Subgroups — C₂²×C₄×D₈×C₃₇ #22681401

Order 9472.

3694 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₃₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈×C₃₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#107×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#102×C₃₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₃₇	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₃₇	—	2	2	no	—
9	C₂²×D₈×C₃₇	C₄	1	16	yes	direct product factor
10	C₂²×D₈×C₃₇	C₂²	1	12	yes	—
11	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₇	—	2	12	no	—
14	C₂×C₄×D₄×C₃₇	C₂²	1	24	yes	complemented
15	C₂²×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₃₇	—	2	2	no	—
17	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	C₄×D₈×C₃₇	C₂²	1	64	yes	direct product factor
19	C₂×#107×C₃₇	C₂²	1	12	yes	complemented
20	C₂×#102×C₃₇	C₂²	1	24	yes	complemented
21	C₂×C₄×C₈×C₃₇	C₂²	1	12	yes	complemented
22	C₂⁵×C₃₇	—	2	2	no	—
23	C₂²×D₄×C₃₇	—	2	6	no	—
24	C₂²×D₄×C₃₇	—	2	8	no	—
25	C₂²×D₄×C₃₇	C₂×C₄	1	16	yes	complemented
26	C₂²×D₄×C₃₇	C₂³	1	12	yes	—
27	C₂³×C₄×C₃₇	—	2	2	no	—
28	C₂³×C₄×C₃₇	—	2	24	no	—
29	C₂³×C₄×C₃₇	—	4	16	no	—
30	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
33	C₂×D₈×C₃₇	C₂×C₄	1	96	yes	direct product factor
34	C₂×D₈×C₃₇	C₂³	1	16	yes	—
35	C₂²×C₈×C₃₇	—	2	4	no	—
36	C₂²×C₈×C₃₇	C₂×C₄	1	8	yes	complemented
37	C₂²×C₈×C₃₇	C₂³	1	6	yes	—
38	C₂²×C₈×C₃₇	C₂³	1	6	yes	—
39	C₄×D₄×C₃₇	—	2	48	no	—
40	C₄×D₄×C₃₇	C₂³	1	32	yes	complemented
41	C₂×C₄⋊C₄×C₃₇	—	2	6	no	—
42	C₂×C₄⋊C₄×C₃₇	C₂³	1	12	yes	—
43	C₂×C₂²⋊C₄×C₃₇	—	2	24	no	—
44	C₂×C₄²×C₃₇	D₄	1	6	yes	—
45	C₂×C₄²×C₃₇	C₂³	1	6	yes	—
46	#107×C₃₇	C₂³	1	16	yes	complemented
47	#102×C₃₇	C₂³	1	32	yes	complemented
48	C₄×C₈×C₃₇	C₂³	1	16	yes	complemented
49	C₂⁴×C₃₇	—	2	16	no	—
50	C₂⁴×C₃₇	—	2	12	no	—
51	C₂⁴×C₃₇	—	4	8	no	—
52	C₂⁴×C₃₇	—	4	8	no	—
53	C₂⁴×C₃₇	C₂×D₄	1	1	yes	characteristic
54	C₂×D₄×C₃₇	—	2	48	no	—
55	C₂×D₄×C₃₇	—	2	12	no	—
56	C₂×D₄×C₃₇	—	2	24	no	—
57	C₂×D₄×C₃₇	C₂²×C₄	1	48	yes	complemented
58	C₂×D₄×C₃₇	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₃₇	—	2	4	no	—
60	C₂²×C₄×C₃₇	—	2	12	no	—
61	C₂²×C₄×C₃₇	—	2	32	no	—
62	C₂²×C₄×C₃₇	—	4	8	no	—
63	C₂²×C₄×C₃₇	—	4	96	no	—
64	C₂²×C₄×C₃₇	D₈	1	8	yes	direct product factor
