Group #27186492 — subgroups

Subgroups — #27186492

Order 12928.

246 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 60 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	#20266199	C₂	1	1	yes	maximal, characteristic, complemented
2	#20266195	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11717581	C₂	1	4	yes	maximal, complemented
4	#20265725	C₂	1	1	yes	maximal, characteristic
5	#15692322	—	2	1	no	—
6	C₄⋊#11717581	C₂²	1	2	yes	—
7	#15692312	—	2	1	no	—
8	#15692312	—	2	1	no	—
9	#15692310	—	2	1	no	—
10	C₄×#11717581	C₂²	1	2	yes	—
11	C₂×#13108073	—	2	1	no	—
12	#15692255	C₄	1	1	yes	characteristic
13	#15692242	—	2	1	no	—
14	#15692226	—	2	1	no	—
15	#15692226	C₄	1	2	yes	—
16	D₄×Dic₁₀₁	—	2	1	no	—
17	D₄×Dic₁₀₁	C₂²	1	2	yes	—
18	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
19	#127×C₁₀₁	C₄	1	1	yes	characteristic
20	C₂×#11717581	—	2	2	no	—
21	C₂×#11717581	—	2	2	no	—
22	#13108073	—	4	4	no	—
23	C₂×Dic₂₀₂	—	2	1	no	—
24	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
25	C₂²×Dic₁₀₁	—	2	2	no	—
26	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
28	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₀₁	—	2	1	no	—
30	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
31	C₂²⋊Dic₁₀₁	—	2	2	no	—
32	C₂²⋊Dic₁₀₁	—	2	2	no	—
33	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
34	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
35	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
36	C₂×D₄×C₁₀₁	—	2	1	no	—
37	C₂×D₄×C₁₀₁	—	2	1	no	—
38	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
39	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₁₀₁	—	2	2	no	—
41	C₂²×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
42	D₄×C₁₀₁	—	2	4	no	—
43	D₄×C₁₀₁	—	4	2	no	—
44	D₄×C₁₀₁	—	4	2	no	—
45	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²⋊D₁₀₁	—	4	4	no	—
49	C₂×Dic₁₀₁	—	2	2	no	—
50	C₂×Dic₁₀₁	—	4	4	no	—
51	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
54	C₄×D₁₀₁	—	2	2	no	—
55	C₄×D₁₀₁	—	2	2	no	—
56	C₄×D₁₀₁	—	4	1	no	—
57	Dic₂₀₂	—	4	2	no	—
58	#11717581	—	4	2	no	—
59	#11717581	—	4	2	no	—
60	C₂²×C₁₀₁	—	4	2	no	—
61	C₂²×C₁₀₁	—	4	2	no	—
62	C₂²×C₁₀₁	—	4	2	no	—
63	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₁₀₁	—	2	1	no	—
65	C₂×D₁₀₁	—	2	1	no	—
66	C₂×D₁₀₁	—	2	1	no	—
67	C₄×C₁₀₁	—	2	1	no	—
68	C₄×C₁₀₁	—	2	1	no	—
69	C₄×C₁₀₁	#131	1	2	yes	—
70	Dic₁₀₁	—	2	1	no	—
71	Dic₁₀₁	—	2	1	no	—
72	Dic₁₀₁	—	4	2	no	—
73	Dic₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₁₀₁	—	8	2	no	—
75	C₂×C₁₀₁	#408	1	1	yes	characteristic
76	C₂×C₁₀₁	#418	1	1	yes	characteristic
77	C₂×C₁₀₁	#421	1	1	yes	characteristic
78	D₁₀₁	—	4	1	no	—
79	#1496	—	101	1	no	maximal, Sylow
80	C₁₀₁	#1496	1	1	yes	characteristic, complemented, Sylow
81	#532	—	101	1	no	—
82	#432	—	101	1	no	—
83	#431	—	101	1	no	—
84	D₄⋊C₈	—	101	4	no	—
85	C₂×#41	—	202	1	no	—
86	C₄⋊Q₈	—	101	1	no	—
87	#127	#87938	1	1	yes	characteristic
88	#124	—	202	1	no	—
89	#121	—	101	2	no	—
90	#121	—	202	1	no	—
91	C₄×D₄	—	101	2	no	—
92	C₄×D₄	—	202	1	no	—
93	C₂×C₄²	—	101	1	no	—
94	C₄⋊C₈	—	101	2	no	—
95	C₄⋊C₈	—	202	1	no	—
96	C₂²⋊C₈	—	202	1	no	—
97	C₂²⋊C₈	—	202	1	no	—
98	#97	—	202	1	no	—
99	C₄×C₈	—	101	2	no	—
100	#41	—	404	4	no	—
101	C₂×Q₈	—	202	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	202	2	no	—
105	C₂×D₄	C₂×#87938	1	2	yes	—
106	C₂²×C₄	—	101	1	no	—
107	C₂²×C₄	—	101	1	no	—
108	C₂²×C₄	—	101	1	no	—
109	C₂²×C₄	—	202	2	no	—
110	C₂×C₈	—	202	2	no	—
111	C₂×C₈	—	202	2	no	—
112	C₄⋊C₄	—	101	2	no	—
113	C₄⋊C₄	—	202	1	no	—
114	C₂²⋊C₄	—	202	2	no	—
115	C₂²⋊C₄	—	202	2	no	—
116	C₄²	—	101	1	no	—
117	C₄²	—	101	1	no	—
118	C₄²	—	101	1	no	—
119	C₄²	C₂×#87938	1	1	yes	characteristic
120	C₂³	—	2	2	no	—
121	C₂³	—	101	1	no	—
122	Q₈	—	404	2	no	—
123	D₄	—	2	4	no	—
124	D₄	—	4	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	404	4	no	—
127	C₂×C₄	—	101	1	no	—
128	C₂×C₄	—	101	1	no	—
129	C₂×C₄	—	101	1	no	—
130	C₂×C₄	—	404	1	no	—
131	C₂×C₄	—	404	4	no	—
132	C₂×C₄	—	202	2	no	—
133	C₂×C₄	—	202	2	no	—
134	C₂×C₄	—	202	2	no	—
135	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
136	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
137	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
138	C₈	—	404	2	no	—
139	C₈	—	404	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	4	2	no	—
143	C₂²	—	202	1	no	—
144	C₂²	—	202	1	no	—
145	C₂²	—	202	1	no	—
146	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	101	2	no	—
150	C₄	—	404	2	no	—
151	C₄	—	202	1	no	—
152	C₄	—	202	1	no	—
153	C₄	#15692342	1	2	yes	—
154	C₂	—	8	2	no	—
155	C₂	—	404	1	no	—
156	C₂	#20266151	1	1	yes	characteristic
157	C₂	#20266169	1	1	yes	characteristic
158	C₂	#20266175	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20266199

