Group #27181655 — subgroups

Subgroups — #27181655

Order 12928.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#20265730	C₂	1	2	yes	maximal
2	#20265585	C₂	1	4	yes	maximal, complemented
3	#20265584	C₂	1	4	yes	maximal, complemented
4	#20265215	C₂	1	1	yes	maximal, characteristic
5	#20265213	C₂	1	2	yes	maximal
6	C₂×#15692165	C₂	1	1	yes	maximal, characteristic
7	#427×C₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11717579	C₂²	1	1	yes	characteristic
9	#15692254	C₂²	1	4	yes	—
10	#15692246	C₂²	1	4	yes	—
11	#15692232	—	2	4	no	—
12	C₄×Dic₂₀₂	—	2	4	no	—
13	#15692214	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	#15692189	C₂²	1	2	yes	—
16	#15692188	C₂²	1	4	yes	—
17	#15692187	C₂²	1	2	yes	—
18	#15692186	C₂²	1	4	yes	—
19	#15692165	C₂²	1	2	yes	—
20	#15692165	C₂²	1	2	yes	—
21	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
22	#117×C₁₀₁	C₂²	1	2	yes	—
23	#107×C₁₀₁	C₂²	1	2	yes	—
24	#106×C₁₀₁	C₂²	1	2	yes	—
25	C₂×Dic₂₀₂	—	2	4	no	—
26	C₂²×Dic₁₀₁	D₄	1	2	yes	—
27	C₂×#11717579	Q₈	1	4	yes	complemented
28	C₂×#11717579	C₂³	1	2	yes	—
29	C₄⋊Dic₁₀₁	—	2	4	no	—
30	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
31	#13108053	—	2	4	no	—
32	#13108053	—	2	4	no	—
33	C₂²⋊Dic₁₀₁	—	2	2	no	—
34	C₄×Dic₁₀₁	—	2	2	no	—
35	C₄×Dic₁₀₁	D₄	1	2	yes	—
36	C₄×Dic₁₀₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₀₁	Q₈	1	4	yes	complemented
39	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
43	C₄²×C₁₀₁	—	2	2	no	—
44	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	—	2	4	no	—
46	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₁	—	2	4	no	—
51	C₂×Dic₁₀₁	—	4	2	no	—
52	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
53	Dic₂₀₂	—	4	8	no	—
54	C₈×C₁₀₁	C₂×Q₈	1	2	yes	—
55	C₈×C₁₀₁	C₂×Q₈	1	2	yes	—
56	#11717579	C₂×Q₈	1	4	yes	—
57	C₂²×C₁₀₁	—	2	2	no	—
58	C₂²×C₁₀₁	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₁₀₁	—	4	4	no	—
61	C₄×C₁₀₁	C₄⋊Q₈	1	2	yes	—
62	C₄×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
64	Dic₁₀₁	—	2	4	no	—
65	Dic₁₀₁	—	4	4	no	—
66	C₂×C₁₀₁	—	2	2	no	—
67	C₂×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₁₀₁	#575	1	2	yes	—
69	#3162	—	101	1	no	maximal, Sylow
70	C₁₀₁	#3162	1	1	yes	characteristic, complemented, Sylow
71	#533	—	101	2	no	—
72	#501	—	101	4	no	—
73	#501	—	101	4	no	—
74	#427	—	101	1	no	—
75	#427	—	101	2	no	—
76	#427	D₁₀₁	1	1	yes	characteristic
77	C₂×#97	—	101	1	no	—
78	C₂²×C₈	—	101	1	no	—
79	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
80	C₄⋊Q₈	—	101	4	no	—
81	#125	—	101	4	no	—
82	C₂²⋊Q₈	—	202	4	no	—
83	C₄×Q₈	—	202	4	no	—
84	#117	—	101	2	no	—
85	#117	C₂×D₁₀₁	1	2	yes	—
86	C₂×C₄²	—	101	1	no	—
87	#107	—	101	2	no	—
88	#107	—	101	4	no	—
89	#107	C₂×D₁₀₁	1	2	yes	—
90	#106	—	101	2	no	—
91	#106	—	101	4	no	—
92	#106	C₂×D₁₀₁	1	2	yes	—
93	#97	—	101	2	no	—
94	#97	—	101	2	no	—
95	C₂×Q₈	—	202	4	no	—
96	C₂²×C₄	—	101	2	no	—
97	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
98	C₂×C₈	—	101	4	no	—
99	C₂×C₈	—	101	2	no	—
100	C₂×C₈	Dic₂₀₂	1	4	yes	complemented
101	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
103	C₄⋊C₄	—	101	4	no	—
104	C₄⋊C₄	—	202	4	no	—
105	C₄⋊C₄	—	202	4	no	—
106	C₄⋊C₄	—	202	4	no	—
107	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	202	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	101	2	no	—
112	C₄²	—	101	2	no	—
113	C₄²	—	202	2	no	—
114	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
115	Q₈	—	404	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	101	4	no	—
118	C₂×C₄	—	404	2	no	—
119	C₂×C₄	—	202	4	no	—
120	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
122	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
123	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
124	C₈	—	101	4	no	—
125	C₈	C₂×Dic₂₀₂	1	2	yes	—
126	C₈	C₂×Dic₂₀₂	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#15692254	1	2	yes	—
129	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	404	4	no	—
132	C₄	—	202	4	no	—
133	C₄	#15692254	1	2	yes	—
134	C₄	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
135	C₄	C₂²×Dic₂₀₂	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#15692254	1	1	yes	characteristic
138	C₂	#20265974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265730

