Group #27187088 — subgroups

Subgroups — #27187088

Order 12928.

272 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 68 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	#20266245	C₂	1	1	yes	maximal, characteristic
2	#20266188	C₂	1	2	yes	maximal
3	(C₂×#87938)⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	#20266049	C₂	1	2	yes	maximal
5	#101×D₁₀₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×#87938	—	2	2	no	—
7	Q₈×#87938	—	2	2	no	—
8	#15692334	—	2	2	no	—
9	#15692334	C₂²	1	2	yes	—
10	C₂×C₄⋊#87938	C₂²	1	1	yes	characteristic
11	C₂×C₄×#87938	C₂²	1	1	yes	characteristic
12	#15692324	—	2	2	no	—
13	#15692323	—	2	2	no	—
14	(C₂×#87938)⋊C₄	—	2	2	no	—
15	C₂×Q₈×D₁₀₁	C₂²	1	1	yes	characteristic
16	M₁₆×D₁₀₁	C₂²	1	2	yes	—
17	#15692174	C₄	1	1	yes	characteristic, complemented
18	#15692173	C₄	1	2	yes	complemented
19	#101×C₁₀₁	C₄	1	1	yes	characteristic, complemented
20	C₂²×#87938	—	2	1	no	—
21	C₂²×#87938	—	2	1	no	—
22	C₄⋊#87938	—	2	2	no	—
23	C₄⋊#87938	—	2	2	no	—
24	C₄⋊#87938	—	2	2	no	—
25	C₄⋊#87938	D₄	1	2	yes	—
26	C₄⋊#87938	D₄	1	2	yes	—
27	C₂²⋊#87938	—	2	2	no	—
28	C₄×#87938	—	2	4	no	—
29	C₄×#87938	—	2	2	no	—
30	C₄×#87938	—	2	2	no	—
31	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
32	C₂×Dic₂₀₂	Q₈	1	2	yes	—
33	Q₈×D₁₀₁	—	2	2	no	—
34	Q₈×D₁₀₁	—	2	2	no	—
35	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
36	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
37	#13108058	C₂×C₄	1	2	yes	—
38	#13108057	—	2	2	no	—
39	C₈×D₁₀₁	—	2	2	no	—
40	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
41	M₁₆×C₁₀₁	C₂×C₄	1	2	yes	—
42	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
43	C₂×#87938	—	2	2	no	—
44	C₂×#87938	—	2	2	no	—
45	C₂×#87938	—	2	2	no	—
46	C₂×#87938	—	4	1	no	—
47	C₂×#87938	—	4	1	no	—
48	C₂×#87938	—	4	2	no	—
49	C₂×#87938	—	4	1	no	—
50	Q₈×C₁₀₁	—	2	2	no	—
51	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
54	C₂×Dic₁₀₁	C₂×Q₈	1	1	yes	characteristic
55	C₄×D₁₀₁	—	2	2	no	—
56	C₄×D₁₀₁	—	2	2	no	—
57	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
58	C₄×D₁₀₁	#41	1	2	yes	—
59	Dic₂₀₂	—	2	2	no	—
60	Dic₂₀₂	—	2	4	no	—
61	C₈×C₁₀₁	—	2	2	no	—
62	#11717579	—	2	2	no	—
63	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×D₁₀₁	—	2	1	no	—
65	C₂×D₁₀₁	—	2	1	no	—
66	C₂×D₁₀₁	#120	1	1	yes	characteristic
67	C₂×D₁₀₁	#123	1	1	yes	characteristic
68	#87938	—	4	2	no	—
69	#87938	—	4	2	no	—
70	#87938	—	4	2	no	—
