Group #37534522 — subgroups

Subgroups — #37534522

Order 23168.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19153816	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153739	C₂	1	4	yes	maximal
3	C₂×C₄×#13089678	C₂	1	1	yes	maximal, characteristic
4	#384×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₆₂	C₂²	1	2	yes	—
6	C₂²×#13089678	C₂²	1	2	yes	—
7	#19153816	—	2	2	no	—
8	#19153816	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₈₁	—	2	2	no	—
10	#19153739	—	2	8	no	—
11	C₄×#13089678	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₈₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₆₂	—	2	2	no	—
16	C₂×Dic₃₆₂	—	2	4	no	—
17	C₂×Dic₃₆₂	—	2	4	no	—
18	C₂×Dic₃₆₂	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₈₁	—	2	2	no	—
20	C₂×#13089678	—	2	2	no	—
21	C₂×#13089678	—	2	2	no	—
22	C₂×#13089678	D₄	1	4	yes	—
23	C₄⋊Dic₁₈₁	—	2	8	no	—
24	C₂²×C₄×C₁₈₁	—	2	2	no	—
25	C₂²×C₄×C₁₈₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₈₁	—	2	4	no	—
28	C₄⋊C₄×C₁₈₁	—	4	4	no	—
29	C₄²×C₁₈₁	C₂×C₄	1	2	yes	—
30	C₄²×C₁₈₁	D₄	1	2	yes	—
31	Dic₃₆₂	—	2	8	no	—
32	Dic₃₆₂	—	4	4	no	—
33	C₂×Dic₁₈₁	—	2	4	no	—
34	C₂×Dic₁₈₁	—	4	2	no	—
35	C₂×Dic₁₈₁	—	4	2	no	—
36	#13089678	—	2	4	no	—
37	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₈₁	—	4	2	no	—
39	C₂×C₄×C₁₈₁	—	4	4	no	—
40	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₈₁	SD₁₆	1	4	yes	—
42	C₂×C₄×C₁₈₁	Q₁₆	1	4	yes	—
43	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₈₁	#41	1	2	yes	—
47	C₂²×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₈₁	#124	1	1	yes	characteristic
49	C₂²×C₁₈₁	#127	1	1	yes	characteristic
50	C₂²×C₁₈₁	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₁₈₁	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₈₁	—	8	2	no	—
53	C₄×C₁₈₁	#103	1	8	yes	complemented
54	C₄×C₁₈₁	C₄×D₄	1	2	yes	—
55	C₄×C₁₈₁	#121	1	2	yes	—
56	Dic₁₈₁	—	4	4	no	—
57	C₂×C₁₈₁	#390	1	1	yes	characteristic
58	C₂×C₁₈₁	C₂×#103	1	2	yes	—
59	C₂×C₁₈₁	#487	1	2	yes	—
60	C₂×C₁₈₁	#492	1	1	yes	characteristic
61	C₂×C₁₈₁	#494	1	1	yes	characteristic
62	C₁₈₁	#1956	1	1	yes	characteristic, complemented, Sylow
63	#1956	—	181	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	181	1	no	—
65	C₂×#103	—	181	4	no	—
66	C₂×C₄×C₈	—	181	1	no	—
67	#384	D₁₈₁	1	1	yes	characteristic
68	C₂²×Q₈	—	181	2	no	—
69	C₂²×C₈	—	181	2	no	—
70	C₄⋊Q₈	—	362	2	no	—
71	C₄⋊Q₈	—	181	4	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	362	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₈₁	1	2	yes	—
75	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
76	#103	—	362	8	no	—
77	C₄×C₈	—	362	2	no	—
78	C₂×Q₈	—	362	2	no	—
79	C₂×Q₈	—	362	4	no	—
80	C₂×Q₈	—	362	4	no	—
81	C₂×Q₈	—	181	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	362	2	no	—
84	C₂²×C₄	C₂²⋊D₁₈₁	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
86	C₂×C₈	—	362	2	no	—
87	C₂×C₈	—	362	2	no	—
88	C₂×C₈	—	181	4	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	362	8	no	—
92	C₄²	C₄×D₁₈₁	1	2	yes	—
93	C₄²	D₇₂₄	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
95	Q₈	—	362	8	no	—
96	Q₈	—	724	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	362	4	no	—
100	C₂×C₄	—	724	2	no	—
101	C₂×C₄	—	724	2	no	—
102	C₂×C₄	#15668426	1	4	yes	—
103	C₂×C₄	#15668437	1	4	yes	—
104	C₂×C₄	#15668439	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₁₈₁	1	1	yes	characteristic
106	C₂×C₄	D₄×D₁₈₁	1	2	yes	—
107	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
108	C₂×C₄	#15668450	1	2	yes	—
109	C₈	—	362	4	no	—
110	C₂²	C₂×#15668426	1	1	yes	characteristic
111	C₂²	#19153805	1	1	yes	characteristic
112	C₂²	#19153812	1	1	yes	characteristic
113	C₂²	C₂×#15668437	1	2	yes	—
114	C₂²	C₂×#15668439	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	724	4	no	—
117	C₄	#19153739	1	8	yes	complemented
118	C₄	#19153778	1	2	yes	—
119	C₄	#19153791	1	2	yes	—
120	C₂	#26041180	1	1	yes	characteristic
121	C₂	C₂×#19153739	1	2	yes	—
122	C₂	#26041631	1	2	yes	—
123	C₂	#26041651	1	1	yes	characteristic
124	C₂	#26041658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153816

