Group #37549840 — subgroups

Subgroups — #37549840

Order 23168.

184 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 66 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	#26042294	C₂	1	2	yes	maximal
2	C₂×#19153886	C₂	1	1	yes	maximal, characteristic
3	#26042242	C₂	1	1	yes	maximal, characteristic
4	#26042162	C₂	1	2	yes	maximal
5	#108×D₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668459	C₂²	1	2	yes	—
7	D₁₈₁⋊M₁₆	C₂²	1	1	yes	characteristic
8	#19153895	C₂²	1	1	yes	characteristic
9	#19153887	—	2	2	no	—
10	#19153886	—	2	2	no	—
11	#19153886	—	2	1	no	—
12	#19153886	—	2	1	no	—
13	#19153870	—	2	1	no	—
14	C₈×#10624382	—	2	1	no	—
15	M₁₆×D₁₈₁	C₂²	1	2	yes	—
16	C₂×C₈×D₁₈₁	C₂²	1	1	yes	characteristic
17	#19153752	C₄	1	1	yes	characteristic, complemented
18	#19153751	C₄	1	2	yes	complemented
19	#108×C₁₈₁	C₄	1	1	yes	characteristic, complemented
20	#15668459	—	2	2	no	—
21	#15668459	—	2	2	no	—
22	#15668459	D₄	1	2	yes	—
23	#15668458	—	2	1	no	—
24	D₁₈₁⋊C₈	—	2	1	no	—
25	C₂×#13089685	—	2	2	no	—
26	C₄⋊#10624382	—	2	1	no	—
27	C₂²⋊#10624382	Q₈	1	2	yes	—
28	C₄×#10624382	—	2	1	no	—
29	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
30	#15668433	C₂×C₄	1	2	yes	—
31	#15668432	—	2	2	no	—
32	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
33	C₈×D₁₈₁	—	2	2	no	—
34	C₈×D₁₈₁	D₄	1	1	yes	characteristic
35	C₈×D₁₈₁	D₄	1	1	yes	characteristic
36	C₈×D₁₈₁	Q₈	1	1	yes	characteristic
37	C₈×D₁₈₁	Q₈	1	1	yes	characteristic
38	M₁₆×C₁₈₁	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
40	C₂×#10624382	—	2	2	no	—
41	#13089685	—	2	2	no	—
42	#13089685	—	4	2	no	—
43	C₂²×D₁₈₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×Dic₁₈₁	#41	1	1	yes	characteristic
45	C₄×D₁₈₁	C₂×D₄	1	1	yes	characteristic
46	C₄×D₁₈₁	C₂×D₄	1	1	yes	characteristic
47	C₄×D₁₈₁	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₁₈₁	#41	1	1	yes	characteristic
49	#13089678	—	2	2	no	—
50	#13089678	C₄⋊C₄	1	1	yes	characteristic
51	#13089678	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₁₈₁	—	2	2	no	—
54	C₈×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
55	C₈×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₈₁	C₄×Q₈	1	1	yes	characteristic
57	C₂×D₁₈₁	—	4	1	no	—
