Group #27171127 — subgroups

Subgroups — #27171127

Order 12928.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15692220	C₂	1	1	yes	maximal, characteristic
2	#20265200	C₂	1	2	yes	maximal
3	C₂×#15692173	C₂	1	2	yes	maximal
4	#20264946	C₂	1	1	yes	maximal, characteristic
5	#343×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
7	C₂×#13108058	C₂²	1	2	yes	—
8	#15692220	—	2	2	no	—
9	#15692220	—	2	2	no	—
10	C₂×#13108053	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₀₁	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	#15692184	—	2	2	no	—
14	#15692184	—	2	2	no	—
15	#15692173	—	2	4	no	—
16	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₀₂	—	2	2	no	—
19	C₂×Dic₂₀₂	—	2	2	no	—
20	C₂×Dic₂₀₂	D₄	1	2	yes	—
21	C₂×Dic₂₀₂	Q₈	1	2	yes	—
22	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
25	#13108058	—	2	2	no	—
26	#13108058	—	2	2	no	—
27	C₂×#11717579	—	2	2	no	—
28	#13108053	—	2	4	no	—
29	C₄×Dic₁₀₁	—	2	2	no	—
30	C₄×Dic₁₀₁	—	2	2	no	—
31	C₄×Dic₁₀₁	D₄	1	2	yes	—
32	C₄×Dic₁₀₁	D₄	1	2	yes	—
33	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₀₁	—	2	2	no	—
36	M₁₆×C₁₀₁	—	2	2	no	—
37	C₂×C₈×C₁₀₁	—	2	2	no	—
38	C₄⋊C₄×C₁₀₁	—	2	2	no	—
39	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	—	2	2	no	—
41	C₂×C₄×C₁₀₁	—	4	2	no	—
42	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	#41	1	2	yes	—
46	C₂×Dic₁₀₁	—	2	2	no	—
47	C₂×Dic₁₀₁	—	2	2	no	—
48	C₂×Dic₁₀₁	—	2	2	no	—
49	C₂×Dic₁₀₁	—	4	1	no	—
50	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
52	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₁₀₁	C₄⋊C₄	1	4	yes	—
54	Dic₂₀₂	—	2	4	no	—
55	Dic₂₀₂	—	2	4	no	—
56	C₈×C₁₀₁	—	4	2	no	—
57	#11717579	—	4	2	no	—
58	C₂²×C₁₀₁	—	2	1	no	—
59	C₂²×C₁₀₁	—	2	1	no	—
60	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₀₁	#117	1	1	yes	characteristic
63	C₄×C₁₀₁	—	4	2	no	—
64	C₄×C₁₀₁	#121	1	2	yes	—
65	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
66	Dic₁₀₁	—	2	4	no	—
67	Dic₁₀₁	—	2	1	no	—
68	Dic₁₀₁	—	2	2	no	—
69	Dic₁₀₁	#101	1	2	yes	complemented
70	C₂×C₁₀₁	—	2	1	no	—
71	C₂×C₁₀₁	—	2	1	no	—
72	C₂×C₁₀₁	#380	1	1	yes	characteristic
73	C₂×C₁₀₁	C₂×#101	1	1	yes	characteristic
74	C₂×C₁₀₁	#413	1	1	yes	characteristic
75	#1801	—	101	1	no	maximal, Sylow
76	C₁₀₁	#1801	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₄×Q₈	—	101	1	no	—
78	#423	—	101	2	no	—
79	C₂×#101	—	101	2	no	—
80	#343	—	101	1	no	—
81	#343	D₁₀₁	1	1	yes	characteristic
82	C₂²×Q₈	—	101	1	no	—
83	C₂×M₁₆	—	101	2	no	—
84	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
85	C₄×Q₈	—	202	2	no	—
86	C₄×Q₈	—	202	2	no	—
87	C₂×C₄⋊C₄	—	101	2	no	—
88	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
89	C₂×C₄²	—	101	1	no	—
90	C₂×C₄²	—	101	2	no	—
91	C₄⋊C₈	—	202	2	no	—
92	C₄⋊C₈	—	202	2	no	—
93	#101	—	202	4	no	—
94	C₂×Q₈	—	101	2	no	—
95	C₂×Q₈	—	101	2	no	—
96	C₂×Q₈	—	202	2	no	—
97	C₂×Q₈	—	202	2	no	—
98	C₂²×C₄	—	101	1	no	—
99	C₂²×C₄	—	101	1	no	—
100	C₂²×C₄	—	101	2	no	—
101	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	202	2	no	—
106	M₁₆	—	202	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	202	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	202	4	no	—
111	C₄²	—	101	2	no	—
112	C₄²	—	101	2	no	—
113	C₄²	—	202	2	no	—
114	C₄²	—	202	2	no	—
115	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
116	Q₈	—	202	4	no	—
117	Q₈	—	202	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	101	1	no	—
121	C₂×C₄	—	101	1	no	—
122	C₂×C₄	—	101	2	no	—
123	C₂×C₄	—	101	4	no	—
124	C₂×C₄	—	404	1	no	—
125	C₂×C₄	—	202	2	no	—
126	C₂×C₄	—	202	2	no	—
127	C₂×C₄	—	202	2	no	—
128	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
129	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
130	C₂×C₄	Q₈×D₁₀₁	1	1	yes	characteristic
131	C₂×C₄	#13108074	1	2	yes	—
132	C₈	—	4	2	no	—
133	C₈	—	404	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
137	C₂²	C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
138	C₂²	#15692212	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	101	2	no	—
141	C₄	—	202	4	no	—
142	C₄	—	202	1	no	—
143	C₄	—	202	2	no	—
144	C₄	#15692228	1	2	yes	—
145	C₄	#15692233	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#20265036	1	1	yes	characteristic
149	C₂	#101×D₁₀₁	1	1	yes	characteristic
150	C₂	#20265158	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692220

