Group #18010783 — subgroups

Subgroups — #18010783

Order 4736.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14558537	C₂	1	1	yes	maximal, characteristic
2	#14558469	C₂	1	1	yes	maximal, characteristic
3	#14558403	C₂	1	4	yes	maximal
4	#14558089	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596953	C₂²	1	2	yes	—
6	C₂×C₄×#3866	C₂²	1	2	yes	—
7	C₄⋊#64438	—	2	2	no	—
8	(C₂×#3866)⋊C₄	—	2	2	no	—
9	#11924535	C₄	1	2	yes	complemented
10	#11924528	C₄	1	1	yes	characteristic, complemented
11	#11924514	—	2	2	no	—
12	#11924501	—	2	2	no	—
13	#11924494	C₂²	1	2	yes	—
14	C₄²×D₃₇	C₂²	1	1	yes	characteristic
15	#125×C₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂²×#3866	—	2	2	no	—
17	C₂×C₄×D₃₇	D₄	1	2	yes	—
18	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₇₄	—	2	2	no	—
20	C₂×#64438	—	2	2	no	—
21	C₄×#3866	—	2	2	no	—
22	C₄×#3866	Q₈	1	4	yes	complemented
23	#10596953	—	2	4	no	—
24	C₄⋊C₄×C₃₇	—	2	2	no	—
25	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
26	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
27	#10596938	—	2	2	no	—
28	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
29	#10596936	—	2	2	no	—
30	#10596936	—	2	2	no	—
31	C₄×Dic₃₇	—	2	2	no	—
32	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₃₇	D₄	1	2	yes	—
34	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂×#3866	—	2	4	no	—
36	C₂×#3866	—	4	4	no	—
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₃₇	—	2	2	no	—
41	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
43	C₄×D₃₇	—	4	2	no	—
44	C₄×D₃₇	C₂×Q₈	1	2	yes	—
45	C₄×D₃₇	#41	1	2	yes	—
46	Dic₇₄	—	4	4	no	—
47	#64438	—	4	2	no	—
48	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₃₇	—	2	2	no	—
50	C₂×D₃₇	C₂²⋊Q₈	1	2	yes	—
51	#3866	—	4	4	no	—
52	C₄×C₃₇	—	2	2	no	—
53	C₄×C₃₇	—	4	2	no	—
54	C₄×C₃₇	C₄×Q₈	1	2	yes	—
55	Dic₃₇	—	2	2	no	—
56	Dic₃₇	—	4	2	no	—
57	Dic₃₇	#124	1	1	yes	characteristic
58	Dic₃₇	C₄⋊Q₈	1	1	yes	characteristic
59	#1995	—	37	1	no	maximal, Sylow
60	C₂×C₃₇	#391	1	1	yes	characteristic
61	C₂×C₃₇	#421	1	2	yes	—
62	D₃₇	—	2	2	no	—
63	#533	—	37	1	no	—
64	#423	—	37	1	no	—
65	#376	—	37	1	no	—
66	#339	—	37	4	no	—
67	C₃₇	#1995	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₁₆	—	37	2	no	—
69	C₄⋊Q₈	—	37	2	no	—
70	#125	—	37	1	no	—
71	#125	#3866	1	1	yes	characteristic, complemented
72	C₂²⋊Q₈	—	74	2	no	—
73	C₄×Q₈	—	74	2	no	—
74	#117	—	37	2	no	—
75	C₂×C₄²	—	37	1	no	—
76	C₂×C₄²	—	37	2	no	—
77	C₄⋊C₈	—	74	2	no	—
78	#95	—	74	2	no	—
79	C₂×Q₈	—	74	2	no	—
80	C₂²×C₄	—	37	1	no	—
81	C₂²×C₄	—	37	2	no	—
82	C₂²×C₄	—	74	2	no	—
83	M₁₆	—	74	4	no	—
84	C₂×C₈	—	74	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	37	2	no	—
87	C₄⋊C₄	—	74	2	no	—
88	C₄⋊C₄	—	74	2	no	—
89	C₄⋊C₄	C₂×#3866	1	2	yes	—
90	C₂²⋊C₄	—	74	2	no	—
91	C₄²	—	37	2	no	—
92	C₄²	—	37	1	no	—
93	C₄²	—	37	4	no	—
94	C₄²	—	74	2	no	—
95	C₄²	—	74	2	no	—
96	C₄²	C₂×#3866	1	1	yes	characteristic
97	C₂³	—	37	1	no	—
98	Q₈	—	148	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	37	1	no	—
101	C₂×C₄	—	37	2	no	—
102	C₂×C₄	—	37	2	no	—
103	C₂×C₄	—	37	2	no	—
104	C₂×C₄	—	74	4	no	—
105	C₂×C₄	—	74	2	no	—
106	C₂×C₄	—	148	2	no	—
107	C₂×C₄	—	148	4	no	—
108	C₂×C₄	C₂²⋊#3866	1	2	yes	—
109	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
110	C₈	—	148	2	no	—
111	C₂²	—	37	2	no	—
112	C₂²	—	74	2	no	—
113	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	37	1	no	—
117	C₄	—	37	1	no	—
118	C₄	—	74	2	no	—
119	C₄	—	148	4	no	—
120	C₄	—	148	2	no	—
121	C₄	Q₈×#3866	1	2	yes	—
122	C₂	—	74	2	no	—
123	C₂	(C₂×#3866)⋊Q₈	1	1	yes	characteristic
124	C₂	#14558530	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558537

