	2
#28707	37

Subgroups — C37×#28707 #22649960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28707 #22649960