58	C₂×D₁₈₁	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×D₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
60	#10624382	—	4	2	no	—
61	C₄×C₁₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
63	Dic₁₈₁	#121	1	1	yes	characteristic
64	Dic₁₈₁	#125	1	1	yes	characteristic
65	C₂×C₁₈₁	—	2	1	no	—
66	C₂×C₁₈₁	#389	1	1	yes	characteristic
67	D₁₈₁	—	2	1	no	—
68	D₁₈₁	—	2	1	no	—
69	C₁₈₁	#1953	1	1	yes	characteristic, complemented, Sylow
70	#1953	—	181	1	no	maximal, Sylow
71	#430	—	181	2	no	—
72	C₂×#108	—	181	1	no	—
73	C₂×#108	—	181	1	no	—
74	#405	—	181	1	no	—
75	#344	—	181	2	no	—
76	C₂×M₁₆	—	181	2	no	—
77	C₂×M₁₆	—	181	1	no	—
78	C₂×M₁₆	—	181	2	no	—
79	C₂²×C₈	—	181	1	no	—
80	#117	—	181	1	no	—
81	#108	—	362	2	no	—
82	#108	—	362	2	no	—
83	#108	—	362	1	no	—
84	#108	—	362	1	no	—
85	#108	—	181	2	no	—
86	#108	—	181	1	no	—
87	#108	#10624382	1	1	yes	characteristic, complemented
88	#97	—	362	1	no	—
89	C₄×C₈	—	362	1	no	—
90	C₂²×C₄	—	181	1	no	—
91	M₁₆	—	362	2	no	—
92	M₁₆	—	362	1	no	—
93	M₁₆	—	362	2	no	—
94	M₁₆	—	362	2	no	—
95	M₁₆	—	181	2	no	—
96	M₁₆	—	181	2	no	—
97	M₁₆	C₂×#10624382	1	2	yes	—
98	C₂×C₈	—	362	2	no	—
99	C₂×C₈	—	362	1	no	—
100	C₂×C₈	—	362	2	no	—
101	C₂×C₈	—	181	1	no	—
102	C₂×C₈	—	181	1	no	—
103	C₂×C₈	—	181	1	no	—
104	C₂×C₈	—	181	1	no	—
105	C₂×C₈	—	181	1	no	—
106	C₂×C₈	C₂×#10624382	1	1	yes	characteristic
107	C₄⋊C₄	—	362	1	no	—
108	C₂²⋊C₄	—	181	2	no	—
109	C₄²	—	362	1	no	—
110	C₂³	—	181	1	no	—
111	C₂×C₄	—	362	2	no	—
112	C₂×C₄	—	181	1	no	—
113	C₂×C₄	—	181	1	no	—
114	C₂×C₄	—	181	1	no	—
115	C₂×C₄	—	181	1	no	—
116	C₂×C₄	—	181	1	no	—
117	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	362	2	no	—
120	C₈	—	362	2	no	—
121	C₈	—	181	1	no	—
122	C₈	—	181	1	no	—
123	C₈	—	724	2	no	—
124	C₈	C₄⋊#10624382	1	1	yes	characteristic
125	C₈	C₄⋊#10624382	1	1	yes	characteristic
126	C₂²	—	181	1	no	—
127	C₂²	—	181	1	no	—
128	C₂²	—	724	1	no	—
129	C₂²	Q₈×#10624382	1	1	yes	characteristic
130	C₄	—	181	1	no	—
131	C₄	—	181	1	no	—
132	C₄	—	724	2	no	—
133	C₄	C₂×C₄⋊#10624382	1	1	yes	characteristic
134	C₄	D₄×#10624382	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	362	1	no	—
137	C₂	—	362	1	no	—
138	C₂	#26042230	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26042294