C₂

yes

maximal, characteristic

#20265200

C₂

yes

maximal

C₂×#15692173

C₂

yes

maximal

#20264946

C₂

yes

maximal, characteristic

#343×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

characteristic

C₂×#13108058

C₂²

yes

—

#15692220

—

#15692220

—

C₂×#13108053

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

—

—

C₂×M₁₆×C₁₀₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

D₄

yes

—

C₂×Dic₂₀₂

Q₈

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

—

—

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

#117

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

#121

yes

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#101

yes

complemented

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1801

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

101

—

#423

—

101

—

C₂×#101

—

101

—

#343

—

101

—

#343

D₁₀₁

yes

characteristic

C₂²×Q₈

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₄×Q₈

—

202

—

C₄×Q₈

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

characteristic

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

C₄⋊C₈

—

202

—

#101

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

100

C₂²×C₄

—

101

—

101

C₂²×C₄

C₄×D₁₀₁

yes

—

102

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

202

—

106

M₁₆

—

202

—

107

C₂×C₈

—

108

C₂×C₈

—

202

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

202

—

111

C₄²

—

101

—

112

C₄²

—

101

—

113

C₄²

—

202

—

114

C₄²

—

202

—

115

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

116

Q₈

—

202

—

117

Q₈

—

202

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

101

—

121

C₂×C₄

—

101

—

122

C₂×C₄

—

101

—

123

C₂×C₄

—

101

—

124

C₂×C₄

—

404

—

125

C₂×C₄

—

202

—

126

C₂×C₄

—

202

—

127

C₂×C₄

—

202

—

128

C₂×C₄

D₄×D₁₀₁

yes

characteristic

129

C₂×C₄

D₄×D₁₀₁

yes

—

130

C₂×C₄

Q₈×D₁₀₁

yes

characteristic

131

C₂×C₄

#13108074

yes

—

132

C₈

—

133

C₈

—

404

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂²⋊C₄×D₁₀₁

yes

characteristic

137

C₂²

C₄⋊C₄×D₁₀₁

yes

characteristic

138

C₂²

#15692212

yes

characteristic

139

C₄

—

140

C₄

—

101

—

141

C₄

—

202

—

142

C₄

—

202

—

143

C₄

—

202

—

144

C₄

#15692228

yes

—

145

C₄

#15692233

yes

—

146

C₂

—

147

C₂

—

148

C₂

#20265036

yes

characteristic

149

C₂

#101×D₁₀₁

yes

characteristic

150

C₂

#20265158

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₀₁
commutator, prime radical	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#343×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#343	D₁₀₁
hypocenter	C₁₀₁	#1801
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₀₁

commutator, prime radical

C₂³×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#343×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
69. Dic₁₀₁	#101	semidirect
76. C₁₀₁	#1801	semidirect

H ≅ G/N

type

69. Dic₁₀₁

#101

semidirect

76. C₁₀₁

#1801

semidirect

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

subgroup	index
C₂×#15692220	2
#20265200	2
C₂×#15692173	2
#20264946	2
#343×C₁₀₁	2
#1801	101

subgroup

index

#343×C₁₀₁

101

Subgroups — #27171127

Conjugacy classes of subgroups

Complements

Maximal subgroups