C₂

yes

maximal, characteristic

#14558469

C₂

yes

maximal, characteristic

#14558403

C₂

yes

maximal

#14558089

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₂×C₄×#3866

C₂²

yes

—

C₄⋊#64438

—

(C₂×#3866)⋊C₄

—

#11924535

C₄

yes

complemented

#11924528

C₄

yes

characteristic, complemented

#11924514

—

#11924501

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#125×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×#64438

—

C₄×#3866

—

C₄×#3866

Q₈

yes

complemented

#10596953

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₂²×D₃₇

C₂×D₄

yes

characteristic

C₂×#3866

—

C₂×#3866

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

C₂×Q₈

yes

—

C₄×D₃₇

#41

yes

—

Dic₇₄

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₂²⋊Q₈

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#124

yes

characteristic

Dic₃₇

C₄⋊Q₈

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

—

—

—

—

C₃₇

#1995

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

#3866

yes

characteristic, complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#3866

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#3866

yes

characteristic

C₂³

—

Q₈

—

148

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

148

—

107

C₂×C₄

—

148

—

108

C₂×C₄

C₂²⋊#3866

yes

—

109

C₂×C₄

C₂²×#3866

yes

characteristic

110

C₈

—

148

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₂²⋊#3866

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

148

—

120

C₄

—

148

—

121

C₄

Q₈×#3866

yes

—

122

C₂

—

123

C₂

(C₂×#3866)⋊Q₈

yes

characteristic

124

C₂

#14558530

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#125×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₃₇	C₂×D₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#125	#3866
hypocenter	C₃₇	#1995
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#125×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₃₇

C₂×D₄

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #11924535	C₄	semidirect
10. #11924528	C₄	semidirect
15. #125×C₃₇	C₄	semidirect
22. C₄×#3866	Q₈	semidirect
67. C₃₇	#1995	semidirect
71. #125	#3866	semidirect

H ≅ G/N

type

9. #11924535

C₄

semidirect

10. #11924528

C₄

semidirect

15. #125×C₃₇

C₄

semidirect

22. C₄×#3866

Q₈

semidirect

67. C₃₇

#1995

semidirect

71. #125

#3866

semidirect

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

subgroup	index
#14558537	2
#14558469	2
#14558403	2
#14558089	2
#1995	37

subgroup

index

Subgroups — #18010783

Conjugacy classes of subgroups

Complements

Maximal subgroups