Group #18010825 — subgroups

Subgroups — #18010825

Order 4736.

260 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924610	C₂	1	2	yes	maximal, complemented
2	#14558428	C₂	1	4	yes	maximal, complemented
3	C₂²×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
4	#11924610	—	2	4	no	—
5	C₂×#10596927	C₂²	1	4	yes	complemented
6	C₂²×C₄×D₃₇	C₄	1	1	yes	characteristic
7	C₂²×#64436	C₂²	1	1	yes	characteristic
8	C₂×C₈×D₃₇	—	2	4	no	—
9	C₂×C₈×D₃₇	C₄	1	2	yes	—
10	C₂×C₈×D₃₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₃₇	C₄	1	1	yes	characteristic
12	C₂³×D₃₇	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₃₇	C₈	1	1	yes	characteristic
15	C₂×C₄×D₃₇	—	2	2	no	—
16	C₂×C₄×D₃₇	—	2	2	no	—
17	C₂×C₄×D₃₇	C₈	1	2	yes	—
18	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₃₇	—	2	2	no	—
20	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
21	C₂×#64436	C₂×C₄	1	1	yes	characteristic
22	C₂×#64436	D₄	1	4	yes	—
23	C₂×#64436	C₂³	1	1	yes	characteristic
24	C₈×D₃₇	—	2	4	no	—
25	C₈×D₃₇	—	2	4	no	—
26	#10596927	—	2	4	no	—
27	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×D₃₇	—	2	2	no	—
29	C₂²×D₃₇	—	2	4	no	—
30	C₂²×D₃₇	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₃₇	M₁₆	1	2	yes	—
34	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₃₇	—	2	2	no	—
36	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
37	C₄×D₃₇	—	2	4	no	—
38	C₄×D₃₇	—	2	2	no	—
39	C₄×D₃₇	—	2	2	no	—
40	C₈×C₃₇	—	2	2	no	—
41	#64436	C₂²⋊C₄	1	2	yes	—
42	#64436	C₂×D₄	1	2	yes	—
43	C₂²×C₃₇	—	2	2	no	—
44	C₂²×C₃₇	C₂²⋊C₈	1	2	yes	—
45	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
46	C₂×D₃₇	—	2	8	no	—
47	C₂×D₃₇	—	2	2	no	—
48	C₂×D₃₇	—	2	2	no	—
49	C₂×D₃₇	M₃₂	1	4	yes	complemented
50	C₄×C₃₇	C₂²⋊C₈	1	2	yes	—
51	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
53	Dic₃₇	—	2	2	no	—
54	#2112	—	37	1	no	maximal, Sylow
55	C₂×C₃₇	—	2	2	no	—
56	C₂×C₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
57	C₂×C₃₇	C₂×M₃₂	1	2	yes	—
58	D₃₇	—	2	4	no	—
59	C₂³×C₈	—	37	1	no	—
60	C₂×M₃₂	—	37	2	no	—
61	C₂²⋊C₁₆	—	37	4	no	—
62	C₃₇	#2112	1	1	yes	characteristic, complemented, Sylow
63	C₂³×C₄	—	37	1	no	—
64	C₂²×C₈	—	37	1	no	—
65	C₂²×C₈	—	37	2	no	—
66	C₂²×C₈	—	37	2	no	—
67	C₂²×C₈	—	74	4	no	—
68	C₂²×C₈	#3866	1	1	yes	characteristic
69	M₃₂	—	74	4	no	—
70	C₂×C₁₆	—	37	4	no	—
71	C₂⁴	—	37	1	no	—
72	C₂²×C₄	—	37	1	no	—
73	C₂²×C₄	—	37	2	no	—
74	C₂²×C₄	—	37	2	no	—
75	C₂²×C₄	—	74	2	no	—
76	C₂²×C₄	—	74	2	no	—
77	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	37	4	no	—
80	C₂×C₈	—	37	1	no	—
81	C₂×C₈	—	37	1	no	—
82	C₂×C₈	—	74	4	no	—
83	C₂×C₈	—	74	4	no	—
84	C₂×C₈	C₂×#3866	1	2	yes	—
85	C₁₆	—	74	4	no	—
86	C₂³	—	37	2	no	—
87	C₂³	—	74	4	no	—
88	C₂³	—	74	2	no	—
89	C₂³	D₃₇⋊C₈	1	1	yes	characteristic
90	C₂×C₄	—	37	2	no	—
91	C₂×C₄	—	74	2	no	—
92	C₂×C₄	—	74	4	no	—
93	C₂×C₄	—	74	2	no	—
94	C₂×C₄	—	74	2	no	—
95	C₂×C₄	D₃₇⋊C₈	1	1	yes	characteristic
96	C₂×C₄	#10596953	1	2	yes	—
97	C₂×C₄	C₂²⋊#3866	1	2	yes	—
98	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	37	2	no	—
101	C₈	—	37	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	37	4	no	—
104	C₂²	—	74	8	no	—
105	C₂²	—	74	2	no	—
106	C₂²	—	74	2	no	—
107	C₂²	#11924593	1	2	yes	—
108	C₂²	C₂×D₃₇⋊C₈	1	1	yes	characteristic
109	C₄	—	74	2	no	—
110	C₄	#11924593	1	2	yes	—
111	C₄	C₂×C₂²⋊#3866	1	1	yes	characteristic
112	C₄	C₂×#10596953	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	—	74	4	no	—
115	C₂	C₂×#11924593	1	1	yes	characteristic
116	C₂	D₃₇⋊M₃₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924610

