Subgroups — C₃₇×#21938 #22643191

Order 9472.

1266 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#380	C₂	1	2	yes	maximal
3	C₃₇×#1853	C₂	1	8	yes	maximal, complemented
4	C₃₇×#1807	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₃₇	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₃₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
8	C₂²×C₂²⋊C₄×C₃₇	—	2	2	no	—
9	C₂×#106×C₃₇	C₂²	1	8	yes	complemented
10	C₂×C₂²⋊C₈×C₃₇	C₄	1	8	yes	complemented
11	C₂×C₂²⋊C₈×C₃₇	C₂²	1	4	yes	—
12	C₃₇×#380	—	2	2	no	—
13	C₃₇×#380	—	2	2	no	—
14	C₃₇×#380	C₂²	1	4	yes	—
15	C₃₇×#380	C₂²	1	4	yes	—
16	C₂×#95×C₃₇	—	2	2	no	—
17	C₃₇×#340	C₂²	1	8	yes	complemented
18	C₂⁵×C₃₇	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₃₇	—	2	4	no	—
20	C₂³×C₄×C₃₇	—	4	4	no	—
21	C₂³×C₄×C₃₇	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₃₇	D₄	1	2	yes	—
23	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₇	Q₈	1	4	yes	—
25	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₃₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₃₇	—	4	4	no	—
28	C₂²×C₈×C₃₇	C₂×C₄	1	8	yes	complemented
29	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
30	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₃₇	—	4	8	no	—
32	C₂×C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
33	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₃₇	—	4	8	no	—
36	#106×C₃₇	—	4	8	no	—
37	C₂²⋊C₈×C₃₇	C₂×C₄	1	16	yes	complemented
38	#95×C₃₇	—	2	4	no	—
39	#95×C₃₇	—	2	4	no	—
40	C₂⁴×C₃₇	—	4	2	no	—
41	C₂⁴×C₃₇	—	4	2	no	—
42	C₂⁴×C₃₇	C₄⋊C₄	1	4	yes	—
43	C₂⁴×C₃₇	SD₁₆	1	8	yes	—
44	C₂⁴×C₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂⁴×C₃₇	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₃₇	—	2	8	no	—
47	C₂²×C₄×C₃₇	—	2	4	no	—
48	C₂²×C₄×C₃₇	—	2	8	no	—
49	C₂²×C₄×C₃₇	—	4	8	no	—
50	C₂²×C₄×C₃₇	—	4	8	no	—
51	C₂²×C₄×C₃₇	—	4	2	no	—
52	C₂²×C₄×C₃₇	—	4	4	no	—
53	C₂²×C₄×C₃₇	—	4	8	no	—
54	C₂²×C₄×C₃₇	—	4	8	no	—
55	C₂²×C₄×C₃₇	—	4	8	no	—
56	C₂²×C₄×C₃₇	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₃₇	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₃₇	—	2	8	no	—
66	C₂×C₈×C₃₇	—	4	8	no	—
67	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
68	C₄⋊C₄×C₃₇	—	2	16	no	—
69	C₄⋊C₄×C₃₇	—	4	8	no	—
70	C₂²⋊C₄×C₃₇	—	4	8	no	—
71	C₂²⋊C₄×C₃₇	—	4	8	no	—
72	C₂³×C₃₇	—	2	4	no	—
73	C₂³×C₃₇	—	2	2	no	—
74	C₂³×C₃₇	—	2	4	no	—
75	C₂³×C₃₇	—	2	2	no	—
76	C₂³×C₃₇	—	4	4	no	—
77	C₂³×C₃₇	—	4	4	no	—
78	C₂³×C₃₇	—	4	16	no	—
79	C₂³×C₃₇	#106	1	16	yes	complemented
80	C₂³×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
81	C₂³×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₃₇	C₂×SD₁₆	1	4	yes	—
83	C₂³×C₃₇	C₂×SD₁₆	1	4	yes	—
84	C₂³×C₃₇	C₂×SD₁₆	1	4	yes	—
85	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₃₇	—	2	4	no	—
87	C₂×C₄×C₃₇	—	2	2	no	—
88	C₂×C₄×C₃₇	—	2	4	no	—
