Group #37535274 — subgroups

Subgroups — #37535274

Order 23168.

210 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 65 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	#26041328	C₂	1	1	yes	maximal, characteristic, complemented
2	#26041305	C₂	1	2	yes	maximal, complemented
3	#26041296	C₂	1	2	yes	maximal
4	#26041036	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668433	C₂²	1	2	yes	—
7	#19153775	C₂²	1	2	yes	—
8	#19153752	—	2	2	no	—
9	#19153745	—	2	4	no	—
10	#19153743	—	2	4	no	—
11	C₂×#41×C₁₈₁	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₁₈₁	—	2	2	no	—
15	#15668433	—	2	2	no	—
16	#15668433	—	2	2	no	—
17	C₂×#13089678	—	2	2	no	—
18	C₄⋊Dic₁₈₁	—	2	2	no	—
19	C₂²⋊Dic₁₈₁	—	2	2	no	—
20	C₄×Dic₁₈₁	D₄	1	4	yes	—
21	#41×C₁₈₁	—	2	2	no	—
22	#41×C₁₈₁	—	2	2	no	—
23	C₂×Q₈×C₁₈₁	Q₈	1	1	yes	characteristic
24	C₂×D₄×C₁₈₁	—	2	1	no	—
25	C₂×D₄×C₁₈₁	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₈₁	—	2	1	no	—
27	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₈₁	—	2	2	no	—
29	C₂×C₈×C₁₈₁	—	2	1	no	—
30	C₂×C₈×C₁₈₁	—	2	2	no	—
31	C₂×C₈×C₁₈₁	D₄	1	2	yes	—
32	C₂×Dic₁₈₁	—	2	4	no	—
33	#13089678	—	4	2	no	—
34	C₂³×C₁₈₁	—	2	1	no	—
35	C₂³×C₁₈₁	#41	1	1	yes	characteristic
36	Q₈×C₁₈₁	—	2	2	no	—
37	D₄×C₁₈₁	—	2	2	no	—
38	D₄×C₁₈₁	—	4	2	no	—
39	C₂×C₄×C₁₈₁	—	2	1	no	—
40	C₂×C₄×C₁₈₁	—	4	1	no	—
41	C₂×C₄×C₁₈₁	—	4	1	no	—
42	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₈₁	#41	1	2	yes	—
46	C₈×C₁₈₁	—	2	2	no	—
47	C₈×C₁₈₁	—	4	1	no	—
48	C₂²×C₁₈₁	—	4	1	no	—
49	C₂²×C₁₈₁	—	4	2	no	—
50	C₂²×C₁₈₁	—	4	1	no	—
51	C₂²×C₁₈₁	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²×C₁₈₁	#123	1	2	yes	—
53	C₄×C₁₈₁	—	4	1	no	—
54	C₄×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₈₁	#123	1	1	yes	characteristic
56	C₄×C₁₈₁	#127	1	1	yes	characteristic
57	Dic₁₈₁	—	4	4	no	—
58	C₂×C₁₈₁	—	2	2	no	—
59	C₂×C₁₈₁	—	2	1	no	—
60	C₂×C₁₈₁	—	8	1	no	—
61	C₂×C₁₈₁	#399	1	1	yes	characteristic
62	C₁₈₁	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	181	1	no	maximal, Sylow
64	#430	—	181	1	no	—
65	#424	—	181	2	no	—
66	#421	—	181	2	no	—
67	#408	D₁₈₁	1	1	yes	characteristic
68	H(3,ℤ/4)	—	181	1	no	—
69	C₂×#41	C₂×D₁₈₁	1	1	yes	characteristic
70	C₂×M₁₆	—	181	2	no	—
71	C₂×M₁₆	C₂×D₁₈₁	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₁₈₁	1	1	yes	characteristic
73	#117	—	181	2	no	—
74	#108	—	362	2	no	—
75	C₄⋊C₈	—	362	4	no	—
76	#104	—	362	4	no	—
77	C₂²⋊C₈	—	2	2	no	—
78	#41	—	2	2	no	—
79	#41	—	2	2	no	—
80	C₂×Q₈	Dic₃₆₂	1	1	yes	characteristic
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	Dic₃₆₂	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	362	2	no	—
87	M₁₆	—	362	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	362	2	no	—
91	C₂×C₈	C₂²⋊D₁₈₁	1	2	yes	—
92	C₄⋊C₄	—	362	2	no	—
93	C₂²⋊C₄	—	362	2	no	—
94	C₄²	—	181	4	no	—
95	C₂³	—	2	1	no	—
96	C₂³	#15668449	1	1	yes	characteristic
97	Q₈	—	2	2	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	362	4	no	—
104	C₂×C₄	D₄×D₁₈₁	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₃₆₂	1	1	yes	characteristic
107	C₂×C₄	#15668448	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	4	1	no	—
110	C₈	—	724	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	#19153795	1	1	yes	characteristic
115	C₂²	#19153798	1	2	yes	—
116	C₄	—	4	1	no	—
117	C₄	—	724	4	no	—
118	C₄	#19153793	1	2	yes	—
119	C₄	#19153797	1	1	yes	characteristic
120	C₄	#19153812	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	8	1	no	—
124	C₂	#26041215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26041328

