Group #28401310 — subgroups

Subgroups — #28401310

Order 14464.

446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 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	#20299146	C₂	1	2	yes	maximal, complemented
2	#20298996	C₂	1	1	yes	maximal, characteristic, complemented
3	#20298944	C₂	1	2	yes	maximal, complemented
4	#20298848	C₂	1	4	yes	maximal, complemented
5	#20298710	C₂	1	4	yes	maximal, complemented
6	C₄×#14268630	C₂	1	1	yes	maximal, characteristic, complemented
7	#493×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14268646	—	2	2	no	—
9	C₂×#14268630	C₂²	1	2	yes	—
10	#16813622	C₂²	1	2	yes	—
11	#16813609	—	2	2	no	—
12	#16813593	—	2	2	no	—
13	#16813593	C₂²	1	4	yes	—
14	D₄×Dic₁₁₃	—	2	2	no	—
15	D₄×Dic₁₁₃	C₂²	1	4	yes	—
16	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
17	#16813554	C₂²	1	4	yes	—
18	#16813543	C₂²	1	8	yes	—
19	#16813532	C₂²	1	2	yes	—
20	C₄×#11744751	C₂²	1	1	yes	characteristic
21	C₂×D₈×C₁₁₃	C₂²	1	4	yes	—
22	#127×C₁₁₃	C₂²	1	2	yes	—
23	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
24	#14268646	—	4	8	no	—
25	C₂×Dic₂₂₆	—	2	2	no	—
26	C₂×C₂²⋊D₁₁₃	—	2	4	no	—
27	C₂²×Dic₁₁₃	—	2	4	no	—
28	C₂×C₄×D₁₁₃	D₄	1	2	yes	—
29	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
30	#14268630	—	2	4	no	—
31	C₂×#11744751	C₂³	1	2	yes	—
32	C₄⋊Dic₁₁₃	—	2	2	no	—
33	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
34	C₂²⋊Dic₁₁₃	—	2	4	no	—
35	C₂²⋊Dic₁₁₃	—	2	4	no	—
36	C₄×Dic₁₁₃	D₄	1	2	yes	—
37	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₁₁₃	—	2	2	no	—
39	C₂×D₄×C₁₁₃	—	2	2	no	—
40	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
41	D₈×C₁₁₃	—	2	8	no	—
42	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
43	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
44	C₂³×C₁₁₃	—	2	4	no	—
45	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
46	D₄×C₁₁₃	—	2	8	no	—
47	D₄×C₁₁₃	—	4	4	no	—
48	D₄×C₁₁₃	—	4	4	no	—
49	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
51	C₂²⋊D₁₁₃	—	4	8	no	—
52	C₂×Dic₁₁₃	—	2	4	no	—
53	C₂×Dic₁₁₃	—	4	8	no	—
54	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
56	C₄×D₁₁₃	—	2	4	no	—
57	C₄×D₁₁₃	—	2	2	no	—
58	Dic₂₂₆	—	4	4	no	—
59	C₈×C₁₁₃	#41	1	4	yes	complemented
60	#11744751	—	2	2	no	—
61	C₂²×C₁₁₃	—	4	4	no	—
62	C₂²×C₁₁₃	—	4	4	no	—
63	C₂²×C₁₁₃	—	4	4	no	—
64	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₁₁₃	—	2	1	no	—
66	C₂×D₁₁₃	—	2	2	no	—
67	C₄×C₁₁₃	AGL(1,ℤ/8)	1	4	yes	—
68	C₄×C₁₁₃	C₂×#41	1	2	yes	—
69	Dic₁₁₃	—	2	1	no	—
70	Dic₁₁₃	—	2	2	no	—
71	Dic₁₁₃	—	4	4	no	—
72	C₂×C₁₁₃	—	8	4	no	—
73	C₂×C₁₁₃	#532	1	1	yes	characteristic
74	C₂×C₁₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
75	D₁₁₃	—	4	1	no	—
76	#3184	—	113	1	no	maximal, Sylow
77	C₁₁₃	#3184	1	1	yes	characteristic, complemented, Sylow
78	#532	—	113	2	no	—
79	#498	—	113	1	no	—
80	#493	D₁₁₃	1	1	yes	characteristic, complemented
81	#486	—	113	2	no	—
82	#469	—	113	4	no	—
83	#442	—	113	4	no	—
84	C₄×M₁₆	—	113	1	no	—
85	C₂×#41	—	226	2	no	—
86	C₂×D₈	C₂×D₁₁₃	1	4	yes	—
87	C₂×M₁₆	—	113	2	no	—
88	C₄⋊Q₈	—	113	2	no	—
89	#127	C₂×D₁₁₃	1	2	yes	—
90	#124	—	226	2	no	—
91	#121	—	113	4	no	—
92	#121	—	226	2	no	—
93	C₄×D₄	—	113	4	no	—
94	C₄×D₄	—	226	2	no	—
95	C₂×C₄²	—	113	1	no	—
96	#106	—	113	4	no	—
97	#102	—	113	8	no	—
98	#97	—	113	2	no	—
99	C₄×C₈	—	113	1	no	—
100	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
101	#41	—	452	8	no	—
102	C₂×Q₈	—	226	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	226	4	no	—
106	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
107	C₂²×C₄	—	113	2	no	—
108	C₂²×C₄	—	113	1	no	—
109	C₂²×C₄	—	226	4	no	—
110	D₈	—	2	8	no	—
111	M₁₆	—	226	4	no	—
112	C₂×C₈	—	113	2	no	—
113	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
114	C₄⋊C₄	—	113	4	no	—
115	C₄⋊C₄	—	226	2	no	—
116	C₂²⋊C₄	—	226	4	no	—
117	C₂²⋊C₄	—	226	4	no	—
118	C₄²	—	113	2	no	—
119	C₄²	—	113	1	no	—
120	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
121	C₂³	—	2	4	no	—
122	C₂³	—	113	1	no	—
123	Q₈	—	452	4	no	—
124	D₄	—	2	8	no	—
125	D₄	—	4	4	no	—
126	D₄	—	4	4	no	—
127	D₄	—	452	8	no	—
128	C₂×C₄	—	113	1	no	—
129	C₂×C₄	—	113	2	no	—
130	C₂×C₄	—	452	8	no	—
131	C₂×C₄	—	226	4	no	—
132	C₂×C₄	—	226	4	no	—
133	C₂×C₄	—	226	2	no	—
134	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
135	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
136	C₈	—	226	2	no	—
137	C₈	#14268646	1	4	yes	complemented
138	C₂²	—	4	4	no	—
139	C₂²	—	4	4	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	226	2	no	—
142	C₂²	—	226	1	no	—
143	C₂²	C₂×D₄×D₁₁₃	1	1	yes	characteristic
144	C₄	—	452	4	no	—
145	C₄	—	226	2	no	—
146	C₄	—	226	1	no	—
147	C₄	#16813646	1	4	yes	—
148	C₄	C₂×#14268646	1	2	yes	—
149	C₂	—	8	4	no	—
150	C₂	—	452	1	no	—
151	C₂	#20299146	1	1	yes	characteristic
152	C₂	C₂×#16813646	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20299146

