Group #18010782 — subgroups

Subgroups — #18010782

Order 4736.

284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 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	#14558538	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3866	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	—	2	2	no	—
7	C₂×C₄×#3866	C₂²	1	2	yes	—
8	#11924603	—	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	8	no	—
22	C₄×#3866	—	2	2	no	—
23	C₄×#3866	Q₈	1	4	yes	complemented
24	#10596953	—	2	4	no	—
25	C₄⋊C₄×C₃₇	—	2	2	no	—
26	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
27	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
28	#10596938	—	2	2	no	—
29	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
30	#10596936	—	2	2	no	—
31	#10596936	—	2	2	no	—
32	C₄×Dic₃₇	—	2	2	no	—
33	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₃₇	D₄	1	2	yes	—
35	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
36	C₂×#3866	—	2	8	no	—
37	C₂×#3866	—	2	4	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₃₇	—	2	2	no	—
42	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
43	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
44	C₄×D₃₇	—	2	4	no	—
45	C₄×D₃₇	C₂×Q₈	1	2	yes	—
46	C₄×D₃₇	#41	1	2	yes	—
47	Dic₇₄	—	4	4	no	—
48	#64438	—	4	2	no	—
49	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₃₇	—	2	2	no	—
51	C₂×D₃₇	C₂²⋊Q₈	1	2	yes	—
52	#3866	—	2	8	no	—
53	C₄×C₃₇	—	2	2	no	—
54	C₄×C₃₇	—	4	2	no	—
55	C₄×C₃₇	C₄×Q₈	1	2	yes	—
56	Dic₃₇	—	4	2	no	—
57	Dic₃₇	#104	1	4	yes	—
58	Dic₃₇	#124	1	1	yes	characteristic
59	Dic₃₇	C₄⋊Q₈	1	1	yes	characteristic
60	#1994	—	37	1	no	maximal, Sylow
61	C₂×C₃₇	#391	1	1	yes	characteristic
62	C₂×C₃₇	C₂×#104	1	2	yes	—
63	D₃₇	—	2	2	no	—
64	#533	—	37	1	no	—
65	#423	—	37	1	no	—
66	C₄³	—	37	1	no	—
67	#339	—	37	4	no	—
68	C₃₇	#1994	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	37	2	no	—
70	C₄⋊Q₈	—	37	2	no	—
71	#125	—	37	1	no	—
72	#125	#3866	1	1	yes	characteristic, complemented
73	C₂²⋊Q₈	—	74	2	no	—
74	C₄×Q₈	—	74	2	no	—
75	#117	—	37	2	no	—
76	C₂×C₄²	—	37	1	no	—
77	C₂×C₄²	—	37	2	no	—
78	C₂×C₄²	—	74	2	no	—
79	C₄⋊C₈	—	74	2	no	—
80	C₂×Q₈	—	74	2	no	—
81	C₂²×C₄	—	37	1	no	—
82	C₂²×C₄	—	37	2	no	—
83	C₂²×C₄	—	74	2	no	—
84	M₁₆	—	74	4	no	—
85	C₂×C₈	—	74	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	37	2	no	—
88	C₄⋊C₄	—	74	2	no	—
89	C₄⋊C₄	—	74	2	no	—
90	C₄⋊C₄	C₂×#3866	1	2	yes	—
91	C₂²⋊C₄	—	74	2	no	—
92	C₄²	—	37	2	no	—
93	C₄²	—	37	1	no	—
94	C₄²	—	37	4	no	—
95	C₄²	—	74	2	no	—
96	C₄²	—	74	2	no	—
97	C₄²	—	74	8	no	—
98	C₄²	C₂×#3866	1	1	yes	characteristic
99	C₂³	—	37	1	no	—
100	Q₈	—	148	4	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	37	1	no	—
103	C₂×C₄	—	37	2	no	—
104	C₂×C₄	—	37	2	no	—
105	C₂×C₄	—	37	2	no	—
106	C₂×C₄	—	74	4	no	—
107	C₂×C₄	—	74	4	no	—
108	C₂×C₄	—	74	8	no	—
109	C₂×C₄	—	74	2	no	—
110	C₂×C₄	C₂²⋊#3866	1	2	yes	—
111	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
112	C₈	—	148	2	no	—
113	C₂²	—	37	2	no	—
114	C₂²	—	74	2	no	—
115	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	37	1	no	—
119	C₄	—	37	1	no	—
120	C₄	—	37	4	no	—
121	C₄	—	74	8	no	—
122	C₄	—	148	2	no	—
123	C₄	Q₈×#3866	1	2	yes	—
124	C₂	—	74	2	no	—
125	C₂	(C₂×#3866)⋊Q₈	1	1	yes	characteristic
126	C₂	D₃₇⋊#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558538

C₂

yes

maximal, characteristic

C₄²×#3866

C₂

yes

maximal, characteristic

#14558403

C₂

yes

maximal

#14558089

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₂×C₄×#3866

—

C₂×C₄×#3866

C₂²

yes

—

#11924603

—

#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

—

—

—

—

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

#104

yes

—

Dic₃₇

#124

yes

characteristic

Dic₃₇

C₄⋊Q₈

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

—

#533

—

#423

—

C₄³

—

#339

—

C₃₇

#1994

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

#3866

yes

characteristic, complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

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

—

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

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²⋊#3866

yes

—

111

C₂×C₄

C₂²×#3866

yes

characteristic

112

C₈

—

148

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₂²⋊#3866

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

148

—

123

C₄

Q₈×#3866

yes

—

124

C₂

—

125

C₂

(C₂×#3866)⋊Q₈

yes

characteristic

126

C₂

D₃₇⋊#104

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	#1994
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
23. C₄×#3866	Q₈	semidirect
68. C₃₇	#1994	semidirect
72. #125	#3866	semidirect

H ≅ G/N

type

9. #11924535

C₄

semidirect

10. #11924528

C₄

semidirect

15. #125×C₃₇

C₄

semidirect

23. C₄×#3866

Q₈

semidirect

68. C₃₇

#1994

semidirect

72. #125

#3866

semidirect

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

subgroup	index
#14558538	2
C₄²×#3866	2
#14558403	2
#14558089	2
#1994	37

subgroup

index

C₄²×#3866

Subgroups — #18010782

Conjugacy classes of subgroups

Complements

Maximal subgroups