Subgroups — C₃₇×#21442 #22642695

Order 9472.

1506 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₄×C₃₇	C₂	1	3	yes	maximal, complemented
3	C₃₇×#2388	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1870	C₂	1	3	yes	maximal, complemented
5	C₃₇×#1851	C₂	1	3	yes	maximal, complemented
6	C₃₇×#1780	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊C₄×C₃₇	C₂	1	3	yes	maximal, complemented
8	C₂²×#41×C₃₇	C₂²	1	1	yes	characteristic
9	C₂×#131×C₃₇	—	2	12	no	—
10	C₂²×M₁₆×C₃₇	C₂²	1	3	yes	—
11	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₇	C₂²	1	3	yes	—
13	C₂×C₄×D₄×C₃₇	C₂²	1	3	yes	—
14	C₈×D₄×C₃₇	—	2	12	no	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	3	yes	—
16	C₂×C₂²⋊C₈×C₃₇	C₂²	1	3	yes	—
17	C₂×C₂²⋊C₈×C₃₇	C₂²	1	3	yes	—
18	C₂×C₂²⋊C₈×C₃₇	C₂²	1	3	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	3	yes	—
20	C₃₇×#395	C₄	1	1	yes	characteristic
21	C₃₇×#394	C₄	1	3	yes	—
22	C₃₇×#393	C₄	1	1	yes	characteristic
23	C₃₇×#392	C₄	1	3	yes	—
24	C₄×C₂²⋊C₄×C₃₇	C₂²	1	3	yes	—
25	C₃₇×#336	C₂²	1	6	yes	—
26	C₂×#41×C₃₇	—	2	6	no	—
27	C₂×#41×C₃₇	—	2	6	no	—
28	C₂²×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₄×C₃₇	C₂×C₄	1	3	yes	—
30	C₂²×D₄×C₃₇	C₂×C₄	1	3	yes	—
31	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₃₇	C₂³	1	3	yes	—
33	#131×C₃₇	—	2	32	no	—
34	C₂×M₁₆×C₃₇	—	2	6	no	—
35	C₂×M₁₆×C₃₇	—	2	12	no	—
36	C₂²×C₈×C₃₇	—	2	12	no	—
37	C₂²×C₈×C₃₇	—	2	6	no	—
38	C₂²×C₈×C₃₇	D₄	1	12	yes	complemented
39	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₇	C₂³	1	3	yes	—
41	C₂²×C₈×C₃₇	C₂³	1	3	yes	—
42	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₃₇	—	2	6	no	—
44	C₄×D₄×C₃₇	—	2	6	no	—
45	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	3	yes	—
46	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	6	yes	—
47	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	3	yes	—
48	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	3	yes	—
49	C₂×C₄²×C₃₇	C₂³	1	3	yes	—
50	C₄⋊C₈×C₃₇	—	2	6	no	—
51	C₂²⋊C₈×C₃₇	—	2	6	no	—
52	C₂²⋊C₈×C₃₇	—	2	6	no	—
53	C₂²⋊C₈×C₃₇	—	2	6	no	—
54	C₄×C₈×C₃₇	—	2	6	no	—
55	#95×C₃₇	C₂×C₄	1	3	yes	—
56	#95×C₃₇	C₂×C₄	1	3	yes	—
57	C₂⁴×C₃₇	C₂²×C₄	1	3	yes	—
58	C₂⁴×C₃₇	C₂²×C₄	1	1	yes	characteristic
59	#41×C₃₇	—	2	8	no	—
60	#41×C₃₇	—	2	24	no	—
61	C₂×Q₈×C₃₇	—	2	6	no	—
62	C₂×D₄×C₃₇	—	2	6	no	—
63	C₂×D₄×C₃₇	—	2	12	no	—
64	C₂×D₄×C₃₇	—	2	6	no	—
65	C₂×D₄×C₃₇	—	2	6	no	—
66	C₂×D₄×C₃₇	—	2	6	no	—
67	C₂²×C₄×C₃₇	—	2	6	no	—
68	C₂²×C₄×C₃₇	—	2	6	no	—
69	C₂²×C₄×C₃₇	—	2	3	no	—
70	C₂²×C₄×C₃₇	—	2	3	no	—
71	C₂²×C₄×C₃₇	C₂²×C₄	1	3	yes	—