71	#87938	—	8	1	no	—
72	C₄×C₁₀₁	—	2	2	no	—
73	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
74	Dic₁₀₁	—	2	2	no	—
75	Dic₁₀₁	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₁₀₁	—	2	1	no	—
77	C₂×C₁₀₁	#385	1	1	yes	characteristic
78	D₁₀₁	—	2	1	no	—
79	D₁₀₁	#454	1	1	yes	characteristic
80	D₁₀₁	#456	1	1	yes	characteristic
81	#1857	—	101	1	no	maximal, Sylow
82	C₁₀₁	#1857	1	1	yes	characteristic, complemented, Sylow
83	#519	—	101	1	no	—
84	#428	—	101	2	no	—
85	C₂×#101	—	101	1	no	—
86	#391	—	101	1	no	—
87	#339	—	101	2	no	—
88	C₂²×Q₈	—	101	1	no	—
89	C₂×M₁₆	—	101	2	no	—
90	C₄×Q₈	—	202	2	no	—
91	C₄×Q₈	—	202	2	no	—
92	#117	—	101	2	no	—
93	#117	—	202	2	no	—
94	C₂×C₄⋊C₄	—	101	1	no	—
95	C₂×C₄²	—	101	1	no	—
96	#107	—	202	2	no	—
97	#106	—	202	2	no	—
98	#101	—	101	1	no	—
99	#101	—	101	2	no	—
100	#101	#87938	1	1	yes	characteristic, complemented
101	#95	—	202	2	no	—
102	C₂×Q₈	—	101	2	no	—
103	C₂×Q₈	—	101	1	no	—
104	C₂×Q₈	—	202	2	no	—
105	C₂×Q₈	—	202	2	no	—
106	C₂×Q₈	C₂×#87938	1	1	yes	characteristic
107	C₂²×C₄	—	101	1	no	—
108	C₂²×C₄	—	101	2	no	—
109	C₂²×C₄	—	202	1	no	—
110	C₂²×C₄	—	202	1	no	—
111	M₁₆	—	101	2	no	—
112	M₁₆	—	202	2	no	—
113	M₁₆	C₂×#87938	1	2	yes	—
114	C₂×C₈	—	202	2	no	—
115	C₄⋊C₄	—	101	2	no	—
116	C₄⋊C₄	—	101	2	no	—
117	C₄⋊C₄	—	202	2	no	—
118	C₄⋊C₄	—	202	2	no	—
119	C₄⋊C₄	—	202	2	no	—
120	C₂²⋊C₄	—	202	2	no	—
121	C₄²	—	202	2	no	—
122	C₄²	—	202	2	no	—
123	C₄²	—	202	4	no	—
124	C₂³	—	101	1	no	—
125	Q₈	—	2	2	no	—
126	Q₈	—	202	4	no	—
127	Q₈	—	202	2	no	—
128	C₂×C₄	—	101	1	no	—
129	C₂×C₄	—	101	2	no	—
130	C₂×C₄	—	101	2	no	—
131	C₂×C₄	—	101	2	no	—
132	C₂×C₄	—	404	1	no	—
133	C₂×C₄	—	404	2	no	—
134	C₂×C₄	—	404	1	no	—
135	C₂×C₄	—	404	1	no	—
136	C₂×C₄	—	202	2	no	—
137	C₂×C₄	—	202	2	no	—
138	C₂×C₄	—	202	2	no	—
139	C₂×C₄	—	202	2	no	—
140	C₂×C₄	—	202	2	no	—
141	C₂×C₄	C₄⋊#87938	1	2	yes	—
142	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	202	2	no	—
145	C₂²	—	101	1	no	—
146	C₂²	—	101	1	no	—
147	C₂²	—	202	1	no	—
148	C₂²	—	202	1	no	—
149	C₂²	C₂×C₄⋊#87938	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	808	1	no	—
152	C₄	—	101	2	no	—
153	C₄	—	404	2	no	—
154	C₄	—	404	2	no	—
155	C₄	—	404	2	no	—
156	C₄	—	202	2	no	—
157	C₄	D₄×#87938	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	101	1	no	—
160	C₂	—	101	1	no	—
161	C₂	—	202	1	no	—
162	C₂	#20266124	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20266245

