Group #27171964 — subgroups

Subgroups — #27171964

Order 12928.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20265727	C₂	1	1	yes	maximal, characteristic
2	#20265201	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692183	C₂	1	4	yes	maximal, complemented
4	#376×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13108074	C₂²	1	2	yes	—
6	C₂×#13108058	C₂²	1	2	yes	—
7	#15692255	C₄	1	1	yes	characteristic, complemented
8	#15692252	C₄	1	1	yes	characteristic, complemented
9	#15692243	C₄	1	2	yes	complemented
10	#15692229	—	2	2	no	—
11	C₄×D₄₀₄	—	2	2	no	—
12	C₄⋊#11717579	—	2	2	no	—
13	#15692183	—	2	8	no	—
14	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
16	#95×C₁₀₁	—	2	2	no	—
17	#13108074	—	2	4	no	—
18	#13108074	—	2	4	no	—
19	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
20	C₂×D₄₀₄	—	2	2	no	—
21	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
23	C₂×C₄×D₁₀₁	—	2	2	no	—
24	#13108058	—	2	4	no	—
25	C₂×#11717579	—	2	2	no	—
26	C₄⋊Dic₁₀₁	—	2	2	no	—
27	#13108051	—	2	4	no	—
28	C₂²×C₄×C₁₀₁	—	2	2	no	—
29	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₄²×C₁₀₁	—	2	2	no	—
32	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₀₁	D₄	1	2	yes	—
35	C₄²×C₁₀₁	D₄	1	4	yes	complemented
36	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₁₀₁	—	2	2	no	—
38	C₂×C₄×C₁₀₁	—	2	4	no	—
39	C₂×C₄×C₁₀₁	—	4	2	no	—
40	C₂×C₄×C₁₀₁	—	4	4	no	—
41	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₁	#41	1	2	yes	—
47	C₂²⋊D₁₀₁	—	4	4	no	—
48	C₂×Dic₁₀₁	—	2	2	no	—
49	D₄₀₄	—	2	4	no	—
50	D₄₀₄	—	4	4	no	—
51	C₄×D₁₀₁	—	4	2	no	—
52	C₄×D₁₀₁	—	4	2	no	—
53	Dic₂₀₂	—	2	4	no	—
54	#11717579	—	4	2	no	—
55	C₂²×C₁₀₁	—	2	2	no	—
56	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₀₁	#121	1	2	yes	—
58	C₂×D₁₀₁	—	4	2	no	—
59	C₂×D₁₀₁	—	4	4	no	—
60	C₄×C₁₀₁	—	2	2	no	—
61	C₄×C₁₀₁	—	2	2	no	—
62	C₄×C₁₀₁	—	4	4	no	—
63	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
64	C₄×C₁₀₁	#124	1	1	yes	characteristic
65	C₄×C₁₀₁	#127	1	1	yes	characteristic
66	Dic₁₀₁	—	4	2	no	—
67	C₂×C₁₀₁	—	2	2	no	—
68	C₂×C₁₀₁	#390	1	1	yes	characteristic
69	C₂×C₁₀₁	#421	1	2	yes	—
70	D₁₀₁	—	8	2	no	—
71	#1963	—	101	1	no	maximal, Sylow
72	C₁₀₁	#1963	1	1	yes	characteristic, complemented, Sylow
73	#532	—	101	1	no	—
74	#423	—	101	1	no	—
75	C₂×#104	—	101	4	no	—
76	#376	D₁₀₁	1	1	yes	characteristic, complemented
77	C₂×#41	—	101	2	no	—
78	C₂×M₁₆	—	101	2	no	—
79	C₄⋊Q₈	—	101	1	no	—
80	#127	—	101	1	no	—
81	#124	—	101	2	no	—
82	#121	—	202	2	no	—
83	C₄×D₄	—	202	2	no	—
84	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
85	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
86	C₄⋊C₈	—	202	2	no	—
87	#104	—	202	8	no	—
88	#95	—	2	2	no	—
89	#41	—	202	4	no	—
90	#41	—	202	4	no	—
91	C₂×Q₈	—	101	2	no	—
92	C₂×D₄	—	101	2	no	—
93	C₂×D₄	—	202	2	no	—
94	C₂×D₄	—	202	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	202	2	no	—
97	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
99	M₁₆	—	202	4	no	—
100	C₂×C₈	—	202	2	no	—
101	C₄⋊C₄	—	202	2	no	—
102	C₂²⋊C₄	—	202	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
105	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
106	C₄²	D₄₀₄	1	2	yes	—
107	C₄²	D₄₀₄	1	4	yes	complemented
108	C₂³	—	202	2	no	—
109	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
110	Q₈	—	202	4	no	—
111	D₄	—	404	4	no	—
112	D₄	—	404	4	no	—
113	D₄	—	202	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	404	2	no	—
118	C₂×C₄	—	404	2	no	—
119	C₂×C₄	—	202	2	no	—
120	C₂×C₄	#13108051	1	2	yes	—
121	C₂×C₄	#13108051	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
124	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
125	C₂×C₄	#13108074	1	2	yes	—
126	C₈	—	404	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	404	2	no	—
129	C₂²	—	404	4	no	—
130	C₂²	C₂×#13108051	1	1	yes	characteristic
131	C₂²	#15692229	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	404	2	no	—
136	C₄	C₄×D₄₀₄	1	2	yes	—
137	C₄	#15692243	1	1	yes	characteristic
138	C₄	#15692252	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	808	2	no	—
141	C₂	#20265079	1	1	yes	characteristic
142	C₂	#20265195	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265727