65	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₃₇	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₃₇	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₃₇	#41	1	4	yes	—
70	C₂²×C₄×C₃₇	C₂⁴	1	3	yes	—
71	D₈×C₃₇	C₂²×C₄	1	64	yes	direct product factor
72	C₂×C₈×C₃₇	—	2	12	no	—
73	C₂×C₈×C₃₇	C₂²×C₄	1	24	yes	complemented
74	C₂×C₈×C₃₇	C₂⁴	1	4	yes	—
75	C₂×C₈×C₃₇	C₂⁴	1	4	yes	—
76	C₄⋊C₄×C₃₇	—	2	12	no	—
77	C₄⋊C₄×C₃₇	C₂⁴	1	8	yes	—
78	C₂²⋊C₄×C₃₇	—	2	32	no	—
79	C₄²×C₃₇	C₂×D₄	1	12	yes	—
80	C₄²×C₃₇	C₂⁴	1	4	yes	—
81	C₂³×C₃₇	—	2	48	no	—
82	C₂³×C₃₇	—	2	8	no	—
83	C₂³×C₃₇	—	4	24	no	—
84	C₂³×C₃₇	—	4	24	no	—
85	C₂³×C₃₇	—	4	64	no	—
86	C₂³×C₃₇	C₄×D₄	1	4	yes	—
87	C₂³×C₃₇	C₂×D₈	1	4	yes	—
88	C₂³×C₃₇	#135	1	4	yes	—
89	C₂³×C₃₇	C₂²×D₄	1	3	yes	—
90	D₄×C₃₇	—	2	48	no	—
91	D₄×C₃₇	—	2	8	no	—
92	D₄×C₃₇	C₂³×C₄	1	16	yes	complemented
93	C₂×C₄×C₃₇	—	2	12	no	—
94	C₂×C₄×C₃₇	—	2	8	no	—
95	C₂×C₄×C₃₇	—	4	24	no	—
96	C₂×C₄×C₃₇	—	4	64	no	—
97	C₂×C₄×C₃₇	C₄×D₄	1	12	yes	—
98	C₂×C₄×C₃₇	C₂×D₈	1	24	yes	direct product factor
99	C₂×C₄×C₃₇	C₂³×C₄	1	6	yes	—
100	C₂×C₄×C₃₇	C₂²×D₄	1	4	yes	—
101	C₂×C₄×C₃₇	C₂²×D₄	1	3	yes	—
102	C₂×C₄×C₃₇	C₂²×D₄	1	4	yes	—
103	C₂×C₄×C₃₇	C₂×#41	1	6	yes	—
104	C₂×C₄×C₃₇	C₂⁵	1	1	yes	characteristic
105	C₈×C₃₇	—	2	4	no	—
106	C₈×C₃₇	C₂³×C₄	1	8	yes	complemented
107	C₂²×C₄×D₈	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₃₇	—	2	16	no	—
109	C₂²×C₃₇	—	4	96	no	—
110	C₂²×C₃₇	—	4	8	no	—
111	C₂²×C₃₇	—	4	8	no	—
112	C₂²×C₃₇	C₄×D₈	1	16	yes	direct product factor
113	C₂²×C₃₇	C₂×C₄×D₄	1	6	yes	—
114	C₂²×C₃₇	C₂²×D₈	1	6	yes	—
115	C₂²×C₃₇	C₂×#135	1	6	yes	—
116	C₂²×C₃₇	C₂³×D₄	1	1	yes	characteristic
117	C₄×C₃₇	—	2	4	no	—
118	C₄×C₃₇	—	4	8	no	—
119	C₄×C₃₇	C₂×C₄×D₄	1	6	yes	—
120	C₄×C₃₇	C₂²×D₈	1	8	yes	direct product factor
121	C₄×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
122	C₄×C₃₇	C₂²×#41	1	1	yes	characteristic
123	C₂³×D₈	C₂×C₃₇	1	1	yes	characteristic
124	C₂²×C₄×D₄	C₂×C₃₇	1	2	yes	complemented
125	C₂×C₄×D₈	C₂×C₃₇	1	24	yes	direct product factor
126	C₂²×#107	C₂×C₃₇	1	1	yes	characteristic, complemented
127	C₂²×#102	C₂×C₃₇	1	2	yes	complemented
128	C₂²×C₄×C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
129	C₂×C₃₇	—	4	16	no	—