C₂

yes

maximal, complemented

#14558428

C₂

yes

maximal, complemented

C₂²×C₈×D₃₇

C₂

yes

maximal, characteristic

#11924610

—

C₂×#10596927

C₂²

yes

complemented

C₂²×C₄×D₃₇

C₄

yes

characteristic

C₂²×#64436

C₂²

yes

characteristic

C₂×C₈×D₃₇

—

C₂×C₈×D₃₇

C₄

yes

—

C₂×C₈×D₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₄

yes

characteristic

C₂³×D₃₇

C₈

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₈

yes

characteristic

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₈

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

D₄

yes

—

C₂×#64436

C₂³

yes

characteristic

C₈×D₃₇

—

C₈×D₃₇

—

#10596927

—

C₂³×C₃₇

C₂×C₈

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

M₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×C₈

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

—

#64436

C₂²⋊C₄

yes

—

#64436

C₂×D₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²⋊C₈

yes

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

M₃₂

yes

complemented

C₄×C₃₇

C₂²⋊C₈

yes

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

C₂×M₁₆

yes

characteristic

Dic₃₇

—

#2112

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₃₇

C₂×M₃₂

yes

—

D₃₇

—

C₂³×C₈

—

C₂×M₃₂

—

C₂²⋊C₁₆

—

C₃₇

#2112

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

#3866

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

D₃₇⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃₇⋊C₈

yes

characteristic

C₂×C₄

#10596953

yes

—

C₂×C₄

C₂²⋊#3866

yes

—

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

#11924593

yes

—

108

C₂²

C₂×D₃₇⋊C₈

yes

characteristic

109

C₄

—

110

C₄

#11924593

yes

—

111

C₄

C₂×C₂²⋊#3866

yes

characteristic

112

C₄

C₂×#10596953

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×#11924593

yes

characteristic

116

C₂

D₃₇⋊M₃₂

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3866
commutator, socle	C₂²×C₃₇	C₂²×C₈
Fitting	C₂²×C₈×C₃₇	C₄
prime radical	C₂³×D₃₇	C₈
cosocle	C₂×#64436	C₂³
hypercenter	C₂²×C₈	#3866
hypocenter	C₃₇	#2112
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#3866

commutator, socle

C₂²×C₃₇

C₂²×C₈

Fitting

C₂²×C₈×C₃₇

C₄

prime radical

C₂³×D₃₇

C₈

C₂³

C₂²×C₈

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
1. C₂×#11924610	C₂	semidirect
2. #14558428	C₂	semidirect
5. C₂×#10596927	C₂²	semidirect
49. C₂×D₃₇	M₃₂	semidirect
62. C₃₇	#2112	semidirect

H ≅ G/N

type

1. C₂×#11924610

C₂

semidirect

2. #14558428

C₂

semidirect

5. C₂×#10596927

C₂²

semidirect

49. C₂×D₃₇

M₃₂

semidirect

62. C₃₇

#2112

semidirect

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

subgroup	index
C₂×#11924610	2
#14558428	2
C₂²×C₈×D₃₇	2
#2112	37

subgroup

index

C₂×#11924610

#14558428

C₂²×C₈×D₃₇

#2112

Subgroups — #18010825

Conjugacy classes of subgroups

Complements

Maximal subgroups