C₂

yes

maximal, characteristic, complemented

#20266195

C₂

yes

maximal, characteristic, complemented

D₄⋊#11717581

C₂

yes

maximal, complemented

#20265725

C₂

yes

maximal, characteristic

#15692322

—

C₄⋊#11717581

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×#13108073

—

#15692255

C₄

yes

—

—

C₄

yes

—

D₄×Dic₁₀₁

—

D₄×Dic₁₀₁

C₂²

yes

—

C₄²×D₁₀₁

C₂²

yes

characteristic

#127×C₁₀₁

C₄

yes

—

—

—

C₂×Dic₂₀₂

—

C₂×C₂²⋊D₁₀₁

—

C₂²×Dic₁₀₁

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₁

D₄

yes

characteristic

C₂×C₄×D₁₀₁

D₄

yes

characteristic

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

characteristic

C₄×Dic₁₀₁

C₂³

yes

characteristic

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂³×C₁₀₁

—

C₂²×D₁₀₁

C₂²⋊C₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

#11717581

—

#11717581

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#131

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

D₁₀₁

—

#1496

—

101

maximal, Sylow

C₁₀₁

#1496

yes

characteristic, complemented, Sylow

#532

—

101

—

#432

—

101

—

#431

—

101

—

D₄⋊C₈

—

101

—

C₂×#41

—

202

—

C₄⋊Q₈

—

101

—

yes

—

202

—

#121

—

101

—

#121

—

202

—

C₄×D₄

—

101

—

C₄×D₄

—

202

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

#97

—

202

—

C₄×C₈

—

101

—

100

#41

—

404

—

101

C₂×Q₈

—

202

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

202

—

105

C₂×D₄

C₂×#87938

yes

—

106

C₂²×C₄

—

101

—

107

C₂²×C₄

—

101

—

108

C₂²×C₄

—

101

—

109

C₂²×C₄

—

202

—

110

C₂×C₈

—

202

—

111

C₂×C₈

—

202

—

112

C₄⋊C₄

—

101

—

113

C₄⋊C₄

—

202

—

114

C₂²⋊C₄

—

202

—

115

C₂²⋊C₄

—

202

—

116

C₄²

—

101

—

117

C₄²

—

101

—

118

C₄²

—

101

—

119

C₄²

C₂×#87938

yes

characteristic

120

C₂³

—

121

C₂³

—

101

—

122

Q₈

—

404

—

123

D₄

—

124

D₄

—

125

D₄

—

126

D₄

—

404

—

127

C₂×C₄

—

101

—

128

C₂×C₄

—

101

—

129

C₂×C₄

—

101

—

130

C₂×C₄

—

404

—

131

C₂×C₄

—

404

—

132

C₂×C₄

—

202

—

133

C₂×C₄

—

202

—

134

C₂×C₄

—

202

—

135

C₂×C₄

C₂²⋊#87938

yes

characteristic

136

C₂×C₄

C₂²⋊#87938

yes

characteristic

137

C₂×C₄

C₂²×#87938

yes

characteristic

138

C₈

—

404

—

139

C₈

—

404

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

202

—

144

C₂²

—

202

—

145

C₂²

—

202

—

146

C₂²

C₂×C₂²⋊#87938

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

101

—

150

C₄

—

404

—

151

C₄

—

202

—

152

C₄

—

202

—

153

C₄

#15692342

yes

—

154

C₂

—

155

C₂

—

404

—

156

C₂

#20266151

yes

characteristic

157

C₂

#20266169

yes

characteristic

158

C₂

#20266175

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	#127×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	#20265725	C₂
cosocle	C₄×Dic₁₀₁	C₂³
hypercenter	#127	#87938
hypocenter	C₁₀₁	#1496
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting

#127×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

#20265725

C₂

cosocle

C₄×Dic₁₀₁

C₂³

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
1. #20266199	C₂	semidirect
2. #20266195	C₂	semidirect
3. D₄⋊#11717581	C₂	semidirect
80. C₁₀₁	#1496	semidirect

H ≅ G/N

type

1. #20266199

C₂

semidirect

2. #20266195

C₂

semidirect

3. D₄⋊#11717581

C₂

semidirect

80. C₁₀₁

#1496

semidirect

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

subgroup	index
#20266199	2
#20266195	2
D₄⋊#11717581	2
#20265725	2
#1496	101

subgroup

index

101

Subgroups — #27186492

Conjugacy classes of subgroups

Complements

Maximal subgroups