Group #27187263 — subgroups

Subgroups — #27187263

Order 12928.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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×#15692318	C₂	1	4	yes	maximal
2	#20266136	C₂	1	1	yes	maximal, characteristic
3	#20266121	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₀₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#87938	C₂²	1	2	yes	—
7	#15692318	—	2	8	no	—
8	#15692310	—	2	2	no	—
9	(C₂×#87938)⋊C₄	—	2	2	no	—
10	C₂×Q₈×D₁₀₁	C₂²	1	2	yes	—
11	#15692255	C₄	1	1	yes	characteristic, complemented
12	#15692254	—	2	2	no	—
13	#15692253	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₁₀₁	—	2	2	no	—
15	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic, complemented
17	C₂²×#87938	—	2	2	no	—
18	D₁₀₁⋊C₈	—	2	2	no	—
19	D₁₀₁⋊C₈	D₄	1	4	yes	—
20	C₂×#11717581	—	2	2	no	—
21	C₄⋊#87938	—	2	4	no	—
22	C₂×Dic₂₀₂	—	2	2	no	—
23	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
24	Q₈×D₁₀₁	—	2	4	no	—
25	Q₈×D₁₀₁	—	2	4	no	—
26	C₂×C₄×D₁₀₁	—	2	2	no	—
27	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
28	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₀₁	—	2	2	no	—
30	#13108053	—	2	4	no	—
31	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₀₁	D₄	1	2	yes	—
33	C₂×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₁₀₁	—	2	2	no	—
35	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
37	C₂×#87938	—	4	2	no	—
38	C₂×#87938	—	4	4	no	—
39	Q₈×C₁₀₁	—	2	4	no	—
40	C₂×C₄×C₁₀₁	—	2	2	no	—
41	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₁	—	2	2	no	—
44	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
46	C₄×D₁₀₁	—	2	4	no	—
47	C₄×D₁₀₁	—	4	2	no	—
48	C₄×D₁₀₁	—	4	2	no	—
49	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
50	C₄×D₁₀₁	#41	1	2	yes	—
51	Dic₂₀₂	—	2	4	no	—
52	Dic₂₀₂	—	4	4	no	—
53	#11717581	—	2	4	no	—
54	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×D₁₀₁	#124	1	1	yes	characteristic
56	C₂×D₁₀₁	#127	1	1	yes	characteristic
57	C₂×D₁₀₁	#137	1	4	yes	—
58	#87938	—	8	2	no	—
59	C₄×C₁₀₁	—	2	2	no	—
60	C₄×C₁₀₁	—	4	2	no	—
61	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
62	Dic₁₀₁	—	2	2	no	—
63	Dic₁₀₁	—	4	2	no	—
64	Dic₁₀₁	#121	1	2	yes	—
65	C₂×C₁₀₁	#390	1	1	yes	characteristic
66	C₂×C₁₀₁	#419	1	2	yes	—
67	D₁₀₁	#491	1	2	yes	—
68	D₁₀₁	#497	1	2	yes	—
69	#1960	—	101	1	no	maximal, Sylow
70	C₁₀₁	#1960	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	101	1	no	—
72	C₂×#103	—	101	4	no	—
73	C₂×#97	—	101	1	no	—
74	#382	—	101	1	no	—
75	C₂²×Q₈	—	101	2	no	—
76	C₂²×C₈	—	101	2	no	—
77	C₄⋊Q₈	—	101	2	no	—
78	C₄⋊Q₈	—	101	1	no	—
79	C₄⋊Q₈	—	202	2	no	—
80	C₄⋊Q₈	#87938	1	1	yes	characteristic, complemented
81	C₂×C₄⋊C₄	—	101	2	no	—
82	C₂×C₄⋊C₄	—	202	2	no	—
83	C₂×C₄²	—	101	1	no	—
84	#103	—	202	8	no	—
85	#97	—	202	2	no	—
86	#95	—	202	2	no	—
87	C₂×Q₈	—	101	2	no	—
88	C₂×Q₈	—	202	2	no	—
89	C₂×Q₈	—	202	4	no	—
90	C₂×Q₈	—	202	4	no	—
91	C₂×Q₈	C₂×#87938	1	2	yes	—
92	C₂²×C₄	—	101	1	no	—
93	C₂²×C₄	—	101	2	no	—
94	C₂²×C₄	—	202	2	no	—
95	C₂²×C₄	—	202	2	no	—
96	C₂×C₈	—	101	4	no	—
97	C₂×C₈	—	202	2	no	—
98	C₂×C₈	—	202	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	202	4	no	—
101	C₄⋊C₄	—	202	4	no	—
102	C₄⋊C₄	—	202	2	no	—
103	C₄²	—	101	2	no	—
104	C₄²	—	101	1	no	—
105	C₄²	C₂×#87938	1	1	yes	characteristic
106	C₂³	—	101	1	no	—
107	Q₈	—	2	4	no	—
108	Q₈	—	404	4	no	—
109	Q₈	—	202	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	101	1	no	—
112	C₂×C₄	—	101	2	no	—
113	C₂×C₄	—	101	2	no	—
114	C₂×C₄	—	101	2	no	—
115	C₂×C₄	—	404	2	no	—
116	C₂×C₄	—	404	2	no	—
117	C₂×C₄	—	404	2	no	—
118	C₂×C₄	—	404	4	no	—
119	C₂×C₄	—	202	4	no	—
120	C₂×C₄	—	202	2	no	—
121	C₂×C₄	C₂²⋊#87938	1	2	yes	—
122	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
123	C₈	—	202	4	no	—
124	C₂²	—	101	1	no	—
125	C₂²	—	101	1	no	—
126	C₂²	—	101	4	no	—
127	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	808	2	no	—
131	C₄	—	101	2	no	—
132	C₄	—	404	2	no	—
133	C₄	—	202	2	no	—
134	C₄	D₄×#87938	1	2	yes	—
135	C₂	—	101	2	no	—
136	C₂	—	101	2	no	—
137	C₂	#20266129	1	1	yes	characteristic
138	C₂	#20266170	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692318

