Group #27172385 — subgroups

Subgroups — #27172385

Order 12928.

296 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20265203	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265182	C₂	1	2	yes	maximal, complemented
3	C₂×#15692172	C₂	1	2	yes	maximal, complemented
4	#20264942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	1	yes	characteristic
7	C₂×#13108058	C₂²	1	2	yes	—
8	#15692214	C₂²	1	1	yes	characteristic
9	#15692184	—	2	2	no	—
10	#15692176	—	2	4	no	—
11	#15692172	—	2	4	no	—
12	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
13	C₂×D₈×C₁₀₁	—	2	4	no	—
14	C₂×D₈×C₁₀₁	—	2	2	no	—
15	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
16	#13108058	—	2	2	no	—
17	#13108058	—	2	2	no	—
18	#13108058	—	2	2	no	—
19	C₂×#11717579	—	2	2	no	—
20	C₂×#11717579	—	2	1	no	—
21	C₄⋊Dic₁₀₁	D₄	1	2	yes	—
22	C₂²⋊Dic₁₀₁	—	2	1	no	—
23	C₄×Dic₁₀₁	—	2	1	no	—
24	C₂⁴×C₁₀₁	—	2	2	no	—
25	C₂×D₄×C₁₀₁	—	2	2	no	—
26	C₂×D₄×C₁₀₁	—	2	2	no	—
27	C₂×D₄×C₁₀₁	—	2	2	no	—
28	C₂×D₄×C₁₀₁	—	2	2	no	—
29	C₂×D₄×C₁₀₁	—	2	2	no	—
30	C₂×D₄×C₁₀₁	D₄	1	4	yes	—
31	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
32	D₈×C₁₀₁	—	2	8	no	—
33	C₂×C₈×C₁₀₁	—	2	2	no	—
34	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
35	C₂³×C₁₀₁	—	2	4	no	—
36	C₂³×C₁₀₁	—	4	2	no	—
37	C₂³×C₁₀₁	—	4	2	no	—
38	C₂³×C₁₀₁	—	8	4	no	—
39	C₂³×C₁₀₁	#41	1	1	yes	characteristic
40	D₄×C₁₀₁	—	2	4	no	—
41	D₄×C₁₀₁	—	2	4	no	—
42	D₄×C₁₀₁	—	4	2	no	—
43	D₄×C₁₀₁	—	4	2	no	—
44	C₂×C₄×C₁₀₁	C₂×D₄	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	C₂×Dic₁₀₁	—	2	2	no	—
49	C₈×C₁₀₁	—	2	2	no	—
50	#11717579	—	4	1	no	—
51	#11717579	—	4	2	no	—
52	C₂²×C₁₀₁	—	2	2	no	—
53	C₂²×C₁₀₁	—	4	4	no	—
54	C₂²×C₁₀₁	—	8	4	no	—
55	C₂²×C₁₀₁	—	8	4	no	—
56	C₂²×C₁₀₁	—	8	4	no	—
57	C₂²×C₁₀₁	#121	1	1	yes	characteristic
58	C₂²×C₁₀₁	#123	1	2	yes	—
59	C₄×C₁₀₁	#120	1	1	yes	characteristic
60	C₄×C₁₀₁	#121	1	2	yes	—
61	C₄×C₁₀₁	#124	1	1	yes	characteristic
62	Dic₁₀₁	—	4	2	no	—
63	C₂×C₁₀₁	—	2	2	no	—
64	C₂×C₁₀₁	—	8	4	no	—
65	C₂×C₁₀₁	#394	1	1	yes	characteristic
66	C₂×C₁₀₁	#472	1	2	yes	—
67	#2045	—	101	1	no	maximal, Sylow
68	C₁₀₁	#2045	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₈	D₁₀₁	1	1	yes	characteristic
70	#424	—	101	1	no	—
71	#418	—	101	2	no	—
72	C₂×#100	—	101	2	no	—
73	#341	—	101	1	no	—
74	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
75	C₂×D₈	—	2	4	no	—
76	C₂×D₈	—	2	2	no	—
77	C₂×M₁₆	—	101	2	no	—
78	C₂×M₁₆	—	101	1	no	—
79	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
80	#117	—	101	1	no	—
81	C₄⋊C₈	—	202	2	no	—
82	#102	—	202	4	no	—
83	#100	—	202	4	no	—
84	C₂⁴	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	C₂²⋊D₁₀₁	1	4	yes	—
91	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
92	D₈	—	2	8	no	—
93	M₁₆	—	202	2	no	—
94	M₁₆	—	202	2	no	—
95	M₁₆	—	202	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	202	2	no	—
98	C₂×C₈	—	202	1	no	—
99	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
100	C₄⋊C₄	—	101	2	no	—
101	C₂²⋊C₄	—	202	1	no	—
102	C₄²	—	202	1	no	—
103	C₂³	—	2	4	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	—	8	4	no	—
107	C₂³	#13108073	1	1	yes	characteristic
108	D₄	—	2	4	no	—
109	D₄	—	2	4	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	C₂×C₄	—	202	2	no	—
113	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
116	C₂×C₄	#13108073	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	404	1	no	—
119	C₈	—	404	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	4	4	no	—
122	C₂²	—	8	4	no	—
123	C₂²	—	8	4	no	—
124	C₂²	—	8	4	no	—
125	C₂²	#15692226	1	1	yes	characteristic
126	C₂²	#15692236	1	2	yes	—
127	C₄	—	404	2	no	—
128	C₄	#15692224	1	1	yes	characteristic
129	C₄	#15692227	1	2	yes	—
130	C₄	#15692242	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	8	4	no	—
133	C₂	#20265091	1	1	yes	characteristic
134	C₂	#20265444	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265203

