Group #18011509 — subgroups

Subgroups — #18011509

Order 4736.

314 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 144 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	#14558585	C₂	1	4	yes	maximal, complemented
2	#14558583	C₂	1	4	yes	maximal, complemented
3	#14558558	C₂	1	4	yes	maximal
4	#14558555	C₂	1	2	yes	maximal
5	#14558089	C₂	1	1	yes	maximal, characteristic
6	#11924603	C₂²	1	4	yes	—
7	C₄⋊#64438	C₂²	1	2	yes	—
8	C₄⋊#64438	C₂²	1	2	yes	—
9	C₄⋊#64438	C₂²	1	4	yes	—
10	#11924593	C₂²	1	4	yes	—
11	#11924591	C₂²	1	4	yes	—
12	#11924591	C₂²	1	4	yes	—
13	C₄×#64438	C₂²	1	4	yes	—
14	#11924535	C₄	1	2	yes	—
15	#11924528	C₄	1	1	yes	characteristic
16	#11924514	C₄	1	4	yes	—
17	#11924501	C₂²	1	4	yes	—
18	#11924494	C₂²	1	2	yes	—
19	C₄²×D₃₇	C₂²	1	1	yes	characteristic
20	#125×C₃₇	C₄	1	1	yes	characteristic
21	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₇₄	C₂×C₄	1	4	yes	—
24	C₂×#64438	C₂³	1	4	yes	—
25	C₂×#64438	C₂³	1	4	yes	—
26	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
28	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
29	#10596938	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
31	#10596936	C₂×C₄	1	4	yes	—
32	#10596936	C₂×C₄	1	4	yes	—
33	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₃₇	C₂³	1	2	yes	—
35	C₄×Dic₃₇	C₂³	1	4	yes	—
36	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
40	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
41	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
42	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
43	C₄×D₃₇	—	2	1	no	—
44	C₄×D₃₇	—	2	1	no	—
45	C₄×D₃₇	—	4	2	no	—
46	Dic₇₄	—	4	4	no	—
47	#64438	—	2	4	no	—
48	#64438	—	2	4	no	—
49	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
50	C₂×D₃₇	—	2	1	no	—
51	C₂×D₃₇	—	2	2	no	—
52	C₄×C₃₇	—	2	1	no	—
53	C₄×C₃₇	—	2	4	no	—
54	C₄×C₃₇	—	2	2	no	—
55	Dic₃₇	—	2	2	no	—
56	Dic₃₇	—	2	4	no	—
57	Dic₃₇	2₋¹⁺⁴	1	1	yes	characteristic
58	Dic₃₇	2₋¹⁺⁴	1	1	yes	characteristic
59	#2987	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	#519	1	1	yes	characteristic
61	C₂×C₃₇	#568	1	2	yes	—
62	D₃₇	—	4	1	no	—
63	#533	—	37	1	no	—
64	#446	—	37	4	no	—
65	#446	—	37	4	no	—
66	#433	—	37	4	no	—
67	#432	—	37	2	no	—
68	C₃₇	#2987	1	1	yes	characteristic, complemented, Sylow
69	C₄⋊Q₈	—	37	2	no	—
70	#125	—	37	1	no	—
71	#125	#3866	1	1	yes	characteristic
72	C₂²⋊Q₈	—	37	4	no	—
73	C₄×Q₈	—	37	4	no	—
74	#117	—	37	2	no	—
75	C₂×C₄²	—	37	1	no	—
76	C₄⋊C₈	—	37	2	no	—
77	C₄⋊C₈	—	37	2	no	—
78	C₄⋊C₈	—	37	4	no	—
79	C₄⋊C₈	—	37	4	no	—
80	C₂²⋊C₈	—	37	4	no	—
81	#97	—	37	4	no	—
82	#97	—	37	4	no	—
83	C₄×C₈	—	37	4	no	—
84	C₂×Q₈	—	37	4	no	—
85	C₂²×C₄	—	37	1	no	—
86	C₂²×C₄	—	37	2	no	—
87	C₂×C₈	—	37	4	no	—
88	C₂×C₈	—	37	4	no	—
89	C₄⋊C₄	—	37	4	no	—
90	C₄⋊C₄	—	37	4	no	—
91	C₄⋊C₄	—	37	2	no	—
92	C₄⋊C₄	C₂×#3866	1	2	yes	—
93	C₄⋊C₄	C₂×#3866	1	4	yes	—
94	C₂²⋊C₄	—	37	4	no	—
95	C₄²	—	37	1	no	—
96	C₄²	—	37	4	no	—
97	C₄²	—	37	2	no	—
98	C₄²	C₂×#3866	1	1	yes	characteristic
99	C₂³	—	37	1	no	—
100	Q₈	—	148	4	no	—
101	C₂×C₄	—	37	1	no	—
102	C₂×C₄	—	37	2	no	—
103	C₂×C₄	—	37	4	no	—
104	C₂×C₄	—	74	1	no	—
105	C₂×C₄	—	74	1	no	—
106	C₂×C₄	—	148	2	no	—
107	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
108	C₂×C₄	C₂²×#3866	1	2	yes	—
109	C₂×C₄	C₂²×#3866	1	4	yes	—
110	C₈	—	74	4	no	—
111	C₈	—	74	4	no	—
112	C₂²	—	74	1	no	—
113	C₂²	—	74	2	no	—
114	C₂²	C₂³×#3866	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	2	4	no	—
118	C₄	—	37	1	no	—
119	C₄	—	37	1	no	—
120	C₄	—	74	2	no	—
121	C₄	—	74	4	no	—
122	C₂	—	148	1	no	—
123	C₂	#14558601	1	1	yes	characteristic
124	C₂	#14558612	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558585