C₂

yes

maximal, characteristic

C₂×#19153739

C₂

yes

maximal

C₂×C₄×#13089678

C₂

yes

maximal, characteristic

#384×C₁₈₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₆₂

C₂²

yes

—

C₂²×#13089678

C₂²

yes

—

#19153816

—

#19153816

C₄

yes

complemented

C₂×C₄⋊Dic₁₈₁

—

#19153739

—

C₄×#13089678

—

C₂×C₄⋊C₄×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

C₂×C₄

yes

—

C₂²×Dic₁₈₁

—

C₂×#13089678

—

C₂×#13089678

—

C₂×#13089678

D₄

yes

—

C₄⋊Dic₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

D₄

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

C₂×C₄

yes

—

C₄²×C₁₈₁

D₄

yes

—

Dic₃₆₂

—

Dic₃₆₂

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₈₁

SD₁₆

yes

—

C₂×C₄×C₁₈₁

Q₁₆

yes

—

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

—

C₂²×C₁₈₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₈₁

yes

C₂²×C₁₈₁

yes

C₂²×C₁₈₁

C₂×SD₁₆

yes

—

C₂²×C₁₈₁

C₂×Q₁₆

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

#103

yes

complemented

C₄×C₁₈₁

C₄×D₄

yes

—

C₄×C₁₈₁

#121

yes

—

Dic₁₈₁

—

C₂×C₁₈₁

#390

yes

characteristic

C₂×C₁₈₁

C₂×#103

yes

—

C₂×C₁₈₁

#487

yes

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

#1956

—

181

maximal, Sylow

C₂×C₄⋊Q₈

—

181

—

C₂×#103

—

181

—

C₂×C₄×C₈

—

181

—

#384

D₁₈₁

yes

characteristic

C₂²×Q₈

—

181

—

C₂²×C₈

—

181

—

C₄⋊Q₈

—

362

—

C₄⋊Q₈

—

181

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

362

—

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

#103

—

362

—

C₄×C₈

—

362

—

C₂×Q₈

—

362

—

C₂×Q₈

—

362

—

C₂×Q₈

—

362

—

C₂×Q₈

—

181

—

C₂²×C₄

—

C₂²×C₄

—

362

—

C₂²×C₄

C₂²⋊D₁₈₁

yes

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

181

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

362

—

C₄²

C₄×D₁₈₁

yes

—

C₄²

D₇₂₄

yes

—

C₂³

C₂×C₂²⋊D₁₈₁

yes

characteristic

Q₈

—

362

—

Q₈

—

724

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

362

—

100

C₂×C₄

—

724

—

101

C₂×C₄

—

724

—

102

C₂×C₄

#15668426

yes

—

103

C₂×C₄

#15668437

yes

—

104

C₂×C₄

#15668439

yes

—

105

C₂×C₄

C₂×C₄×D₁₈₁

yes

characteristic

106

C₂×C₄

D₄×D₁₈₁

yes

—

107

C₂×C₄

C₂×D₇₂₄

yes

characteristic

108

C₂×C₄

#15668450

yes

—

109

C₈

—

362

—

110

C₂²

C₂×#15668426

yes

characteristic

111

C₂²

#19153805

yes

characteristic

112

C₂²

#19153812

yes

characteristic

113

C₂²

C₂×#15668437

yes

—

114

C₂²

C₂×#15668439

yes

—

115

C₄

—

116

C₄

—

724

—

117

C₄

#19153739

yes

complemented

118

C₄

#19153778

yes

—

119

C₄

#19153791

yes

—

120

C₂

#26041180

yes

characteristic

121

C₂

C₂×#19153739

yes

—

122

C₂

#26041631

yes

—

123

C₂

#26041651

yes

characteristic

124

C₂

#26041658

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₈₁
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	#384×C₁₈₁	C₂
socle, prime radical	C₂³×C₁₈₁	C₂×D₄
cosocle	C₂²×C₄×C₁₈₁	C₂³
hypercenter	#384	D₁₈₁
hypocenter	C₁₈₁	#1956
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₈₁

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

#384×C₁₈₁

C₂

socle, prime radical

C₂³×C₁₈₁

C₂×D₄

cosocle

C₂²×C₄×C₁₈₁

C₂³

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #19153816	C₄	semidirect
53. C₄×C₁₈₁	#103	semidirect
62. C₁₈₁	#1956	semidirect
117. C₄	#19153739	semidirect

H ≅ G/N

type

8. #19153816

C₄

semidirect

53. C₄×C₁₈₁

#103

semidirect

62. C₁₈₁

#1956

semidirect

117. C₄

#19153739

semidirect

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

subgroup	index
C₂×#19153816	2
C₂×#19153739	2
C₂×C₄×#13089678	2
#384×C₁₈₁	2
#1956	181

subgroup

index

#384×C₁₈₁

181

Subgroups — #37534522

Conjugacy classes of subgroups

Complements

Maximal subgroups