Group #27186057 — subgroups

Subgroups — #27186057

Order 12928.

294 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 34 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	#20266020	C₂	1	2	yes	maximal, complemented
2	#20265032	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11717581	—	2	2	no	—
4	C₂⁴×D₁₀₁	C₄	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
6	C₂×#13108051	—	2	2	no	—
7	C₂×C₂²⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
8	C₂×#11717581	—	4	2	no	—
9	C₂³×D₁₀₁	—	2	1	no	—
10	C₂³×D₁₀₁	—	2	1	no	—
11	C₂³×D₁₀₁	—	4	4	no	—
12	C₂³×D₁₀₁	—	4	1	no	—
13	C₂³×D₁₀₁	—	4	1	no	—
14	C₂³×D₁₀₁	C₈	1	2	yes	complemented
15	C₂²×Dic₁₀₁	D₄	1	2	yes	—
16	C₂²⋊Dic₁₀₁	—	4	1	no	—
17	#13108051	—	4	4	no	—
18	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₁	—	2	1	no	—
20	C₂²⋊C₄×C₁₀₁	—	4	1	no	—
21	C₂³×C₁₀₁	—	2	1	no	—
22	C₂³×C₁₀₁	—	2	1	no	—
23	C₂³×C₁₀₁	—	4	2	no	—
24	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂³×C₁₀₁	C₂×C₈	1	1	yes	characteristic
26	C₂³×C₁₀₁	M₁₆	1	1	yes	characteristic
27	C₂²×D₁₀₁	—	2	4	no	—
28	C₂²×D₁₀₁	—	2	2	no	—
29	C₂²×D₁₀₁	—	4	4	no	—
30	C₂²×D₁₀₁	—	4	4	no	—
31	C₂²×D₁₀₁	—	4	2	no	—
32	C₂²×D₁₀₁	—	4	2	no	—
33	C₂²×D₁₀₁	—	4	8	no	—
34	C₂²×D₁₀₁	—	4	4	no	—
35	C₂²×D₁₀₁	—	4	1	no	—
36	C₂²×D₁₀₁	—	4	1	no	—
37	C₂²×D₁₀₁	—	4	2	no	—
38	C₂²×D₁₀₁	—	4	4	no	—
39	C₂×C₄×C₁₀₁	—	4	2	no	—
40	C₂×C₄×C₁₀₁	—	4	1	no	—
41	C₂×Dic₁₀₁	—	2	2	no	—
42	C₂×Dic₁₀₁	—	4	2	no	—
43	#11717581	—	8	2	no	—
44	C₂²×C₁₀₁	—	2	2	no	—
45	C₂²×C₁₀₁	—	2	1	no	—
46	C₂²×C₁₀₁	—	2	1	no	—
47	C₂²×C₁₀₁	—	4	1	no	—
48	C₂²×C₁₀₁	—	4	1	no	—
49	C₂²×C₁₀₁	—	4	2	no	—
50	C₂²×C₁₀₁	—	4	2	no	—
51	C₂²×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
52	C₂²×C₁₀₁	#99	1	1	yes	characteristic
53	C₂²×C₁₀₁	#100	1	1	yes	characteristic
54	C₂×D₁₀₁	—	2	4	no	—
55	C₂×D₁₀₁	—	4	4	no	—
56	C₂×D₁₀₁	—	4	2	no	—
57	C₂×D₁₀₁	—	4	8	no	—
58	C₂×D₁₀₁	—	4	4	no	—
59	C₂×D₁₀₁	—	4	2	no	—
60	C₂×D₁₀₁	—	4	4	no	—
61	C₂×D₁₀₁	—	4	4	no	—
62	C₄×C₁₀₁	—	8	1	no	—