89	C₂×C₄×C₃₇	—	2	8	no	—
90	C₂×C₄×C₃₇	—	2	8	no	—
91	C₂×C₄×C₃₇	—	2	4	no	—
92	C₂×C₄×C₃₇	—	4	4	no	—
93	C₂×C₄×C₃₇	—	4	8	no	—
94	C₂×C₄×C₃₇	—	4	8	no	—
95	C₂×C₄×C₃₇	—	4	8	no	—
96	C₂×C₄×C₃₇	—	4	8	no	—
97	C₂×C₄×C₃₇	—	8	16	no	—
98	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₃₇	H(5,𝔽₂)	1	4	yes	—
103	C₈×C₃₇	—	2	8	no	—
104	#21938	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₃₇	—	2	2	no	—
106	C₂²×C₃₇	—	2	4	no	—
107	C₂²×C₃₇	—	2	4	no	—
108	C₂²×C₃₇	—	2	2	no	—
109	C₂²×C₃₇	—	2	8	no	—
110	C₂²×C₃₇	—	2	8	no	—
111	C₂²×C₃₇	—	2	8	no	—
112	C₂²×C₃₇	—	4	16	no	—
113	C₂²×C₃₇	C₂×#106	1	4	yes	—
114	C₂²×C₃₇	C₂×#106	1	8	yes	—
115	C₂²×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₃₇	#534	1	2	yes	—
117	C₂²×C₃₇	C₂²×SD₁₆	1	2	yes	—
118	C₂²×C₃₇	#575	1	2	yes	—
119	C₄×C₃₇	—	2	4	no	—
120	C₄×C₃₇	—	2	2	no	—
121	C₄×C₃₇	—	8	8	no	—
122	C₄×C₃₇	#518	1	2	yes	—
123	C₄×C₃₇	#543	1	2	yes	—
124	C₂²×C₂²⋊C₈	C₂×C₃₇	1	1	yes	characteristic
125	C₂×#380	C₂×C₃₇	1	2	yes	—
126	#1853	C₂×C₃₇	1	8	yes	complemented
127	#1807	C₂×C₃₇	1	4	yes	complemented
128	C₂×C₃₇	—	2	8	no	—
129	C₂×C₃₇	—	2	4	no	—
130	C₂×C₃₇	#2315	1	1	yes	characteristic
131	C₂×C₃₇	C₂²×#106	1	1	yes	characteristic
132	C₂×C₃₇	#2911	1	1	yes	characteristic
133	C₂×C₃₇	#3187	1	4	yes	—
134	C₂⁴×C₄	C₂²×C₃₇	1	1	yes	characteristic
135	C₂³×C₈	C₂²×C₃₇	1	2	yes	—
136	C₂²×C₄⋊C₄	C₂²×C₃₇	1	4	yes	—
137	C₂²×C₂²⋊C₄	—	2	2	no	—
138	C₂×#106	C₂²×C₃₇	1	8	yes	complemented
139	C₂×C₂²⋊C₈	C₄×C₃₇	1	8	yes	complemented
140	C₂×C₂²⋊C₈	C₂²×C₃₇	1	4	yes	—
141	#380	—	2	2	no	—
142	#380	—	2	2	no	—
143	#380	C₂²×C₃₇	1	4	yes	—
144	#380	C₂²×C₃₇	1	4	yes	—
145	C₂×#95	—	2	2	no	—
146	#340	C₂²×C₃₇	1	8	yes	complemented
147	C₃₇	#21938	1	1	yes	characteristic, direct product factor, Sylow
148	C₂⁵	D₄×C₃₇	1	1	yes	characteristic
149	C₂³×C₄	—	2	4	no	—
150	C₂³×C₄	—	4	4	no	—
151	C₂³×C₄	C₂×C₄×C₃₇	1	4	yes	—
152	C₂³×C₄	D₄×C₃₇	1	2	yes	—
153	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
154	C₂³×C₄	Q₈×C₃₇	1	4	yes	—
155	C₂³×C₄	C₂³×C₃₇	1	2	yes	—
156	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
157	C₂²×C₈	—	4	4	no	—
158	C₂²×C₈	C₂×C₄×C₃₇	1	8	yes	complemented
159	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	—	4	8	no	—
162	C₂×C₄⋊C₄	C₂³×C₃₇	1	8	yes	—
163	C₂×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₂²⋊C₄	—	2	4	no	—
165	C₂×C₂²⋊C₄	—	4	8	no	—
166	#106	—	4	8	no	—
167	C₂²⋊C₈	C₂×C₄×C₃₇	1	16	yes	complemented
168	#95	—	2	4	no	—
169	#95	—	2	4	no	—
170	C₂⁴	—	4	2	no	—
171	C₂⁴	—	4	2	no	—
172	C₂⁴	C₄⋊C₄×C₃₇	1	4	yes	—
173	C₂⁴	SD₁₆×C₃₇	1	8	yes	—
174	C₂⁴	C₂×D₄×C₃₇	1	1	yes	characteristic
175	C₂⁴	C₂×D₄×C₃₇	1	2	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	4	8	no	—
180	C₂²×C₄	—	4	8	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	—	4	8	no	—
184	C₂²×C₄	—	4	8	no	—
185	C₂²×C₄	—	4	8	no	—
186	C₂²×C₄	C₄⋊C₄×C₃₇	1	8	yes	—
