Subgroups — C₃₇×#28622 #22649875

Order 9472.

902 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#492	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#490	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2642	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2328	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2055	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1991	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#1960	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1877	C₂	1	4	yes	maximal, complemented
9	C₃₇×#1768	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₃₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#492	—	2	4	no	—
16	C₃₇×#490	—	2	4	no	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×#103×C₃₇	C₂²	1	4	yes	—
19	C₂×#102×C₃₇	C₂²	1	4	yes	—
20	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₃₇×#400	—	2	4	no	—
23	C₃₇×#391	C₂²	1	2	yes	—
24	C₃₇×#391	C₂²	1	2	yes	—
25	C₃₇×#390	C₂²	1	2	yes	—
26	C₃₇×#386	—	2	4	no	—
27	C₃₇×#382	C₂²	1	1	yes	characteristic
28	C₃₇×#376	C₂²	1	1	yes	characteristic
29	C₃₇×#340	C₂²	1	4	yes	—
30	C₃₇×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×D₄×C₃₇	—	2	2	no	—
33	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₃₇	—	2	2	no	—
35	C₂×SD₁₆×C₃₇	—	2	8	no	—
36	C₂×SD₁₆×C₃₇	—	2	8	no	—
37	C₂×SD₁₆×C₃₇	—	4	4	no	—
38	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₃₇	—	2	2	no	—
41	C₄⋊Q₈×C₃₇	—	2	2	no	—
42	#127×C₃₇	—	2	2	no	—
43	#127×C₃₇	D₄	1	4	yes	—
44	#125×C₃₇	—	2	2	no	—
45	#125×C₃₇	—	2	2	no	—
46	C₄×D₄×C₃₇	—	4	4	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₃₇	—	2	4	no	—
57	#103×C₃₇	—	4	4	no	—
58	#102×C₃₇	—	2	8	no	—
59	#97×C₃₇	—	2	2	no	—
60	C₄×C₈×C₃₇	—	2	2	no	—
61	#95×C₃₇	—	2	2	no	—
62	#95×C₃₇	—	2	4	no	—
63	#95×C₃₇	—	2	4	no	—
64	#95×C₃₇	D₄	1	4	yes	—
65	C₂⁴×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	2	2	no	—
68	C₂×Q₈×C₃₇	—	2	4	no	—
69	C₂×Q₈×C₃₇	—	2	4	no	—
70	C₂×D₄×C₃₇	—	2	8	no	—
71	C₂×D₄×C₃₇	—	2	4	no	—
72	C₂×D₄×C₃₇	—	2	2	no	—
73	C₂×D₄×C₃₇	—	2	4	no	—
74	C₂×D₄×C₃₇	—	8	4	no	—
75	C₂×D₄×C₃₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₃₇	—	2	2	no	—
77	C₂²×C₄×C₃₇	—	2	2	no	—
78	C₂²×C₄×C₃₇	—	2	2	no	—
79	C₂²×C₄×C₃₇	—	4	4	no	—
80	C₂²×C₄×C₃₇	—	8	4	no	—
81	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
85	SD₁₆×C₃₇	—	4	16	no	—
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	—	2	2	no	—
88	C₂×C₈×C₃₇	—	2	2	no	—
89	C₂×C₈×C₃₇	—	2	2	no	—
90	C₂×C₈×C₃₇	—	2	2	no	—
91	C₂×C₈×C₃₇	#41	1	4	yes	—
92	C₄⋊C₄×C₃₇	—	2	2	no	—
93	C₄⋊C₄×C₃₇	—	2	2	no	—
94	C₄⋊C₄×C₃₇	—	2	4	no	—
95	C₄⋊C₄×C₃₇	—	4	4	no	—
96	C₄⋊C₄×C₃₇	—	4	4	no	—
97	C₂²⋊C₄×C₃₇	—	4	4	no	—
98	C₂²⋊C₄×C₃₇	—	8	4	no	—
99	C₄²×C₃₇	—	2	2	no	—
100	C₄²×C₃₇	—	2	2	no	—
101	C₄²×C₃₇	C₂×D₄	1	2	yes	—
102	C₄²×C₃₇	#41	1	2	yes	—
103	C₂³×C₃₇	—	2	4	no	—
104	C₂³×C₃₇	—	4	2	no	—
105	C₂³×C₃₇	—	4	2	no	—
106	C₂³×C₃₇	—	4	4	no	—
107	C₂³×C₃₇	—	4	4	no	—
108	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₃₇	—	4	4	no	—
110	Q₈×C₃₇	—	4	4	no	—
111	D₄×C₃₇	—	2	8	no	—
112	D₄×C₃₇	—	4	4	no	—
113	D₄×C₃₇	—	8	8	no	—
114	C₂×C₄×C₃₇	—	2	4	no	—
115	C₂×C₄×C₃₇	—	2	2	no	—
116	C₂×C₄×C₃₇	—	2	2	no	—
117	C₂×C₄×C₃₇	—	2	4	no	—
118	C₂×C₄×C₃₇	—	2	4	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	—	4	2	no	—
122	C₂×C₄×C₃₇	—	4	2	no	—
123	C₂×C₄×C₃₇	—	4	2	no	—
124	C₂×C₄×C₃₇	—	4	4	no	—
125	C₂×C₄×C₃₇	—	4	4	no	—
126	C₂×C₄×C₃₇	—	8	8	no	—
127	C₂×C₄×C₃₇	#121	1	4	yes	—
128	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
132	C₈×C₃₇	—	2	4	no	—
133	C₈×C₃₇	—	4	2	no	—
134	#28622	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₃₇	—	4	8	no	—
136	C₂²×C₃₇	—	4	4	no	—
137	C₂²×C₃₇	—	8	4	no	—
138	C₂²×C₃₇	—	8	2	no	—
139	C₂²×C₃₇	—	8	2	no	—
140	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
141	C₂²×C₃₇	#532	1	1	yes	characteristic
142	C₂²×C₃₇	#535	1	1	yes	characteristic
143	C₂²×C₃₇	#577	1	4	yes	—
144	C₄×C₃₇	—	2	4	no	—
145	C₄×C₃₇	—	4	2	no	—
146	C₄×C₃₇	—	4	2	no	—
147	C₄×C₃₇	—	8	2	no	—
148	C₄×C₃₇	—	8	2	no	—
149	C₄×C₃₇	—	8	2	no	—
150	C₄×C₃₇	#546	1	2	yes	—
151	C₄×C₃₇	#559	1	2	yes	—