72	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₇	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₃₇	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	6	yes	—
76	C₂²×C₄×C₃₇	#41	1	6	yes	—
77	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₃₇	—	2	24	no	—
79	C₂×C₈×C₃₇	—	2	6	no	—
80	C₂×C₈×C₃₇	—	2	24	no	—
81	C₂×C₈×C₃₇	—	2	3	no	—
82	C₂×C₈×C₃₇	—	2	3	no	—
83	C₂×C₈×C₃₇	—	2	6	no	—
84	C₂×C₈×C₃₇	—	4	4	no	—
85	C₂×C₈×C₃₇	C₂×D₄	1	6	yes	—
86	C₂×C₈×C₃₇	C₂×D₄	1	6	yes	—
87	C₄⋊C₄×C₃₇	—	2	6	no	—
88	C₂²⋊C₄×C₃₇	—	2	6	no	—
89	C₂²⋊C₄×C₃₇	—	2	6	no	—
90	C₂²⋊C₄×C₃₇	—	2	12	no	—
91	C₄²×C₃₇	—	2	3	no	—
92	C₄²×C₃₇	—	2	3	no	—
93	C₂³×C₃₇	—	2	3	no	—
94	C₂³×C₃₇	—	2	6	no	—
95	C₂³×C₃₇	—	2	6	no	—
96	C₂³×C₃₇	—	2	3	no	—
97	C₂³×C₃₇	—	2	3	no	—
98	C₂³×C₃₇	—	2	3	no	—
99	C₂³×C₃₇	C₄×D₄	1	6	yes	—
100	C₂³×C₃₇	#131	1	2	yes	complemented
101	C₂³×C₃₇	C₂³×C₄	1	1	yes	characteristic
102	Q₈×C₃₇	—	2	8	no	—
103	D₄×C₃₇	—	2	24	no	—
104	D₄×C₃₇	—	2	24	no	—
105	C₂×C₄×C₃₇	—	2	3	no	—
106	C₂×C₄×C₃₇	—	2	6	no	—
107	C₂×C₄×C₃₇	—	2	3	no	—
108	C₂×C₄×C₃₇	—	2	6	no	—
109	C₂×C₄×C₃₇	—	2	6	no	—
110	C₂×C₄×C₃₇	—	2	12	no	—
111	C₂×C₄×C₃₇	—	2	3	no	—
112	C₂×C₄×C₃₇	—	2	3	no	—
113	C₂×C₄×C₃₇	—	2	6	no	—
114	C₂×C₄×C₃₇	—	4	1	no	—
115	C₂×C₄×C₃₇	—	4	3	no	—
116	C₂×C₄×C₃₇	C₄×D₄	1	6	yes	—
117	C₂×C₄×C₃₇	#131	1	6	yes	complemented
118	C₂×C₄×C₃₇	C₂²×D₄	1	3	yes	—
119	C₂×C₄×C₃₇	C₂×#41	1	3	yes	—
120	C₈×C₃₇	—	2	6	no	—
121	C₈×C₃₇	—	4	1	no	—
122	C₈×C₃₇	—	4	3	no	—
123	C₈×C₃₇	#120	1	4	yes	—
124	#21442	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₃₇	—	2	6	no	—
126	C₂²×C₃₇	—	2	6	no	—
127	C₂²×C₃₇	—	2	6	no	—
128	C₂²×C₃₇	—	2	6	no	—
129	C₂²×C₃₇	—	4	6	no	—
130	C₂²×C₃₇	—	4	3	no	—
131	C₂²×C₃₇	—	4	6	no	—
132	C₂²×C₃₇	—	4	1	no	—
133	C₂²×C₃₇	C₂×C₄×D₄	1	3	yes	—
134	C₂²×C₃₇	C₂×#131	1	1	yes	characteristic
135	C₂²×C₃₇	C₂×#131	1	3	yes	—
136	C₄×C₃₇	—	2	6	no	—
137	C₄×C₃₇	—	2	6	no	—
138	C₄×C₃₇	—	4	3	no	—
139	C₄×C₃₇	—	4	1	no	—
140	C₄×C₃₇	C₂×#120	1	1	yes	characteristic
141	C₄×C₃₇	#525	1	3	yes	—
142	C₂²×#131	C₂×C₃₇	1	1	yes	characteristic, complemented
143	C₂×C₈×D₄	C₂×C₃₇	1	3	yes	complemented
144	#2388	C₂×C₃₇	1	1	yes	characteristic
145	#1870	C₂×C₃₇	1	3	yes	complemented
146	#1851	C₂×C₃₇	1	3	yes	complemented
147	#1780	C₂×C₃₇	1	1	yes	characteristic, complemented
148	C₈×C₂²⋊C₄	C₂×C₃₇	1	3	yes	complemented
149	C₂×C₃₇	—	4	6	no	—
150	C₂×C₃₇	—	4	2	no	—
151	C₂×C₃₇	C₄×#120	1	1	yes	characteristic
152	C₂×C₃₇	#2929	1	3	yes	—
153	C₂×C₃₇	#2931	1	3	yes	—
154	C₂²×#41	C₂²×C₃₇	1	1	yes	characteristic
155	C₂×#131	—	2	12	no	—
