Group #27172083 — subgroups

Subgroups — #27172083

Order 12928.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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×#15692255	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
3	#20264940	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀₂	—	2	1	no	—
6	#15692255	—	2	2	no	—
7	#15692255	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
9	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₀₁	—	2	1	no	—
12	#15692162	—	2	1	no	—
13	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₁	—	2	2	no	—
16	C₂×Dic₂₀₂	—	2	4	no	—
17	C₂×Dic₂₀₂	—	4	2	no	—
18	C₂×Dic₂₀₂	—	4	2	no	—
19	C₂²×Dic₁₀₁	—	2	2	no	—
20	C₂²×Dic₁₀₁	—	2	2	no	—
21	C₄⋊Dic₁₀₁	—	2	4	no	—
22	C₄⋊Dic₁₀₁	—	2	2	no	—
23	C₄⋊Dic₁₀₁	—	2	2	no	—
24	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₀₁	Q₈	1	4	yes	complemented
26	C₄×Dic₁₀₁	—	4	2	no	—
27	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₀₁	—	2	2	no	—
31	C₂×C₈×C₁₀₁	—	2	4	no	—
32	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
33	C₄²×C₁₀₁	D₄	1	2	yes	—
34	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₁	—	2	2	no	—
36	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₀₁	D₈	1	2	yes	—
39	C₂×C₄×C₁₀₁	SD₁₆	1	2	yes	—
40	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₀₁	#41	1	2	yes	—
44	C₂×Dic₁₀₁	—	2	4	no	—
45	C₂×Dic₁₀₁	—	2	4	no	—
46	C₂×Dic₁₀₁	—	4	2	no	—
47	C₂×Dic₁₀₁	—	4	2	no	—
48	C₂×Dic₁₀₁	—	4	2	no	—
49	C₂×Dic₁₀₁	—	4	2	no	—
50	Dic₂₀₂	—	4	8	no	—
51	C₈×C₁₀₁	—	4	2	no	—
52	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₁₀₁	#137	1	2	yes	—
57	C₄×C₁₀₁	—	2	2	no	—
58	C₄×C₁₀₁	#102	1	4	yes	—
59	C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
60	C₄×C₁₀₁	#124	1	1	yes	characteristic
61	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₁₀₁	—	4	4	no	—
63	Dic₁₀₁	—	4	4	no	—
64	C₂×C₁₀₁	#391	1	1	yes	characteristic
65	C₂×C₁₀₁	C₂×#102	1	1	yes	characteristic
66	C₂×C₁₀₁	#419	1	1	yes	characteristic
67	C₂×C₁₀₁	#474	1	2	yes	—
68	C₂×C₁₀₁	#475	1	2	yes	—
69	#1988	—	101	1	no	maximal, Sylow
70	C₁₀₁	#1988	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	101	1	no	—
72	C₂×C₄⋊C₈	D₁₀₁	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	101	1	no	—
74	#340	—	101	4	no	—
75	C₂²×Q₈	—	202	1	no	—
76	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
77	C₄⋊Q₈	—	101	4	no	—
78	C₄⋊Q₈	—	202	2	no	—
79	C₂×C₄⋊C₄	—	101	2	no	—
80	C₂×C₄⋊C₄	—	101	2	no	—
81	C₂×C₄⋊C₄	—	202	1	no	—
82	C₂×C₄²	—	202	1	no	—
83	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	#95	—	202	1	no	—
86	C₂×Q₈	—	404	2	no	—
87	C₂×Q₈	—	404	2	no	—
88	C₂×Q₈	—	202	4	no	—
89	C₂²×C₄	—	202	2	no	—
90	C₂²×C₄	—	202	2	no	—
91	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
92	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₄⋊C₄	—	101	4	no	—
97	C₄⋊C₄	—	101	4	no	—
98	C₄⋊C₄	—	202	4	no	—
99	C₄⋊C₄	—	202	2	no	—
100	C₄⋊C₄	—	202	2	no	—
101	C₄²	—	404	2	no	—
102	C₄²	C₄×D₁₀₁	1	2	yes	—
103	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
104	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
105	Q₈	—	404	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	404	2	no	—
108	C₂×C₄	—	404	2	no	—
109	C₂×C₄	—	404	2	no	—
110	C₂×C₄	—	404	2	no	—
111	C₂×C₄	—	202	4	no	—
112	C₂×C₄	—	202	4	no	—
113	C₂×C₄	#13108051	1	2	yes	—
114	C₂×C₄	#13108051	1	2	yes	—
115	C₂×C₄	D₈₀₈	1	2	yes	—
116	C₂×C₄	#13108063	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
120	C₂×C₄	#13108073	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₂²	C₂×#13108051	1	1	yes	characteristic
123	C₂²	#15692235	1	2	yes	—
124	C₂²	C₂×D₈₀₈	1	1	yes	characteristic
125	C₂²	C₂×#13108063	1	1	yes	characteristic
126	C₂²	#15692288	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	404	4	no	—
129	C₄	—	404	4	no	—
130	C₄	#15692177	1	4	yes	—
131	C₄	#15692220	1	2	yes	—
132	C₄	#15692242	1	1	yes	characteristic
133	C₄	#15692253	1	1	yes	characteristic
134	C₂	#20265082	1	1	yes	characteristic
135	C₂	C₂×#15692177	1	1	yes	characteristic
136	C₂	#20265187	1	1	yes	characteristic
137	C₂	#20265458	1	2	yes	—
138	C₂	#20265463	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692255

