Group #18010834 — subgroups

Subgroups — #18010834

Order 4736.

220 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 78 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	#14558430	C₂	1	4	yes	maximal, complemented
3	C₂×#11924544	C₂	1	1	yes	maximal, characteristic
4	#11924610	—	2	2	no	—
5	#11924610	C₂²	1	4	yes	—
6	C₂×#10596927	—	2	2	no	—
7	C₂×#10596962	C₄	1	1	yes	characteristic
8	#11924544	—	2	2	no	—
9	#11924544	—	2	2	no	—
10	C₂×#10596929	C₂²	1	2	yes	—
11	C₂²×#64436	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₃₇	C₄	1	2	yes	—
13	C₂×M₁₆×C₃₇	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₂²⋊D₃₇	C₈	1	2	yes	—
16	#10596962	—	2	2	no	—
17	#10596962	—	2	2	no	—
18	C₂×D₁₄₈	C₈	1	1	yes	characteristic
19	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
20	C₂×Dic₇₄	C₈	1	1	yes	characteristic
21	M₁₆×C₃₇	—	2	2	no	—
22	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
23	C₂×#64436	C₂×C₄	1	1	yes	characteristic
24	C₂×#64436	D₄	1	2	yes	—
25	C₂×#64436	D₄	1	2	yes	—
26	C₂×#64436	C₂³	1	1	yes	characteristic
27	#10596929	—	2	4	no	—
28	C₈×D₃₇	—	2	2	no	—
29	C₈×D₃₇	—	2	2	no	—
30	#10596927	—	2	4	no	—
31	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂²×D₃₇	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₃₇	M₁₆	1	2	yes	—
36	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²⋊D₃₇	—	2	4	no	—
38	C₂×Dic₃₇	C₂×C₈	1	2	yes	—
39	D₁₄₈	—	2	2	no	—
40	C₄×D₃₇	—	2	2	no	—
41	C₄×D₃₇	—	2	2	no	—
42	Dic₇₄	—	2	2	no	—
43	C₈×C₃₇	—	2	2	no	—
44	#64436	C₂²⋊C₄	1	1	yes	characteristic
45	#64436	C₂²⋊C₄	1	1	yes	characteristic
46	#64436	C₂×D₄	1	2	yes	—
47	C₂²×C₃₇	—	4	1	no	—
48	C₂²×C₃₇	C₂²⋊C₈	1	2	yes	—
49	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
50	C₂×D₃₇	—	2	2	no	—
51	C₂×D₃₇	—	4	2	no	—
52	C₄×C₃₇	C₂²⋊C₈	1	2	yes	—
53	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
55	Dic₃₇	—	2	2	no	—
56	#2115	—	37	1	no	maximal, Sylow
57	C₂×C₃₇	—	2	1	no	—
58	C₂×C₃₇	—	2	2	no	—
59	C₂×C₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
60	D₃₇	—	4	2	no	—
61	C₂×#131	—	37	1	no	—
62	C₂×M₃₂	—	37	2	no	—
63	#349	—	37	4	no	—
64	C₃₇	#2115	1	1	yes	characteristic, complemented, Sylow
65	C₂×#41	—	37	1	no	—
66	#131	—	74	2	no	—
67	#131	—	74	2	no	—
68	C₂×M₁₆	—	37	2	no	—
69	C₂×M₁₆	#3866	1	1	yes	characteristic
70	C₂²×C₈	—	37	1	no	—
71	C₂²×C₈	—	37	2	no	—
72	M₃₂	—	37	4	no	—
73	M₃₂	—	74	2	no	—
74	C₂×C₁₆	—	74	2	no	—
75	#41	—	74	2	no	—
76	#41	—	74	2	no	—
77	C₂×Q₈	—	37	1	no	—
78	C₂×D₄	—	37	2	no	—
79	C₂×D₄	—	37	1	no	—
80	C₂²×C₄	—	37	2	no	—
81	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	74	4	no	—
84	C₂×C₈	—	37	2	no	—
85	C₂×C₈	—	37	2	no	—
86	C₂×C₈	—	37	1	no	—
87	C₂×C₈	—	37	1	no	—
88	C₂×C₈	—	74	2	no	—
89	C₂×C₈	—	74	2	no	—
90	C₂×C₈	C₂×#3866	1	2	yes	—
91	C₁₆	—	74	4	no	—
92	C₂³	—	37	2	no	—
93	C₂³	D₃₇⋊C₈	1	1	yes	characteristic
94	Q₈	—	74	2	no	—
95	D₄	—	74	4	no	—
96	D₄	—	74	2	no	—
97	C₂×C₄	—	37	2	no	—
98	C₂×C₄	—	74	2	no	—
99	C₂×C₄	—	74	2	no	—
100	C₂×C₄	D₃₇⋊C₈	1	1	yes	characteristic
101	C₂×C₄	#10596953	1	2	yes	—
102	C₂×C₄	C₂²⋊#3866	1	2	yes	—
103	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	37	2	no	—
106	C₈	—	37	1	no	—
107	C₈	—	37	1	no	—
108	C₂²	—	4	1	no	—
109	C₂²	—	74	2	no	—
110	C₂²	—	148	2	no	—
111	C₂²	#11924593	1	2	yes	—
112	C₂²	C₂×D₃₇⋊C₈	1	1	yes	characteristic
113	C₄	—	74	2	no	—
114	C₄	#11924593	1	2	yes	—
115	C₄	C₂×C₂²⋊#3866	1	1	yes	characteristic
116	C₄	C₂×#10596953	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	2	no	—
119	C₂	—	148	2	no	—
120	C₂	C₂×#11924593	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924610

