Subgroups — C₃₇×#8268 #22629521

Order 9472.

390 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#97×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#1564	C₂	1	2	yes	maximal
3	C₃₇×#1450	C₂	1	2	yes	maximal
4	C₃₇×#1276	C₂	1	2	yes	maximal
5	C₃₇×#431	C₂²	1	4	yes	—
6	C₂×#97×C₃₇	C₄	1	2	yes	—
7	C₂×#97×C₃₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₄³×C₃₇	C₂²	1	1	yes	characteristic
11	C₃₇×#335	C₄	1	4	yes	complemented
12	C₃₇×#334	C₄	1	4	yes	complemented
13	C₃₇×#322	—	2	2	no	—
14	C₈²×C₃₇	—	2	2	no	—
15	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₃₇	—	2	2	no	—
19	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
22	C₂²⋊C₈×C₃₇	—	2	4	no	—
23	#97×C₃₇	—	2	4	no	—
24	#97×C₃₇	C₈	1	8	yes	complemented
25	C₄×C₈×C₃₇	—	2	4	no	—
26	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₃₇	D₄	1	2	yes	—
29	C₄×C₈×C₃₇	Q₈	1	2	yes	—
30	C₂²×C₄×C₃₇	—	2	2	no	—
31	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₇	#41	1	2	yes	—
33	C₂×C₈×C₃₇	—	2	2	no	—
34	C₂×C₈×C₃₇	—	2	8	no	—
35	C₂×C₈×C₃₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₃₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₃₇	C₄²	1	2	yes	—
38	C₂×C₈×C₃₇	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₃₇	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₃₇	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₃₇	M₁₆	1	4	yes	—
42	C₄²×C₃₇	—	2	2	no	—
43	C₄²×C₃₇	—	4	4	no	—
44	C₄²×C₃₇	C₂×C₈	1	4	yes	—
45	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
46	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₃₇	C₂×Q₈	1	1	yes	characteristic
48	C₂³×C₃₇	#117	1	1	yes	characteristic
49	C₂×C₄×C₃₇	—	2	4	no	—
50	C₂×C₄×C₃₇	—	2	4	no	—
51	C₂×C₄×C₃₇	—	4	4	no	—
52	C₂×C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₇	C₄×D₄	1	2	yes	—
55	C₂×C₄×C₃₇	C₄×Q₈	1	2	yes	—
56	C₂×C₄×C₃₇	C₂²×C₈	1	2	yes	—
57	C₂×C₄×C₃₇	C₂×M₁₆	1	2	yes	—
58	C₈×C₃₇	—	2	1	no	—
59	C₈×C₃₇	—	2	1	no	—
60	C₈×C₃₇	—	4	8	no	—
61	C₈×C₃₇	C₄×C₈	1	4	yes	—
62	C₈×C₃₇	C₄⋊C₈	1	8	yes	—
63	#8268	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₃₇	—	2	2	no	—
65	C₂²×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₃₇	#431	1	2	yes	—
67	C₄×C₃₇	—	2	2	no	—
68	C₄×C₃₇	—	4	4	no	—
69	C₄×C₃₇	C₂×C₄×C₈	1	1	yes	characteristic
70	C₄×C₃₇	C₄×M₁₆	1	1	yes	characteristic
71	C₄×C₃₇	C₂×C₄⋊C₈	1	2	yes	—
72	C₄×C₃₇	#444	1	4	yes	—
73	C₄×#97	C₂×C₃₇	1	1	yes	characteristic
74	#1564	C₂×C₃₇	1	2	yes	—
75	#1450	C₂×C₃₇	1	2	yes	—
76	#1276	C₂×C₃₇	1	2	yes	—
77	C₂×C₃₇	—	2	2	no	—
78	C₂×C₃₇	C₄×C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₃₇	#1773	1	2	yes	—
80	#431	C₂²×C₃₇	1	4	yes	—
81	C₂×#97	C₄×C₃₇	1	2	yes	—
82	C₂×#97	C₄×C₃₇	1	2	yes	—
83	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
85	C₄³	C₂²×C₃₇	1	1	yes	characteristic
86	#335	C₄×C₃₇	1	4	yes	complemented
87	#334	C₄×C₃₇	1	4	yes	complemented
88	#322	—	2	2	no	—
89	C₈²	—	2	2	no	—
90	C₃₇	#8268	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
93	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
94	C₂×C₄²	—	2	2	no	—
95	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
96	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
97	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
98	C₂²⋊C₈	—	2	4	no	—
99	#97	—	2	4	no	—
100	#97	C₈×C₃₇	1	8	yes	complemented
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
103	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
104	C₄×C₈	D₄×C₃₇	1	2	yes	—
105	C₄×C₈	Q₈×C₃₇	1	2	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
108	C₂²×C₄	#41×C₃₇	1	2	yes	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	C₄²×C₃₇	1	1	yes	characteristic
112	C₂×C₈	C₄²×C₃₇	1	1	yes	characteristic
113	C₂×C₈	C₄²×C₃₇	1	2	yes	—
114	C₂×C₈	C₄⋊C₄×C₃₇	1	4	yes	—
115	C₂×C₈	C₂×C₈×C₃₇	1	4	yes	—
116	C₂×C₈	C₂×C₈×C₃₇	1	4	yes	—
117	C₂×C₈	M₁₆×C₃₇	1	4	yes	—
118	C₄²	—	2	2	no	—
119	C₄²	—	4	4	no	—
120	C₄²	C₂×C₈×C₃₇	1	4	yes	—
121	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
122	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
123	C₄²	C₂×Q₈×C₃₇	1	1	yes	characteristic
124	C₂³	#117×C₃₇	1	1	yes	characteristic
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	C₂×C₄²×C₃₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
130	C₂×C₄	C₄×D₄×C₃₇	1	2	yes	—
131	C₂×C₄	C₄×Q₈×C₃₇	1	2	yes	—
132	C₂×C₄	C₂²×C₈×C₃₇	1	2	yes	—
133	C₂×C₄	C₂×M₁₆×C₃₇	1	2	yes	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	4	8	no	—
137	C₈	C₄×C₈×C₃₇	1	4	yes	—
138	C₈	C₄⋊C₈×C₃₇	1	8	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₄×C₄⋊C₄×C₃₇	1	1	yes	characteristic
141	C₂²	C₃₇×#431	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	4	4	no	—
144	C₄	C₂×C₄×C₈×C₃₇	1	1	yes	characteristic
145	C₄	C₄×M₁₆×C₃₇	1	1	yes	characteristic
146	C₄	C₂×C₄⋊C₈×C₃₇	1	2	yes	—
147	C₄	C₃₇×#444	1	4	yes	—
148	C₂	—	2	2	no	—
149	C₂	C₄×C₄⋊C₈×C₃₇	1	1	yes	characteristic
150	C₂	C₃₇×#1773	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₇	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₃₇
socle	C₂²×C₃₇	C₄×C₄⋊C₄
prime radical	C₂³×C₃₇	#117
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₇ ◁ C₂×C₄×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₃₇×#335	C₄	semidirect
12. C₃₇×#334	C₄	semidirect
24. #97×C₃₇	C₈	semidirect
63. #8268	C₃₇	direct (N × H)
86. #335	C₄×C₃₇	semidirect
87. #334	C₄×C₃₇	semidirect
90. C₃₇	#8268	direct (N × H)
100. #97	C₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#97×C₃₇	2
C₃₇×#1564	2
C₃₇×#1450	2
C₃₇×#1276	2
#8268	37

Subgroups — C37×#8268 #22629521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#8268 #22629521