C₂

yes

maximal, complemented

#20298996

C₂

yes

maximal, characteristic, complemented

#20298944

C₂

yes

maximal, complemented

#20298848

C₂

yes

maximal, complemented

#20298710

C₂

yes

maximal, complemented

C₄×#14268630

C₂

yes

maximal, characteristic, complemented

#493×C₁₁₃

C₂

yes

maximal, characteristic, complemented

C₂×#14268646

—

C₂×#14268630

C₂²

yes

—

#16813622

C₂²

yes

—

#16813609

—

#16813593

—

#16813593

C₂²

yes

—

D₄×Dic₁₁₃

—

D₄×Dic₁₁₃

C₂²

yes

—

C₄²×D₁₁₃

C₂²

yes

characteristic

#16813554

C₂²

yes

—

#16813543

C₂²

yes

—

#16813532

C₂²

yes

—

C₄×#11744751

C₂²

yes

characteristic

C₂×D₈×C₁₁₃

C₂²

yes

—

#127×C₁₁₃

C₂²

yes

—

C₄×C₈×C₁₁₃

C₂²

yes

characteristic

#14268646

—

C₂×Dic₂₂₆

—

C₂×C₂²⋊D₁₁₃

—

C₂²×Dic₁₁₃

—

C₂×C₄×D₁₁₃

D₄

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

characteristic

#14268630

—

C₂×#11744751

C₂³

yes

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

C₂³

yes

—

C₂²⋊Dic₁₁₃

—

C₂²⋊Dic₁₁₃

—

C₄×Dic₁₁₃

D₄

yes

—

C₄×Dic₁₁₃

C₂³

yes

characteristic

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

—

C₂×D₄×C₁₁₃

C₂³

yes

—

D₈×C₁₁₃

—

C₂×C₈×C₁₁₃

C₂³

yes

—

C₄²×C₁₁₃

C₂³

yes

characteristic

C₂³×C₁₁₃

—

C₂²×D₁₁₃

C₂×D₄

yes

characteristic

D₄×C₁₁₃

—

D₄×C₁₁₃

—

D₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂⁴

yes

characteristic

C₂²⋊D₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₁₃

C₂×D₄

yes

—

C₄×D₁₁₃

—

C₄×D₁₁₃

—

Dic₂₂₆

—

C₈×C₁₁₃

#41

yes

complemented

#11744751

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂²×D₄

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

—

C₄×C₁₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₁₃

C₂×#41

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

#532

yes

characteristic

C₂×C₁₁₃

C₂×AGL(1,ℤ/8)

