Group #27170184 — subgroups

Subgroups — #27170184

Order 12928.

336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20265812	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265369	C₂	1	2	yes	maximal, complemented
3	#20264921	C₂	1	1	yes	maximal, characteristic, complemented
4	#20264906	C₂	1	2	yes	maximal, complemented
5	#493×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₁₀₁	—	2	2	no	—
7	C₂×#13108059	—	2	2	no	—
8	#15692226	—	2	2	no	—
9	#15692223	—	2	2	no	—
10	C₄×D₄₀₄	C₂²	1	2	yes	—
11	C₄⋊#11717579	C₂²	1	2	yes	complemented
12	#15692176	—	2	2	no	—
13	C₂×D₈×C₁₀₁	—	2	2	no	—
14	#127×C₁₀₁	C₂²	1	1	yes	characteristic
15	#127×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
16	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₁₀₁	—	2	2	no	—
18	C₂×D₄₀₄	D₄	1	2	yes	—
19	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
20	D₄×D₁₀₁	—	4	4	no	—
21	D₄×D₁₀₁	—	4	4	no	—
22	C₂×C₄×D₁₀₁	—	2	2	no	—
23	#13108059	—	4	4	no	—
24	C₂×#11717579	—	2	2	no	—
25	C₄⋊Dic₁₀₁	D₄	1	2	yes	—
26	C₂²⋊Dic₁₀₁	—	2	2	no	—
27	#13108051	—	2	2	no	—
28	C₂×D₄×C₁₀₁	—	2	2	no	—
29	C₂×D₄×C₁₀₁	—	2	1	no	—
30	C₂×D₄×C₁₀₁	—	2	2	no	—
31	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
32	D₈×C₁₀₁	—	4	4	no	—
33	C₂×C₈×C₁₀₁	—	2	1	no	—
34	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
35	C₂³×C₁₀₁	—	2	2	no	—
36	C₂³×C₁₀₁	—	4	1	no	—
37	C₂²×D₁₀₁	—	2	2	no	—
38	C₂²×D₁₀₁	—	4	8	no	—
39	C₂²×D₁₀₁	—	8	2	no	—
40	D₄×C₁₀₁	—	4	2	no	—
41	D₄×C₁₀₁	—	4	2	no	—
42	D₄×C₁₀₁	—	4	2	no	—
43	D₄×C₁₀₁	—	4	2	no	—
44	D₄×C₁₀₁	—	4	2	no	—
45	D₄×C₁₀₁	—	4	2	no	—
46	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₁₀₁	—	4	8	no	—
49	C₂×Dic₁₀₁	—	2	2	no	—
50	D₄₀₄	—	2	2	no	—
51	D₄₀₄	D₈	1	4	yes	complemented
52	C₄×D₁₀₁	—	4	2	no	—
53	C₄×D₁₀₁	—	4	2	no	—
54	C₈×C₁₀₁	—	2	2	no	—
55	#11717579	—	4	2	no	—
56	C₂²×C₁₀₁	—	4	4	no	—
57	C₂²×C₁₀₁	—	4	2	no	—
58	C₂²×C₁₀₁	—	8	2	no	—
59	C₂²×C₁₀₁	—	8	1	no	—
60	C₂²×C₁₀₁	#120	1	1	yes	characteristic
61	C₂×D₁₀₁	—	2	4	no	—
62	C₂×D₁₀₁	—	4	2	no	—
63	C₂×D₁₀₁	—	4	2	no	—
64	C₂×D₁₀₁	—	8	8	no	—
