Group #27171347 — subgroups

Subgroups — #27171347

Order 12928.

272 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20265220	C₂	1	2	yes	maximal
2	C₂×#15692174	C₂	1	1	yes	maximal, characteristic
3	#20265084	C₂	1	1	yes	maximal, characteristic
4	#20264936	C₂	1	2	yes	maximal
5	#519×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
9	#15692188	—	2	2	no	—
10	#15692186	—	2	2	no	—
11	#15692174	C₄	1	4	yes	complemented
12	#15692162	—	2	2	no	—
13	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₀₁	—	2	2	no	—
15	C₄×Q₈×C₁₀₁	—	2	2	no	—
16	#117×C₁₀₁	—	2	2	no	—
17	#117×C₁₀₁	C₂²	1	2	yes	—
18	C₂²×Dic₁₀₁	—	2	1	no	—
19	C₂²×Dic₁₀₁	—	2	1	no	—
20	#13108058	—	2	2	no	—
21	#13108058	C₂×C₄	1	4	yes	—
22	C₂×#11717579	—	2	2	no	—
23	C₄⋊Dic₁₀₁	—	2	2	no	—
24	C₄×Dic₁₀₁	—	2	2	no	—
25	C₂×Q₈×C₁₀₁	—	2	2	no	—
26	C₂×Q₈×C₁₀₁	—	2	2	no	—
27	C₂×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
28	C₂×Q₈×C₁₀₁	Q₈	1	2	yes	—
29	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₁	—	2	2	no	—
32	C₄⋊C₄×C₁₀₁	—	2	2	no	—
33	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
34	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
35	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
36	C₄²×C₁₀₁	—	2	4	no	—
37	C₄²×C₁₀₁	—	2	2	no	—
38	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₀₁	—	2	4	no	—
40	Q₈×C₁₀₁	—	2	4	no	—
41	C₂×C₄×C₁₀₁	—	2	2	no	—
42	C₂×C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄×C₁₀₁	—	2	2	no	—
44	C₂×C₄×C₁₀₁	—	2	2	no	—
45	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
46	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₀₁	#41	1	2	yes	—
50	C₂×Dic₁₀₁	—	2	2	no	—
51	C₂×Dic₁₀₁	—	4	1	no	—
52	C₂×Dic₁₀₁	—	4	1	no	—
53	C₂×Dic₁₀₁	—	4	2	no	—
54	C₂×Dic₁₀₁	—	4	1	no	—
55	#11717579	—	2	4	no	—
56	C₂²×C₁₀₁	—	2	1	no	—
57	C₂²×C₁₀₁	—	2	1	no	—
58	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₁	#120	1	1	yes	characteristic
60	C₂²×C₁₀₁	#123	1	1	yes	characteristic
61	C₄×C₁₀₁	—	2	4	no	—
62	C₄×C₁₀₁	—	4	2	no	—
63	C₄×C₁₀₁	—	4	2	no	—
64	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
65	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
66	Dic₁₀₁	—	4	2	no	—
67	Dic₁₀₁	—	8	1	no	—
68	C₂×C₁₀₁	—	2	1	no	—
69	C₂×C₁₀₁	—	2	1	no	—
70	C₂×C₁₀₁	#385	1	1	yes	characteristic
71	C₂×C₁₀₁	#454	1	1	yes	characteristic
72	C₂×C₁₀₁	#456	1	1	yes	characteristic
73	#1857	—	101	1	no	maximal, Sylow
74	C₁₀₁	#1857	1	1	yes	characteristic, complemented, Sylow
75	#519	D₁₀₁	1	1	yes	characteristic
76	#428	—	101	2	no	—
77	C₂×#101	—	101	1	no	—
78	#391	—	101	1	no	—
79	#339	—	101	2	no	—
80	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
81	C₂×M₁₆	—	101	2	no	—
82	C₄×Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	—	2	2	no	—
85	#117	C₂×D₁₀₁	1	2	yes	—
86	C₂×C₄⋊C₄	—	101	1	no	—
87	C₂×C₄²	—	101	1	no	—
88	#107	—	202	2	no	—
89	#106	—	202	2	no	—
90	#101	—	101	4	no	—
91	#95	—	202	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	Dic₂₀₂	1	2	yes	—
95	C₂×Q₈	C₄×D₁₀₁	1	2	yes	—
96	C₂²×C₄	—	202	1	no	—
97	C₂²×C₄	—	202	1	no	—
98	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
100	M₁₆	—	101	4	no	—
101	M₁₆	—	202	2	no	—
102	C₂×C₈	—	202	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	202	2	no	—
106	C₄⋊C₄	C₂²⋊D₁₀₁	1	2	yes	—
107	C₄⋊C₄	C₂²⋊D₁₀₁	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	4	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	202	2	no	—
112	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	404	2	no	—
120	C₂×C₄	—	404	1	no	—
121	C₂×C₄	—	404	1	no	—
122	C₂×C₄	—	404	1	no	—
123	C₂×C₄	—	202	2	no	—
124	C₂×C₄	#13108053	1	4	yes	—
125	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
127	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
128	C₂×C₄	#13108074	1	2	yes	—
129	C₈	—	202	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×#13108053	1	1	yes	characteristic
133	C₂²	#15692224	1	1	yes	characteristic
134	C₂²	#15692238	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	808	1	no	—
139	C₄	—	404	2	no	—
140	C₄	C₄×C₂²⋊D₁₀₁	1	2	yes	—
141	C₄	#15692234	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#20265050	1	1	yes	characteristic
145	C₂	#20265350	1	1	yes	characteristic
146	C₂	#20265360	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265220

