Group #27187924 — subgroups

Subgroups — #27187924

Order 12928.

348 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 162 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	#20266229	C₂	1	4	yes	maximal, complemented
2	#20266227	C₂	1	4	yes	maximal, complemented
3	#20266184	C₂	1	2	yes	maximal
4	#20266181	C₂	1	1	yes	maximal, characteristic
5	#20266142	C₂	1	2	yes	maximal
6	#20266136	C₂	1	1	yes	maximal, characteristic
7	#20265733	C₂	1	1	yes	maximal, characteristic
8	C₂×#13108084	C₂²	1	2	yes	—
9	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
11	#15692322	C₂²	1	4	yes	—
12	C₄⋊#11717581	C₂²	1	4	yes	—
13	C₄⋊#11717581	C₂²	1	4	yes	—
14	#15692310	C₂²	1	2	yes	—
15	#15692310	C₂²	1	2	yes	—
16	#15692310	C₂²	1	4	yes	—
17	C₄×#11717581	C₂²	1	4	yes	—
18	#15692254	C₄	1	2	yes	—
19	#15692247	C₄	1	1	yes	characteristic
20	#15692233	C₄	1	4	yes	—
21	#15692220	C₂²	1	4	yes	—
22	#15692213	C₂²	1	2	yes	—
23	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
24	#125×C₁₀₁	C₄	1	1	yes	characteristic
25	#13108084	—	2	4	no	—
26	D₁₀₁⋊C₈	—	2	2	no	—
27	D₁₀₁⋊C₈	Q₈	1	4	yes	complemented
28	C₂×#11717581	C₂³	1	4	yes	—
29	C₂×#11717581	C₂³	1	1	yes	characteristic
30	C₂×#11717581	C₂³	1	1	yes	characteristic
31	C₂×#11717581	C₂³	1	2	yes	—
32	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
36	#13108053	C₂×C₄	1	4	yes	—
37	#13108053	C₂×C₄	1	4	yes	—
38	#13108051	C₂×C₄	1	4	yes	—
39	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₀₁	C₂³	1	4	yes	—
41	C₄×Dic₁₀₁	C₂³	1	2	yes	—
42	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
44	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
50	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
51	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
52	C₄×D₁₀₁	—	4	2	no	—
53	C₄×D₁₀₁	C₂×Q₈	1	2	yes	—
54	C₄×D₁₀₁	#41	1	2	yes	—
55	Dic₂₀₂	—	4	4	no	—
56	#11717581	—	2	1	no	—
57	#11717581	—	2	1	no	—
58	#11717581	—	2	4	no	—
59	#11717581	C₂×Q₈	1	4	yes	—
60	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
61	C₂×D₁₀₁	—	2	2	no	—
62	C₂×D₁₀₁	C₄×Q₈	1	2	yes	—
63	C₄×C₁₀₁	—	2	4	no	—
64	C₄×C₁₀₁	—	2	2	no	—
65	C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
66	Dic₁₀₁	—	2	2	no	—
67	Dic₁₀₁	—	2	4	no	—
68	Dic₁₀₁	C₂²×Q₈	1	1	yes	characteristic
69	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₁₀₁	#568	1	2	yes	—
72	D₁₀₁	—	2	2	no	—
73	#2961	—	101	1	no	maximal, Sylow
74	C₁₀₁	#2961	1	1	yes	characteristic, complemented, Sylow
75	#533	—	101	1	no	—
76	#446	—	101	4	no	—
77	#446	—	101	4	no	—
78	#424	—	101	2	no	—
79	#423	—	101	1	no	—
80	#405	—	101	2	no	—
81	C₂×#97	—	101	1	no	—
82	C₂×M₁₆	—	101	2	no	—
83	C₂²×C₈	—	101	1	no	—
84	C₂²×C₈	—	101	1	no	—
85	C₄⋊Q₈	—	101	2	no	—
86	#125	—	101	1	no	—
87	#125	#87938	1	1	yes	characteristic
88	C₂²⋊Q₈	—	101	4	no	—
89	C₄×Q₈	—	101	4	no	—
90	#117	—	101	2	no	—
91	C₂×C₄²	—	101	1	no	—
92	C₄⋊C₈	—	101	4	no	—
93	C₄⋊C₈	—	101	4	no	—
94	C₄⋊C₈	—	101	4	no	—
95	#97	—	101	2	no	—
96	#97	—	101	2	no	—
97	#97	—	101	4	no	—
98	C₄×C₈	—	101	4	no	—
99	C₂×Q₈	—	101	4	no	—
100	C₂²×C₄	—	101	1	no	—
101	C₂²×C₄	—	101	2	no	—
102	M₁₆	—	202	4	no	—
103	C₂×C₈	—	101	2	no	—
104	C₂×C₈	—	101	4	no	—
105	C₂×C₈	—	101	4	no	—
106	C₂×C₈	—	101	1	no	—
107	C₂×C₈	—	101	1	no	—
108	C₂×C₈	—	202	2	no	—
109	C₄⋊C₄	—	101	4	no	—
110	C₄⋊C₄	—	101	4	no	—
111	C₄⋊C₄	—	101	2	no	—
112	C₄⋊C₄	C₂×#87938	1	2	yes	—
113	C₄⋊C₄	C₂×#87938	1	4	yes	—
114	C₂²⋊C₄	—	101	4	no	—
115	C₄²	—	101	1	no	—
116	C₄²	—	101	4	no	—
117	C₄²	—	101	2	no	—
118	C₄²	C₂×#87938	1	1	yes	characteristic
119	C₂³	—	101	1	no	—
120	Q₈	—	404	4	no	—
121	C₂×C₄	—	101	1	no	—
122	C₂×C₄	—	101	2	no	—
123	C₂×C₄	—	101	2	no	—
124	C₂×C₄	—	101	2	no	—
125	C₂×C₄	—	101	4	no	—
126	C₂×C₄	—	404	2	no	—
127	C₂×C₄	C₂²×#87938	1	2	yes	—
128	C₂×C₄	C₂²×#87938	1	4	yes	—
129	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
130	C₈	—	101	4	no	—
131	C₈	—	202	1	no	—
132	C₈	—	202	1	no	—
133	C₈	—	202	4	no	—
134	C₂²	—	101	2	no	—
135	C₂²	—	202	2	no	—
136	C₂²	C₂³×#87938	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	101	1	no	—
140	C₄	—	101	1	no	—
141	C₄	—	202	2	no	—
142	C₄	—	202	4	no	—
143	C₄	Q₈×#87938	1	2	yes	—
144	C₂	—	202	2	no	—
145	C₂	C₂×Q₈×#87938	1	1	yes	characteristic
146	C₂	#20266256	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20266229

