Group #27187859 — subgroups

Subgroups — #27187859

Order 12928.

484 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 162 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×#15692343	C₂	1	2	yes	maximal, complemented
2	#20266207	C₂	1	8	yes	maximal, complemented
3	#20266182	C₂	1	1	yes	maximal, characteristic, complemented
4	#20266152	C₂	1	2	yes	maximal, complemented
5	#20266136	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265728	C₂	1	1	yes	maximal, characteristic
7	#15692343	—	2	8	no	—
8	C₂×#13108084	C₂²	1	2	yes	—
9	C₂×#13108084	C₂²	1	4	yes	—
10	C₂×D₁₀₁⋊C₈	C₂²	1	4	yes	—
11	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
13	#15692322	C₂²	1	4	yes	—
14	#15692312	C₂²	1	8	yes	—
15	#15692310	C₂²	1	4	yes	—
16	C₂×#13108075	C₂²	1	2	yes	—
17	#15692252	C₄	1	1	yes	characteristic
18	#15692244	C₄	1	2	yes	—
19	#15692230	C₄	1	4	yes	—
20	#15692217	C₂²	1	4	yes	—
21	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic
23	#13108084	—	2	4	no	—
24	#13108084	—	2	8	no	—
25	D₁₀₁⋊C₈	—	2	2	no	—
26	D₁₀₁⋊C₈	—	2	8	no	—
27	D₁₀₁⋊C₈	D₄	1	4	yes	complemented
28	C₂×#11717581	C₂³	1	2	yes	—
29	C₂×#11717581	C₂³	1	4	yes	—
30	C₂×#11717581	C₂³	1	2	yes	—
31	#13108075	—	2	4	no	—
32	#13108075	—	2	4	no	—
33	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
34	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
35	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
38	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
39	#13108051	C₂×C₄	1	8	yes	—
40	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₁₀₁	C₂³	1	2	yes	—
42	C₂×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
44	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
46	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₁₀₁	—	2	4	no	—
48	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
52	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
53	D₄₀₄	—	2	8	no	—
54	D₄₀₄	—	2	4	no	—
55	D₄₀₄	—	4	4	no	—
56	C₄×D₁₀₁	—	2	4	no	—
57	C₄×D₁₀₁	—	2	4	no	—
58	C₄×D₁₀₁	—	4	2	no	—
59	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
60	C₄×D₁₀₁	#41	1	2	yes	—
61	#11717581	—	2	4	no	—
62	#11717581	—	2	2	no	—
63	#11717581	C₂×D₄	1	4	yes	—
64	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₀₁	—	2	2	no	—
66	C₂×D₁₀₁	—	2	4	no	—
67	C₂×D₁₀₁	—	2	8	no	—
68	C₂×D₁₀₁	C₄×D₄	1	2	yes	—
69	C₄×C₁₀₁	—	2	4	no	—
70	C₄×C₁₀₁	—	2	2	no	—