C₂

yes

maximal

C₂×#15692174

C₂

yes

maximal, characteristic

#20265084

C₂

yes

maximal, characteristic

#20264936

C₂

yes

maximal

#519×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₁

C₂²

yes

—

—

C₄

yes

complemented

#15692162

—

C₂²×Q₈×C₁₀₁

C₂²

yes

characteristic

C₄×Q₈×C₁₀₁

—

C₄×Q₈×C₁₀₁

—

#117×C₁₀₁

—

#117×C₁₀₁

C₂²

yes

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

#13108058

—

#13108058

C₂×C₄

yes

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₂×Q₈×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

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

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1857

yes

characteristic, complemented, Sylow

#519

D₁₀₁

yes

characteristic

#428

—

101

—

C₂×#101

—

101

—

#391

—

101

—

#339

—

101

—

C₂²×Q₈

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₁₀₁

yes

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄²

—

101

—

#107

—

202

—

#106

—

202

—

#101

—

101

—

#95

—

202

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₂₀₂

yes

—

C₂×Q₈

C₄×D₁₀₁

yes

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

100

M₁₆

—

101

—

101

M₁₆

—

202

—

102

C₂×C₈

—

202

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

202

—

106

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

107

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

202

—

112

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

404

—

120

C₂×C₄

—

404

—

121

C₂×C₄

—

404

—

122

C₂×C₄

—

404

—

123

C₂×C₄

—

202

—

124

C₂×C₄

#13108053

yes

—

125

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

127

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

128

C₂×C₄

#13108074

yes

—

129

C₈

—

202

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#13108053

yes

characteristic

133

C₂²

#15692224

yes

characteristic

134

C₂²

#15692238

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

808

—

139

C₄

—

404

—

140

C₄

C₄×C₂²⋊D₁₀₁

yes

—

141

C₄

#15692234

yes

—

142

C₂

—

143

C₂

—

144

C₂

#20265050

yes

characteristic

145

C₂

#20265350

yes

characteristic

146

C₂

#20265360

yes

characteristic

	subgroup	G/H
center	C₂²	#15692224
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#519×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#519	D₁₀₁
hypocenter	C₁₀₁	#1857
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #519

subgroup

G/H

center

C₂²

#15692224

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#519×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#120

prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #519

N	H ≅ G/N	type
11. #15692174	C₄	semidirect
74. C₁₀₁	#1857	semidirect

H ≅ G/N

type

11. #15692174

C₄

semidirect

74. C₁₀₁

#1857

semidirect

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

subgroup	index
#20265220	2
C₂×#15692174	2
#20265084	2
#20264936	2
#519×C₁₀₁	2
#1857	101

subgroup

index

#519×C₁₀₁

101

Subgroups — #27171347

Conjugacy classes of subgroups

Complements

Maximal subgroups