C₂

yes

maximal, complemented

#14558583

C₂

yes

maximal, complemented

#14558558

C₂

yes

maximal

#14558555

C₂

yes

maximal

#14558089

C₂

yes

maximal, characteristic

#11924603

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

#11924593

C₂²

yes

—

#11924591

C₂²

yes

—

#11924591

C₂²

yes

—

C₄×#64438

C₂²

yes

—

#11924535

C₄

yes

—

#11924528

C₄

yes

characteristic

#11924514

C₄

yes

—

#11924501

C₂²

yes

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#125×C₃₇

C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂³

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂²×D₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂⁴

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

#64438

—

#64438

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

2₋¹⁺⁴

yes

characteristic

Dic₃₇

2₋¹⁺⁴

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

—

—

—

—

—

C₃₇

#2987

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

yes

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#3866

yes

—

C₄⋊C₄

C₂×#3866

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

100

Q₈

—

148

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

148

—

107

C₂×C₄

C₂²×#3866

yes

characteristic

108

C₂×C₄

C₂²×#3866

yes

—

109

C₂×C₄

C₂²×#3866

yes

—

110

C₈

—

111

C₈

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂³×#3866

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₂

—

148

—

123

C₂

#14558601

yes

characteristic

124

C₂

#14558612

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	#125×C₃₇	C₄
prime radical	C₂²×D₃₇	C₂²×C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	#125	#3866
hypocenter	C₃₇	#2987
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#3866

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

#125×C₃₇

C₄

prime radical

C₂²×D₃₇

C₂²×C₄

cosocle

C₂×Dic₃₇

C₂⁴

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #14558585	C₂	semidirect
2. #14558583	C₂	semidirect
68. C₃₇	#2987	semidirect

H ≅ G/N

type

1. #14558585

C₂

semidirect

2. #14558583

C₂

semidirect

68. C₃₇

#2987

semidirect

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

subgroup	index
#14558585	2
#14558583	2
#14558558	2
#14558555	2
#14558089	2
#2987	37

subgroup

index

Subgroups — #18011509

Conjugacy classes of subgroups

Complements

Maximal subgroups