Group #27171352 — subgroups

Subgroups — #27171352

Order 12928.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20265225	C₂	1	2	yes	maximal, complemented
2	#20265161	C₂	1	1	yes	maximal, characteristic
3	#20265144	C₂	1	1	yes	maximal, characteristic
4	#20264941	C₂	1	2	yes	maximal, complemented
5	#568×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	1	yes	characteristic
7	C₂×#13108058	C₂²	1	2	yes	—
8	C₂²×#11717579	C₂²	1	1	yes	characteristic
9	#15692190	—	2	4	no	—
10	#15692174	C₄	1	2	yes	—
11	#15692172	C₄	1	2	yes	—
12	C₂²⋊#11717579	—	2	2	no	—
13	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
14	#131×C₁₀₁	—	2	4	no	—
15	C₂×M₁₆×C₁₀₁	—	2	2	no	—
16	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
17	#13108058	—	2	2	no	—
18	#13108058	—	2	2	no	—
19	#13108058	C₂×C₄	1	4	yes	—
20	C₂×#11717579	—	2	2	no	—
21	C₂×#11717579	—	2	1	no	—
22	C₂×#11717579	—	2	1	no	—
23	C₂×#11717579	—	2	2	no	—
24	#41×C₁₀₁	—	2	2	no	—
25	#41×C₁₀₁	—	2	2	no	—
26	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₁₀₁	Q₈	1	2	yes	—
29	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₀₁	—	2	4	no	—
32	M₁₆×C₁₀₁	—	2	2	no	—
33	M₁₆×C₁₀₁	D₄	1	4	yes	—
34	C₂×C₈×C₁₀₁	—	2	4	no	—
35	C₂×C₈×C₁₀₁	—	2	2	no	—
36	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₁₀₁	—	2	2	no	—
39	D₄×C₁₀₁	—	2	4	no	—
40	D₄×C₁₀₁	—	2	2	no	—
41	C₂×C₄×C₁₀₁	—	2	2	no	—
42	C₂×C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	#41	1	2	yes	—
48	C₈×C₁₀₁	—	2	4	no	—
49	#11717579	—	2	4	no	—
50	#11717579	—	4	1	no	—
51	#11717579	—	4	1	no	—
52	C₂²×C₁₀₁	—	2	2	no	—
53	C₂²×C₁₀₁	—	4	1	no	—
54	C₂²×C₁₀₁	—	4	2	no	—
55	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₀₁	—	2	2	no	—
58	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
59	C₄×C₁₀₁	#120	1	1	yes	characteristic
60	C₄×C₁₀₁	#123	1	1	yes	characteristic
61	C₂×C₁₀₁	—	2	1	no	—
62	C₂×C₁₀₁	—	2	2	no	—
63	C₂×C₁₀₁	—	4	2	no	—
64	C₂×C₁₀₁	#385	1	1	yes	characteristic
65	#1858	—	101	1	no	maximal, Sylow
66	C₁₀₁	#1858	1	1	yes	characteristic, complemented, Sylow
67	#568	D₁₀₁	1	1	yes	characteristic
68	#430	—	101	2	no	—
69	#413	—	101	1	no	—
70	#408	—	101	1	no	—
71	#341	—	101	2	no	—
72	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	101	2	no	—
76	C₂×M₁₆	—	101	1	no	—
77	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
78	C₂²×C₈	—	101	1	no	—
79	#108	—	202	4	no	—
80	#101	—	101	2	no	—
81	#100	—	101	2	no	—
82	C₂²⋊C₈	—	202	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₁₀₁	1	1	yes	characteristic
86	C₂×D₄	Dic₂₀₂	1	2	yes	—
87	C₂×D₄	C₄×D₁₀₁	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	101	4	no	—
93	M₁₆	—	202	2	no	—
94	M₁₆	—	202	2	no	—
95	M₁₆	C₂²⋊D₁₀₁	1	4	yes	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	202	1	no	—
99	C₂×C₈	—	202	1	no	—
100	C₂×C₈	—	202	2	no	—
101	C₂×C₈	—	202	2	no	—
102	C₂³	#13108053	1	2	yes	—
103	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#13108053	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
113	C₂×C₄	#13108074	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	404	1	no	—
116	C₈	—	404	1	no	—
117	C₈	—	202	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×#13108053	1	1	yes	characteristic
122	C₂²	#15692234	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	C₄×C₂²⋊D₁₀₁	1	2	yes	—
125	C₄	#15692224	1	1	yes	characteristic
126	C₄	#15692238	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#20265050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265225