65	C₄×C₁₀₁	—	2	1	no	—
66	C₄×C₁₀₁	C₂×D₈	1	2	yes	—
67	C₄×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
68	Dic₁₀₁	—	4	2	no	—
69	C₂×C₁₀₁	—	8	2	no	—
70	C₂×C₁₀₁	—	16	1	no	—
71	C₂×C₁₀₁	#447	1	2	yes	—
72	C₂×C₁₀₁	#453	1	1	yes	characteristic
73	D₁₀₁	—	4	4	no	—
74	#1619	—	101	1	no	maximal, Sylow
75	C₁₀₁	#1619	1	1	yes	characteristic, complemented, Sylow
76	D₄²	—	101	1	no	—
77	#493	D₁₀₁	1	1	yes	characteristic, complemented
78	#459	—	101	2	no	—
79	#331	—	101	1	no	—
80	D₄⋊C₈	—	101	2	no	—
81	C₂²×D₄	—	202	2	no	—
82	C₂×D₈	—	2	2	no	—
83	C₂×D₈	—	202	2	no	—
84	#127	C₂×D₁₀₁	1	1	yes	characteristic
85	#127	C₂×D₁₀₁	1	1	yes	characteristic, complemented
86	#121	—	202	2	no	—
87	#120	—	202	2	no	—
88	C₄×D₄	—	101	2	no	—
89	C₄⋊C₈	—	101	2	no	—
90	#102	—	202	2	no	—
91	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
92	C₂⁴	—	202	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	101	2	no	—
97	C₂×D₄	—	404	4	no	—
98	C₂×D₄	—	404	4	no	—
99	C₂×D₄	—	202	4	no	—
100	C₂×D₄	C₂²⋊D₁₀₁	1	2	yes	—
101	C₂²×C₄	—	202	2	no	—
102	D₈	—	4	4	no	—
103	D₈	—	404	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	202	2	no	—
106	C₄⋊C₄	—	101	2	no	—
107	C₂²⋊C₄	—	202	2	no	—
108	C₂²⋊C₄	—	202	2	no	—
109	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	808	2	no	—
113	C₂³	—	404	8	no	—
114	C₂³	—	202	2	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	101	4	no	—
122	D₄	—	404	8	no	—
123	D₄	—	202	2	no	—
124	C₂×C₄	—	404	2	no	—
125	C₂×C₄	—	404	2	no	—
126	C₂×C₄	—	202	2	no	—
127	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
128	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	404	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	808	8	no	—
136	C₂²	—	404	2	no	—
137	C₂²	—	404	2	no	—
138	C₂²	—	202	4	no	—
139	C₂²	#15692223	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	404	2	no	—
142	C₄	D₈×D₁₀₁	1	2	yes	—
143	C₄	#15692279	1	2	yes	—
144	C₂	—	8	2	no	—
145	C₂	—	16	1	no	—
146	C₂	—	404	4	no	—
147	C₂	#20265309	1	2	yes	—
148	C₂	#20265344	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265812

