Group #27187302 — subgroups

Subgroups — #27187302

Order 12928.

292 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 110 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×C₄⋊#11717581	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11717581	C₂	1	1	yes	maximal, characteristic
3	#20266043	C₂	1	4	yes	maximal
4	C₂×#15692220	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717581	C₂²	1	2	yes	—
6	C₂²×#11717581	C₂²	1	2	yes	—
7	C₄⋊#11717581	—	2	2	no	—
8	C₄×#11717581	—	2	2	no	—
9	C₂²×Dic₂₀₂	C₄	1	1	yes	characteristic
10	#15692220	—	2	2	no	—
11	#15692220	—	2	2	no	—
12	C₂×#13108053	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
16	C₂×#11717581	—	2	2	no	—
17	C₂×#11717581	—	2	2	no	—
18	C₂×#11717581	—	2	2	no	—
19	C₂×#11717581	—	2	4	no	—
20	C₂×#11717581	Q₈	1	4	yes	complemented
21	C₂×Dic₂₀₂	—	2	2	no	—
22	C₂×Dic₂₀₂	—	2	2	no	—
23	C₂×Dic₂₀₂	C₈	1	4	yes	complemented
24	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
26	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
27	#13108053	—	2	4	no	—
28	C₄×Dic₁₀₁	—	2	2	no	—
29	C₄×Dic₁₀₁	—	2	2	no	—
30	C₄×Dic₁₀₁	D₄	1	2	yes	—
31	C₄×Dic₁₀₁	D₄	1	2	yes	—
32	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₁	—	2	2	no	—
35	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄×C₁₀₁	—	2	4	no	—
38	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₁	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₁₀₁	M₁₆	1	2	yes	—
41	C₂×Dic₁₀₁	—	2	2	no	—
42	C₂×Dic₁₀₁	—	2	2	no	—
43	C₂×Dic₁₀₁	—	2	2	no	—
44	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₁₀₁	C₂×C₈	1	4	yes	—
46	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
49	C₂×Dic₁₀₁	#41	1	2	yes	—
50	Dic₂₀₂	—	2	8	no	—
51	#11717581	—	2	4	no	—
52	#11717581	—	4	2	no	—
53	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
55	C₂²×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₁₀₁	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₁₀₁	#131	1	2	yes	—
58	C₄×C₁₀₁	—	4	2	no	—
59	C₄×C₁₀₁	C₂²⋊C₈	1	4	yes	—
60	Dic₁₀₁	—	2	4	no	—
61	Dic₁₀₁	—	2	2	no	—
62	Dic₁₀₁	C₂²⋊Q₈	1	2	yes	—
63	Dic₁₀₁	#124	1	1	yes	characteristic
64	Dic₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₀₁	#391	1	1	yes	characteristic
66	C₂×C₁₀₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₁₀₁	#408	1	1	yes	characteristic
68	C₂×C₁₀₁	C₈×Q₈	1	2	yes	—
69	C₂×C₁₀₁	#446	1	2	yes	—
70	#1984	—	101	1	no	maximal, Sylow
71	C₁₀₁	#1984	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄×Q₈	—	101	1	no	—
73	C₂×C₄⋊C₈	—	101	1	no	—
74	C₂×C₄×C₈	—	101	1	no	—
75	#336	—	101	4	no	—
76	C₂²×Q₈	—	101	1	no	—
77	C₂²×C₈	—	101	2	no	—
78	C₂²×C₈	—	101	2	no	—
79	C₄×Q₈	—	202	2	no	—
80	C₄×Q₈	—	202	2	no	—
81	C₂×C₄⋊C₄	—	101	2	no	—
82	C₂×C₄⋊C₄	#87938	1	1	yes	characteristic
83	C₂×C₄²	—	101	1	no	—
84	C₂×C₄²	—	101	2	no	—
85	C₄⋊C₈	—	202	2	no	—
86	C₄×C₈	—	202	2	no	—
87	C₂×Q₈	—	101	4	no	—
88	C₂×Q₈	—	202	2	no	—
89	C₂×Q₈	—	202	2	no	—
90	C₂²×C₄	—	101	1	no	—
91	C₂²×C₄	—	101	1	no	—
92	C₂²×C₄	—	101	2	no	—
93	C₂²×C₄	C₂×#87938	1	2	yes	—
94	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
95	C₂×C₈	—	101	4	no	—
96	C₂×C₈	—	202	2	no	—
97	C₂×C₈	—	202	4	no	—
98	C₂×C₈	—	202	2	no	—
99	C₂×C₈	—	202	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	202	4	no	—
102	C₄²	—	101	2	no	—
103	C₄²	—	101	2	no	—
104	C₄²	—	202	2	no	—
105	C₄²	—	202	2	no	—
106	C₂³	C₂²×#87938	1	1	yes	characteristic
107	Q₈	—	202	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	101	1	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	—	101	4	no	—
116	C₂×C₄	—	202	2	no	—
117	C₂×C₄	—	202	2	no	—
118	C₂×C₄	—	202	2	no	—
119	C₂×C₄	C₂²⋊#87938	1	2	yes	—
120	C₂×C₄	D₁₀₁⋊C₈	1	2	yes	—
121	C₂×C₄	#13108084	1	2	yes	—
122	C₈	—	404	2	no	—
123	C₈	—	202	4	no	—
124	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
125	C₂²	Q₈×#87938	1	2	yes	—
126	C₂²	C₂×D₁₀₁⋊C₈	1	1	yes	characteristic
127	C₂²	C₂×#13108084	1	1	yes	characteristic
128	C₂²	#15692342	1	2	yes	—
129	C₄	—	4	2	no	—
130	C₄	—	101	1	no	—
131	C₄	—	101	2	no	—
132	C₄	—	101	1	no	—
133	C₄	—	202	4	no	—
134	C₄	—	202	2	no	—
135	C₄	#15692312	1	4	yes	—
136	C₂	(C₂×#87938)⋊Q₈	1	1	yes	characteristic
137	C₂	C₂×#15692312	1	1	yes	characteristic
138	C₂	#20266151	1	1	yes	characteristic
139	C₂	#20266226	1	2	yes	—
140	C₂	#20266227	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11717581