C₂

yes

maximal, characteristic, complemented

#26041305

C₂

yes

maximal, complemented

#26041296

C₂

yes

maximal

#26041036

C₂

yes

maximal, characteristic, complemented

#408×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668433

C₂²

yes

—

#19153775

C₂²

yes

—

#19153752

—

#19153745

—

#19153743

—

C₂×#41×C₁₈₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₈₁

C₂²

yes

characteristic

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

C₂²⋊C₈×C₁₈₁

—

#15668433

—

#15668433

—

C₂×#13089678

—

C₄⋊Dic₁₈₁

—

C₂²⋊Dic₁₈₁

—

C₄×Dic₁₈₁

D₄

yes

—

#41×C₁₈₁

—

#41×C₁₈₁

—

C₂×Q₈×C₁₈₁

Q₈

yes

characteristic

C₂×D₄×C₁₈₁

—

C₂×D₄×C₁₈₁

Q₈

yes

characteristic

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

M₁₆×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

D₄

yes

—

C₂×Dic₁₈₁

—

#13089678

—

C₂³×C₁₈₁

—

C₂³×C₁₈₁

#41

yes

characteristic

Q₈×C₁₈₁

—

D₄×C₁₈₁

—

D₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

—

C₈×C₁₈₁

—

C₈×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₈₁

#123

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂²⋊Q₈

yes

—

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#399

yes

characteristic

C₁₈₁

#2098

yes

characteristic, complemented, Sylow

#2098

—

181

maximal, Sylow

#430

—

181

—

#424

—

181

—

#421

—

181

—

#408

D₁₈₁

yes

characteristic

H(3,ℤ/4)

—

181

—

C₂×#41

C₂×D₁₈₁

yes

characteristic

C₂×M₁₆

—

181

—

C₂×M₁₆

C₂×D₁₈₁

yes

characteristic

C₂²×C₈

C₂×D₁₈₁

yes

characteristic

#117

—

181

—

#108

—

362

—

C₄⋊C₈

—

362

—

#104

—

362

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₃₆₂

yes

characteristic

C₂×D₄

—

C₂×D₄

Dic₃₆₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

M₁₆

—

362

—

M₁₆

—

362

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

362

—

C₂×C₈

C₂²⋊D₁₈₁

yes

—

C₄⋊C₄

—

362

—

C₂²⋊C₄

—

362

—

C₄²

—

181

—

C₂³

—

C₂³

#15668449

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

362

—

104

C₂×C₄

D₄×D₁₈₁

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

106

C₂×C₄

C₂×Dic₃₆₂

yes

characteristic

107

C₂×C₄

#15668448

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

724

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#19153795

yes

characteristic

115

C₂²

#19153798

yes

—

116

C₄

—

117

C₄

—

724

—

118

C₄

#19153793

yes

—

119

C₄

#19153797

yes

characteristic

120

C₄

#19153812

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

#26041215

yes

characteristic

	subgroup	G/H
center	C₄	#19153812
commutator, cosocle	C₂²×C₄×C₁₈₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	#408×C₁₈₁	C₂
socle	C₂×C₁₈₁	#399
prime radical	C₂×#41×C₁₈₁	C₂²
hypercenter	#408	D₁₈₁
hypocenter	C₁₈₁	#2098
LCS	G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#19153812

commutator, cosocle

C₂²×C₄×C₁₈₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

#408×C₁₈₁

C₂

socle

C₂×C₁₈₁

#399

prime radical

C₂×#41×C₁₈₁

C₂²

D₁₈₁

C₁₈₁

G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. #26041328	C₂	semidirect
2. #26041305	C₂	semidirect
4. #26041036	C₂	semidirect
62. C₁₈₁	#2098	semidirect

H ≅ G/N

type

1. #26041328

C₂

semidirect

2. #26041305

C₂

semidirect

4. #26041036

C₂

semidirect

62. C₁₈₁

#2098

semidirect

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

subgroup	index
#26041328	2
#26041305	2
#26041296	2
#26041036	2
#408×C₁₈₁	2
#2098	181

subgroup

index

#408×C₁₈₁

181

Subgroups — #37535274

Conjugacy classes of subgroups

Complements

Maximal subgroups