Group #13265603 — subgroups

Subgroups — #13265603

Order 1792.

206 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 91 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	#11726802	C₂	1	2	yes	maximal
2	C₈×Dic₂₈	C₂	1	1	yes	maximal, characteristic
3	C₇×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#11725312	C₂	1	2	yes	maximal
5	#11725246	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₈	C₄	1	1	yes	characteristic
7	C₈×Dic₁₄	C₂²	1	2	yes	—
8	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₇×C₈²	C₂²	1	1	yes	characteristic
11	#89297	—	2	2	no	—
12	Dic₇⋊C₁₆	—	2	2	no	—
13	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
14	#89253	C₄	1	1	yes	characteristic
15	#89251	C₄	1	2	yes	—
16	#12054	—	7	1	no	maximal, Sylow
17	C₂×Dic₂₈	C₈	1	1	yes	characteristic
18	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
19	C₂×C₇×C₁₆	—	2	1	no	—
20	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
22	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
25	#6392	C₈	1	1	yes	characteristic
26	#6390	C₈	1	2	yes	—
27	Dic₇⋊C₈	—	2	2	no	—
28	C₈×Dic₇	—	2	2	no	—
29	C₂×C₇⋊C₁₆	—	2	2	no	—
30	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
31	#2183	—	7	2	no	—
32	C₈×Q₁₆	—	7	1	no	—
33	#1370	—	7	1	no	—
34	Q₈⋊C₁₆	—	7	2	no	—
35	C₈×C₁₆	D₇	1	1	yes	characteristic
36	C₂×Dic₁₄	C₂×C₈	1	2	yes	—
37	C₂×C₇×C₈	—	2	1	no	—
38	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
40	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂×C₇×C₈	#41	1	1	yes	characteristic
42	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
43	C₄⋊Dic₇	C₂×C₈	1	2	yes	—
44	Dic₇⋊C₄	—	2	2	no	—
45	C₄×Dic₇	—	2	2	no	—
46	C₂×C₇⋊C₈	—	2	2	no	—
47	Dic₂₈	—	2	2	no	—
48	C₇×C₁₆	—	2	2	no	—
49	C₇×C₁₆	—	2	1	no	—
50	C₇×C₁₆	Q₁₆	1	2	yes	complemented
51	C₇⋊C₁₆	—	4	2	no	—
52	C₈×Q₈	—	7	2	no	—
53	C₄×Q₁₆	—	7	1	no	—
54	C₄⋊C₁₆	—	7	2	no	—
55	C₄⋊C₁₆	—	14	2	no	—
56	#346	—	14	2	no	—
57	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
58	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
59	#335	—	7	1	no	—
60	Q₈⋊C₈	—	7	2	no	—
61	C₈²	C₂×D₇	1	1	yes	characteristic
62	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇	#131	1	1	yes	characteristic
65	C₂×Dic₇	—	2	2	no	—
66	C₂×Dic₇	—	2	2	no	—
67	Dic₁₄	—	2	2	no	—
68	Dic₁₄	—	2	2	no	—
69	C₇×C₈	—	2	2	no	—
70	C₇×C₈	—	2	1	no	—
71	C₇×C₈	—	2	1	no	—
72	C₇×C₈	M₃₂	1	2	yes	—
73	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
74	C₇×C₈	#135	1	1	yes	characteristic
75	C₇⋊C₈	—	4	2	no	—
76	C₂×Q₁₆	—	7	1	no	—
77	C₄×Q₈	—	7	2	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	14	2	no	—
80	C₂×C₁₆	D₂₈	1	1	yes	characteristic
81	C₂×C₁₆	D₂₈	1	1	yes	characteristic
82	#107	—	7	1	no	—
83	C₄⋊C₈	—	7	2	no	—
84	C₄⋊C₈	—	14	2	no	—
85	#103	—	7	2	no	—
86	C₄×C₈	—	14	2	no	—
87	C₄×C₈	C₄×D₇	1	1	yes	characteristic
88	C₄×C₈	C₄×D₇	1	1	yes	characteristic
89	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
90	C₂²×C₇	C₈×D₄	1	1	yes	characteristic
91	C₄×C₇	—	2	1	no	—
92	C₄×C₇	C₄×Q₁₆	1	1	yes	characteristic
93	C₄×C₇	#443	1	1	yes	characteristic
94	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
95	C₄×C₇	#504	1	1	yes	characteristic
96	Dic₇	—	4	2	no	—
97	Dic₇	—	4	2	no	—
98	C₂×Q₈	—	7	2	no	—
99	Q₁₆	—	14	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	14	2	no	—
102	C₂×C₈	C₈×D₇	1	1	yes	characteristic
103	C₂×C₈	C₈×D₇	1	1	yes	characteristic
104	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
105	C₂×C₈	#1157	1	1	yes	characteristic
106	C₄⋊C₄	—	7	2	no	—
107	C₄⋊C₄	—	14	2	no	—
108	C₄²	—	14	2	no	—
109	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	—	28	2	no	—
113	C₁₆	Dic₂₈	1	2	yes	complemented
114	C₂×C₇	C₈×Q₁₆	1	1	yes	characteristic
115	C₂×C₇	#2169	1	1	yes	characteristic
116	C₂×C₇	#2170	1	1	yes	characteristic
117	Q₈	—	14	2	no	—
118	Q₈	—	14	2	no	—
119	C₂×C₄	—	14	2	no	—
120	C₂×C₄	—	14	2	no	—
121	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
122	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
123	C₂×C₄	#6464	1	1	yes	characteristic
124	C₈	—	2	2	no	—
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	28	2	no	—
128	C₈	#6372	1	2	yes	—
129	C₈	#6467	1	1	yes	characteristic
130	C₈	C₂×Dic₂₈	1	1	yes	characteristic
131	C₇	#12054	1	1	yes	characteristic, complemented, Sylow
132	C₂²	C₈×D₂₈	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	28	2	no	—
135	C₄	—	28	2	no	—
136	C₄	C₄×Dic₂₈	1	1	yes	characteristic
137	C₄	#89472	1	1	yes	characteristic
138	C₄	C₂×#6372	1	1	yes	characteristic
139	C₄	#89673	1	1	yes	characteristic
140	C₂	C₈×Dic₂₈	1	1	yes	characteristic
141	C₂	#11726809	1	1	yes	characteristic
142	C₂	#11726810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11726802

