Subgroups — C₃₇×#19596 #22640849

Order 9472.

814 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3167	C₂	1	4	yes	maximal, complemented
2	C₃₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#3147	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1708	C₂	1	2	yes	maximal, complemented
5	C₃₇×#1707	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1706	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1705	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₃₇	—	2	2	no	—
10	C₃₇×#533	C₂²	1	2	yes	—
11	C₃₇×#532	C₂²	1	2	yes	—
12	C₃₇×#500	C₂²	1	2	yes	—
13	C₃₇×#499	—	2	2	no	—
14	C₃₇×#498	C₂²	1	2	yes	—
15	C₃₇×#498	C₂²	1	4	yes	—
16	C₃₇×#495	—	2	2	no	—
17	C₃₇×#494	C₂²	1	1	yes	characteristic
18	C₃₇×#493	C₂²	1	1	yes	characteristic
19	C₃₇×#492	C₂²	1	2	yes	—
20	C₃₇×#492	C₂²	1	4	yes	—
21	C₃₇×#487	C₂²	1	4	yes	—
22	C₃₇×#486	C₂²	1	4	yes	—
23	C₃₇×#465	—	2	8	no	—
24	C₄×SD₁₆×C₃₇	—	2	8	no	—
25	C₃₇×#427	—	2	2	no	—
26	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
28	C₈²×C₃₇	C₂²	1	1	yes	characteristic
29	C₈²×C₃₇	C₂²	1	2	yes	—
30	C₈²×C₃₇	C₂²	1	1	yes	characteristic
31	C₂×#41×C₃₇	—	2	2	no	—
32	#135×C₃₇	—	4	8	no	—
33	C₂×Q₁₆×C₃₇	—	2	2	no	—
34	C₂×SD₁₆×C₃₇	—	2	4	no	—
35	C₂×SD₁₆×C₃₇	—	2	8	no	—
36	C₂×D₈×C₃₇	—	2	2	no	—
37	C₂²×C₈×C₃₇	D₄	1	4	yes	—
38	C₂²×C₈×C₃₇	D₄	1	2	yes	—
39	C₄⋊Q₈×C₃₇	C₂³	1	2	yes	—
40	C₄⋊Q₈×C₃₇	C₂³	1	4	yes	—
41	#127×C₃₇	C₂³	1	2	yes	—
42	#125×C₃₇	—	2	2	no	—
43	#124×C₃₇	—	2	2	no	—
44	C₂²⋊Q₈×C₃₇	—	2	4	no	—
45	#121×C₃₇	—	2	4	no	—
46	C₄×Q₈×C₃₇	—	2	4	no	—
47	C₄×D₄×C₃₇	—	2	4	no	—
48	#117×C₃₇	—	2	2	no	—
49	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	#107×C₃₇	—	2	4	no	—
51	#106×C₃₇	—	2	4	no	—
52	#106×C₃₇	—	2	8	no	—
53	#103×C₃₇	—	2	8	no	—
54	#102×C₃₇	—	2	8	no	—
55	C₄×C₈×C₃₇	D₄	1	4	yes	—
56	C₄×C₈×C₃₇	D₄	1	2	yes	—
57	C₄×C₈×C₃₇	C₂³	1	2	yes	—
58	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₃₇	C₂³	1	2	yes	—
60	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
61	#41×C₃₇	—	4	8	no	—
62	C₂×Q₈×C₃₇	—	2	4	no	—
63	C₂×Q₈×C₃₇	—	2	2	no	—
64	C₂×D₄×C₃₇	—	2	4	no	—
65	C₂×D₄×C₃₇	—	2	2	no	—
66	C₂×D₄×C₃₇	—	4	2	no	—
67	C₂²×C₄×C₃₇	—	4	2	no	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₃₇	—	4	4	no	—
71	SD₁₆×C₃₇	—	4	8	no	—
72	SD₁₆×C₃₇	—	4	16	no	—
73	D₈×C₃₇	—	4	4	no	—
74	C₂×C₈×C₃₇	—	2	8	no	—
75	C₂×C₈×C₃₇	SD₁₆	1	8	yes	complemented
76	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
80	C₄⋊C₄×C₃₇	—	2	4	no	—
81	C₄⋊C₄×C₃₇	—	2	4	no	—
82	C₄⋊C₄×C₃₇	—	2	4	no	—
83	C₄⋊C₄×C₃₇	—	4	4	no	—
84	C₂²⋊C₄×C₃₇	—	4	2	no	—
85	C₂²⋊C₄×C₃₇	—	4	4	no	—
86	C₄²×C₃₇	—	4	2	no	—
87	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₃₇	C₂×D₄	1	2	yes	—
89	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
