Group #27171223 — subgroups

Subgroups — #27171223

Order 12928.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#20265037	C₂	1	1	yes	maximal, characteristic
3	#20264940	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₁	—	2	1	no	—
6	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
7	C₂×C₄⋊Dic₁₀₁	—	2	2	no	—
8	C₂×C₄⋊Dic₁₀₁	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₁₀₁	—	2	1	no	—
12	#15692162	—	2	1	no	—
13	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
16	C₂²×Dic₁₀₁	—	2	4	no	—
17	C₂²×Dic₁₀₁	—	2	2	no	—
18	C₂²×Dic₁₀₁	—	2	2	no	—
19	C₂²×Dic₁₀₁	—	4	4	no	—
20	C₄⋊Dic₁₀₁	—	2	4	no	—
21	C₄⋊Dic₁₀₁	—	2	2	no	—
22	C₄⋊Dic₁₀₁	—	2	4	no	—
23	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
24	C₂²⋊Dic₁₀₁	—	4	2	no	—
25	C₂⁴×C₁₀₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₁	—	2	4	no	—
27	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₀₁	—	2	2	no	—
32	C₂×C₈×C₁₀₁	—	2	4	no	—
33	C₂³×C₁₀₁	—	2	2	no	—
34	C₂³×C₁₀₁	—	2	2	no	—
35	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₁₀₁	SD₁₆	1	2	yes	—
37	C₂³×C₁₀₁	Q₁₆	1	2	yes	—
38	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	—	2	2	no	—
40	C₂×C₄×C₁₀₁	—	2	4	no	—
41	C₂×C₄×C₁₀₁	—	2	4	no	—
42	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	#41	1	4	yes	—
46	C₂×Dic₁₀₁	—	2	4	no	—
47	C₂×Dic₁₀₁	—	4	8	no	—
48	C₂×Dic₁₀₁	—	4	4	no	—
49	C₂×Dic₁₀₁	—	4	2	no	—
50	C₂×Dic₁₀₁	—	4	2	no	—
51	C₂×Dic₁₀₁	—	4	2	no	—
52	C₈×C₁₀₁	—	4	2	no	—
53	C₂²×C₁₀₁	—	2	2	no	—
54	C₂²×C₁₀₁	—	2	2	no	—
55	C₂²×C₁₀₁	—	2	4	no	—
56	C₂²×C₁₀₁	—	2	4	no	—
57	C₂²×C₁₀₁	#103	1	4	yes	—
58	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₁	#123	1	2	yes	—
60	C₂²×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₀₁	—	2	2	no	—
64	C₄×C₁₀₁	#117	1	2	yes	—
65	C₄×C₁₀₁	#123	1	2	yes	—
66	Dic₁₀₁	—	4	4	no	—
67	Dic₁₀₁	—	8	2	no	—
68	C₂×C₁₀₁	—	2	4	no	—
69	C₂×C₁₀₁	#381	1	1	yes	characteristic
70	C₂×C₁₀₁	C₂×#103	1	1	yes	characteristic
71	C₂×C₁₀₁	#418	1	1	yes	characteristic
72	C₂×C₁₀₁	#481	1	2	yes	—
73	C₂×C₁₀₁	#484	1	2	yes	—
74	#1824	—	101	1	no	maximal, Sylow
75	C₁₀₁	#1824	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₄⋊C₄	—	101	1	no	—
77	C₂×C₂²⋊C₈	D₁₀₁	1	1	yes	characteristic
78	#380	—	101	1	no	—
79	#340	—	101	4	no	—
80	C₂³×C₄	—	202	1	no	—
81	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
83	C₂×C₄⋊C₄	—	101	4	no	—
84	C₂×C₄⋊C₄	—	101	2	no	—
85	C₂×C₄⋊C₄	—	101	2	no	—
86	C₂×C₄⋊C₄	—	202	1	no	—
87	C₂×C₄⋊C₄	—	202	2	no	—
88	C₂×C₂²⋊C₄	—	202	1	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	#95	—	202	1	no	—
91	C₂⁴	D₄₀₄	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	404	4	no	—
94	C₂²×C₄	—	202	2	no	—
95	C₂²×C₄	—	202	4	no	—
96	C₂²×C₄	—	202	2	no	—
97	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
98	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
100	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₄⋊C₄	—	101	4	no	—
104	C₄⋊C₄	—	202	4	no	—
105	C₄⋊C₄	—	202	4	no	—
106	C₄⋊C₄	—	202	2	no	—
107	C₂²⋊C₄	—	404	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	#13108051	1	2	yes	—
111	C₂³	#13108063	1	2	yes	—
112	C₂³	Dic₄₀₄	1	2	yes	—
113	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	404	2	no	—
118	C₂×C₄	—	404	2	no	—
119	C₂×C₄	—	404	2	no	—
120	C₂×C₄	—	404	8	no	—
121	C₂×C₄	—	404	4	no	—
122	C₂×C₄	—	202	4	no	—
123	C₂×C₄	#13108051	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
126	C₂×C₄	#13108073	1	4	yes	—
127	C₈	—	4	2	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#15692180	1	4	yes	—
133	C₂²	C₂×#13108051	1	1	yes	characteristic
134	C₂²	#15692239	1	2	yes	—
135	C₂²	C₂×#13108063	1	1	yes	characteristic
136	C₂²	C₂×Dic₄₀₄	1	1	yes	characteristic
137	C₂²	#15692283	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	808	2	no	—
140	C₄	—	404	4	no	—
141	C₄	#15692211	1	2	yes	—
142	C₄	#15692236	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#20265038	1	1	yes	characteristic
145	C₂	C₂×#15692180	1	1	yes	characteristic
146	C₂	#20265183	1	1	yes	characteristic
147	C₂	#20265496	1	2	yes	—
148	C₂	#20265513	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₁₀₁