156	C₂²×M₁₆	C₂²×C₃₇	1	3	yes	—
157	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
158	C₂³×C₈	C₂²×C₃₇	1	3	yes	—
159	C₂×C₄×D₄	C₂²×C₃₇	1	3	yes	—
160	C₈×D₄	—	2	12	no	—
161	C₂×C₄⋊C₈	C₂²×C₃₇	1	3	yes	—
162	C₂×C₂²⋊C₈	C₂²×C₃₇	1	3	yes	—
163	C₂×C₂²⋊C₈	C₂²×C₃₇	1	3	yes	—
164	C₂×C₂²⋊C₈	C₂²×C₃₇	1	3	yes	—
165	C₂×C₄×C₈	C₂²×C₃₇	1	3	yes	—
166	#395	C₄×C₃₇	1	1	yes	characteristic
167	#394	C₄×C₃₇	1	3	yes	—
168	#393	C₄×C₃₇	1	1	yes	characteristic
169	#392	C₄×C₃₇	1	3	yes	—
170	C₄×C₂²⋊C₄	C₂²×C₃₇	1	3	yes	—
171	#336	C₂²×C₃₇	1	6	yes	—
172	C₃₇	#21442	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×#41	—	2	6	no	—
174	C₂×#41	—	2	6	no	—
175	C₂²×Q₈	C₂×C₄×C₃₇	1	1	yes	characteristic
176	C₂²×D₄	C₂×C₄×C₃₇	1	3	yes	—
177	C₂²×D₄	C₂×C₄×C₃₇	1	3	yes	—
178	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
179	C₂³×C₄	C₂³×C₃₇	1	3	yes	—
180	#131	—	2	32	no	—
181	C₂×M₁₆	—	2	6	no	—
182	C₂×M₁₆	—	2	12	no	—
183	C₂²×C₈	—	2	12	no	—
184	C₂²×C₈	—	2	6	no	—
185	C₂²×C₈	D₄×C₃₇	1	12	yes	complemented
186	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₃₇	1	3	yes	—
188	C₂²×C₈	C₂³×C₃₇	1	3	yes	—
189	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
190	C₄×D₄	—	2	6	no	—
191	C₄×D₄	—	2	6	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	3	yes	—
193	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	3	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	3	yes	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	6	yes	—
196	C₂×C₄²	C₂³×C₃₇	1	3	yes	—
197	C₄⋊C₈	—	2	6	no	—
198	C₂²⋊C₈	—	2	6	no	—
199	C₂²⋊C₈	—	2	6	no	—
200	C₂²⋊C₈	—	2	6	no	—
201	C₄×C₈	—	2	6	no	—
202	#95	C₂×C₄×C₃₇	1	3	yes	—
203	#95	C₂×C₄×C₃₇	1	3	yes	—
204	C₂⁴	C₂²×C₄×C₃₇	1	3	yes	—
205	C₂⁴	C₂²×C₄×C₃₇	1	1	yes	characteristic
206	#41	—	2	8	no	—
207	#41	—	2	24	no	—
208	C₂×Q₈	—	2	6	no	—
209	C₂×D₄	—	2	6	no	—
210	C₂×D₄	—	2	12	no	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	2	6	no	—
213	C₂×D₄	—	2	6	no	—
214	C₂²×C₄	—	2	6	no	—
215	C₂²×C₄	—	2	6	no	—
216	C₂²×C₄	—	2	3	no	—
217	C₂²×C₄	—	2	3	no	—
218	C₂²×C₄	C₂²×C₄×C₃₇	1	3	yes	—
219	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₃₇	1	3	yes	—
221	C₂²×C₄	C₂²×C₄×C₃₇	1	3	yes	—
222	C₂²×C₄	C₂×D₄×C₃₇	1	6	yes	—
223	C₂²×C₄	#41×C₃₇	1	6	yes	—
224	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
225	M₁₆	—	2	24	no	—
226	C₂×C₈	—	2	6	no	—
227	C₂×C₈	—	2	24	no	—
228	C₂×C₈	—	2	3	no	—
229	C₂×C₈	—	2	3	no	—
230	C₂×C₈	—	2	6	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	C₂×D₄×C₃₇	1	6	yes	—
233	C₂×C₈	C₂×D₄×C₃₇	1	6	yes	—
234	C₄⋊C₄	—	2	6	no	—