C₂

yes

maximal, characteristic

C₂×C₄×#11717581

C₂

yes

maximal, characteristic

#20266043

C₂

yes

maximal

C₂×#15692220

C₂

yes

maximal, characteristic

C₂²×#11717581

C₂²

yes

—

C₂²×#11717581

C₂²

yes

—

C₄⋊#11717581

—

C₄×#11717581

—

C₂²×Dic₂₀₂

C₄

yes

—

—

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

—

—

—

—

Q₈

yes

complemented

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₈

yes

complemented

C₂²×Dic₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂³

yes

characteristic

#13108053

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

—

C₂×C₄×C₁₀₁

M₁₆

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×C₈

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

C₂×Q₈

yes

—

C₂×Dic₁₀₁

#41

yes

—

Dic₂₀₂

—

#11717581

—

#11717581

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×Q₈

yes

—

C₂²×C₁₀₁

C₂²×C₈

yes

characteristic

C₂²×C₁₀₁

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₁

#131

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂²⋊C₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

#124

yes

characteristic

Dic₁₀₁

C₄⋊Q₈

yes

characteristic

C₂×C₁₀₁

#391

yes

characteristic

C₂×C₁₀₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₀₁

#408

yes

characteristic

C₂×C₁₀₁

C₈×Q₈

yes

—

C₂×C₁₀₁

#446

yes

—

#1984

—

101

maximal, Sylow

C₁₀₁

#1984

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

101

—

C₂×C₄⋊C₈

—

101

—

C₂×C₄×C₈

—

101

—

#336

—

101

—

C₂²×Q₈

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₄×Q₈

—

202

—

C₄×Q₈

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

#87938

yes

characteristic

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

C₄×C₈

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×#87938

yes

—

C₂²×C₄

C₂×#87938

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

202

—

102

C₄²

—

101

—

103

C₄²

—

101

—

104

C₄²

—

202

—

105

C₄²

—

202

—

106

C₂³

C₂²×#87938

yes

characteristic

107

Q₈

—

202

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

101

—

111

C₂×C₄

—

101

—

112

C₂×C₄

—

101

—

113

C₂×C₄

—

101

—

114

C₂×C₄

—

101

—

115

C₂×C₄

—

101

—

116

C₂×C₄

—

202

—

117

C₂×C₄

—

202

—

118

C₂×C₄

—

202

—

119

C₂×C₄

C₂²⋊#87938

yes

—

120

C₂×C₄

D₁₀₁⋊C₈

yes

—

121

C₂×C₄

#13108084

yes

—

122

C₈

—

404

—

123

C₈

—

202

—

124

C₂²

C₂×C₂²⋊#87938

yes

characteristic

125

C₂²

Q₈×#87938

yes

—

126

C₂²

C₂×D₁₀₁⋊C₈

yes

characteristic

127

C₂²

C₂×#13108084

yes

characteristic

128

C₂²

#15692342

yes

—

129

C₄

—

130

C₄

—

101

—

131

C₄

—

101

—

132

C₄

—

101

—

133

C₄

—

202

—

134

C₄

—

202

—

135

C₄

#15692312

yes

—

136

C₂

(C₂×#87938)⋊Q₈

yes

characteristic

137

C₂

C₂×#15692312

yes

characteristic

138

C₂

#20266151

yes

characteristic

139

C₂

#20266226

yes

—

140

C₂

#20266227

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87938
commutator	C₂²×C₁₀₁	C₂²×C₈
Fitting	C₂×C₄⋊C₄×C₁₀₁	C₄
socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	C₂×C₄⋊C₄	#87938
hypocenter	C₁₀₁	#1984
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#87938

commutator

C₂²×C₁₀₁

C₂²×C₈

Fitting

C₂×C₄⋊C₄×C₁₀₁

C₄

socle, prime radical

C₂³×C₁₀₁

C₂²×C₄

cosocle

C₂²×Dic₁₀₁

C₂³

C₂×C₄⋊C₄

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
20. C₂×#11717581	Q₈	semidirect
23. C₂×Dic₂₀₂	C₈	semidirect
71. C₁₀₁	#1984	semidirect

H ≅ G/N

type

20. C₂×#11717581

Q₈

semidirect

23. C₂×Dic₂₀₂

C₈

semidirect

71. C₁₀₁

#1984

semidirect

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

subgroup	index
C₂×C₄⋊#11717581	2
C₂×C₄×#11717581	2
#20266043	2
C₂×#15692220	2
#1984	101

subgroup

index

101

Subgroups — #27187302

Conjugacy classes of subgroups

Complements

Maximal subgroups