130	C₂×C₃₇	C₂×C₄×D₈	1	12	yes	direct product factor
131	C₂×C₃₇	C₂²×C₄×D₄	1	1	yes	characteristic
132	C₂×C₃₇	C₂³×D₈	1	1	yes	characteristic
133	C₂×C₃₇	C₂²×#135	1	1	yes	characteristic
134	C₂³×D₄	C₂²×C₃₇	1	2	yes	—
135	C₂⁴×C₄	—	2	2	no	—
136	C₂²×D₈	C₄×C₃₇	1	16	yes	direct product factor
137	C₂²×D₈	C₂²×C₃₇	1	12	yes	—
138	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
139	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
140	C₂×C₄×D₄	—	2	12	no	—
141	C₂×C₄×D₄	C₂²×C₃₇	1	24	yes	complemented
142	C₂²×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
143	C₂²×C₂²⋊C₄	—	2	2	no	—
144	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
145	C₄×D₈	C₂²×C₃₇	1	64	yes	direct product factor
146	C₂×#107	C₂²×C₃₇	1	12	yes	complemented
147	C₂×#102	C₂²×C₃₇	1	24	yes	complemented
148	C₂×C₄×C₈	C₂²×C₃₇	1	12	yes	complemented
149	C₃₇	C₂²×C₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
150	C₂⁵	—	2	2	no	—
151	C₂²×D₄	—	2	6	no	—
152	C₂²×D₄	—	2	8	no	—
153	C₂²×D₄	C₂×C₄×C₃₇	1	16	yes	complemented
154	C₂²×D₄	C₂³×C₃₇	1	12	yes	—
155	C₂³×C₄	—	2	2	no	—
156	C₂³×C₄	—	2	24	no	—
157	C₂³×C₄	—	4	16	no	—
158	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
159	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
160	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
161	C₂×D₈	C₂×C₄×C₃₇	1	96	yes	direct product factor
162	C₂×D₈	C₂³×C₃₇	1	16	yes	—
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	C₂×C₄×C₃₇	1	8	yes	complemented
165	C₂²×C₈	C₂³×C₃₇	1	6	yes	—
166	C₂²×C₈	C₂³×C₃₇	1	6	yes	—
167	C₄×D₄	—	2	48	no	—
168	C₄×D₄	C₂³×C₃₇	1	32	yes	complemented
169	C₂×C₄⋊C₄	—	2	6	no	—
170	C₂×C₄⋊C₄	C₂³×C₃₇	1	12	yes	—
171	C₂×C₂²⋊C₄	—	2	24	no	—
172	C₂×C₄²	D₄×C₃₇	1	6	yes	—
173	C₂×C₄²	C₂³×C₃₇	1	6	yes	—
174	#107	C₂³×C₃₇	1	16	yes	complemented
175	#102	C₂³×C₃₇	1	32	yes	complemented
176	C₄×C₈	C₂³×C₃₇	1	16	yes	complemented
177	C₂⁴	—	2	16	no	—
178	C₂⁴	—	2	12	no	—
179	C₂⁴	—	4	8	no	—
180	C₂⁴	—	4	8	no	—
181	C₂⁴	C₂×D₄×C₃₇	1	1	yes	characteristic
182	C₂×D₄	—	2	48	no	—
183	C₂×D₄	—	2	12	no	—
184	C₂×D₄	—	2	24	no	—
185	C₂×D₄	C₂²×C₄×C₃₇	1	48	yes	complemented
186	C₂×D₄	C₂⁴×C₃₇	1	8	yes	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	12	no	—
189	C₂²×C₄	—	2	32	no	—
190	C₂²×C₄	—	4	8	no	—