C₂

yes

maximal

#20266136

C₂

yes

maximal, characteristic

#20266121

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₀₁

C₂

yes

maximal, characteristic

C₂×D₁₀₁⋊C₈

C₂²

yes

—

C₂×C₄⋊#87938

C₂²

yes

—

#15692318

—

#15692310

—

(C₂×#87938)⋊C₄

—

C₂×Q₈×D₁₀₁

C₂²

yes

—

#15692255

C₄

yes

characteristic, complemented

#15692254

—

#15692253

C₄

yes

complemented

C₄⋊C₄×D₁₀₁

—

C₄²×D₁₀₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₁

C₄

yes

characteristic, complemented

C₂²×#87938

—

D₁₀₁⋊C₈

—

D₁₀₁⋊C₈

D₄

yes

—

C₂×#11717581

—

C₄⋊#87938

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

Q₈×D₁₀₁

—

Q₈×D₁₀₁

—

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

characteristic

C₄⋊Dic₁₀₁

—

#13108053

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×D₁₀₁

C₂×D₄

yes

characteristic

C₂×#87938

—

C₂×#87938

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

C₂×D₄

yes

—

C₄×D₁₀₁

#41

yes

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717581

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

C₂×D₁₀₁

yes

C₂×D₁₀₁

yes

C₂×D₁₀₁

yes

—

#87938

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#121

yes

—

C₂×C₁₀₁

#390

yes

characteristic

C₂×C₁₀₁

#419

yes

—

D₁₀₁

#491

yes

—

D₁₀₁

#497

yes

—

#1960

—

101

maximal, Sylow

C₁₀₁

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

101

—

C₂×#103

—

101

—

C₂×#97

—

101

—

#382

—

101

—

C₂²×Q₈

—

101

—

C₂²×C₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

202

—

C₄⋊Q₈

#87938

yes

characteristic, complemented

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄²

—

101

—

#103

—

202

—

#97

—

202

—

#95

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

C₂×#87938

yes

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

100

C₄⋊C₄

—

202

—

101

C₄⋊C₄

—

202

—

102

C₄⋊C₄

—

202

—

103

C₄²

—

101

—

104

C₄²

—

101

—

105

C₄²

C₂×#87938

yes

characteristic

106

C₂³

—

101

—

107

Q₈

—

108

Q₈

—

404

—

109

Q₈

—

202

—

110

C₂×C₄

—

111

C₂×C₄

—

101

—

112

C₂×C₄

—

101

—

113

C₂×C₄

—

101

—

114

C₂×C₄

—

101

—

115

C₂×C₄

—

404

—

116

C₂×C₄

—

404

—

117

C₂×C₄

—

404

—

118

C₂×C₄

—

404

—

119

C₂×C₄

—

202

—

120

C₂×C₄

—

202

—

121

C₂×C₄

C₂²⋊#87938

yes

—

122

C₂×C₄

C₂²×#87938

yes

characteristic

123

C₈

—

202

—

124

C₂²

—

101

—

125

C₂²

—

101

—

126

C₂²

—

101

—

127

C₂²

C₂×C₂²⋊#87938

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

808

—

131

C₄

—

101

—

132

C₄

—

404

—

133

C₄

—

202

—

134

C₄

D₄×#87938

yes

—

135

C₂

—

101

—

136

C₂

—

101

—

137

C₂

#20266129

yes

characteristic

138

C₂

#20266170

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	C₄⋊Q₈×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₁	C₂×D₄
cosocle	C₂×C₄×D₁₀₁	C₂³
hypercenter	C₄⋊Q₈	#87938
hypocenter	C₁₀₁	#1960
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting

C₄⋊Q₈×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₁

C₂×D₄

cosocle

C₂×C₄×D₁₀₁

C₂³

C₄⋊Q₈

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #15692255	C₄	semidirect
13. #15692253	C₄	semidirect
16. C₄⋊Q₈×C₁₀₁	C₄	semidirect
70. C₁₀₁	#1960	semidirect
80. C₄⋊Q₈	#87938	semidirect

H ≅ G/N

type

11. #15692255

C₄

semidirect

13. #15692253

C₄

semidirect

16. C₄⋊Q₈×C₁₀₁

C₄

semidirect

70. C₁₀₁

#1960

semidirect

80. C₄⋊Q₈

#87938

semidirect

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

subgroup	index
C₂×#15692318	2
#20266136	2
#20266121	2
C₄⋊Q₈×D₁₀₁	2
#1960	101

subgroup

index

C₄⋊Q₈×D₁₀₁

101

Subgroups — #27187263

Conjugacy classes of subgroups

Complements

Maximal subgroups