63	Dic₁₀₁	—	4	2	no	—
64	C₂×C₁₀₁	—	2	1	no	—
65	C₂×C₁₀₁	—	2	1	no	—
66	C₂×C₁₀₁	—	4	1	no	—
67	C₂×C₁₀₁	—	4	1	no	—
68	C₂×C₁₀₁	#323	1	1	yes	characteristic
69	C₂×C₁₀₁	AΣL(1,𝔽₁₆)	1	2	yes	—
70	D₁₀₁	—	4	4	no	—
71	#1316	—	101	1	no	maximal, Sylow
72	C₁₀₁	#1316	1	1	yes	characteristic, complemented, Sylow
73	#379	—	101	1	no	—
74	#323	—	101	2	no	—
75	C₂⁵	—	101	1	no	—
76	C₂×C₂²⋊C₄	—	101	1	no	—
77	C₂×C₂²⋊C₄	—	202	2	no	—
78	C₂×C₂²⋊C₄	#87938	1	1	yes	characteristic
79	C₂²⋊C₈	—	202	2	no	—
80	C₂⁴	—	101	2	no	—
81	C₂⁴	—	404	1	no	—
82	C₂⁴	—	404	1	no	—
83	C₂⁴	—	404	4	no	—
84	C₂⁴	—	202	1	no	—
85	C₂⁴	—	202	1	no	—
86	C₂⁴	C₂×#87938	1	1	yes	characteristic
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	101	2	no	—
89	C₂×C₈	—	404	2	no	—
90	C₂²⋊C₄	—	4	1	no	—
91	C₂²⋊C₄	—	404	4	no	—
92	C₂²⋊C₄	—	404	1	no	—
93	C₂³	—	2	1	no	—
94	C₂³	—	2	1	no	—
95	C₂³	—	4	2	no	—
96	C₂³	—	404	2	no	—
97	C₂³	—	404	2	no	—
98	C₂³	—	404	2	no	—
99	C₂³	—	404	4	no	—
100	C₂³	—	404	8	no	—
101	C₂³	—	404	4	no	—
102	C₂³	—	404	4	no	—
103	C₂³	—	404	4	no	—
104	C₂³	—	404	1	no	—
105	C₂³	—	404	1	no	—
106	C₂³	—	202	4	no	—
107	C₂³	—	202	2	no	—
108	C₂³	C₂²⋊#87938	1	1	yes	characteristic
109	C₂³	D₁₀₁⋊C₈	1	1	yes	characteristic
110	C₂³	#13108084	1	1	yes	characteristic
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	404	2	no	—
114	C₂×C₄	—	202	2	no	—
115	C₈	—	808	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	—	4	2	no	—
123	C₂²	—	404	4	no	—
124	C₂²	—	404	2	no	—
125	C₂²	—	404	4	no	—
126	C₂²	—	404	4	no	—
127	C₂²	—	404	8	no	—
128	C₂²	—	404	2	no	—
129	C₂²	—	404	4	no	—
130	C₂²	—	202	4	no	—
131	C₂²	#15692312	1	1	yes	characteristic
132	C₂²	#15692314	1	1	yes	characteristic
133	C₂²	#15692315	1	1	yes	characteristic
134	C₄	—	8	1	no	—
135	C₄	—	404	2	no	—
136	C₂	—	2	1	no	—
137	C₂	—	2	1	no	—
138	C₂	—	4	1	no	—
139	C₂	—	4	1	no	—
140	C₂	—	404	4	no	—
141	C₂	#20266021	1	1	yes	characteristic
142	C₂	#20266078	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20266020