90	C₂³×C₃₇	—	4	2	no	—
91	C₂³×C₃₇	#127	1	1	yes	characteristic
92	Q₈×C₃₇	—	4	4	no	—
93	Q₈×C₃₇	—	4	8	no	—
94	D₄×C₃₇	—	4	4	no	—
95	D₄×C₃₇	—	4	8	no	—
96	D₄×C₃₇	—	8	4	no	—
97	C₂×C₄×C₃₇	—	2	4	no	—
98	C₂×C₄×C₃₇	—	4	4	no	—
99	C₂×C₄×C₃₇	—	4	2	no	—
100	C₂×C₄×C₃₇	—	8	4	no	—
101	C₂×C₄×C₃₇	#127	1	2	yes	—
102	C₂×C₄×C₃₇	#127	1	1	yes	characteristic
103	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
104	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
106	C₈×C₃₇	—	2	4	no	—
107	C₈×C₃₇	C₂×SD₁₆	1	4	yes	—
108	C₈×C₃₇	C₂×SD₁₆	1	4	yes	—
109	C₈×C₃₇	#135	1	8	yes	complemented
110	#19596	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₃₇	—	2	2	no	—
112	C₂²×C₃₇	—	8	2	no	—
113	C₂²×C₃₇	—	8	4	no	—
114	C₂²×C₃₇	#492	1	2	yes	—
115	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
116	C₄×C₃₇	—	2	2	no	—
117	C₄×C₃₇	—	8	2	no	—
118	C₄×C₃₇	—	8	4	no	—
119	C₄×C₃₇	#492	1	2	yes	—
120	C₄×C₃₇	C₂²×SD₁₆	1	2	yes	—
121	C₄×C₃₇	C₂×#135	1	4	yes	—
122	#3167	C₂×C₃₇	1	4	yes	complemented
123	#3160	C₂×C₃₇	1	1	yes	characteristic, complemented
124	#3147	C₂×C₃₇	1	1	yes	characteristic
125	#1708	C₂×C₃₇	1	2	yes	complemented
126	#1707	C₂×C₃₇	1	2	yes	complemented
127	#1706	C₂×C₃₇	1	2	yes	complemented
128	#1705	C₂×C₃₇	1	2	yes	complemented
129	C₂×C₈²	C₂×C₃₇	1	1	yes	characteristic, complemented
130	C₂×C₃₇	—	2	2	no	—
131	C₂×C₃₇	—	16	2	no	—
132	C₂×C₃₇	C₂×#492	1	1	yes	characteristic
133	C₂×C₃₇	#3147	1	2	yes	—
134	C₂×#135	—	2	2	no	—
135	#533	C₂²×C₃₇	1	2	yes	—
136	#532	C₂²×C₃₇	1	2	yes	—
137	#500	C₂²×C₃₇	1	2	yes	—
138	#499	—	2	2	no	—
139	#498	C₂²×C₃₇	1	2	yes	—
140	#498	C₂²×C₃₇	1	4	yes	—
141	#495	—	2	2	no	—
142	#494	C₂²×C₃₇	1	1	yes	characteristic
143	#493	C₂²×C₃₇	1	1	yes	characteristic
144	#492	C₂²×C₃₇	1	2	yes	—
145	#492	C₂²×C₃₇	1	4	yes	—
146	#487	C₂²×C₃₇	1	4	yes	—
147	#486	C₂²×C₃₇	1	4	yes	—
148	#465	—	2	8	no	—
149	C₄×SD₁₆	—	2	8	no	—
150	#427	—	2	2	no	—
151	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
153	C₈²	C₂²×C₃₇	1	1	yes	characteristic
154	C₈²	C₂²×C₃₇	1	2	yes	—
155	C₈²	C₂²×C₃₇	1	1	yes	characteristic
156	C₃₇	#19596	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×#41	—	2	2	no	—
158	#135	—	4	8	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×SD₁₆	—	2	4	no	—
161	C₂×SD₁₆	—	2	8	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂²×C₈	D₄×C₃₇	1	4	yes	—
164	C₂²×C₈	D₄×C₃₇	1	2	yes	—
165	C₄⋊Q₈	C₂³×C₃₇	1	2	yes	—
166	C₄⋊Q₈	C₂³×C₃₇	1	4	yes	—
167	#127	C₂³×C₃₇	1	2	yes	—
168	#125	—	2	2	no	—
169	#124	—	2	2	no	—
170	C₂²⋊Q₈	—	2	4	no	—
171	#121	—	2	4	no	—
172	C₄×Q₈	—	2	4	no	—
173	C₄×D₄	—	2	4	no	—
174	#117	—	2	2	no	—
175	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
176	#107	—	2	4	no	—
177	#106	—	2	4	no	—
178	#106	—	2	8	no	—