C₂

yes

maximal, characteristic

#20266188

C₂

yes

maximal

(C₂×#87938)⋊Q₈

C₂

yes

maximal, characteristic

#20266049

C₂

yes

maximal

#101×D₁₀₁

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₄⋊#87938

C₂²

yes

characteristic

C₂×C₄×#87938

C₂²

yes

—

—

—

C₂×Q₈×D₁₀₁

C₂²

yes

characteristic

M₁₆×D₁₀₁

C₂²

yes

—

#15692174

C₄

yes

characteristic, complemented

#15692173

C₄

yes

complemented

#101×C₁₀₁

C₄

yes

characteristic, complemented

C₂²×#87938

—

C₂²×#87938

—

—

—

—

D₄

yes

—

C₄⋊#87938

D₄

yes

—

—

—

—

—

C₂×Dic₂₀₂

C₂×C₄

yes

characteristic

C₂×Dic₂₀₂

Q₈

yes

—

Q₈×D₁₀₁

—

Q₈×D₁₀₁

—

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

characteristic

#13108058

C₂×C₄

yes

—

#13108057

—

C₈×D₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

characteristic

M₁₆×C₁₀₁

C₂×C₄

yes

—

C₂²×D₁₀₁

C₂×D₄

yes

—

—

—

—

—

—

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×Q₈

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

C₂×D₄

yes

—

C₄×D₁₀₁

#41

yes

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

yes

C₂×D₁₀₁

yes

—

—

—

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

C₂²⋊Q₈

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#385

yes

characteristic

D₁₀₁

—

D₁₀₁

yes

D₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1857

yes

characteristic, complemented, Sylow

#519

—

101

—

#428

—

101

—

C₂×#101

—

101

—

#391

—

101

—

#339

—

101

—

C₂²×Q₈

—

101

—

C₂×M₁₆

—

101

—

C₄×Q₈

—

202

—

C₄×Q₈

—

202

—

#117

—

101

—

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄²

—

101

—

#107

—

202

—

#106

—

202

—

#101

—

101

—

#101

—

101

—

100

#101

#87938

yes

characteristic, complemented

101

#95

—

202

—

102

C₂×Q₈

—

101

—

103

C₂×Q₈

—

101

—

104

C₂×Q₈

—

202

—

105

C₂×Q₈

—

202

—

106

C₂×Q₈

C₂×#87938

yes

characteristic

107

C₂²×C₄

—

101

—

108

C₂²×C₄

—

101

—

109

C₂²×C₄

—

202

—

110

C₂²×C₄

—

202

—

111

M₁₆

—

101

—

112

M₁₆

—

202

—

113

M₁₆

C₂×#87938

yes

—

114

C₂×C₈

—

202

—

115

C₄⋊C₄

—

101

—

116

C₄⋊C₄

—

101

—

117

C₄⋊C₄

—

202

—

118

C₄⋊C₄

—

202

—

119

C₄⋊C₄

—

202

—

120

C₂²⋊C₄

—

202

—

121

C₄²

—

202

—

122

C₄²

—

202

—

123

C₄²

—

202

—

124

C₂³

—

101

—

125

Q₈

—

126

Q₈

—

202

—

127

Q₈

—

202

—

128

C₂×C₄

—

101

—

129

C₂×C₄

—

101

—

130

C₂×C₄

—

101

—

131

C₂×C₄

—

101

—

132

C₂×C₄

—

404

—

133

C₂×C₄

—

404

—

134

C₂×C₄

—

404

—

135

C₂×C₄

—

404

—

136

C₂×C₄

—

202

—

137

C₂×C₄

—

202

—

138

C₂×C₄

—

202

—

139

C₂×C₄

—

202

—

140

C₂×C₄

—

202

—

141

C₂×C₄

C₄⋊#87938

yes

—

142

C₂×C₄

C₂²×#87938

yes

characteristic

143

C₈

—

144

C₈

—

202

—

145

C₂²

—

101

—

146

C₂²

—

101

—

147

C₂²

—

202

—

148

C₂²

—

202

—

149

C₂²

C₂×C₄⋊#87938

yes

characteristic

150

C₄

—

151

C₄

—

808

—

152

C₄

—

101

—

153

C₄

—

404

—

154

C₄

—

404

—

155

C₄

—

404

—

156

C₄

—

202

—

157

C₄

D₄×#87938

yes

—

158

C₂

—

159

C₂

—

101

—

160

C₂

—

101

—

161

C₂

—

202

—

162

C₂

#20266124

yes

characteristic

	subgroup	G/H
center	C₂	#20266124
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	#101×C₁₀₁	C₄
socle	C₂×C₁₀₁	#385
prime radical	C₂²×D₁₀₁	C₂×D₄
cosocle	C₂×C₄×D₁₀₁	C₂³
hypercenter	#101	#87938
hypocenter	C₁₀₁	#1857
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

#20266124

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting

#101×C₁₀₁

C₄

socle

C₂×C₁₀₁

#385

prime radical

C₂²×D₁₀₁

C₂×D₄

cosocle

C₂×C₄×D₁₀₁

C₂³

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
17. #15692174	C₄	semidirect
18. #15692173	C₄	semidirect
19. #101×C₁₀₁	C₄	semidirect
82. C₁₀₁	#1857	semidirect
100. #101	#87938	semidirect

H ≅ G/N

type

17. #15692174

C₄

semidirect

18. #15692173

C₄

semidirect

19. #101×C₁₀₁

C₄

semidirect

82. C₁₀₁

#1857

semidirect

100. #101

#87938

semidirect

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

subgroup	index
#20266245	2
#20266188	2
(C₂×#87938)⋊Q₈	2
#20266049	2
#101×D₁₀₁	2
#1857	101

subgroup

index

#101×D₁₀₁

101

Subgroups — #27187088

Conjugacy classes of subgroups

Complements

Maximal subgroups