C₂

yes

maximal, complemented

#20266227

C₂

yes

maximal, complemented

#20266184

C₂

yes

maximal

#20266181

C₂

yes

maximal, characteristic

#20266142

C₂

yes

maximal

#20266136

C₂

yes

maximal, characteristic

#20265733

C₂

yes

maximal, characteristic

C₂×#13108084

C₂²

yes

—

C₂×D₁₀₁⋊C₈

C₂²

yes

characteristic

C₂×D₁₀₁⋊C₈

C₂²

yes

characteristic

#15692322

C₂²

yes

—

C₄⋊#11717581

C₂²

yes

—

C₄⋊#11717581

C₂²

yes

—

#15692310

C₂²

yes

—

#15692310

C₂²

yes

—

#15692310

C₂²

yes

—

C₄×#11717581

C₂²

yes

—

#15692254

C₄

yes

—

#15692247

C₄

yes

characteristic

#15692233

C₄

yes

—

#15692220

C₂²

yes

—

#15692213

C₂²

yes

—

C₄²×D₁₀₁

C₂²

yes

characteristic

#125×C₁₀₁

C₄

yes

characteristic

#13108084

—

D₁₀₁⋊C₈

—

D₁₀₁⋊C₈

Q₈

yes

complemented

C₂×#11717581

C₂³

yes

—

C₂×#11717581

C₂³

yes

characteristic

C₂×#11717581

C₂³

yes

characteristic

C₂×#11717581

C₂³

yes

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

characteristic

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

#13108053

C₂×C₄

yes

—

#13108053

C₂×C₄

yes

—

#13108051

C₂×C₄

yes

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

C₂³

yes

—

C₄×Dic₁₀₁

C₂³

yes

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×D₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