152	C₂×#492	C₂×C₃₇	1	1	yes	characteristic
153	C₂×#490	C₂×C₃₇	1	1	yes	characteristic
154	#2642	C₂×C₃₇	1	1	yes	characteristic, complemented
155	#2328	C₂×C₃₇	1	1	yes	characteristic
156	#2055	C₂×C₃₇	1	4	yes	complemented
157	#1991	C₂×C₃₇	1	1	yes	characteristic, complemented
158	#1960	C₂×C₃₇	1	1	yes	characteristic, complemented
159	#1877	C₂×C₃₇	1	4	yes	complemented
160	#1768	C₂×C₃₇	1	1	yes	characteristic, complemented
161	C₂×C₃₇	—	8	4	no	—
162	C₂×C₃₇	#2544	1	1	yes	characteristic
163	C₂×C₃₇	#3042	1	2	yes	—
164	C₂×C₃₇	#3174	1	2	yes	—
165	C₂×C₃₇	#3222	1	2	yes	—
166	C₂²×SD₁₆	C₂²×C₃₇	1	4	yes	—
167	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
168	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
169	C₂×#125	C₂²×C₃₇	1	1	yes	characteristic
170	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
171	#492	—	2	4	no	—
172	#490	—	2	4	no	—
173	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
174	C₂×#103	C₂²×C₃₇	1	4	yes	—
175	C₂×#102	C₂²×C₃₇	1	4	yes	—
176	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
178	#400	—	2	4	no	—
179	#391	C₂²×C₃₇	1	2	yes	—
180	#391	C₂²×C₃₇	1	2	yes	—
181	#390	C₂²×C₃₇	1	2	yes	—
182	#386	—	2	4	no	—
183	#382	C₂²×C₃₇	1	1	yes	characteristic
184	#376	C₂²×C₃₇	1	1	yes	characteristic
185	#340	C₂²×C₃₇	1	4	yes	—
186	#336	C₂²×C₃₇	1	2	yes	—
187	C₃₇	#28622	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂×SD₁₆	—	2	8	no	—
193	C₂×SD₁₆	—	2	8	no	—
194	C₂×SD₁₆	—	4	4	no	—
195	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
196	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	#127	—	2	2	no	—
200	#127	D₄×C₃₇	1	4	yes	—
201	#125	—	2	2	no	—
202	#125	—	2	2	no	—
203	C₄×D₄	—	4	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	4	no	—
210	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
213	C₄⋊C₈	—	2	4	no	—
214	#103	—	4	4	no	—
215	#102	—	2	8	no	—
216	#97	—	2	2	no	—
217	C₄×C₈	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	4	no	—
220	#95	—	2	4	no	—
221	#95	D₄×C₃₇	1	4	yes	—
222	C₂⁴	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	8	4	no	—
232	C₂×D₄	C₂×D₄×C₃₇	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	8	4	no	—
238	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
242	SD₁₆	—	4	16	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	#41×C₃₇	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	8	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	C₂×D₄×C₃₇	1	2	yes	—
259	C₄²	#41×C₃₇	1	2	yes	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	4	no	—
265	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	D₄	—	2	8	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	8	8	no	—
284	C₂×C₄	#121×C₃₇	1	4	yes	—
285	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
287	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
288	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
289	C₈	—	2	4	no	—
290	C₈	—	4	2	no	—
291	C₂²	—	4	8	no	—
292	C₂²	—	4	4	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
297	C₂²	C₃₇×#532	1	1	yes	characteristic
298	C₂²	C₃₇×#535	1	1	yes	characteristic
299	C₂²	C₃₇×#577	1	4	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₃₇×#546	1	2	yes	—
307	C₄	C₃₇×#559	1	2	yes	—
308	C₂	—	8	4	no	—
309	C₂	C₃₇×#2544	1	1	yes	characteristic
310	C₂	C₃₇×#3042	1	2	yes	—
311	C₂	C₃₇×#3174	1	2	yes	—
312	C₂	C₃₇×#3222	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₃₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#2642	C₂	semidirect
5. C₃₇×#2055	C₂	semidirect
6. C₃₇×#1991	C₂	semidirect
7. C₃₇×#1960	C₂	semidirect
8. C₃₇×#1877	C₂	semidirect
9. C₃₇×#1768	C₂	semidirect
134. #28622	C₃₇	direct (N × H)
154. #2642	C₂×C₃₇	semidirect
156. #2055	C₂×C₃₇	semidirect
157. #1991	C₂×C₃₇	semidirect
158. #1960	C₂×C₃₇	semidirect
159. #1877	C₂×C₃₇	semidirect
160. #1768	C₂×C₃₇	semidirect
187. C₃₇	#28622	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×#492	2
C₂×C₃₇×#490	2
C₃₇×#2642	2
C₃₇×#2328	2
C₃₇×#2055	2
C₃₇×#1991	2
C₃₇×#1960	2
C₃₇×#1877	2
C₃₇×#1768	2
#28622	37

Subgroups — C37×#28622 #22649875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28622 #22649875