179	#103	—	2	8	no	—
180	#102	—	2	8	no	—
181	C₄×C₈	D₄×C₃₇	1	4	yes	—
182	C₄×C₈	D₄×C₃₇	1	2	yes	—
183	C₄×C₈	C₂³×C₃₇	1	2	yes	—
184	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
185	C₄×C₈	C₂³×C₃₇	1	2	yes	—
186	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
187	#41	—	4	8	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
196	Q₁₆	—	4	4	no	—
197	SD₁₆	—	4	8	no	—
198	SD₁₆	—	4	16	no	—
199	D₈	—	4	4	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	SD₁₆×C₃₇	1	8	yes	complemented
202	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
203	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
204	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
205	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	4	2	no	—
213	C₄²	C₂×D₄×C₃₇	1	2	yes	—
214	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
215	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
216	C₂³	—	4	2	no	—
217	C₂³	#127×C₃₇	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	Q₈	—	4	8	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	8	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	#127×C₃₇	1	2	yes	—
228	C₂×C₄	#127×C₃₇	1	1	yes	characteristic
229	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
232	C₈	—	2	4	no	—
233	C₈	C₂×SD₁₆×C₃₇	1	4	yes	—
234	C₈	C₂×SD₁₆×C₃₇	1	4	yes	—
235	C₈	#135×C₃₇	1	8	yes	complemented
236	C₂²	—	2	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	4	no	—
239	C₂²	C₃₇×#492	1	2	yes	—
240	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	4	no	—
244	C₄	C₃₇×#492	1	2	yes	—
245	C₄	C₂²×SD₁₆×C₃₇	1	2	yes	—
246	C₄	C₂×#135×C₃₇	1	4	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	16	2	no	—
249	C₂	C₂×C₃₇×#492	1	1	yes	characteristic
250	C₂	C₃₇×#3147	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂×#127
prime radical	C₃₇×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3167	C₂	semidirect
2. C₃₇×#3160	C₂	semidirect
4. C₃₇×#1708	C₂	semidirect
5. C₃₇×#1707	C₂	semidirect
6. C₃₇×#1706	C₂	semidirect
7. C₃₇×#1705	C₂	semidirect
8. C₂×C₈²×C₃₇	C₂	semidirect
75. C₂×C₈×C₃₇	SD₁₆	semidirect
109. C₈×C₃₇	#135	semidirect
110. #19596	C₃₇	direct (N × H)
122. #3167	C₂×C₃₇	semidirect
123. #3160	C₂×C₃₇	semidirect
125. #1708	C₂×C₃₇	semidirect
126. #1707	C₂×C₃₇	semidirect
127. #1706	C₂×C₃₇	semidirect
128. #1705	C₂×C₃₇	semidirect
129. C₂×C₈²	C₂×C₃₇	semidirect
156. C₃₇	#19596	direct (N × H)
201. C₂×C₈	SD₁₆×C₃₇	semidirect
235. C₈	#135×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3167	2
C₃₇×#3160	2
C₃₇×#3147	2
C₃₇×#1708	2
C₃₇×#1707	2
C₃₇×#1706	2
C₃₇×#1705	2
C₂×C₈²×C₃₇	2
#19596	37

Subgroups — C37×#19596 #22640849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#19596 #22640849