C₂

yes

maximal

C₂×#19153886

C₂

yes

maximal, characteristic

#26042242

C₂

yes

maximal, characteristic

#26042162

C₂

yes

maximal

#108×D₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668459

C₂²

yes

—

D₁₈₁⋊M₁₆

C₂²

yes

characteristic

#19153895

C₂²

yes

—

—

—

—

—

—

M₁₆×D₁₈₁

C₂²

yes

—

C₂×C₈×D₁₈₁

C₂²

yes

characteristic

#19153752

C₄

yes

characteristic, complemented

#19153751

C₄

yes

complemented

#108×C₁₈₁

C₄

yes

characteristic, complemented

#15668459

—

#15668459

—

#15668459

D₄

yes

—

#15668458

—

D₁₈₁⋊C₈

—

C₂×#13089685

—

C₄⋊#10624382

—

C₂²⋊#10624382

Q₈

yes

—

C₄×#10624382

—

C₂×C₄×D₁₈₁

C₂³

yes

characteristic

#15668433

C₂×C₄

yes

—

#15668432

—

C₂×#13089678

C₂×C₄

yes

characteristic

C₈×D₁₈₁

—

C₈×D₁₈₁

D₄

yes

characteristic

C₈×D₁₈₁

D₄

yes

characteristic

C₈×D₁₈₁

Q₈

yes

characteristic

C₈×D₁₈₁

Q₈

yes

characteristic

M₁₆×C₁₈₁

C₂×C₄

yes

—

C₂×C₈×C₁₈₁

C₂×C₄

yes

—

—

—

C₂²×D₁₈₁

C₂×Q₈

yes

characteristic

C₂×Dic₁₈₁

#41

yes

characteristic

C₄×D₁₈₁

C₂×D₄

yes

characteristic

C₄×D₁₈₁

C₂×D₄

yes

characteristic

C₄×D₁₈₁

C₂×Q₈

yes

C₄×D₁₈₁

yes

—

C₄⋊C₄

yes

characteristic

#13089678

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₈×C₁₈₁

—

C₈×C₁₈₁

C₄⋊C₄

yes

characteristic

C₈×C₁₈₁

C₄⋊C₄

yes

characteristic

C₂²×C₁₈₁

C₄×Q₈

yes

characteristic

C₂×D₁₈₁

—

C₂×D₁₈₁

C₂²⋊Q₈

yes

characteristic

C₂×D₁₈₁

C₄⋊Q₈

yes

characteristic

#10624382

—

C₄×C₁₈₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₈₁

C₄×D₄

yes

Dic₁₈₁

yes

Dic₁₈₁

yes

C₂×C₁₈₁

—

C₂×C₁₈₁

#389

yes

characteristic

D₁₈₁

—

D₁₈₁

—

C₁₈₁

#1953

yes

characteristic, complemented, Sylow

#1953

—

181

maximal, Sylow

#430

—

181

—

C₂×#108

—

181

—

C₂×#108

—

181

—

#405

—

181

—

#344

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂²×C₈

—

181

—

#117

—

181

—

#108

—

362

—

#108

—

362

—

#108

—

362

—

#108

—

362

—

#108

—

181

—

#108

—

181

—

#108

#10624382

yes

characteristic, complemented

#97

—

362

—

C₄×C₈

—

362

—

C₂²×C₄

—

181

—

M₁₆

—

362

—

M₁₆

—

362

—

M₁₆

—

362

—

M₁₆

—

362

—

M₁₆

—

181

—

M₁₆

—

181

—

M₁₆

C₂×#10624382

yes

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

100

C₂×C₈

—

362

—

101

C₂×C₈

—

181

—

102

C₂×C₈

—

181

—

103

C₂×C₈

—

181

—

104

C₂×C₈

—

181

—

105

C₂×C₈

—

181

—

106

C₂×C₈

C₂×#10624382

yes

characteristic

107

C₄⋊C₄

—

362

—

108

C₂²⋊C₄

—

181

—

109

C₄²

—

362

—

110

C₂³

—

181

—

111

C₂×C₄

—

362

—

112

C₂×C₄

—

181

—

113

C₂×C₄

—

181

—

114

C₂×C₄

—

181

—

115

C₂×C₄

—

181

—

116

C₂×C₄

—

181

—

117

C₂×C₄

C₂²×#10624382

yes

characteristic

118

C₈

—

119

C₈

—

362

—

120

C₈

—

362

—

121

C₈

—

181

—

122

C₈

—

181

—

123

C₈

—

724

—

124

C₈

C₄⋊#10624382

yes

characteristic

125

C₈

C₄⋊#10624382

yes

characteristic

126

C₂²

—

181

—

127

C₂²

—

181

—

128

C₂²

—

724

—

129

C₂²

Q₈×#10624382

yes

characteristic

130

C₄

—

181

—

131

C₄

—

181

—

132

C₄

—

724

—

133

C₄

C₂×C₄⋊#10624382

yes

characteristic

134

C₄

D₄×#10624382

yes

characteristic

135

C₂

—

136

C₂

—

362

—

137

C₂

—

362

—

138

C₂

#26042230

yes

characteristic

	subgroup	G/H
center	C₂	#26042230
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10624382
Fitting	#108×C₁₈₁	C₄
socle	C₂×C₁₈₁	#389
prime radical	C₂²×D₁₈₁	C₂×Q₈
cosocle	C₂×C₄×D₁₈₁	C₂³
hypercenter	#108	#10624382
hypocenter	C₁₈₁	#1953
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#26042230

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10624382

Fitting

#108×C₁₈₁

C₄

socle

C₂×C₁₈₁

#389

prime radical

C₂²×D₁₈₁

C₂×Q₈

cosocle

C₂×C₄×D₁₈₁

C₂³

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
17. #19153752	C₄	semidirect
18. #19153751	C₄	semidirect
19. #108×C₁₈₁	C₄	semidirect
69. C₁₈₁	#1953	semidirect
87. #108	#10624382	semidirect

H ≅ G/N

type

17. #19153752

C₄

semidirect

18. #19153751

C₄

semidirect

19. #108×C₁₈₁

C₄

semidirect

69. C₁₈₁

#1953

semidirect

87. #108

#10624382

semidirect

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

subgroup	index
#26042294	2
C₂×#19153886	2
#26042242	2
#26042162	2
#108×D₁₈₁	2
#1953	181

subgroup

index

#108×D₁₈₁

181

Subgroups — #37549840

Conjugacy classes of subgroups

Complements

Maximal subgroups