191	C₂²×C₄	—	4	96	no	—
192	C₂²×C₄	D₈×C₃₇	1	8	yes	direct product factor
193	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
194	C₂²×C₄	C₂×D₄×C₃₇	1	6	yes	—
195	C₂²×C₄	C₂×D₄×C₃₇	1	3	yes	—
196	C₂²×C₄	C₂×D₄×C₃₇	1	6	yes	—
197	C₂²×C₄	#41×C₃₇	1	4	yes	—
198	C₂²×C₄	C₂⁴×C₃₇	1	3	yes	—
199	D₈	C₂²×C₄×C₃₇	1	64	yes	direct product factor
200	C₂×C₈	—	2	12	no	—
201	C₂×C₈	C₂²×C₄×C₃₇	1	24	yes	complemented
202	C₂×C₈	C₂⁴×C₃₇	1	4	yes	—
203	C₂×C₈	C₂⁴×C₃₇	1	4	yes	—
204	C₄⋊C₄	—	2	12	no	—
205	C₄⋊C₄	C₂⁴×C₃₇	1	8	yes	—
206	C₂²⋊C₄	—	2	32	no	—
207	C₄²	C₂×D₄×C₃₇	1	12	yes	—
208	C₄²	C₂⁴×C₃₇	1	4	yes	—
209	C₂³	—	2	8	no	—
210	C₂³	—	2	48	no	—
211	C₂³	—	4	24	no	—
212	C₂³	—	4	24	no	—
213	C₂³	—	4	64	no	—
214	C₂³	C₄×D₄×C₃₇	1	4	yes	—
215	C₂³	C₂×D₈×C₃₇	1	4	yes	—
216	C₂³	#135×C₃₇	1	4	yes	—
217	C₂³	C₂²×D₄×C₃₇	1	3	yes	—
218	D₄	—	2	8	no	—
219	D₄	—	2	48	no	—
220	D₄	C₂³×C₄×C₃₇	1	16	yes	complemented
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	4	24	no	—
224	C₂×C₄	—	4	64	no	—
225	C₂×C₄	C₄×D₄×C₃₇	1	12	yes	—
226	C₂×C₄	C₂×D₈×C₃₇	1	24	yes	direct product factor
227	C₂×C₄	C₂³×C₄×C₃₇	1	6	yes	—
228	C₂×C₄	C₂²×D₄×C₃₇	1	4	yes	—
229	C₂×C₄	C₂²×D₄×C₃₇	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₃₇	1	3	yes	—
231	C₂×C₄	C₂×#41×C₃₇	1	6	yes	—
232	C₂×C₄	C₂⁵×C₃₇	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	C₂³×C₄×C₃₇	1	8	yes	complemented
235	C₂²	—	2	16	no	—
236	C₂²	—	4	96	no	—
237	C₂²	—	4	8	no	—
238	C₂²	—	4	8	no	—
239	C₂²	C₄×D₈×C₃₇	1	16	yes	direct product factor
240	C₂²	C₂×C₄×D₄×C₃₇	1	6	yes	—
241	C₂²	C₂²×D₈×C₃₇	1	6	yes	—
242	C₂²	C₂×#135×C₃₇	1	6	yes	—
243	C₂²	C₂³×D₄×C₃₇	1	1	yes	characteristic
244	C₄	—	2	4	no	—
245	C₄	—	4	8	no	—
246	C₄	C₂×C₄×D₄×C₃₇	1	6	yes	—
247	C₄	C₂²×D₈×C₃₇	1	8	yes	direct product factor
248	C₄	C₂⁴×C₄×C₃₇	1	1	yes	characteristic
249	C₄	C₂²×#41×C₃₇	1	1	yes	characteristic
250	C₂	—	4	16	no	—
251	C₂	C₂×C₄×D₈×C₃₇	1	12	yes	direct product factor
252	C₂	C₂²×C₄×D₄×C₃₇	1	1	yes	characteristic
253	C₂	C₂³×D₈×C₃₇	1	1	yes	characteristic
254	C₂	C₂²×#135×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₃₇	D₄
commutator	C₄	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₇
socle	C₂⁴×C₃₇	C₂×D₄