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₀₁

C₂

yes

maximal, characteristic

#20264940

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

—

#15692255

—

#15692255

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

—

#15692162

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

C₄⋊Dic₁₀₁

Q₈

yes

complemented

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

D₈

yes

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

—

C₂²×C₁₀₁

C₂×D₈

yes

characteristic

C₂²×C₁₀₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₁

#137

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#102

yes

—

C₄×C₁₀₁

C₄×Q₈

yes

—

C₄×C₁₀₁

#124

yes

characteristic

C₄×C₁₀₁

C₄⋊Q₈

yes

characteristic

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#475

yes

—

#1988

—

101

maximal, Sylow

C₁₀₁

#1988

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

101

—

C₂×C₄⋊C₈

D₁₀₁

yes

characteristic

C₄×C₄⋊C₄

—

101

—

#340

—

101

—

C₂²×Q₈

—

202

—

C₂²×C₈

C₂×D₁₀₁

yes

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄²

—

202

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

#95

—

202

—

C₂×Q₈

—

404

—

C₂×Q₈

—

404

—

C₂×Q₈

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

D₄₀₄

yes

characteristic

C₂²×C₄

D₄₀₄

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

100

C₄⋊C₄

—

202

—

101

C₄²

—

404

—

102

C₄²

C₄×D₁₀₁

yes

—

103

C₄²

C₂²⋊D₁₀₁

yes

—

104

C₂³

C₂×D₄₀₄

yes

characteristic

105

Q₈

—

404

—

106

C₂×C₄

—

107

C₂×C₄

—

404

—

108

C₂×C₄

—

404

—

109

C₂×C₄

—

404

—

110

C₂×C₄

—

404

—

111

C₂×C₄

—

202

—

112

C₂×C₄

—

202

—

113

C₂×C₄

#13108051

yes

—

114

C₂×C₄

#13108051

yes

—

115

C₂×C₄

D₈₀₈

yes

—

116

C₂×C₄

#13108063

yes

—

117

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

119

C₂×C₄

Q₈×D₁₀₁

yes

—

120

C₂×C₄

#13108073

yes

—

121

C₈

—

122

C₂²

C₂×#13108051

yes

characteristic

123

C₂²

#15692235

yes

—

124

C₂²

C₂×D₈₀₈

yes

characteristic

125

C₂²

C₂×#13108063

yes

characteristic

126

C₂²

#15692288

yes

—

127

C₄

—

128

C₄

—

404

—

129

C₄

—

404

—

130

C₄

#15692177

yes

—

131

C₄

#15692220

yes

—

132

C₄

#15692242

yes

characteristic

133

C₄

#15692253

yes

characteristic

134

C₂

#20265082

yes

characteristic

135

C₂

C₂×#15692177

yes

characteristic

136

C₂

#20265187

yes

characteristic

137

C₂

#20265458

yes

—

138

C₂

#20265463

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₀₄
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×C₄⋊C₈×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₁
hypocenter	C₁₀₁	#1988
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₄₀₄

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×C₄⋊C₈×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₁

hypocenter

C₁₀₁

#1988

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #15692255	C₄	semidirect
25. C₄⋊Dic₁₀₁	Q₈	semidirect
70. C₁₀₁	#1988	semidirect

H ≅ G/N

type

7. #15692255

C₄

semidirect

25. C₄⋊Dic₁₀₁

Q₈

semidirect

70. C₁₀₁

#1988

semidirect

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

subgroup	index
C₂×#15692255	2
C₄×C₄⋊Dic₁₀₁	2
#20264940	2
C₂×C₄⋊C₈×C₁₀₁	2
#1988	101

subgroup

index

C₂×#15692255

C₄×C₄⋊Dic₁₀₁

#20264940

C₂×C₄⋊C₈×C₁₀₁

#1988

101

Subgroups — #27172083

Conjugacy classes of subgroups

Complements

Maximal subgroups