C₂⁴

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

C₂×Q₈

yes

—

C₄×D₁₀₁

#41

yes

—

Dic₂₀₂

—

—

—

—

C₂×Q₈

yes

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

C₄×Q₈

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

C₂²×Q₈

yes

characteristic

Dic₁₀₁

C₂×#41

yes

characteristic

C₂×C₁₀₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₀₁

#568

yes

—

D₁₀₁

—

#2961

—

101

maximal, Sylow

C₁₀₁

#2961

yes

characteristic, complemented, Sylow

#533

—

101

—

#446

—

101

—

#446

—

101

—

#424

—

101

—

#423

—

101

—

#405

—

101

—

C₂×#97

—

101

—

C₂×M₁₆

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₄⋊Q₈

—

101

—

#125

—

101

—

#125

#87938

yes

characteristic

C₂²⋊Q₈

—

101

—

C₄×Q₈

—

101

—

#117

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

#97

—

101

—

#97

—

101

—

#97

—

101

—

C₄×C₈

—

101

—

C₂×Q₈

—

101

—

100

C₂²×C₄

—

101

—

101

C₂²×C₄

—

101

—

102

M₁₆

—

202

—

103

C₂×C₈

—

101

—

104

C₂×C₈

—

101

—

105

C₂×C₈

—

101

—

106

C₂×C₈

—

101

—

107

C₂×C₈

—

101

—

108

C₂×C₈

—

202

—

109

C₄⋊C₄

—

101

—

110

C₄⋊C₄

—

101

—

111

C₄⋊C₄

—

101

—

112

C₄⋊C₄

C₂×#87938

yes

—

113

C₄⋊C₄

C₂×#87938

yes

—

114

C₂²⋊C₄

—

101

—

115

C₄²

—

101

—

116

C₄²

—

101

—

117

C₄²

—

101

—

118

C₄²

C₂×#87938

yes

characteristic

119

C₂³

—

101

—

120

Q₈

—

404

—

121

C₂×C₄

—

101

—

122

C₂×C₄

—

101

—

123

C₂×C₄

—

101

—

124

C₂×C₄

—

101

—

125

C₂×C₄

—

101

—

126

C₂×C₄

—

404

—

127

C₂×C₄

C₂²×#87938

yes

—

128

C₂×C₄

C₂²×#87938

yes

—

129

C₂×C₄

C₂²×#87938

yes

characteristic

130

C₈

—

101

—

131

C₈

—

202

—

132

C₈

—

202

—

133

C₈

—

202

—

134

C₂²

—

101

—

135

C₂²

—

202

—

136

C₂²

C₂³×#87938

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

101

—

140

C₄

—

101

—

141

C₄

—

202

—

142

C₄

—

202

—

143

C₄

Q₈×#87938

yes

—

144

C₂

—

202

—

145

C₂

C₂×Q₈×#87938

yes

characteristic

146

C₂

#20266256

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#125×C₁₀₁	C₄
prime radical	C₂²×D₁₀₁	C₂²×C₄
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	#125	#87938
hypocenter	C₁₀₁	#2961
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#87938

commutator, socle

C₂²×C₁₀₁

C₂³×C₄

Fitting

#125×C₁₀₁

C₄

prime radical

C₂²×D₁₀₁

C₂²×C₄

cosocle

C₂×Dic₁₀₁

C₂⁴

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #20266229	C₂	semidirect
2. #20266227	C₂	semidirect
27. D₁₀₁⋊C₈	Q₈	semidirect
74. C₁₀₁	#2961	semidirect

H ≅ G/N

type

1. #20266229

C₂

semidirect

2. #20266227

C₂

semidirect

27. D₁₀₁⋊C₈

Q₈

semidirect

74. C₁₀₁

#2961

semidirect

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

subgroup	index
#20266229	2
#20266227	2
#20266184	2
#20266181	2
#20266142	2
#20266136	2
#20265733	2
#2961	101

subgroup

index

101

Subgroups — #27187924

Conjugacy classes of subgroups

Complements

Maximal subgroups