C₂

yes

maximal, characteristic, complemented

#20265369

C₂

yes

maximal, complemented

#20264921

C₂

yes

maximal, characteristic, complemented

#20264906

C₂

yes

maximal, complemented

#493×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₁₀₁

—

C₂×#13108059

—

#15692226

—

#15692223

—

C₄×D₄₀₄

C₂²

yes

—

C₄⋊#11717579

C₂²

yes

complemented

#15692176

—

C₂×D₈×C₁₀₁

—

#127×C₁₀₁

C₂²

yes

characteristic

#127×C₁₀₁

C₂²

yes

characteristic, complemented

C₄×C₈×C₁₀₁

C₂²

yes

characteristic, complemented

C₂³×D₁₀₁

—

C₂×D₄₀₄

D₄

yes

—

C₂×C₂²⋊D₁₀₁

—

D₄×D₁₀₁

—

D₄×D₁₀₁

—

C₂×C₄×D₁₀₁

—

#13108059

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

D₄

yes

—

C₂²⋊Dic₁₀₁

—

#13108051

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

D₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

D₄₀₄

—

D₄₀₄

D₈

yes

complemented

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#120

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×D₈

yes

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#447

yes

—

C₂×C₁₀₁

#453

yes

characteristic

D₁₀₁

—

#1619

—

101

maximal, Sylow

C₁₀₁

#1619

yes

characteristic, complemented, Sylow

D₄²

—

101

—

#493

D₁₀₁

yes

characteristic, complemented

#459

—

101

—

#331

—

101

—

D₄⋊C₈

—

101

—

C₂²×D₄

—

202

—

C₂×D₈

—

C₂×D₈

—

202

—

#127

C₂×D₁₀₁

yes

characteristic

#127

C₂×D₁₀₁

yes

characteristic, complemented

#121

—

202

—

#120

—

202

—

C₄×D₄

—

101

—

C₄⋊C₈

—

101

—

#102

—

202

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic, complemented

C₂⁴

—

202

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

101

—

C₂×D₄

—

404

—

C₂×D₄

—

404

—

C₂×D₄

—

202

—

100

C₂×D₄

C₂²⋊D₁₀₁

yes

—

101

C₂²×C₄

—

202

—

102

D₈

—

103

D₈

—

404

—

104

C₂×C₈

—

105

C₂×C₈

—

202

—

106

C₄⋊C₄

—

101

—

107

C₂²⋊C₄

—

202

—

108

C₂²⋊C₄

—

202

—

109

C₄²

C₂²×D₁₀₁

yes

characteristic

110

C₂³

—

111

C₂³

—

112

C₂³

—

808

—

113

C₂³

—

404

—

114

C₂³

—

202

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

101

—

122

D₄

—

404

—

123

D₄

—

202

—

124

C₂×C₄

—

404

—

125

C₂×C₄

—

404

—

126

C₂×C₄

—

202

—

127

C₂×C₄

D₄×D₁₀₁

yes

—

128

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

129

C₈

—

130

C₈

—

404

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

808

—

136

C₂²

—

404

—

137

C₂²

—

404

—

138

C₂²

—

202

—

139

C₂²

#15692223

yes

characteristic

140

C₄

—

141

C₄

—

404

—

142

C₄

D₈×D₁₀₁

yes

—

143

C₄

#15692279

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

404

—

147

C₂

#20265309

yes

—

148

C₂

#20265344

yes

characteristic

	subgroup	G/H
center	C₂²	#15692223
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#493×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
hypercenter	#493	D₁₀₁
hypocenter	C₁₀₁	#1619
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#15692223

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#493×C₁₀₁

C₂

C₂²×C₁₀₁

D₁₀₁

C₁₀₁

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #20265812	C₂	semidirect
2. #20265369	C₂	semidirect
3. #20264921	C₂	semidirect
4. #20264906	C₂	semidirect
5. #493×C₁₀₁	C₂	semidirect
11. C₄⋊#11717579	C₂²	semidirect
15. #127×C₁₀₁	C₂²	semidirect
16. C₄×C₈×C₁₀₁	C₂²	semidirect
51. D₄₀₄	D₈	semidirect
75. C₁₀₁	#1619	semidirect
77. #493	D₁₀₁	semidirect
85. #127	C₂×D₁₀₁	semidirect
91. C₄×C₈	C₂×D₁₀₁	semidirect

H ≅ G/N

type

1. #20265812

C₂

semidirect

2. #20265369

C₂

semidirect

3. #20264921

C₂

semidirect

4. #20264906

C₂

semidirect

5. #493×C₁₀₁

C₂

semidirect

11. C₄⋊#11717579

C₂²

semidirect

15. #127×C₁₀₁

C₂²

semidirect

16. C₄×C₈×C₁₀₁

C₂²

semidirect

51. D₄₀₄

D₈

semidirect

75. C₁₀₁

#1619

semidirect

77. #493

D₁₀₁

semidirect

85. #127

C₂×D₁₀₁

semidirect

91. C₄×C₈

C₂×D₁₀₁

semidirect

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

subgroup	index
#20265812	2
#20265369	2
#20264921	2
#20264906	2
#493×C₁₀₁	2
#1619	101

subgroup

index

#493×C₁₀₁

101

Subgroups — #27170184

Conjugacy classes of subgroups

Complements

Maximal subgroups