C₂

yes

maximal, characteristic

#20265037

C₂

yes

maximal, characteristic

#20264940

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂³×Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

—

#15692162

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂²⋊C₈×C₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

C₂²⋊Dic₁₀₁

—

C₂⁴×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

SD₁₆

yes

—

C₂³×C₁₀₁

Q₁₆

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#103

yes

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#123

yes

—

C₂²×C₁₀₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₁

C₂×Q₁₆

yes

characteristic

C₂²×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#117

yes

—

C₄×C₁₀₁

#123

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#484

yes

—

#1824

—

101

maximal, Sylow

C₁₀₁

#1824

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₈

D₁₀₁

yes

characteristic

#380

—

101

—

#340

—

101

—

C₂³×C₄

—

202

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

C₂×D₁₀₁

yes

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

—

202

—

C₂×C₂²⋊C₄

—

202

—

C₂²⋊C₈

—

#95

—

202

—

C₂⁴

D₄₀₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

404

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

C₄×D₁₀₁

yes

—

C₂²×C₄

D₄₀₄

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

100

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

101

—

104

C₄⋊C₄

—

202

—

105

C₄⋊C₄

—

202

—

106

C₄⋊C₄

—

202

—

107

C₂²⋊C₄

—

404

—

108

C₂³

—

109

C₂³

—

110

C₂³

#13108051

yes

—

111

C₂³

#13108063

yes

—

112

C₂³

Dic₄₀₄

yes

—

113

C₂³

C₂×D₄₀₄

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

404

—

118

C₂×C₄

—

404

—

119

C₂×C₄

—

404

—

120

C₂×C₄

—

404

—

121

C₂×C₄

—

404

—

122

C₂×C₄

—

202

—

123

C₂×C₄

#13108051

yes

—

124

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

126

C₂×C₄

#13108073

yes

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#15692180

yes

—

133

C₂²

C₂×#13108051

yes

characteristic

134

C₂²

#15692239

yes

—

135

C₂²

C₂×#13108063

yes

characteristic

136

C₂²

C₂×Dic₄₀₄

yes

characteristic

137

C₂²

#15692283

yes

—

138

C₄

—

139

C₄

—

808

—

140

C₄

—

404

—

141

C₄

#15692211

yes

—

142

C₄

#15692236

yes

—

143

C₂

—

144

C₂

#20265038

yes

characteristic

145

C₂

C₂×#15692180

yes

characteristic

146

C₂

#20265183

yes

characteristic

147

C₂

#20265496

yes

—

148

C₂

#20265513

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₀₄
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×C₂²⋊C₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
prime radical	C₂⁴×C₁₀₁	D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₀₁
hypocenter	C₁₀₁	#1824
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₄₀₄

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×C₂²⋊C₈×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

prime radical

C₂⁴×C₁₀₁

D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₀₁

hypocenter

C₁₀₁

#1824

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #20264940	C₂	semidirect
8. C₂×C₄⋊Dic₁₀₁	C₄	semidirect
75. C₁₀₁	#1824	semidirect

H ≅ G/N

type

3. #20264940

C₂

semidirect

8. C₂×C₄⋊Dic₁₀₁

C₄

semidirect

75. C₁₀₁

#1824

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₁₀₁	2
#20265037	2
#20264940	2
C₂×C₂²⋊C₈×C₁₀₁	2
#1824	101

subgroup

index

C₂²×C₄⋊Dic₁₀₁

#20265037

#20264940

C₂×C₂²⋊C₈×C₁₀₁

#1824

101

Subgroups — #27171223

Conjugacy classes of subgroups

Complements

Maximal subgroups