C₂

yes

maximal, complemented

#14558430

C₂

yes

maximal, complemented

C₂×#11924544

C₂

yes

maximal, characteristic

#11924610

—

#11924610

C₂²

yes

—

C₂×#10596927

—

C₂×#10596962

C₄

yes

—

—

C₂²

yes

—

C₂²×#64436

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₄

yes

—

C₂×M₁₆×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₃₇

C₈

yes

—

#10596962

—

#10596962

—

C₂×D₁₄₈

C₈

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×Dic₇₄

C₈

yes

characteristic

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

D₄

yes

—

C₂×#64436

D₄

yes

—

C₂×#64436

C₂³

yes

characteristic

#10596929

—

C₈×D₃₇

—

C₈×D₃₇

—

#10596927

—

C₂³×C₃₇

C₂×C₈

yes

characteristic

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

C₂²⋊D₃₇

—

C₂×Dic₃₇

C₂×C₈

yes

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

—

#64436

C₂²⋊C₄

yes

characteristic

#64436

C₂²⋊C₄

yes

characteristic

#64436

C₂×D₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²⋊C₈

yes

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

C₂²⋊C₈

yes

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

C₂×M₁₆

yes

characteristic

Dic₃₇

—

#2115

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂×C₂²⋊C₈

yes

characteristic

D₃₇

—

C₂×#131

—

C₂×M₃₂

—

#349

—

C₃₇

#2115

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

#3866

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

—

C₁₆

—

C₂³

—

C₂³

D₃₇⋊C₈

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

D₃₇⋊C₈

yes

characteristic

101

C₂×C₄

#10596953

yes

—

102

C₂×C₄

C₂²⋊#3866

yes

—

103

C₂×C₄

C₂²×#3866

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

148

—

111

C₂²

#11924593

yes

—

112

C₂²

C₂×D₃₇⋊C₈

yes

characteristic

113

C₄

—

114

C₄

#11924593

yes

—

115

C₄

C₂×C₂²⋊#3866

yes

characteristic

116

C₄

C₂×#10596953

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

148

—

120

C₂

C₂×#11924593

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#3866
commutator	C₂²×C₃₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂×M₁₆×C₃₇	C₄
socle	C₂×C₃₇	C₂×C₂²⋊C₈
prime radical	C₂×#10596962	C₄
cosocle	C₂×#64436	C₂³
hypercenter	C₂×M₁₆	#3866
hypocenter	C₃₇	#2115
LCS	G ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₂×C₂²⋊#3866

commutator

C₂²×C₃₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₂×M₁₆×C₃₇

C₄

socle

C₂×C₃₇

C₂×C₂²⋊C₈

C₄

C₂³

C₂×M₁₆

C₃₇

G ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#11924610	C₂	semidirect
2. #14558430	C₂	semidirect
64. C₃₇	#2115	semidirect

H ≅ G/N

type

1. C₂×#11924610

C₂

semidirect

2. #14558430

C₂

semidirect

64. C₃₇

#2115

semidirect

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

subgroup	index
C₂×#11924610	2
#14558430	2
C₂×#11924544	2
#2115	37

subgroup

index

Subgroups — #18010834

Conjugacy classes of subgroups

Complements

Maximal subgroups