Group #27170500 — subgroups

Subgroups — #27170500

Order 12928.

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

normal subgroups only

This group has 156 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	#20265556	C₂	1	1	yes	maximal, characteristic
2	#20265462	C₂	1	1	yes	maximal, characteristic
3	Q₈×#11717579	C₂	1	1	yes	maximal, characteristic
4	#20264929	C₂	1	1	yes	maximal, characteristic
5	#20264923	C₂	1	1	yes	maximal, characteristic
6	#20264922	C₂	1	1	yes	maximal, characteristic
7	#475×C₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#13108064	—	2	1	no	—
9	C₂×#13108064	—	2	1	no	—
10	#15692255	C₂²	1	1	yes	characteristic
11	#15692186	—	2	1	no	—
12	C₄⋊#11717579	—	2	1	no	—
13	C₄⋊#11717579	C₂²	1	1	yes	characteristic
14	C₄⋊#11717579	C₂²	1	1	yes	characteristic
15	#15692179	—	2	1	no	—
16	C₄×#11717579	—	2	1	no	—
17	C₄×#11717579	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
20	#106×C₁₀₁	—	2	1	no	—
21	C₄⋊C₈×C₁₀₁	C₂²	1	1	yes	characteristic
22	#103×C₁₀₁	—	2	1	no	—
23	C₂×Dic₂₀₂	—	2	1	no	—
24	#13108064	—	4	2	no	—
25	#13108064	—	4	2	no	—
26	C₂×#11717579	—	2	1	no	—
27	C₂×#11717579	—	2	1	no	—
28	C₂×#11717579	D₄	1	1	yes	characteristic
29	C₂×#11717579	D₄	1	1	yes	characteristic
30	C₄⋊Dic₁₀₁	—	2	1	no	—
31	C₄⋊Dic₁₀₁	—	2	1	no	—
32	C₂×Q₈×C₁₀₁	—	2	1	no	—
33	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₁	—	2	1	no	—
35	C₄⋊C₄×C₁₀₁	—	2	1	no	—
36	C₄⋊C₄×C₁₀₁	—	2	1	no	—
37	C₄⋊C₄×C₁₀₁	—	2	1	no	—
38	C₄⋊C₄×C₁₀₁	D₄	1	1	yes	characteristic
39	C₄²×C₁₀₁	—	2	1	no	—
40	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
41	Q₈×C₁₀₁	—	2	1	no	—
42	Q₈×C₁₀₁	—	4	1	no	—
43	Q₈×C₁₀₁	—	4	1	no	—
44	Q₈×C₁₀₁	Q₁₆	1	2	yes	complemented
45	C₂×C₄×C₁₀₁	—	2	1	no	—
46	C₂×C₄×C₁₀₁	—	2	1	no	—
47	C₂×C₄×C₁₀₁	—	4	1	no	—
48	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
51	C₂×Dic₁₀₁	—	4	1	no	—
52	Dic₂₀₂	—	4	1	no	—
53	Dic₂₀₂	—	4	1	no	—
54	C₈×C₁₀₁	—	4	1	no	—
55	#11717579	—	2	1	no	—
56	#11717579	—	4	1	no	—
57	#11717579	—	4	1	no	—
58	#11717579	SD₁₆	1	2	yes	—
59	C₂²×C₁₀₁	#121	1	1	yes	characteristic
60	C₄×C₁₀₁	—	2	2	no	—
61	C₄×C₁₀₁	—	2	1	no	—
62	C₄×C₁₀₁	—	4	1	no	—
63	C₄×C₁₀₁	—	8	1	no	—
64	C₄×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
66	C₄×C₁₀₁	#135	1	1	yes	characteristic
67	C₄×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	Dic₁₀₁	—	8	1	no	—
69	C₂×C₁₀₁	#460	1	1	yes	characteristic
70	C₂×C₁₀₁	#467	1	1	yes	characteristic
71	C₂×C₁₀₁	#473	1	1	yes	characteristic
72	#1670	—	101	1	no	maximal, Sylow
73	C₁₀₁	#1670	1	1	yes	characteristic, complemented, Sylow
74	#494	—	101	1	no	—
75	#475	—	101	1	no	—
76	#475	D₁₀₁	1	1	yes	characteristic
77	C₈×Q₈	—	101	1	no	—
78	#335	—	101	1	no	—
79	#332	—	101	1	no	—
80	#332	—	101	1	no	—
81	C₂×Q₁₆	—	202	1	no	—
82	C₂×Q₁₆	—	202	1	no	—
83	C₄⋊Q₈	—	101	1	no	—
84	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
85	C₄×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
86	#106	—	2	1	no	—
87	#106	—	202	1	no	—
88	C₄⋊C₈	—	101	1	no	—
89	C₄⋊C₈	—	101	1	no	—
90	C₄⋊C₈	—	202	1	no	—
91	C₄⋊C₈	C₂×D₁₀₁	1	1	yes	characteristic
92	#103	—	2	1	no	—
93	#103	—	202	1	no	—
94	C₄×C₈	—	101	1	no	—
95	C₄×C₈	—	202	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	202	1	no	—
98	C₂×Q₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
99	Q₁₆	—	404	2	no	—
100	Q₁₆	—	404	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	101	1	no	—
103	C₂×C₈	—	101	1	no	—
104	C₂×C₈	—	202	1	no	—
105	C₂×C₈	—	202	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	202	1	no	—
110	C₄⋊C₄	—	202	1	no	—
111	C₄⋊C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	404	1	no	—
118	Q₈	—	404	1	no	—
119	Q₈	#13108064	1	2	yes	complemented
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	404	1	no	—
124	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
126	C₂×C₄	#13108073	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	101	2	no	—
129	C₈	—	404	1	no	—
130	C₈	—	404	1	no	—
131	C₈	—	202	1	no	—
132	C₂²	#15692227	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	808	1	no	—
138	C₄	SD₁₆×D₁₀₁	1	1	yes	characteristic
139	C₄	C₂×#13108064	1	1	yes	characteristic
140	C₄	#15692275	1	1	yes	characteristic
141	C₄	#15692282	1	1	yes	characteristic
142	C₂	#20265375	1	1	yes	characteristic
143	C₂	#20265416	1	1	yes	characteristic
144	C₂	#20265450	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265556