C₂

yes

maximal, characteristic, complemented

#20265182

C₂

yes

maximal, complemented

C₂×#15692172

C₂

yes

maximal, complemented

#20264942

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

characteristic

C₂×#13108058

C₂²

yes

—

#15692214

C₂²

yes

—

—

—

C₂²×D₄×C₁₀₁

C₂²

yes

—

C₂×D₈×C₁₀₁

—

C₂×D₈×C₁₀₁

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

—

—

—

—

C₄⋊Dic₁₀₁

D₄

yes

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

D₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

#41

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#121

yes

characteristic

C₂²×C₁₀₁

#123

yes

—

C₄×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

#121

yes

—

C₄×C₁₀₁

#124

yes

characteristic

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#394

yes

characteristic

C₂×C₁₀₁

#472

yes

—

#2045

—

101

maximal, Sylow

C₁₀₁

#2045

yes

characteristic, complemented, Sylow

C₂²×D₈

D₁₀₁

yes

characteristic

#424

—

101

—

#418

—

101

—

C₂×#100

—

101

—

#341

—

101

—

C₂²×D₄

C₂×D₁₀₁

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

#117

—

101

—

C₄⋊C₈

—

202

—

#102

—

202

—

#100

—

202

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

D₈

—

M₁₆

—

202

—

M₁₆

—

202

—

M₁₆

—

202

—

C₂×C₈

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

C₂²⋊D₁₀₁

yes

—

100

C₄⋊C₄

—

101

—

101

C₂²⋊C₄

—

202

—

102

C₄²

—

202

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

#13108073

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

202

—

113

C₂×C₄

D₄×D₁₀₁

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

115

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

116

C₂×C₄

#13108073

yes

—

117

C₈

—

118

C₈

—

404

—

119

C₈

—

404

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

#15692226

yes

characteristic

126

C₂²

#15692236

yes

—

127

C₄

—

404

—

128

C₄

#15692224

yes

characteristic

129

C₄

#15692227

yes

—

130

C₄

#15692242

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

#20265091

yes

characteristic

134

C₂

#20265444

yes

—

	subgroup	G/H
center	C₂²	#15692226
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting, prime radical	C₂²×D₈×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#121
hypercenter	C₂²×D₈	D₁₀₁
hypocenter	C₁₀₁	#2045
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

#15692226

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting, prime radical

C₂²×D₈×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#121

hypercenter

C₂²×D₈

D₁₀₁

hypocenter

C₁₀₁

#2045

LCS

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

N	H ≅ G/N	type
1. #20265203	C₂	semidirect
2. #20265182	C₂	semidirect
3. C₂×#15692172	C₂	semidirect
4. #20264942	C₂	semidirect
68. C₁₀₁	#2045	semidirect

H ≅ G/N

type

1. #20265203

C₂

semidirect

2. #20265182

C₂

semidirect

3. C₂×#15692172

C₂

semidirect

4. #20264942

C₂

semidirect

68. C₁₀₁

#2045

semidirect

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

subgroup	index
#20265203	2
#20265182	2
C₂×#15692172	2
#20264942	2
C₂²×D₈×C₁₀₁	2
#2045	101

subgroup

index

C₂²×D₈×C₁₀₁

101

Subgroups — #27172385

Conjugacy classes of subgroups

Complements

Maximal subgroups