Group #27181477 — subgroups

Subgroups — #27181477

Order 12928.

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

normal subgroups only

This group has 392 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	#20265734	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265545	C₂	1	2	yes	maximal, complemented
3	#20265541	C₂	1	2	yes	maximal, complemented
4	#20265536	C₂	1	4	yes	maximal, complemented
5	#20265173	C₂	1	4	yes	maximal, complemented
6	C₂×#15692166	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11717579	C₂²	1	2	yes	—
9	#15692255	C₂²	1	2	yes	—
10	#15692247	C₂²	1	2	yes	—
11	#15692234	—	2	2	no	—
12	C₄×Dic₂₀₂	—	2	2	no	—
13	#15692214	C₂²	1	2	yes	—
14	#15692179	C₂²	1	8	yes	—
15	#15692176	C₂²	1	8	yes	—
16	#15692166	C₂²	1	2	yes	—
17	#15692166	C₂²	1	2	yes	—
18	C₂×#41×C₁₀₁	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
20	#127×C₁₀₁	C₂²	1	1	yes	characteristic
21	#124×C₁₀₁	C₂²	1	2	yes	—
22	#121×C₁₀₁	—	2	2	no	—
23	C₄×D₄×C₁₀₁	—	2	2	no	—
24	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₂₀₂	—	2	2	no	—
26	C₂×#11717579	—	2	4	no	—
27	C₂×#11717579	C₂³	1	4	yes	—
28	C₄⋊Dic₁₀₁	—	2	2	no	—
29	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
30	#13108053	—	2	4	no	—
31	C₂²⋊Dic₁₀₁	—	2	2	no	—
32	C₄×Dic₁₀₁	—	2	2	no	—
33	#41×C₁₀₁	—	2	8	no	—
34	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
35	C₂×D₄×C₁₀₁	—	2	2	no	—
36	C₂×D₄×C₁₀₁	—	2	2	no	—
37	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₀₁	—	2	2	no	—
39	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
40	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₁	—	2	2	no	—
42	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
43	C₄²×C₁₀₁	D₄	1	2	yes	—
44	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
45	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
46	C₂³×C₁₀₁	—	2	2	no	—
47	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₁₀₁	—	2	4	no	—
49	D₄×C₁₀₁	—	2	4	no	—
50	D₄×C₁₀₁	—	4	4	no	—
51	D₄×C₁₀₁	—	4	4	no	—
52	C₂×C₄×C₁₀₁	—	2	2	no	—
53	C₂×C₄×C₁₀₁	—	4	2	no	—
54	C₂×C₄×C₁₀₁	—	4	4	no	—
55	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₁	#41	1	4	yes	—
59	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₁₀₁	—	2	4	no	—
61	Dic₂₀₂	—	4	4	no	—
62	#11717579	—	2	4	no	—
63	C₂²×C₁₀₁	—	2	2	no	—
64	C₂²×C₁₀₁	—	4	2	no	—
65	C₂²×C₁₀₁	—	4	4	no	—
66	C₂²×C₁₀₁	#124	1	2	yes	—
67	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₁₀₁	—	2	2	no	—
69	C₄×C₁₀₁	—	2	2	no	—
70	C₄×C₁₀₁	—	4	2	no	—
71	C₄×C₁₀₁	#124	1	2	yes	—
72	C₄×C₁₀₁	C₂×#41	1	2	yes	—
73	Dic₁₀₁	—	4	4	no	—
74	C₂×C₁₀₁	—	2	2	no	—
75	C₂×C₁₀₁	—	8	2	no	—
76	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
77	C₂×C₁₀₁	#575	1	2	yes	—
78	#3135	—	101	1	no	maximal, Sylow
79	C₁₀₁	#3135	1	1	yes	characteristic, complemented, Sylow
80	#533	—	101	1	no	—
81	#532	D₁₀₁	1	1	yes	characteristic
82	#491	—	101	2	no	—
83	#490	—	101	2	no	—
84	#489	—	101	4	no	—
85	#416	—	101	4	no	—
86	C₂×#97	—	101	1	no	—
87	C₂×#41	C₂×D₁₀₁	1	2	yes	—
88	C₂²×C₈	—	101	2	no	—
89	C₄⋊Q₈	—	101	2	no	—
90	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
91	#127	C₂×D₁₀₁	1	1	yes	characteristic
92	#125	—	101	2	no	—
93	#124	C₂×D₁₀₁	1	2	yes	—
94	C₂²⋊Q₈	—	202	2	no	—
95	#121	—	2	2	no	—
96	C₄×Q₈	—	202	2	no	—
97	C₄×D₄	—	2	2	no	—
98	#117	—	101	2	no	—
99	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
100	#103	—	101	8	no	—
101	#102	—	101	8	no	—
102	#97	—	101	2	no	—
103	#97	—	101	2	no	—
104	#41	—	2	8	no	—
105	C₂×Q₈	—	202	2	no	—
106	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
113	C₂×C₈	—	101	4	no	—
114	C₂×C₈	—	202	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	101	4	no	—
117	C₄⋊C₄	—	202	4	no	—
118	C₄⋊C₄	—	202	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	202	2	no	—
121	C₄²	—	202	2	no	—
122	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
123	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
124	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	404	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	202	4	no	—
136	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
139	C₂×C₄	#13108073	1	4	yes	—
140	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
141	C₈	—	202	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	4	no	—
145	C₂²	#15692242	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	404	4	no	—
151	C₄	#15692242	1	2	yes	—
152	C₄	C₂×#13108073	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	8	2	no	—
155	C₂	C₂×#15692242	1	1	yes	characteristic
156	C₂	#20265975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265734