C₂

yes

maximal, characteristic

#20265201

C₂

yes

maximal, characteristic, complemented

C₂×#15692183

C₂

yes

maximal, complemented

#376×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108074

C₂²

yes

—

C₂×#13108058

C₂²

yes

—

#15692255

C₄

yes

characteristic, complemented

#15692252

C₄

yes

characteristic, complemented

#15692243

C₄

yes

complemented

#15692229

—

C₄×D₄₀₄

—

C₄⋊#11717579

—

#15692183

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#95×C₁₀₁

—

#13108074

—

#13108074

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂×D₄₀₄

—

C₂×D₄₀₄

C₂×C₄

yes

—

C₂×C₂²⋊D₁₀₁

—

C₂×C₄×D₁₀₁

—

#13108058

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

#13108051

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

D₄

yes

complemented

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂²×D₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

D₄₀₄

—

D₄₀₄

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#121

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#390

yes

characteristic

C₂×C₁₀₁

#421

yes

—

D₁₀₁

—

#1963

—

101

maximal, Sylow

C₁₀₁

#1963

yes

characteristic, complemented, Sylow

#532

—

101

—

#423

—

101

—

C₂×#104

—

101

—

#376

D₁₀₁

yes

characteristic, complemented

C₂×#41

—

101

—

C₂×M₁₆

—

101

—

C₄⋊Q₈

—

101

—

#127

—

101

—

#124

—

101

—

#121

—

202

—

C₄×D₄

—

202

—

C₂×C₄²

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

202

—

#104

—

202

—

#95

—

#41

—

202

—

#41

—

202

—

C₂×Q₈

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

202

—

100

C₂×C₈

—

202

—

101

C₄⋊C₄

—

202

—

102

C₂²⋊C₄

—

202

—

103

C₄²

—

104

C₄²

C₄×D₁₀₁

yes

characteristic

105

C₄²

C₄×D₁₀₁

yes

characteristic

106

C₄²

D₄₀₄

yes

—

107

C₄²

D₄₀₄

yes

complemented

108

C₂³

—

202

—

109

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

110

Q₈

—

202

—

111

D₄

—

404

—

112

D₄

—

404

—

113

D₄

—

202

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

404

—

118

C₂×C₄

—

404

—

119

C₂×C₄

—

202

—

120

C₂×C₄

#13108051

yes

—

121

C₂×C₄

#13108051

yes

—

122

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

123

C₂×C₄

C₂×D₄₀₄

yes

—

124

C₂×C₄

C₂×D₄₀₄

yes

characteristic

125

C₂×C₄

#13108074

yes

—

126

C₈

—

404

—

127

C₂²

—

128

C₂²

—

404

—

129

C₂²

—

404

—

130

C₂²

C₂×#13108051

yes

characteristic

131

C₂²

#15692229

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

404

—

136

C₄

C₄×D₄₀₄

yes

—

137

C₄

#15692243

yes

characteristic

138

C₄

#15692252

yes

characteristic

139

C₂

—

140

C₂

—

808

—

141

C₂

#20265079

yes

characteristic

142

C₂

#20265195

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀₄
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#376×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	#20265727	C₂
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#376	D₁₀₁
hypocenter	C₁₀₁	#1963
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₄₀₄

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#376×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

#20265727

C₂

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #20265201	C₂	semidirect
3. C₂×#15692183	C₂	semidirect
4. #376×C₁₀₁	C₂	semidirect
7. #15692255	C₄	semidirect
8. #15692252	C₄	semidirect
9. #15692243	C₄	semidirect
35. C₄²×C₁₀₁	D₄	semidirect
72. C₁₀₁	#1963	semidirect
76. #376	D₁₀₁	semidirect
107. C₄²	D₄₀₄	semidirect

H ≅ G/N

type

2. #20265201

C₂

semidirect

3. C₂×#15692183

C₂

semidirect

4. #376×C₁₀₁

C₂

semidirect

7. #15692255

C₄

semidirect

8. #15692252

C₄

semidirect

9. #15692243

C₄

semidirect

35. C₄²×C₁₀₁

D₄

semidirect

72. C₁₀₁

#1963

semidirect

76. #376

D₁₀₁

semidirect

107. C₄²

D₄₀₄

semidirect

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

subgroup	index
#20265727	2
#20265201	2
C₂×#15692183	2
#376×C₁₀₁	2
#1963	101

subgroup

index

#376×C₁₀₁

101

Subgroups — #27171964

Conjugacy classes of subgroups

Complements

Maximal subgroups