235	C₂²⋊C₄	—	2	6	no	—
236	C₂²⋊C₄	—	2	6	no	—
237	C₂²⋊C₄	—	2	12	no	—
238	C₄²	—	2	3	no	—
239	C₄²	—	2	3	no	—
240	C₂³	—	2	6	no	—
241	C₂³	—	2	3	no	—
242	C₂³	—	2	3	no	—
243	C₂³	—	2	3	no	—
244	C₂³	—	2	3	no	—
245	C₂³	—	2	6	no	—
246	C₂³	C₄×D₄×C₃₇	1	6	yes	—
247	C₂³	#131×C₃₇	1	2	yes	complemented
248	C₂³	C₂³×C₄×C₃₇	1	1	yes	characteristic
249	Q₈	—	2	8	no	—
250	D₄	—	2	24	no	—
251	D₄	—	2	24	no	—
252	C₂×C₄	—	2	3	no	—
253	C₂×C₄	—	2	3	no	—
254	C₂×C₄	—	2	6	no	—
255	C₂×C₄	—	2	6	no	—
256	C₂×C₄	—	2	6	no	—
257	C₂×C₄	—	2	12	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	2	3	no	—
260	C₂×C₄	—	2	3	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	3	no	—
263	C₂×C₄	C₄×D₄×C₃₇	1	6	yes	—
264	C₂×C₄	#131×C₃₇	1	6	yes	complemented
265	C₂×C₄	C₂²×D₄×C₃₇	1	3	yes	—
266	C₂×C₄	C₂×#41×C₃₇	1	3	yes	—
267	C₈	—	2	6	no	—
268	C₈	—	4	1	no	—
269	C₈	—	4	3	no	—
270	C₈	#120×C₃₇	1	4	yes	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	6	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	2	6	no	—
275	C₂²	—	4	6	no	—
276	C₂²	—	4	6	no	—
277	C₂²	—	4	3	no	—
278	C₂²	—	4	1	no	—
279	C₂²	C₂×C₄×D₄×C₃₇	1	3	yes	—
280	C₂²	C₂×#131×C₃₇	1	1	yes	characteristic
281	C₂²	C₂×#131×C₃₇	1	3	yes	—
282	C₄	—	2	6	no	—
283	C₄	—	2	6	no	—
284	C₄	—	4	3	no	—
285	C₄	—	4	1	no	—
286	C₄	C₂×#120×C₃₇	1	1	yes	characteristic
287	C₄	C₃₇×#525	1	3	yes	—
288	C₂	—	4	6	no	—
289	C₂	—	4	2	no	—
290	C₂	C₄×#120×C₃₇	1	1	yes	characteristic
291	C₂	C₃₇×#2929	1	3	yes	—
292	C₂	C₃₇×#2931	1	3	yes	—

	subgroup	G/H
center	C₂²×C₈×C₃₇	C₂³
commutator	C₂³	C₂³×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂³×C₄
prime radical	C₃₇×#2388	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#131×C₃₇	C₂	semidirect
2. C₂×C₈×D₄×C₃₇	C₂	semidirect
4. C₃₇×#1870	C₂	semidirect
5. C₃₇×#1851	C₂	semidirect
6. C₃₇×#1780	C₂	semidirect
7. C₈×C₂²⋊C₄×C₃₇	C₂	semidirect
38. C₂²×C₈×C₃₇	D₄	semidirect
100. C₂³×C₃₇	#131	semidirect
117. C₂×C₄×C₃₇	#131	semidirect
124. #21442	C₃₇	direct (N × H)
142. C₂²×#131	C₂×C₃₇	semidirect
143. C₂×C₈×D₄	C₂×C₃₇	semidirect
145. #1870	C₂×C₃₇	semidirect
146. #1851	C₂×C₃₇	semidirect
147. #1780	C₂×C₃₇	semidirect
148. C₈×C₂²⋊C₄	C₂×C₃₇	semidirect
172. C₃₇	#21442	direct (N × H)
185. C₂²×C₈	D₄×C₃₇	semidirect
247. C₂³	#131×C₃₇	semidirect
264. C₂×C₄	#131×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#131×C₃₇	2
C₂×C₈×D₄×C₃₇	2
C₃₇×#2388	2
C₃₇×#1870	2
C₃₇×#1851	2
C₃₇×#1780	2
C₈×C₂²⋊C₄×C₃₇	2
#21442	37

Subgroups — C37×#21442 #22642695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#21442 #22642695