Group #27171053 — subgroups

Subgroups — #27171053

Order 12928.

820 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15692172	C₂	1	4	yes	maximal, complemented
2	#20265141	C₂	1	2	yes	maximal, complemented
3	C₂³×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#13108058	C₂²	1	4	yes	complemented
5	#15692172	—	2	8	no	—
6	C₂²⋊#11717579	—	2	4	no	—
7	C₂⁵×C₁₀₁	C₄	1	2	yes	—
8	C₂²×D₄×C₁₀₁	—	2	8	no	—
9	C₂²×D₄×C₁₀₁	—	2	4	no	—
10	C₂²×D₄×C₁₀₁	C₄	1	2	yes	—
11	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
13	#13108058	—	2	8	no	—
14	C₂×#11717579	—	2	4	no	—
15	C₂⁴×C₁₀₁	—	2	8	no	—
16	C₂⁴×C₁₀₁	—	2	4	no	—
17	C₂⁴×C₁₀₁	—	4	8	no	—
18	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
19	C₂⁴×C₁₀₁	C₂×C₄	1	4	yes	—
20	C₂×D₄×C₁₀₁	—	2	32	no	—
21	C₂×D₄×C₁₀₁	—	2	8	no	—
22	C₂×D₄×C₁₀₁	—	2	4	no	—
23	C₂×D₄×C₁₀₁	—	2	4	no	—
24	C₂×D₄×C₁₀₁	—	2	4	no	—
25	C₂×D₄×C₁₀₁	D₄	1	8	yes	—
26	C₂²×C₄×C₁₀₁	—	2	4	no	—
27	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
29	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
30	C₂³×C₁₀₁	—	2	4	no	—
31	C₂³×C₁₀₁	—	2	2	no	—
32	C₂³×C₁₀₁	—	2	16	no	—
33	C₂³×C₁₀₁	—	2	4	no	—
34	C₂³×C₁₀₁	—	2	4	no	—
35	C₂³×C₁₀₁	—	4	8	no	—
36	C₂³×C₁₀₁	—	4	32	no	—
37	C₂³×C₁₀₁	—	4	8	no	—
38	C₂³×C₁₀₁	—	4	8	no	—
39	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
40	C₂³×C₁₀₁	C₂²⋊C₄	1	8	yes	—
41	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
42	D₄×C₁₀₁	—	2	16	no	—
43	D₄×C₁₀₁	—	2	16	no	—
44	C₂×C₄×C₁₀₁	—	2	2	no	—
45	C₂×C₄×C₁₀₁	—	2	8	no	—
46	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
49	#11717579	—	4	4	no	—
50	C₂²×C₁₀₁	—	2	8	no	—
51	C₂²×C₁₀₁	—	2	2	no	—
52	C₂²×C₁₀₁	—	2	2	no	—
53	C₂²×C₁₀₁	—	2	2	no	—
54	C₂²×C₁₀₁	—	2	8	no	—
55	C₂²×C₁₀₁	—	4	8	no	—
56	C₂²×C₁₀₁	—	4	32	no	—
57	C₂²×C₁₀₁	—	4	8	no	—
58	C₂²×C₁₀₁	—	4	8	no	—
59	C₂²×C₁₀₁	#100	1	4	yes	complemented
60	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
61	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₁₀₁	—	2	2	no	—
63	C₄×C₁₀₁	#120	1	4	yes	—
64	C₂×C₁₀₁	—	2	4	no	—
65	C₂×C₁₀₁	—	2	2	no	—
66	C₂×C₁₀₁	—	4	8	no	—
67	C₂×C₁₀₁	#379	1	1	yes	characteristic
68	C₂×C₁₀₁	C₂×#100	1	2	yes	—
69	#1783	—	101	1	no	maximal, Sylow
70	C₁₀₁	#1783	1	1	yes	characteristic, complemented, Sylow
71	C₂³×D₄	D₁₀₁	1	1	yes	characteristic
72	C₂×#100	—	101	4	no	—
73	#407	—	101	2	no	—
74	C₂⁵	Dic₁₀₁	1	2	yes	—
75	C₂²×D₄	—	2	8	no	—
76	C₂²×D₄	—	2	4	no	—
77	C₂²×D₄	Dic₁₀₁	1	2	yes	—
78	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
79	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
80	C₂×M₁₆	—	101	4	no	—
81	#100	—	202	8	no	—
82	C₂²⋊C₈	—	202	4	no	—
83	C₂⁴	—	2	8	no	—
84	C₂⁴	—	2	4	no	—
85	C₂⁴	—	4	8	no	—
86	C₂⁴	C₂×Dic₁₀₁	1	1	yes	characteristic
87	C₂⁴	C₂×Dic₁₀₁	1	4	yes	—
88	C₂×D₄	—	2	32	no	—
89	C₂×D₄	—	2	8	no	—
90	C₂×D₄	—	2	4	no	—
91	C₂×D₄	—	2	4	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂×D₄	C₂²⋊D₁₀₁	1	8	yes	—
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
97	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
98	M₁₆	—	202	8	no	—
99	C₂×C₈	—	202	4	no	—
100	C₂³	—	2	4	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	2	16	no	—
103	C₂³	—	2	4	no	—
104	C₂³	—	2	4	no	—
105	C₂³	—	4	8	no	—
106	C₂³	—	4	32	no	—
107	C₂³	—	4	8	no	—
108	C₂³	—	4	8	no	—
109	C₂³	C₂²⋊Dic₁₀₁	1	2	yes	—
110	C₂³	C₂²⋊Dic₁₀₁	1	8	yes	—
111	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
112	D₄	—	2	16	no	—
113	D₄	—	2	16	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	8	no	—
116	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
119	C₈	—	404	4	no	—
120	C₂²	—	2	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	8	no	—
125	C₂²	—	4	8	no	—
126	C₂²	—	4	32	no	—
127	C₂²	—	4	8	no	—
128	C₂²	—	4	8	no	—
129	C₂²	#15692172	1	4	yes	complemented
130	C₂²	C₂×C₂²⋊Dic₁₀₁	1	2	yes	—
131	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	#15692224	1	4	yes	—
134	C₂	—	2	4	no	—
135	C₂	—	2	2	no	—
136	C₂	—	4	8	no	—
137	C₂	#20265033	1	1	yes	characteristic
138	C₂	C₂×#15692172	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692172