prime radical	C₂³×D₈×C₃₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₇ ◁ C₂²×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₄×C₃₇	C₂	semidirect
3. C₂×C₄×D₈×C₃₇	C₂	direct (N × H)
4. C₂²×#107×C₃₇	C₂	semidirect
5. C₂²×#102×C₃₇	C₂	semidirect
6. C₂²×C₄×C₈×C₃₇	C₂	semidirect
9. C₂²×D₈×C₃₇	C₄	direct (N × H)
14. C₂×C₄×D₄×C₃₇	C₂²	semidirect
18. C₄×D₈×C₃₇	C₂²	direct (N × H)
19. C₂×#107×C₃₇	C₂²	semidirect
20. C₂×#102×C₃₇	C₂²	semidirect
21. C₂×C₄×C₈×C₃₇	C₂²	semidirect
25. C₂²×D₄×C₃₇	C₂×C₄	semidirect
33. C₂×D₈×C₃₇	C₂×C₄	direct (N × H)
36. C₂²×C₈×C₃₇	C₂×C₄	semidirect
40. C₄×D₄×C₃₇	C₂³	semidirect
46. #107×C₃₇	C₂³	semidirect
47. #102×C₃₇	C₂³	semidirect
48. C₄×C₈×C₃₇	C₂³	semidirect
57. C₂×D₄×C₃₇	C₂²×C₄	semidirect
64. C₂²×C₄×C₃₇	D₈	direct (N × H)
71. D₈×C₃₇	C₂²×C₄	direct (N × H)
73. C₂×C₈×C₃₇	C₂²×C₄	semidirect
92. D₄×C₃₇	C₂³×C₄	semidirect
98. C₂×C₄×C₃₇	C₂×D₈	direct (N × H)
106. C₈×C₃₇	C₂³×C₄	semidirect
107. C₂²×C₄×D₈	C₃₇	direct (N × H)
112. C₂²×C₃₇	C₄×D₈	direct (N × H)
120. C₄×C₃₇	C₂²×D₈	direct (N × H)
124. C₂²×C₄×D₄	C₂×C₃₇	semidirect
125. C₂×C₄×D₈	C₂×C₃₇	direct (N × H)
126. C₂²×#107	C₂×C₃₇	semidirect
127. C₂²×#102	C₂×C₃₇	semidirect
128. C₂²×C₄×C₈	C₂×C₃₇	semidirect
130. C₂×C₃₇	C₂×C₄×D₈	direct (N × H)
136. C₂²×D₈	C₄×C₃₇	direct (N × H)
141. C₂×C₄×D₄	C₂²×C₃₇	semidirect
145. C₄×D₈	C₂²×C₃₇	direct (N × H)
146. C₂×#107	C₂²×C₃₇	semidirect
147. C₂×#102	C₂²×C₃₇	semidirect
148. C₂×C₄×C₈	C₂²×C₃₇	semidirect
149. C₃₇	C₂²×C₄×D₈	direct (N × H)
153. C₂²×D₄	C₂×C₄×C₃₇	semidirect
161. C₂×D₈	C₂×C₄×C₃₇	direct (N × H)
164. C₂²×C₈	C₂×C₄×C₃₇	semidirect
168. C₄×D₄	C₂³×C₃₇	semidirect
174. #107	C₂³×C₃₇	semidirect
175. #102	C₂³×C₃₇	semidirect
176. C₄×C₈	C₂³×C₃₇	semidirect
185. C₂×D₄	C₂²×C₄×C₃₇	semidirect
192. C₂²×C₄	D₈×C₃₇	direct (N × H)
199. D₈	C₂²×C₄×C₃₇	direct (N × H)
201. C₂×C₈	C₂²×C₄×C₃₇	semidirect
220. D₄	C₂³×C₄×C₃₇	semidirect
226. C₂×C₄	C₂×D₈×C₃₇	direct (N × H)
234. C₈	C₂³×C₄×C₃₇	semidirect
239. C₂²	C₄×D₈×C₃₇	direct (N × H)
247. C₄	C₂²×D₈×C₃₇	direct (N × H)
251. C₂	C₂×C₄×D₈×C₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₃₇	2
C₂²×C₄×D₄×C₃₇	2
C₂×C₄×D₈×C₃₇	2
C₂²×#107×C₃₇	2
C₂²×#102×C₃₇	2
C₂²×C₄×C₈×C₃₇	2
C₂²×C₄×D₈	37

Subgroups — C22×C4×D8×C37 #22681401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×D₈×C₃₇ #22681401