Group #27172280 — subgroups

Subgroups — #27172280

Order 12928.

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

normal subgroups only

This group has 258 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×#15692178	C₂	1	2	yes	maximal
2	#20265083	C₂	1	1	yes	maximal, characteristic
3	#20264939	C₂	1	2	yes	maximal
4	#20264932	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
7	C₂²×#11717579	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
9	#15692178	—	2	4	no	—
10	#15692162	—	2	2	no	—
11	#125×C₁₀₁	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₀₂	—	2	2	no	—
16	C₂×Dic₂₀₂	—	2	2	no	—
17	C₂×Dic₂₀₂	—	2	1	no	—
18	C₂×Dic₂₀₂	Q₈	1	2	yes	—
19	C₂²×Dic₁₀₁	—	2	1	no	—
20	C₂²×Dic₁₀₁	—	2	1	no	—
21	C₂×#11717579	—	2	2	no	—
22	C₂×#11717579	—	2	2	no	—
23	C₂×#11717579	—	2	2	no	—
24	C₄⋊Dic₁₀₁	—	2	2	no	—
25	C₂²×C₄×C₁₀₁	—	2	2	no	—
26	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₁	—	2	8	no	—
29	C₄⋊C₄×C₁₀₁	—	2	2	no	—
30	C₄⋊C₄×C₁₀₁	D₄	1	4	yes	—
31	C₄²×C₁₀₁	—	2	2	no	—
32	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₁	—	2	2	no	—
34	C₂×C₄×C₁₀₁	—	2	4	no	—
35	C₂×C₄×C₁₀₁	—	2	2	no	—
36	C₂×C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄×C₁₀₁	—	4	2	no	—
38	C₂×C₄×C₁₀₁	—	4	2	no	—
39	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₁	#41	1	2	yes	—
42	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
43	C₂×Dic₁₀₁	—	2	2	no	—
44	C₂×Dic₁₀₁	—	4	1	no	—
45	C₂×Dic₁₀₁	—	4	1	no	—
46	C₂×Dic₁₀₁	—	4	1	no	—
47	C₂×Dic₁₀₁	—	4	2	no	—
48	Dic₂₀₂	—	2	4	no	—
49	Dic₂₀₂	—	4	2	no	—
50	#11717579	—	4	2	no	—
51	C₂²×C₁₀₁	#120	1	1	yes	characteristic
52	C₂²×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₁₀₁	#124	1	1	yes	characteristic
54	C₂²×C₁₀₁	#135	1	2	yes	—
55	C₂²×C₁₀₁	#137	1	2	yes	—
56	C₄×C₁₀₁	—	4	2	no	—
57	C₄×C₁₀₁	—	4	4	no	—
58	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₀₁	#124	1	2	yes	—
60	Dic₁₀₁	—	4	2	no	—
61	Dic₁₀₁	—	8	1	no	—
62	C₂×C₁₀₁	#393	1	1	yes	characteristic
63	C₂×C₁₀₁	#452	1	2	yes	—
64	C₂×C₁₀₁	#479	1	2	yes	—
65	C₂×C₁₀₁	#488	1	1	yes	characteristic
66	C₂×C₁₀₁	#491	1	1	yes	characteristic
67	#2026	—	101	1	no	maximal, Sylow
68	C₁₀₁	#2026	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	D₁₀₁	1	1	yes	characteristic
70	C₂×#103	—	101	2	no	—
71	#391	—	101	1	no	—
72	#340	—	101	2	no	—
73	#336	—	101	1	no	—
74	C₂²×Q₈	—	101	1	no	—
75	C₂²×C₈	—	101	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	101	1	no	—
79	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
80	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
81	#103	—	202	4	no	—
82	#95	—	202	2	no	—
83	C₂×Q₈	—	101	2	no	—
84	C₂×Q₈	—	202	2	no	—
85	C₂×Q₈	—	202	2	no	—
86	C₂×Q₈	—	202	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	202	1	no	—
89	C₂²×C₄	—	202	1	no	—
90	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
92	C₂×C₈	—	202	2	no	—
93	C₂×C₈	—	202	2	no	—
94	C₂×C₈	—	202	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	202	2	no	—
98	C₄⋊C₄	C₂²⋊D₁₀₁	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
101	Q₈	—	404	2	no	—
102	Q₈	—	202	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	404	1	no	—
110	C₂×C₄	—	404	2	no	—
111	C₂×C₄	—	404	1	no	—
112	C₂×C₄	—	404	1	no	—
113	C₂×C₄	—	202	2	no	—
114	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
115	C₂×C₄	Q₈×D₁₀₁	1	1	yes	characteristic
116	C₂×C₄	#13108075	1	2	yes	—
117	C₂×C₄	#13108075	1	1	yes	characteristic
118	C₈	—	404	2	no	—
119	C₂²	#15692224	1	1	yes	characteristic
120	C₂²	#15692231	1	1	yes	characteristic
121	C₂²	#15692244	1	1	yes	characteristic
122	C₂²	#15692275	1	2	yes	—
123	C₂²	#15692285	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	808	1	no	—
127	C₄	—	404	2	no	—
128	C₄	#15692231	1	2	yes	—
129	C₄	#15692244	1	2	yes	—
130	C₂	#20265088	1	1	yes	characteristic
131	C₂	#20265339	1	2	yes	—
132	C₂	#20265481	1	2	yes	—
133	C₂	#20265534	1	1	yes	characteristic
134	C₂	#20265547	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692178