C₂

yes

maximal, complemented

#20265141

C₂

yes

maximal, complemented

C₂³×D₄×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

complemented

#15692172

—

C₂²⋊#11717579

—

C₂⁵×C₁₀₁

C₄

yes

—

C₂²×D₄×C₁₀₁

—

C₂²×D₄×C₁₀₁

—

C₂²×D₄×C₁₀₁

C₄

yes

—

C₂²×D₄×C₁₀₁

C₂²

yes

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

#13108058

—

C₂×#11717579

—

C₂⁴×C₁₀₁

—

C₂⁴×C₁₀₁

—

C₂⁴×C₁₀₁

—

C₂⁴×C₁₀₁

C₂×C₄

yes

characteristic

C₂⁴×C₁₀₁

C₂×C₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#100

yes

complemented

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#379

yes

characteristic

C₂×C₁₀₁

C₂×#100

yes

—

#1783

—

101

maximal, Sylow

C₁₀₁

#1783

yes

characteristic, complemented, Sylow

C₂³×D₄

D₁₀₁

yes

characteristic

C₂×#100

—

101

—

#407

—

101

—

C₂⁵

Dic₁₀₁

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

Dic₁₀₁

yes

—

C₂²×D₄

C₂×D₁₀₁

yes

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

#100

—

202

—

C₂²⋊C₈

—

202

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×Dic₁₀₁

yes

characteristic

C₂⁴

C₂×Dic₁₀₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

202

—

C₂×C₈

—

202

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²⋊Dic₁₀₁

yes

—

110

C₂³

C₂²⋊Dic₁₀₁

yes

—

111

C₂³

C₂²×Dic₁₀₁

yes

characteristic

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

119

C₈

—

404

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#15692172

yes

complemented

130

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

—

131

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

132

C₄

—

133

C₄

#15692224

yes

—

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

#20265033

yes

characteristic

138

C₂

C₂×#15692172

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₁
commutator	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting, prime radical	C₂³×D₄×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂³×D₄	D₁₀₁
hypocenter	C₁₀₁	#1783
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₁

commutator

C₂³×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting, prime radical

C₂³×D₄×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂³×D₄

D₁₀₁

hypocenter

C₁₀₁

#1783

LCS

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×D₄

N	H ≅ G/N	type
1. C₂×#15692172	C₂	semidirect
2. #20265141	C₂	semidirect
4. C₂×#13108058	C₂²	semidirect
59. C₂²×C₁₀₁	#100	semidirect
70. C₁₀₁	#1783	semidirect
129. C₂²	#15692172	semidirect

H ≅ G/N

type

1. C₂×#15692172

C₂

semidirect

2. #20265141

C₂

semidirect

4. C₂×#13108058

C₂²

semidirect

59. C₂²×C₁₀₁

#100

semidirect

70. C₁₀₁

#1783

semidirect

129. C₂²

#15692172

semidirect

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

subgroup	index
C₂×#15692172	2
#20265141	2
C₂³×D₄×C₁₀₁	2
#1783	101

subgroup

index

C₂×#15692172

#20265141

C₂³×D₄×C₁₀₁

#1783

101

Subgroups — #27171053

Conjugacy classes of subgroups

Complements

Maximal subgroups