C₂

yes

maximal, complemented

#20265161

C₂

yes

maximal, characteristic

#20265144

C₂

yes

maximal, characteristic

#20264941

C₂

yes

maximal, complemented

#568×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

characteristic

C₂×#13108058

C₂²

yes

—

C₂²×#11717579

C₂²

yes

characteristic

#15692190

—

#15692174

C₄

yes

—

#15692172

C₄

yes

—

C₂²⋊#11717579

—

C₂×#41×C₁₀₁

C₂²

yes

characteristic

#131×C₁₀₁

—

C₂×M₁₆×C₁₀₁

—

C₂×M₁₆×C₁₀₁

C₂²

yes

—

#13108058

—

#13108058

—

#13108058

C₂×C₄

yes

—

—

—

—

—

#41×C₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

D₄

yes

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

C₄⋊C₄

yes

—

C₂³×C₁₀₁

C₂×Q₈

yes

characteristic

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×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×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₈×C₁₀₁

—

#11717579

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#385

yes

characteristic

#1858

—

101

maximal, Sylow

C₁₀₁

#1858

yes

characteristic, complemented, Sylow

#568

D₁₀₁

yes

characteristic

#430

—

101

—

#413

—

101

—

#408

—

101

—

#341

—

101

—

C₂×#41

C₂×D₁₀₁

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

#108

—

202

—

#101

—

101

—

#100

—

101

—

C₂²⋊C₈

—

202

—

#41

—

#41

—

C₂×Q₈

C₄×D₁₀₁

yes

characteristic

C₂×D₄

Dic₂₀₂

yes

—

C₂×D₄

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

101

—

M₁₆

—

202

—

M₁₆

—

202

—

M₁₆

C₂²⋊D₁₀₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

100

C₂×C₈

—

202

—

101

C₂×C₈

—

202

—

102

C₂³

#13108053

yes

—

103

C₂³

C₂×Dic₂₀₂

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#13108053

yes

—

110

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

113

C₂×C₄

#13108074

yes

—

114

C₈

—

115

C₈

—

404

—

116

C₈

—

404

—

117

C₈

—

202

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×#13108053

yes

characteristic

122

C₂²

#15692234

yes

—

123

C₄

—

124

C₄

C₄×C₂²⋊D₁₀₁

yes

—

125

C₄

#15692224

yes

characteristic

126

C₄

#15692238

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#20265050

yes

characteristic

	subgroup	G/H
center	C₄	#15692224
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#568×C₁₀₁	C₂
socle	C₂×C₁₀₁	#385
prime radical	C₂×#41×C₁₀₁	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#568	D₁₀₁
hypocenter	C₁₀₁	#1858
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#15692224

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#568×C₁₀₁

C₂

socle

C₂×C₁₀₁

#385

prime radical

C₂×#41×C₁₀₁

C₂²

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #20265225	C₂	semidirect
4. #20264941	C₂	semidirect
66. C₁₀₁	#1858	semidirect

H ≅ G/N

type

1. #20265225

C₂

semidirect

4. #20264941

C₂

semidirect

66. C₁₀₁

#1858

semidirect

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

subgroup	index
#20265225	2
#20265161	2
#20265144	2
#20264941	2
#568×C₁₀₁	2
#1858	101

subgroup

index

#568×C₁₀₁

101

Subgroups — #27171352

Conjugacy classes of subgroups

Complements

Maximal subgroups