Subgroups — C₂²×C₃₇×#508 #22683881

Order 9472.

2118 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₇×#508	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Q₃₂×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×SD₃₂×C₃₇	C₂	1	2	yes	maximal, complemented
5	C₂²×D₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₃₇	—	2	2	no	—
8	C₂×#135×C₃₇	—	2	12	no	—
9	C₂×#135×C₃₇	C₂²	1	24	yes	complemented
10	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₃₇	—	2	2	no	—
12	C₂²×D₈×C₃₇	C₂²	1	2	yes	—
13	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₃₇×#508	C₂²	1	64	yes	direct product factor
15	C₂×Q₃₂×C₃₇	C₂²	1	12	yes	complemented
16	C₂×SD₃₂×C₃₇	C₂²	1	24	yes	complemented
17	C₂×D₁₆×C₃₇	C₂²	1	12	yes	complemented
18	C₂²×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₃₇	C₂²	1	12	yes	complemented
21	C₂×#41×C₃₇	—	2	24	no	—
22	C₂×#41×C₃₇	—	4	12	no	—
23	C₂²×Q₈×C₃₇	—	2	2	no	—
24	C₂²×D₄×C₃₇	—	2	2	no	—
25	C₂²×D₄×C₃₇	—	4	2	no	—
26	C₂³×C₄×C₃₇	—	4	2	no	—
27	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
28	#135×C₃₇	—	2	48	no	—
29	#135×C₃₇	C₂³	1	32	yes	complemented
30	C₂×Q₁₆×C₃₇	—	2	6	no	—
31	C₂×Q₁₆×C₃₇	C₂³	1	12	yes	—
32	C₂×SD₁₆×C₃₇	—	2	24	no	—
33	C₂×D₈×C₃₇	—	2	6	no	—
34	C₂×D₈×C₃₇	C₂³	1	12	yes	—
35	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₇	D₄	1	6	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	C₂³	1	6	yes	—
39	Q₃₂×C₃₇	C₂³	1	16	yes	complemented
40	SD₃₂×C₃₇	C₂³	1	32	yes	complemented
41	D₁₆×C₃₇	C₂³	1	16	yes	complemented
42	C₂×C₁₆×C₃₇	C₂³	1	6	yes	—
43	C₂×C₁₆×C₃₇	C₂³	1	6	yes	—
44	C₂×C₁₆×C₃₇	C₂³	1	16	yes	complemented
45	C₂⁴×C₃₇	—	4	2	no	—
46	C₂⁴×C₃₇	D₈	1	1	yes	characteristic
47	#41×C₃₇	—	2	32	no	—
48	#41×C₃₇	—	4	48	no	—
49	C₂×Q₈×C₃₇	—	2	12	no	—
50	C₂×Q₈×C₃₇	—	4	6	no	—
51	C₂×D₄×C₃₇	—	2	12	no	—
52	C₂×D₄×C₃₇	—	4	24	no	—
53	C₂×D₄×C₃₇	—	4	6	no	—
54	C₂²×C₄×C₃₇	—	4	2	no	—
55	C₂²×C₄×C₃₇	—	4	24	no	—
56	C₂²×C₄×C₃₇	D₈	1	6	yes	—
57	C₂²×C₄×C₃₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₇	C₂×D₄	1	6	yes	—
60	Q₁₆×C₃₇	—	2	12	no	—
61	Q₁₆×C₃₇	C₂⁴	1	8	yes	—
62	SD₁₆×C₃₇	—	2	32	no	—
63	D₈×C₃₇	—	2	12	no	—
64	D₈×C₃₇	C₂⁴	1	8	yes	—