71	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
72	Dic₁₀₁	—	2	2	no	—
73	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
74	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
75	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
76	C₂×C₁₀₁	#568	1	2	yes	—
77	D₁₀₁	—	2	2	no	—
78	D₁₀₁	—	4	4	no	—
79	#2932	—	101	1	no	maximal, Sylow
80	C₁₀₁	#2932	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	101	2	no	—
82	#532	—	101	1	no	—
83	#435	—	101	8	no	—
84	#423	—	101	1	no	—
85	#408	—	101	2	no	—
86	C₂×#97	—	101	1	no	—
87	C₂×#41	—	101	2	no	—
88	#131	—	202	8	no	—
89	C₂×M₁₆	—	101	2	no	—
90	C₂×M₁₆	—	101	4	no	—
91	C₂²×C₈	—	101	1	no	—
92	C₂²×C₈	—	101	4	no	—
93	C₂²×C₈	—	101	1	no	—
94	C₄⋊Q₈	#87938	1	1	yes	characteristic
95	#127	—	101	1	no	—
96	#124	—	101	2	no	—
97	#121	—	101	4	no	—
98	C₄×D₄	—	101	4	no	—
99	C₂×C₄²	—	101	1	no	—
100	C₄⋊C₈	—	101	4	no	—
101	C₂²⋊C₈	—	101	8	no	—
102	#97	—	101	4	no	—
103	#41	—	202	4	no	—
104	#41	—	202	4	no	—
105	C₂×Q₈	C₂×#87938	1	2	yes	—
106	C₂×D₄	—	101	4	no	—
107	C₂×D₄	—	101	4	no	—
108	C₂×D₄	—	101	2	no	—
109	C₂²×C₄	—	101	1	no	—
110	C₂²×C₄	—	101	2	no	—
111	C₂²×C₄	—	101	4	no	—
112	M₁₆	—	202	8	no	—
113	M₁₆	—	202	4	no	—
114	C₂×C₈	—	101	2	no	—
115	C₂×C₈	—	101	4	no	—
116	C₂×C₈	—	101	2	no	—
117	C₂×C₈	—	101	4	no	—
118	C₂×C₈	—	202	8	no	—
119	C₂×C₈	—	202	2	no	—
120	C₄⋊C₄	C₂×#87938	1	4	yes	—
121	C₂²⋊C₄	—	101	8	no	—
122	C₄²	—	101	1	no	—
123	C₄²	—	101	2	no	—
124	C₄²	C₂×#87938	1	1	yes	characteristic
125	C₂³	—	101	1	no	—
126	C₂³	—	101	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	404	4	no	—
129	D₄	—	202	8	no	—
130	D₄	—	202	4	no	—
131	C₂×C₄	—	101	1	no	—
132	C₂×C₄	—	101	2	no	—
133	C₂×C₄	—	101	2	no	—
134	C₂×C₄	—	101	2	no	—
135	C₂×C₄	—	404	2	no	—
136	C₂×C₄	—	202	4	no	—
137	C₂×C₄	—	202	4	no	—
138	C₂×C₄	C₂²×#87938	1	2	yes	—
139	C₂×C₄	C₂²×#87938	1	4	yes	—
140	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
141	C₈	—	101	4	no	—
142	C₈	—	202	4	no	—
143	C₈	—	202	2	no	—
144	C₂²	—	101	2	no	—
145	C₂²	—	202	4	no	—
146	C₂²	—	202	8	no	—
147	C₂²	—	202	2	no	—
148	C₂²	C₂³×#87938	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	101	1	no	—
152	C₄	—	101	1	no	—
153	C₄	—	202	2	no	—
154	C₄	D₄×#87938	1	2	yes	—
155	C₂	—	404	4	no	—
156	C₂	—	202	2	no	—
157	C₂	C₂×D₄×#87938	1	1	yes	characteristic
158	C₂	#20266257	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692343