C₂

yes

maximal

#20265585

C₂

yes

maximal, complemented

#20265584

C₂

yes

maximal, complemented

#20265215

C₂

yes

maximal, characteristic

#20265213

C₂

yes

maximal

C₂×#15692165

C₂

yes

maximal, characteristic

#427×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

characteristic

#15692254

C₂²

yes

—

#15692246

C₂²

yes

—

#15692232

—

C₄×Dic₂₀₂

—

#15692214

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

#15692189

C₂²

yes

—

#15692188

C₂²

yes

—

#15692187

C₂²

yes

—

#15692186

C₂²

yes

—

#15692165

C₂²

yes

—

#15692165

C₂²

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

#117×C₁₀₁

C₂²

yes

—

#107×C₁₀₁

C₂²

yes

—

#106×C₁₀₁

C₂²

yes

—

C₂×Dic₂₀₂

—

C₂²×Dic₁₀₁

D₄

yes

—

C₂×#11717579

Q₈

yes

complemented

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

—

#13108053

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

Q₈

yes

complemented

C₂×C₈×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

C₂³

yes

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂×D₄

yes

—

Dic₂₀₂

—

C₈×C₁₀₁

C₂×Q₈

yes

—

C₈×C₁₀₁

C₂×Q₈

yes

—

#11717579

C₂×Q₈

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₄⋊Q₈

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄⋊Q₈

yes

—

C₄×C₁₀₁

C₂²×Q₈

yes

characteristic

C₄×C₁₀₁

C₂²×Q₈

yes

characteristic

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₁

#575

yes

—

#3162

—

101

maximal, Sylow

C₁₀₁

#3162

yes

characteristic, complemented, Sylow

#533

—

101

—

#501

—

101

—

#501

—

101

—

#427

—

101

—

#427

—

101

—

#427

D₁₀₁

yes

characteristic

C₂×#97

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

C₄⋊Q₈

—

101

—

#125

—

101

—

C₂²⋊Q₈

—

202

—

C₄×Q₈

—

202

—

#117

—

101

—

#117

C₂×D₁₀₁

yes

—

C₂×C₄²

—

101

—

#107

—

101

—

#107

—

101

—

#107

C₂×D₁₀₁

yes

—

#106

—

101

—

#106

—

101

—

#106

C₂×D₁₀₁

yes

—

#97

—

101

—

#97

—

101

—

C₂×Q₈

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

101

—

100

C₂×C₈

Dic₂₀₂

yes

complemented

101

C₂×C₈

C₂²×D₁₀₁

yes

characteristic

102

C₂×C₈

C₂²×D₁₀₁

yes

characteristic

103

C₄⋊C₄

—

101

—

104

C₄⋊C₄

—

202

—

105

C₄⋊C₄

—

202

—

106

C₄⋊C₄

—

202

—

107

C₄⋊C₄

C₂²×D₁₀₁

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

202

—

110

C₄²

—

111

C₄²

—

101

—

112

C₄²

—

101

—

113

C₄²

—

202

—

114

C₂³

D₄×D₁₀₁

yes

characteristic

115

Q₈

—

404

—

116

C₂×C₄

—

117

C₂×C₄

—

101

—

118

C₂×C₄

—

404

—

119

C₂×C₄

—

202

—

120

C₂×C₄

D₄×D₁₀₁

yes

characteristic

121

C₂×C₄

Q₈×D₁₀₁

yes

—

122

C₂×C₄

C₂×Dic₂₀₂

yes

—

123

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

124

C₈

—

101

—

125

C₈

C₂×Dic₂₀₂

yes

—

126

C₈

C₂×Dic₂₀₂

yes

—

127

C₂²

—

128

C₂²

#15692254

yes

—

129

C₂²

C₂×D₄×D₁₀₁

yes

characteristic

130

C₄

—

131

C₄

—

404

—

132

C₄

—

202

—

133

C₄

#15692254

yes

—

134

C₄

C₂×Q₈×D₁₀₁

yes

characteristic

135

C₄

C₂²×Dic₂₀₂

yes

characteristic

136

C₂

—

137

C₂

C₂×#15692254

yes

characteristic

138

C₂

#20265974

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#427×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂³×C₁₀₁	C₂×D₄
hypercenter	#427	D₁₀₁
hypocenter	C₁₀₁	#3162
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₁

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

#427×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂²×D₄

prime radical

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #20265585	C₂	semidirect
3. #20265584	C₂	semidirect
27. C₂×#11717579	Q₈	semidirect
38. C₂×C₈×C₁₀₁	Q₈	semidirect
70. C₁₀₁	#3162	semidirect
100. C₂×C₈	Dic₂₀₂	semidirect

H ≅ G/N

type

2. #20265585

C₂

semidirect

3. #20265584

C₂

semidirect

27. C₂×#11717579

Q₈

semidirect

38. C₂×C₈×C₁₀₁

Q₈

semidirect

70. C₁₀₁

#3162

semidirect

100. C₂×C₈

Dic₂₀₂

semidirect

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

subgroup	index
#20265730	2
#20265585	2
#20265584	2
#20265215	2
#20265213	2
C₂×#15692165	2
#427×C₁₀₁	2
#3162	101

subgroup

index

#427×C₁₀₁

101

Subgroups — #27181655

Conjugacy classes of subgroups

Complements

Maximal subgroups