65	C₂×C₈×C₃₇	C₂×D₄	1	3	yes	—
66	C₂×C₈×C₃₇	C₂×D₄	1	6	yes	—
67	C₂×C₈×C₃₇	C₂×D₄	1	12	yes	—
68	C₂×C₈×C₃₇	C₂⁴	1	3	yes	—
69	C₂×C₈×C₃₇	C₂⁴	1	4	yes	—
70	C₁₆×C₃₇	C₂⁴	1	4	yes	—
71	C₁₆×C₃₇	C₂⁴	1	4	yes	—
72	C₂³×C₃₇	—	2	4	no	—
73	C₂³×C₃₇	—	4	12	no	—
74	C₂³×C₃₇	—	8	8	no	—
75	C₂³×C₃₇	C₂×D₈	1	6	yes	—
76	C₂³×C₃₇	C₂×D₈	1	1	yes	characteristic
77	Q₈×C₃₇	—	2	8	no	—
78	Q₈×C₃₇	—	4	12	no	—
79	D₄×C₃₇	—	2	8	no	—
80	D₄×C₃₇	—	4	32	no	—
81	D₄×C₃₇	—	4	12	no	—
82	C₂×C₄×C₃₇	—	4	12	no	—
83	C₂×C₄×C₃₇	—	4	32	no	—
84	C₂×C₄×C₃₇	C₂×D₈	1	6	yes	—
85	C₂×C₄×C₃₇	C₂×D₈	1	3	yes	—
86	C₂×C₄×C₃₇	C₂×D₈	1	12	yes	—
87	C₂×C₄×C₃₇	C₂²×D₄	1	3	yes	—
88	C₂×C₄×C₃₇	C₂²×D₄	1	4	yes	—
89	C₈×C₃₇	C₂²×D₄	1	3	yes	—
90	C₈×C₃₇	C₂²×D₄	1	4	yes	—
91	C₈×C₃₇	C₂⁵	1	1	yes	characteristic
92	C₂²×#508	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₃₇	—	2	12	no	—
94	C₂²×C₃₇	—	4	8	no	—
95	C₂²×C₃₇	—	8	24	no	—
96	C₂²×C₃₇	#508	1	4	yes	direct product factor
97	C₂²×C₃₇	C₂²×D₈	1	4	yes	—
98	C₂²×C₃₇	C₂²×D₈	1	3	yes	—
99	C₄×C₃₇	—	4	8	no	—
100	C₄×C₃₇	C₂²×D₈	1	4	yes	—
101	C₄×C₃₇	C₂²×D₈	1	3	yes	—
102	C₄×C₃₇	C₂³×D₄	1	1	yes	characteristic
103	C₂²×#135	C₂×C₃₇	1	2	yes	complemented
104	C₂×#508	C₂×C₃₇	1	24	yes	direct product factor
105	C₂²×Q₃₂	C₂×C₃₇	1	1	yes	characteristic, complemented
106	C₂²×SD₃₂	C₂×C₃₇	1	2	yes	complemented
107	C₂²×D₁₆	C₂×C₃₇	1	1	yes	characteristic, complemented
108	C₂³×C₁₆	C₂×C₃₇	1	1	yes	characteristic, complemented
109	C₂×C₃₇	—	2	4	no	—
110	C₂×C₃₇	—	8	8	no	—
111	C₂×C₃₇	C₂×#508	1	6	yes	direct product factor
112	C₂×C₃₇	C₂³×D₈	1	1	yes	characteristic
113	C₂²×#41	—	2	2	no	—
114	C₂×#135	—	2	12	no	—
115	C₂×#135	C₂²×C₃₇	1	24	yes	complemented
116	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
117	C₂²×SD₁₆	—	2	2	no	—
118	C₂²×D₈	C₂²×C₃₇	1	2	yes	—
119	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
120	#508	C₂²×C₃₇	1	64	yes	direct product factor
121	C₂×Q₃₂	C₂²×C₃₇	1	12	yes	complemented
122	C₂×SD₃₂	C₂²×C₃₇	1	24	yes	complemented
123	C₂×D₁₆	C₂²×C₃₇	1	12	yes	complemented
124	C₂²×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
125	C₂²×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
126	C₂²×C₁₆	C₂²×C₃₇	1	12	yes	complemented