yes

—

D₁₁₃

—

#3184

—

113

maximal, Sylow

C₁₁₃

#3184

yes

characteristic, complemented, Sylow

#532

—

113

—

#498

—

113

—

#493

D₁₁₃

yes

characteristic, complemented

#486

—

113

—

#469

—

113

—

#442

—

113

—

C₄×M₁₆

—

113

—

C₂×#41

—

226

—

C₂×D₈

C₂×D₁₁₃

yes

—

C₂×M₁₆

—

113

—

C₄⋊Q₈

—

113

—

#127

C₂×D₁₁₃

yes

—

#124

—

226

—

#121

—

113

—

#121

—

226

—

C₄×D₄

—

113

—

C₄×D₄

—

226

—

C₂×C₄²

—

113

—

#106

—

113

—

#102

—

113

—

#97

—

113

—

C₄×C₈

—

113

—

100

C₄×C₈

C₂×D₁₁₃

yes

characteristic

101

#41

—

452

—

102

C₂×Q₈

—

226

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

226

—

106

C₂×D₄

C₂²×D₁₁₃

yes

—

107

C₂²×C₄

—

113

—

108

C₂²×C₄

—

113

—

109

C₂²×C₄

—

226

—

110

D₈

—

111

M₁₆

—

226

—

112

C₂×C₈

—

113

—

113

C₂×C₈

C₂²×D₁₁₃

yes

—

114

C₄⋊C₄

—

113

—

115

C₄⋊C₄

—

226

—

116

C₂²⋊C₄

—

226

—

117

C₂²⋊C₄

—

226

—

118

C₄²

—

113

—

119

C₄²

—

113

—

120

C₄²

C₂²×D₁₁₃

yes

characteristic

121

C₂³

—

122

C₂³

—

113

—

123

Q₈

—

452

—

124

D₄

—

125

D₄

—

126

D₄

—

127

D₄

—

452

—

128

C₂×C₄

—

113

—

129

C₂×C₄

—

113

—

130

C₂×C₄

—

452

—

131

C₂×C₄

—

226

—

132

C₂×C₄

—

226

—

133

C₂×C₄

—

226

—

134

C₂×C₄

D₄×D₁₁₃

yes

—

135

C₂×C₄

C₂³×D₁₁₃

yes

characteristic

136

C₈

—

226

—

137

C₈

#14268646

yes

complemented

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

226

—

142

C₂²

—

226

—

143

C₂²

C₂×D₄×D₁₁₃

yes

characteristic

144

C₄

—

452

—

145

C₄

—

226

—

146

C₄

—

226

—

147

C₄

#16813646

yes

—

148

C₄

C₂×#14268646

yes

—

149

C₂

—

150

C₂

—

452

—

151

C₂

#20299146

yes

characteristic

152

C₂

C₂×#16813646

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₁₃
commutator, cosocle	C₂×C₄×C₁₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	#493×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂²×D₄
hypercenter	#493	D₁₁₃
hypocenter	C₁₁₃	#3184
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₁₁₃

commutator, cosocle

C₂×C₄×C₁₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₁₃

Fitting

#493×C₁₁₃

C₂

socle

C₂²×C₁₁₃

C₂²×D₄

D₁₁₃

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #20299146	C₂	semidirect
2. #20298996	C₂	semidirect
3. #20298944	C₂	semidirect
4. #20298848	C₂	semidirect
5. #20298710	C₂	semidirect
6. C₄×#14268630	C₂	semidirect
7. #493×C₁₁₃	C₂	semidirect
59. C₈×C₁₁₃	#41	semidirect
77. C₁₁₃	#3184	semidirect
80. #493	D₁₁₃	semidirect
137. C₈	#14268646	semidirect

H ≅ G/N

type

1. #20299146

C₂

semidirect

2. #20298996

C₂

semidirect

3. #20298944

C₂

semidirect

4. #20298848

C₂

semidirect

5. #20298710

C₂

semidirect

6. C₄×#14268630

C₂

semidirect

7. #493×C₁₁₃

C₂

semidirect

59. C₈×C₁₁₃

#41

semidirect

77. C₁₁₃

#3184

semidirect

80. #493

D₁₁₃

semidirect

137. C₈

#14268646

semidirect

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

subgroup	index
#20299146	2
#20298996	2
#20298944	2
#20298848	2
#20298710	2
C₄×#14268630	2
#493×C₁₁₃	2
#3184	113

subgroup

index

#493×C₁₁₃

113

Subgroups — #28401310

Conjugacy classes of subgroups

Complements

Maximal subgroups