C₂

yes

maximal, complemented

#20265032

C₂

yes

maximal, characteristic

C₂²⋊#11717581

—

C₂⁴×D₁₀₁

C₄

yes

characteristic

C₂×C₂²⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×#13108051

—

C₂×C₂²⋊C₄×C₁₀₁

C₄

yes

characteristic

C₂×#11717581

—

C₂³×D₁₀₁

—

C₂³×D₁₀₁

—

C₂³×D₁₀₁

—

C₂³×D₁₀₁

—

C₂³×D₁₀₁

—

C₂³×D₁₀₁

C₈

yes

complemented

C₂²×Dic₁₀₁

D₄

yes

—

C₂²⋊Dic₁₀₁

—

#13108051

—

C₂⁴×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂³×C₁₀₁

C₂×C₈

yes

characteristic

C₂³×C₁₀₁

M₁₆

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717581

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²⋊C₈

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#323

yes

characteristic

C₂×C₁₀₁

AΣL(1,𝔽₁₆)

yes

—

D₁₀₁

—

#1316

—

101

maximal, Sylow

C₁₀₁

#1316

yes

characteristic, complemented, Sylow

#379

—

101

—

#323

—

101

—

C₂⁵

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

202

—

C₂×C₂²⋊C₄

#87938

yes

characteristic

C₂²⋊C₈

—

202

—

C₂⁴

—

101

—

C₂⁴

—

404

—

C₂⁴

—

404

—

C₂⁴

—

404

—

C₂⁴

—

202

—

C₂⁴

—

202

—

C₂⁴

C₂×#87938

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

101

—

C₂×C₈

—

404

—

C₂²⋊C₄

—

C₂²⋊C₄

—

404

—

C₂²⋊C₄

—

404

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

404

—

C₂³

—

404

—

C₂³

—

404

—

C₂³

—

404

—

100

C₂³

—

404

—

101

C₂³

—

404

—

102

C₂³

—

404

—

103

C₂³

—

404

—

104

C₂³

—

404

—

105

C₂³

—

404

—

106

C₂³

—

202

—

107

C₂³

—

202

—

108

C₂³

C₂²⋊#87938

yes

characteristic

109

C₂³

D₁₀₁⋊C₈

yes

characteristic

110

C₂³

#13108084

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

404

—

114

C₂×C₄

—

202

—

115

C₈

—

808

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

404

—

124

C₂²

—

404

—

125

C₂²

—

404

—

126

C₂²

—

404

—

127

C₂²

—

404

—

128

C₂²

—

404

—

129

C₂²

—

404

—

130

C₂²

—

202

—

131

C₂²

#15692312

yes

characteristic

132

C₂²

#15692314

yes

characteristic

133

C₂²

#15692315

yes

characteristic

134

C₄

—

135

C₄

—

404

—

136

C₂

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

—

404

—

141

C₂

#20266021

yes

characteristic

142

C₂

#20266078

yes

—

	subgroup	G/H
center	C₂²	#15692314
commutator	C₂³×C₁₀₁	C₂×C₈
Frattini	C₂⁴	C₂×#87938
Fitting	C₂×C₂²⋊C₄×C₁₀₁	C₄
socle	C₂²×C₁₀₁	#99
prime radical	C₂⁴×D₁₀₁	C₄
cosocle	C₂×C₂²⋊Dic₁₀₁	C₂²
hypercenter	C₂×C₂²⋊C₄	#87938
hypocenter	C₁₀₁	#1316
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂⁴ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center

C₂²

#15692314

commutator

C₂³×C₁₀₁

C₂×C₈

Frattini

C₂⁴

C₂×#87938

Fitting

C₂×C₂²⋊C₄×C₁₀₁

C₄

socle

C₂²×C₁₀₁

#99

prime radical

C₂⁴×D₁₀₁

C₄

cosocle

C₂×C₂²⋊Dic₁₀₁

C₂²

C₂×C₂²⋊C₄

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂⁴ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. #20266020	C₂	semidirect
14. C₂³×D₁₀₁	C₈	semidirect
72. C₁₀₁	#1316	semidirect

H ≅ G/N

type

1. #20266020

C₂

semidirect

14. C₂³×D₁₀₁

C₈

semidirect

72. C₁₀₁

#1316

semidirect

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

subgroup	index
#20266020	2
#20265032	2
#1316	101

subgroup

index

#20266020

#20265032

#1316

101

Subgroups — #27186057

Conjugacy classes of subgroups

Complements

Maximal subgroups