127	C₃₇	C₂²×#508	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×#41	—	2	24	no	—
129	C₂×#41	—	4	12	no	—
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×D₄	—	2	2	no	—
132	C₂²×D₄	—	4	2	no	—
133	C₂³×C₄	—	4	2	no	—
134	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
135	#135	—	2	48	no	—
136	#135	C₂³×C₃₇	1	32	yes	complemented
137	C₂×Q₁₆	—	2	6	no	—
138	C₂×Q₁₆	C₂³×C₃₇	1	12	yes	—
139	C₂×SD₁₆	—	2	24	no	—
140	C₂×D₈	—	2	6	no	—
141	C₂×D₈	C₂³×C₃₇	1	12	yes	—
142	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
143	C₂²×C₈	D₄×C₃₇	1	6	yes	—
144	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₃₇	1	6	yes	—
146	Q₃₂	C₂³×C₃₇	1	16	yes	complemented
147	SD₃₂	C₂³×C₃₇	1	32	yes	complemented
148	D₁₆	C₂³×C₃₇	1	16	yes	complemented
149	C₂×C₁₆	C₂³×C₃₇	1	6	yes	—
150	C₂×C₁₆	C₂³×C₃₇	1	6	yes	—
151	C₂×C₁₆	C₂³×C₃₇	1	16	yes	complemented
152	C₂⁴	—	4	2	no	—
153	C₂⁴	D₈×C₃₇	1	1	yes	characteristic
154	#41	—	2	32	no	—
155	#41	—	4	48	no	—
156	C₂×Q₈	—	2	12	no	—
157	C₂×Q₈	—	4	6	no	—
158	C₂×D₄	—	2	12	no	—
159	C₂×D₄	—	4	24	no	—
160	C₂×D₄	—	4	6	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	—	4	24	no	—
163	C₂²×C₄	D₈×C₃₇	1	6	yes	—
164	C₂²×C₄	D₈×C₃₇	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₃₇	1	6	yes	—
167	Q₁₆	—	2	12	no	—
168	Q₁₆	C₂⁴×C₃₇	1	8	yes	—
169	SD₁₆	—	2	32	no	—
170	D₈	—	2	12	no	—
171	D₈	C₂⁴×C₃₇	1	8	yes	—
172	C₂×C₈	C₂×D₄×C₃₇	1	3	yes	—
173	C₂×C₈	C₂×D₄×C₃₇	1	6	yes	—
174	C₂×C₈	C₂×D₄×C₃₇	1	12	yes	—
175	C₂×C₈	C₂⁴×C₃₇	1	3	yes	—
176	C₂×C₈	C₂⁴×C₃₇	1	4	yes	—
177	C₁₆	C₂⁴×C₃₇	1	4	yes	—
178	C₁₆	C₂⁴×C₃₇	1	4	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	—	4	12	no	—
181	C₂³	—	8	8	no	—
182	C₂³	C₂×D₈×C₃₇	1	6	yes	—
183	C₂³	C₂×D₈×C₃₇	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	Q₈	—	4	12	no	—
186	D₄	—	2	8	no	—
187	D₄	—	4	32	no	—
188	D₄	—	4	12	no	—
189	C₂×C₄	—	4	12	no	—
190	C₂×C₄	—	4	32	no	—
191	C₂×C₄	C₂×D₈×C₃₇	1	6	yes	—
192	C₂×C₄	C₂×D₈×C₃₇	1	3	yes	—
193	C₂×C₄	C₂×D₈×C₃₇	1	12	yes	—
194	C₂×C₄	C₂²×D₄×C₃₇	1	3	yes	—
195	C₂×C₄	C₂²×D₄×C₃₇	1	4	yes	—
196	C₈	C₂²×D₄×C₃₇	1	3	yes	—
197	C₈	C₂²×D₄×C₃₇	1	4	yes	—
198	C₈	C₂⁵×C₃₇	1	1	yes	characteristic