C₂

yes

maximal

#20265083

C₂

yes

maximal, characteristic

#20264939

C₂

yes

maximal

#20264932

C₂

yes

maximal, characteristic

C₂×#125×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

characteristic

C₂²×#11717579

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

#15692178

—

#15692162

—

#125×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

Q₈

yes

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717579

—

C₂²×C₁₀₁

#120

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

#137

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

#124

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

#393

yes

characteristic

C₂×C₁₀₁

#452

yes

—

C₂×C₁₀₁

#479

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#2026

yes

characteristic, complemented, Sylow

C₂×#125

D₁₀₁

yes

characteristic

C₂×#103

—

101

—

#391

—

101

—

#340

—

101

—

#336

—

101

—

C₂²×Q₈

—

101

—

C₂²×C₈

—

101

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#103

—

202

—

#95

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

101

Q₈

—

404

—

102

Q₈

—

202

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

404

—

110

C₂×C₄

—

404

—

111

C₂×C₄

—

404

—

112

C₂×C₄

—

404

—

113

C₂×C₄

—

202

—

114

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

115

C₂×C₄

Q₈×D₁₀₁

yes

characteristic

116

C₂×C₄

#13108075

yes

—

117

C₂×C₄

#13108075

yes

characteristic

118

C₈

—

404

—

119

C₂²

#15692224

yes

characteristic

120

C₂²

#15692231

yes

characteristic

121

C₂²

#15692244

yes

characteristic

122

C₂²

#15692275

yes

—

123

C₂²

#15692285

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

808

—

127

C₄

—

404

—

128

C₄

#15692231

yes

—

129

C₄

#15692244

yes

—

130

C₂

#20265088

yes

characteristic

131

C₂

#20265339

yes

—

132

C₂

#20265481

yes

—

133

C₂

#20265534

yes

characteristic

134

C₂

#20265547

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#125×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
hypercenter	C₂×#125	D₁₀₁
hypocenter	C₁₀₁	#2026
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#125×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
68. C₁₀₁	#2026	semidirect

H ≅ G/N

type

68. C₁₀₁

#2026

semidirect

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

subgroup	index
C₂×#15692178	2
#20265083	2
#20264939	2
#20264932	2
C₂×#125×C₁₀₁	2
#2026	101

subgroup

index

C₂×#125×C₁₀₁

101

Subgroups — #27172280

Conjugacy classes of subgroups

Complements

Maximal subgroups