C₂

yes

maximal, characteristic

#20265462

C₂

yes

maximal, characteristic

Q₈×#11717579

C₂

yes

maximal, characteristic

#20264929

C₂

yes

maximal, characteristic

#20264923

C₂

yes

maximal, characteristic

#20264922

C₂

yes

maximal, characteristic

#475×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108064

—

C₂×#13108064

—

#15692255

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#11717579

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₁

C₂²

yes

characteristic

C₄×Q₈×C₁₀₁

C₂²

yes

characteristic

#106×C₁₀₁

—

C₄⋊C₈×C₁₀₁

C₂²

yes

characteristic

#103×C₁₀₁

—

C₂×Dic₂₀₂

—

—

—

—

—

D₄

yes

characteristic

C₂×#11717579

D₄

yes

characteristic

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

characteristic

C₂×C₈×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

characteristic

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

Q₁₆

yes

complemented

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

—

—

—

—

SD₁₆

yes

—

C₂²×C₁₀₁

#121

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₀₁

C₂×Q₁₆

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

Dic₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1670

yes

characteristic, complemented, Sylow

#494

—

101

—

#475

—

101

—

#475

D₁₀₁

yes

characteristic

C₈×Q₈

—

101

—

#335

—

101

—

#332

—

101

—

#332

—

101

—

C₂×Q₁₆

—

202

—

C₂×Q₁₆

—

202

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

C₂×D₁₀₁

yes

characteristic

C₄×Q₈

C₂×D₁₀₁

yes

characteristic

#106

—

#106

—

202

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

202

—

C₄⋊C₈

C₂×D₁₀₁

yes

characteristic

#103

—

#103

—

202

—

C₄×C₈

—

101

—

C₄×C₈

—

202

—

C₂×Q₈

—

C₂×Q₈

—

202

—

C₂×Q₈

C₂²⋊D₁₀₁

yes

characteristic

Q₁₆

—

404

—

100

Q₁₆

—

404

—

101

C₂×C₈

—

102

C₂×C₈

—

101

—

103

C₂×C₈

—

101

—

104

C₂×C₈

—

202

—

105

C₂×C₈

—

202

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

202

—

110

C₄⋊C₄

—

202

—

111

C₄⋊C₄

C₂²⋊D₁₀₁

yes

characteristic

112

C₄²

—

113

C₄²

C₂²×D₁₀₁

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

404

—

118

Q₈

—

404

—

119

Q₈

#13108064

yes

complemented

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

404

—

124

C₂×C₄

D₄×D₁₀₁

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

126

C₂×C₄

#13108073

yes

characteristic

127

C₈

—

128

C₈

—

101

—

129

C₈

—

404

—

130

C₈

—

404

—

131

C₈

—

202

—

132

C₂²

#15692227

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

808

—

138

C₄

SD₁₆×D₁₀₁

yes

characteristic

139

C₄

C₂×#13108064

yes

characteristic

140

C₄

#15692275

yes

characteristic

141

C₄

#15692282

yes

characteristic

142

C₂

#20265375

yes

characteristic

143

C₂

#20265416

yes

characteristic

144

C₂

#20265450

yes

characteristic

	subgroup	G/H
center	C₂²	#15692227
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#475×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#121
hypercenter	#475	D₁₀₁
hypocenter	C₁₀₁	#1670
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#15692227

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#475×C₁₀₁

C₂

C₂²×C₁₀₁

D₁₀₁

C₁₀₁

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
44. Q₈×C₁₀₁	Q₁₆	semidirect
73. C₁₀₁	#1670	semidirect
119. Q₈	#13108064	semidirect

H ≅ G/N

type

44. Q₈×C₁₀₁

Q₁₆

semidirect

73. C₁₀₁

#1670

semidirect

119. Q₈

#13108064

semidirect

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

subgroup	index
#20265556	2
#20265462	2
Q₈×#11717579	2
#20264929	2
#20264923	2
#20264922	2
#475×C₁₀₁	2
#1670	101

subgroup

index

#475×C₁₀₁

101

Subgroups — #27170500

Conjugacy classes of subgroups

Complements

Maximal subgroups