199	C₂²	—	2	12	no	—
200	C₂²	—	4	8	no	—
201	C₂²	—	8	24	no	—
202	C₂²	C₃₇×#508	1	4	yes	direct product factor
203	C₂²	C₂²×D₈×C₃₇	1	4	yes	—
204	C₂²	C₂²×D₈×C₃₇	1	3	yes	—
205	C₄	—	4	8	no	—
206	C₄	C₂²×D₈×C₃₇	1	4	yes	—
207	C₄	C₂²×D₈×C₃₇	1	3	yes	—
208	C₄	C₂³×D₄×C₃₇	1	1	yes	characteristic
209	C₂	—	2	4	no	—
210	C₂	—	8	8	no	—
211	C₂	C₂×C₃₇×#508	1	6	yes	direct product factor
212	C₂	C₂³×D₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₇	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₃₇
socle	C₂³×C₃₇	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ C₂³×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#135×C₃₇	C₂	semidirect
2. C₂×C₃₇×#508	C₂	direct (N × H)
3. C₂²×Q₃₂×C₃₇	C₂	semidirect
4. C₂²×SD₃₂×C₃₇	C₂	semidirect
5. C₂²×D₁₆×C₃₇	C₂	semidirect
6. C₂³×C₁₆×C₃₇	C₂	semidirect
9. C₂×#135×C₃₇	C₂²	semidirect
14. C₃₇×#508	C₂²	direct (N × H)
15. C₂×Q₃₂×C₃₇	C₂²	semidirect
16. C₂×SD₃₂×C₃₇	C₂²	semidirect
17. C₂×D₁₆×C₃₇	C₂²	semidirect
20. C₂²×C₁₆×C₃₇	C₂²	semidirect
29. #135×C₃₇	C₂³	semidirect
39. Q₃₂×C₃₇	C₂³	semidirect
40. SD₃₂×C₃₇	C₂³	semidirect
41. D₁₆×C₃₇	C₂³	semidirect
44. C₂×C₁₆×C₃₇	C₂³	semidirect
92. C₂²×#508	C₃₇	direct (N × H)
96. C₂²×C₃₇	#508	direct (N × H)
103. C₂²×#135	C₂×C₃₇	semidirect
104. C₂×#508	C₂×C₃₇	direct (N × H)
105. C₂²×Q₃₂	C₂×C₃₇	semidirect
106. C₂²×SD₃₂	C₂×C₃₇	semidirect
107. C₂²×D₁₆	C₂×C₃₇	semidirect
108. C₂³×C₁₆	C₂×C₃₇	semidirect
111. C₂×C₃₇	C₂×#508	direct (N × H)
115. C₂×#135	C₂²×C₃₇	semidirect
120. #508	C₂²×C₃₇	direct (N × H)
121. C₂×Q₃₂	C₂²×C₃₇	semidirect
122. C₂×SD₃₂	C₂²×C₃₇	semidirect
123. C₂×D₁₆	C₂²×C₃₇	semidirect
126. C₂²×C₁₆	C₂²×C₃₇	semidirect
127. C₃₇	C₂²×#508	direct (N × H)
136. #135	C₂³×C₃₇	semidirect
146. Q₃₂	C₂³×C₃₇	semidirect
147. SD₃₂	C₂³×C₃₇	semidirect
148. D₁₆	C₂³×C₃₇	semidirect
151. C₂×C₁₆	C₂³×C₃₇	semidirect
202. C₂²	C₃₇×#508	direct (N × H)
211. C₂	C₂×C₃₇×#508	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#135×C₃₇	2
C₂×C₃₇×#508	2
C₂²×Q₃₂×C₃₇	2
C₂²×SD₃₂×C₃₇	2
C₂²×D₁₆×C₃₇	2
C₂³×C₁₆×C₃₇	2
C₂²×#508	37

Subgroups — C22×C37×#508 #22683881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₇×#508 #22683881