187	C₂²×C₄	C₄⋊C₄×C₃₇	1	4	yes	—
188	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
189	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
192	C₂²×C₄	C₂×Q₈×C₃₇	1	4	yes	—
193	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
194	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
195	C₂×C₈	—	2	8	no	—
196	C₂×C₈	—	4	8	no	—
197	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
198	C₄⋊C₄	—	2	16	no	—
199	C₄⋊C₄	—	4	8	no	—
200	C₂²⋊C₄	—	4	8	no	—
201	C₂²⋊C₄	—	4	8	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	4	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	4	4	no	—
207	C₂³	—	4	4	no	—
208	C₂³	—	4	16	no	—
209	C₂³	#106×C₃₇	1	16	yes	complemented
210	C₂³	C₂×C₄⋊C₄×C₃₇	1	4	yes	—
211	C₂³	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
212	C₂³	C₂×SD₁₆×C₃₇	1	4	yes	—
213	C₂³	C₂×SD₁₆×C₃₇	1	4	yes	—
214	C₂³	C₂×SD₁₆×C₃₇	1	4	yes	—
215	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	8	16	no	—
228	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	4	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
230	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
231	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
232	C₂×C₄	H(5,𝔽₂)×C₃₇	1	4	yes	—
233	C₈	—	2	8	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	8	no	—
237	C₂²	—	2	8	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	4	16	no	—
242	C₂²	C₂×#106×C₃₇	1	4	yes	—
243	C₂²	C₂×#106×C₃₇	1	8	yes	—
244	C₂²	C₂²×C₄⋊C₄×C₃₇	1	1	yes	characteristic
245	C₂²	C₃₇×#534	1	2	yes	—
246	C₂²	C₂²×SD₁₆×C₃₇	1	2	yes	—
247	C₂²	C₃₇×#575	1	2	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	8	8	no	—
251	C₄	C₃₇×#518	1	2	yes	—
252	C₄	C₃₇×#543	1	2	yes	—
253	C₂	—	2	8	no	—
254	C₂	—	2	4	no	—
255	C₂	C₃₇×#2315	1	1	yes	characteristic
256	C₂	C₂²×#106×C₃₇	1	1	yes	characteristic
257	C₂	C₃₇×#2911	1	1	yes	characteristic
258	C₂	C₃₇×#3187	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂⁵×C₃₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂⁴×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1853	C₂	semidirect
4. C₃₇×#1807	C₂	semidirect
9. C₂×#106×C₃₇	C₂²	semidirect
10. C₂×C₂²⋊C₈×C₃₇	C₄	semidirect
17. C₃₇×#340	C₂²	semidirect
28. C₂²×C₈×C₃₇	C₂×C₄	semidirect
37. C₂²⋊C₈×C₃₇	C₂×C₄	semidirect
79. C₂³×C₃₇	#106	semidirect
104. #21938	C₃₇	direct (N × H)
126. #1853	C₂×C₃₇	semidirect
127. #1807	C₂×C₃₇	semidirect
138. C₂×#106	C₂²×C₃₇	semidirect
139. C₂×C₂²⋊C₈	C₄×C₃₇	semidirect
146. #340	C₂²×C₃₇	semidirect
147. C₃₇	#21938	direct (N × H)
158. C₂²×C₈	C₂×C₄×C₃₇	semidirect
167. C₂²⋊C₈	C₂×C₄×C₃₇	semidirect
209. C₂³	#106×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊C₈×C₃₇	2
C₂×C₃₇×#380	2
C₃₇×#1853	2
C₃₇×#1807	2
#21938	37

Subgroups — C37×#21938 #22643191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#21938 #22643191