C₂

yes

maximal, complemented

#20266207

C₂

yes

maximal, complemented

#20266182

C₂

yes

maximal, characteristic, complemented

#20266152

C₂

yes

maximal, complemented

#20266136

C₂

yes

maximal, characteristic, complemented

#20265728

C₂

yes

maximal, characteristic

#15692343

—

C₂×#13108084

C₂²

yes

—

C₂×#13108084

C₂²

yes

—

C₂×D₁₀₁⋊C₈

C₂²

yes

—

C₂×D₁₀₁⋊C₈

C₂²

yes

characteristic

C₂×D₁₀₁⋊C₈

C₂²

yes

characteristic

#15692322

C₂²

yes

—

#15692312

C₂²

yes

—

#15692310

C₂²

yes

—

C₂×#13108075

C₂²

yes

—

#15692252

C₄

yes

characteristic

#15692244

C₄

yes

—

#15692230

C₄

yes

—

#15692217

C₂²

yes

—

C₄²×D₁₀₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₁

C₄

yes

characteristic

#13108084

—

#13108084

—

D₁₀₁⋊C₈

—

D₁₀₁⋊C₈

—

D₁₀₁⋊C₈

D₄

yes

complemented

C₂×#11717581

C₂³

yes

—

C₂×#11717581

C₂³

yes

—

C₂×#11717581

C₂³

yes

—

#13108075

—

#13108075

—

C₂×D₄₀₄

C₂×C₄

yes

—

C₂×D₄₀₄

C₂×C₄

yes

—

C₂×D₄₀₄

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

characteristic

#13108051

C₂×C₄

yes

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

C₂³

yes

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×D₁₀₁

C₂²×C₄

yes

—

C₂²×D₁₀₁

C₂²×C₄

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

C₂⁴

yes

characteristic

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

C₂×D₄

yes

—

C₄×D₁₀₁

#41

yes

—

#11717581

—

#11717581

—

#11717581

C₂×D₄

yes

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

C₂²×D₄

yes

characteristic

Dic₁₀₁

C₂×#41

yes

characteristic

C₂×C₁₀₁

C₂×C₄×D₄

yes

characteristic

C₂×C₁₀₁

#568

yes

—

D₁₀₁

—

D₁₀₁

—

#2932

—

101

maximal, Sylow

C₁₀₁

#2932

yes

characteristic, complemented, Sylow

C₂×#131

—

101

—

#532

—

101

—

#435

—

101

—

#423

—

101

—

#408

—

101

—

C₂×#97

—

101

—

C₂×#41

—

101

—

#131

—

202

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₄⋊Q₈

#87938

yes

characteristic

#127

—

101

—

#124

—

101

—

#121

—

101

—

C₄×D₄

—

101

—

C₂×C₄²

—

101

—

100

C₄⋊C₈

—

101

—

101

C₂²⋊C₈

—

101

—

102

#97

—

101

—

103

#41

—

202

—

104

#41

—

202

—

105

C₂×Q₈

C₂×#87938

yes

—

106

C₂×D₄

—

101

—

107

C₂×D₄

—

101

—

108

C₂×D₄

—

101

—

109

C₂²×C₄

—

101

—

110

C₂²×C₄

—

101

—

111

C₂²×C₄

—

101

—

112

M₁₆

—

202

—

113

M₁₆

—

202

—

114

C₂×C₈

—

101

—

115

C₂×C₈

—

101

—

116

C₂×C₈

—

101

—

117

C₂×C₈

—

101

—

118

C₂×C₈

—

202

—

119

C₂×C₈

—

202

—

120

C₄⋊C₄

C₂×#87938

yes

—

121

C₂²⋊C₄

—

101

—

122

C₄²

—

101

—

123

C₄²

—

101

—

124

C₄²

C₂×#87938

yes

characteristic

125

C₂³

—

101

—

126

C₂³

—

101

—

127

Q₈

—

128

D₄

—

404

—

129

D₄

—

202

—

130

D₄

—

202

—

131

C₂×C₄

—

101

—

132

C₂×C₄

—

101

—

133

C₂×C₄

—

101

—

134

C₂×C₄

—

101

—

135

C₂×C₄

—

404

—

136

C₂×C₄

—

202

—

137

C₂×C₄

—

202

—

138

C₂×C₄

C₂²×#87938

yes

—

139

C₂×C₄

C₂²×#87938

yes

—

140

C₂×C₄

C₂²×#87938

yes

characteristic

141

C₈

—

101

—

142

C₈

—

202

—

143

C₈

—

202

—

144

C₂²

—

101

—

145

C₂²

—

202

—

146

C₂²

—

202

—

147

C₂²

—

202

—

148

C₂²

C₂³×#87938

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

101

—

152

C₄

—

101

—

153

C₄

—

202

—

154

C₄

D₄×#87938

yes

—

155

C₂

—

404

—

156

C₂

—

202

—

157

C₂

C₂×D₄×#87938

yes

characteristic

158

C₂

#20266257

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₄⋊Q₈×C₁₀₁	C₄
prime radical	#20265728	C₂
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	C₄⋊Q₈	#87938
hypocenter	C₁₀₁	#2932
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#87938

commutator, socle

C₂²×C₁₀₁

C₂³×C₄

Fitting

C₄⋊Q₈×C₁₀₁

C₄

prime radical

#20265728

C₂

cosocle

C₂×Dic₁₀₁

C₂⁴

C₄⋊Q₈

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×#15692343	C₂	semidirect
2. #20266207	C₂	semidirect
3. #20266182	C₂	semidirect
4. #20266152	C₂	semidirect
5. #20266136	C₂	semidirect
27. D₁₀₁⋊C₈	D₄	semidirect
80. C₁₀₁	#2932	semidirect

H ≅ G/N

type

1. C₂×#15692343

C₂

semidirect

2. #20266207

C₂

semidirect

3. #20266182

C₂

semidirect

4. #20266152

C₂

semidirect

5. #20266136

C₂

semidirect

27. D₁₀₁⋊C₈

D₄

semidirect

80. C₁₀₁

#2932

semidirect

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

subgroup	index
C₂×#15692343	2
#20266207	2
#20266182	2
#20266152	2
#20266136	2
#20265728	2
#2932	101

subgroup

index

101

Subgroups — #27187859

Conjugacy classes of subgroups

Complements

Maximal subgroups