C₂

yes

maximal

C₈×Dic₂₈

C₂

yes

maximal, characteristic

C₇×C₈×C₁₆

C₂

yes

maximal, characteristic

#11725312

C₂

yes

maximal

#11725246

C₂

yes

maximal, characteristic

C₄×Dic₂₈

C₄

yes

characteristic

C₈×Dic₁₄

C₂²

yes

—

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₇×C₈²

C₂²

yes

characteristic

#89297

—

Dic₇⋊C₁₆

—

C₄⋊(C₇⋊C₁₆)

C₂²

yes

—

#89253

C₄

yes

characteristic

#89251

C₄

yes

—

#12054

—

maximal, Sylow

C₂×Dic₂₈

C₈

yes

characteristic

C₄×Dic₁₄

C₂×C₄

yes

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

D₄

yes

characteristic

C₂×C₇×C₁₆

D₄

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

characteristic

#6392

C₈

yes

characteristic

#6390

C₈

yes

—

Dic₇⋊C₈

—

C₈×Dic₇

—

C₂×C₇⋊C₁₆

—

C₄⋊(C₇⋊C₈)

C₂×C₄

yes

—

#2183

—

C₈×Q₁₆

—

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

D₇

yes

characteristic

C₂×Dic₁₄

C₂×C₈

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×C₈

yes

characteristic

C₂×C₇×C₈

C₂×C₈

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

#41

yes

characteristic

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄⋊Dic₇

C₂×C₈

yes

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

Dic₂₈

—

C₇×C₁₆

—

C₇×C₁₆

—

C₇×C₁₆

Q₁₆

yes

complemented

C₇⋊C₁₆

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

C₂×D₇

yes

characteristic

C₄×C₁₆

C₂×D₇

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₇

yes

characteristic

C₂×C₄×C₇

C₄×D₄

yes

characteristic

C₂×C₄×C₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₇

#131

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

M₃₂

yes

—

C₇×C₈

C₂×Q₁₆

yes

characteristic

C₇×C₈

#135

yes

characteristic

C₇⋊C₈

—

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₂₈

yes

characteristic

C₂×C₁₆

D₂₈

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₄×D₇

yes

characteristic

C₄×C₈

C₄×D₇

yes

characteristic

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

C₈×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

C₄×Q₁₆

yes

characteristic

C₄×C₇

#443

yes

characteristic

C₄×C₇

C₂×M₃₂

yes

characteristic

C₄×C₇

#504

yes

characteristic

Dic₇

—

Dic₇

—

C₂×Q₈

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₈×D₇

yes

characteristic

103

C₂×C₈

C₈×D₇

yes

characteristic

104

C₂×C₈

C₂×D₂₈

yes

characteristic

105

C₂×C₈

#1157

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

C₂×C₄×D₇

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

Dic₂₈

yes

complemented

114

C₂×C₇

C₈×Q₁₆

yes

characteristic

115

C₂×C₇

#2169

yes

characteristic

116

C₂×C₇

#2170

yes

characteristic

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₄×D₂₈

yes

characteristic

122

C₂×C₄

C₂×C₈×D₇

yes

characteristic

123

C₂×C₄

#6464

yes

characteristic

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

#6372

yes

—

129

C₈

#6467

yes

characteristic

130

C₈

C₂×Dic₂₈

yes

characteristic

131

C₇

#12054

yes

characteristic, complemented, Sylow

132

C₂²

C₈×D₂₈

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₄×Dic₂₈

yes

characteristic

137

C₄

#89472

yes

characteristic

138

C₄

C₂×#6372

yes

characteristic

139

C₄

#89673

yes

characteristic

140

C₂

C₈×Dic₂₈

yes

characteristic

141

C₂

#11726809

yes

characteristic

142

C₂

#11726810

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×C₈×C₁₆	C₂
socle, prime radical	C₂²×C₇	C₈×D₄
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₈×C₁₆	D₇
hypocenter	C₇	#12054
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆

subgroup

G/H

center

C₂×C₈

C₂×D₂₈

commutator

C₂×C₄×C₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇

Fitting

C₇×C₈×C₁₆

C₂

socle, prime radical

C₂²×C₇

C₈×D₄

cosocle

C₄×C₇×C₈

C₂³

hypercenter

C₈×C₁₆

D₇

hypocenter

C₇

#12054

LCS

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆

N	H ≅ G/N	type
50. C₇×C₁₆	Q₁₆	semidirect
113. C₁₆	Dic₂₈	semidirect
131. C₇	#12054	semidirect

H ≅ G/N

type

50. C₇×C₁₆

Q₁₆

semidirect

113. C₁₆

Dic₂₈

semidirect

131. C₇

#12054

semidirect

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

subgroup	index
#11726802	2
C₈×Dic₂₈	2
C₇×C₈×C₁₆	2
#11725312	2
#11725246	2
#12054	7

subgroup

index

#11726802

C₈×Dic₂₈

C₇×C₈×C₁₆

#11725312

#11725246

#12054

Subgroups — #13265603

Conjugacy classes of subgroups

Complements

Maximal subgroups