C₂

yes

maximal, characteristic, complemented

#20265545

C₂

yes

maximal, complemented

#20265541

C₂

yes

maximal, complemented

#20265536

C₂

yes

maximal, complemented

#20265173

C₂

yes

maximal, complemented

C₂×#15692166

C₂

yes

maximal, characteristic, complemented

#532×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

#15692255

C₂²

yes

—

#15692247

C₂²

yes

—

#15692234

—

C₄×Dic₂₀₂

—

#15692214

C₂²

yes

—

#15692179

C₂²

yes

—

#15692176

C₂²

yes

—

#15692166

C₂²

yes

—

#15692166

C₂²

yes

—

C₂×#41×C₁₀₁

C₂²

yes

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

characteristic

#127×C₁₀₁

C₂²

yes

characteristic

#124×C₁₀₁

C₂²

yes

—

#121×C₁₀₁

—

C₄×D₄×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×#11717579

—

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂³

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

Dic₂₀₂

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#124

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#124

yes

—

C₄×C₁₀₁

C₂×#41

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#124

yes

characteristic

C₂×C₁₀₁

#575

yes

—

#3135

—

101

maximal, Sylow

C₁₀₁

#3135

yes

characteristic, complemented, Sylow

#533

—

101

—

#532

D₁₀₁

yes

characteristic

#491

—

101

—

#490

—

101

—

#489

—

101

—

#416

—

101

—

C₂×#97

—

101

—

C₂×#41

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

C₂×D₁₀₁

yes

characteristic

#127

C₂×D₁₀₁

yes

characteristic

#125

—

101

—

#124

C₂×D₁₀₁

yes

—

C₂²⋊Q₈

—

202

—

#121

—

C₄×Q₈

—

202

—

C₄×D₄

—

#117

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

100

#103

—

101

—

101

#102

—

101

—

102

#97

—

101

—

103

#97

—

101

—

104

#41

—

105

C₂×Q₈

—

202

—

106

C₂×Q₈

C₂²×D₁₀₁

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₁₀₁

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

112

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

113

C₂×C₈

—

101

—

114

C₂×C₈

—

202

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

101

—

117

C₄⋊C₄

—

202

—

118

C₄⋊C₄

—

202

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

202

—

121

C₄²

—

202

—

122

C₄²

C₂²⋊D₁₀₁

yes

—

123

C₄²

C₂²×D₁₀₁

yes

characteristic

124

C₄²

C₂²×D₁₀₁

yes

characteristic

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

127

Q₈

—

128

Q₈

—

404

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

202

—

136

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

139

C₂×C₄

#13108073

yes

—

140

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

141

C₈

—

202

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#15692242

yes

—

146

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

404

—

151

C₄

#15692242

yes

—

152

C₄

C₂×#13108073

yes

—

153

C₂

—

154

C₂

—

155

C₂

C₂×#15692242

yes

characteristic

156

C₂

#20265975

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting, prime radical	#532×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
hypercenter	#532	D₁₀₁
hypocenter	C₁₀₁	#3135
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting, prime radical

#532×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂²×D₄

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #20265734	C₂	semidirect
2. #20265545	C₂	semidirect
3. #20265541	C₂	semidirect
4. #20265536	C₂	semidirect
5. #20265173	C₂	semidirect
6. C₂×#15692166	C₂	semidirect
79. C₁₀₁	#3135	semidirect

H ≅ G/N

type

1. #20265734

C₂

semidirect

2. #20265545

C₂

semidirect

3. #20265541

C₂

semidirect

4. #20265536

C₂

semidirect

5. #20265173

C₂

semidirect

6. C₂×#15692166

C₂

semidirect

79. C₁₀₁

#3135

semidirect

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

subgroup	index
#20265734	2
#20265545	2
#20265541	2
#20265536	2
#20265173	2
C₂×#15692166	2
#532×C₁₀₁	2
#3135	101

subgroup

index

#532×C₁₀₁

101

Subgroups — #27181477

Conjugacy classes of subgroups

Complements

Maximal subgroups