Group #13274281 — subgroups

Subgroups — #13274281

Order 1792.

206 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₂²×C₇×C₃₂	C₂	1	1	yes	maximal, characteristic
2	#11729153	C₂	1	1	yes	maximal, characteristic
3	C₂×#89294	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊C₃₂	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
6	#89873	C₄	1	1	yes	characteristic
7	C₂×#6386	C₄	1	1	yes	characteristic
8	C₂×C₇×C₃₂	—	2	2	no	—
9	C₂×C₇×C₃₂	C₂²	1	2	yes	—
10	#89297	C₄	1	2	yes	—
11	C₁₆×Dic₇	C₂²	1	2	yes	—
12	#89294	—	2	2	no	—
13	C₂×C₇⋊C₃₂	C₂²	1	2	yes	—
14	#13628	—	7	1	no	maximal, Sylow
15	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
16	#6489	C₈	1	1	yes	characteristic
17	C₂×#1136	C₈	1	1	yes	characteristic
18	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₇×C₁₆	D₄	1	2	yes	—
20	C₂×C₇×C₁₆	Q₈	1	2	yes	—
21	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
22	#6389	C₈	1	2	yes	—
23	C₈×Dic₇	C₂×C₄	1	2	yes	—
24	#6386	—	2	2	no	—
25	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
26	C₇×C₃₂	—	2	2	no	—
27	C₇⋊C₃₂	—	2	2	no	—
28	C₂×M₆₄	—	7	1	no	—
29	C₂²×C₃₂	D₇	1	1	yes	characteristic
30	#2108	—	7	1	no	—
31	C₄⋊C₃₂	—	7	4	no	—
32	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
34	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₇×C₈	M₁₆	1	2	yes	—
36	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₂²⋊Dic₇	C₁₆	1	2	yes	—
38	C₄⋊Dic₇	C₁₆	1	2	yes	—
39	C₄×Dic₇	C₂×C₈	1	2	yes	—
40	#1136	—	2	2	no	—
41	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
42	C₇×C₁₆	C₄⋊C₄	1	2	yes	—
43	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
44	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
45	C₇⋊C₁₆	—	2	2	no	—
46	C₂×M₃₂	—	7	1	no	—
47	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
48	#405	—	7	1	no	—
49	M₆₄	—	14	2	no	—
50	C₂×C₃₂	—	2	2	no	—
51	C₂×C₃₂	—	7	2	no	—
52	C₂×C₃₂	C₂×D₇	1	2	yes	—
53	#346	—	7	2	no	—
54	C₄×C₁₆	—	7	2	no	—
55	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇	M₃₂	1	2	yes	—
59	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
60	C₂×Dic₇	C₂×C₁₆	1	4	yes	—
61	C₇×C₈	C₄⋊C₈	1	2	yes	—
62	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
64	C₇⋊C₈	—	2	2	no	—
65	C₂×M₁₆	—	7	1	no	—
66	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
67	#117	—	7	1	no	—
68	M₃₂	—	14	2	no	—
69	C₂×C₁₆	—	7	2	no	—
70	C₂×C₁₆	Dic₁₄	1	2	yes	—
71	C₂×C₁₆	C₄×D₇	1	1	yes	characteristic
72	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
73	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
74	#97	—	7	2	no	—
75	C₄×C₈	—	7	2	no	—
76	C₃₂	—	2	2	no	—
77	C₃₂	—	14	2	no	—
78	C₂²×C₇	—	2	2	no	—
79	C₂²×C₇	C₄⋊C₁₆	1	2	yes	—
80	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
81	C₄×C₇	C₄⋊C₁₆	1	2	yes	—
82	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
84	Dic₇	—	2	4	no	—
85	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
86	M₁₆	—	14	2	no	—
87	C₂×C₈	—	7	2	no	—
88	C₂×C₈	C₈×D₇	1	1	yes	characteristic
89	C₂×C₈	#1131	1	2	yes	—
90	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
91	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
92	C₄⋊C₄	—	7	2	no	—
93	C₂²⋊C₄	—	7	2	no	—
94	C₄²	—	7	2	no	—
95	C₁₆	—	14	2	no	—
96	C₁₆	Dic₇⋊C₄	1	2	yes	—
97	C₁₆	C₂×Dic₁₄	1	1	yes	characteristic
98	C₁₆	C₂×C₂²⋊D₇	1	1	yes	characteristic
99	C₂×C₇	—	2	2	no	—
100	C₂×C₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
101	C₂×C₇	#2258	1	2	yes	—
102	C₂³	D₇×C₁₆	1	1	yes	characteristic
103	C₂×C₄	—	7	4	no	—
104	C₂×C₄	D₇×C₁₆	1	1	yes	characteristic
105	C₂×C₄	#6372	1	2	yes	—
106	C₂×C₄	Dic₇⋊C₈	1	2	yes	—
107	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
108	C₈	—	14	2	no	—
109	C₈	Dic₇⋊C₈	1	2	yes	—
110	C₈	C₂×#1131	1	1	yes	characteristic
111	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
112	C₇	#13628	1	1	yes	characteristic, complemented, Sylow
113	C₂²	—	2	2	no	—
114	C₂²	Dic₇⋊C₁₆	1	2	yes	—
115	C₂²	C₂×D₇×C₁₆	1	1	yes	characteristic
116	C₄	—	14	4	no	—
117	C₄	Dic₇⋊C₁₆	1	2	yes	—
118	C₄	C₂×#6372	1	1	yes	characteristic
119	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#11727120	1	2	yes	—
122	C₂	C₂×Dic₇⋊C₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇×C₃₂

C₂

yes

maximal, characteristic

#11729153

C₂

yes

maximal, characteristic

C₂×#89294

C₂

yes

maximal, characteristic

Dic₇⋊C₃₂

C₂

yes

maximal, complemented

C₂²×C₇×C₁₆

C₂²

yes

characteristic

#89873

C₄

yes

characteristic

C₂×#6386

C₄

yes

characteristic

C₂×C₇×C₃₂

—

C₂×C₇×C₃₂

C₂²

yes

—

#89297

C₄

yes

—

C₁₆×Dic₇

C₂²

yes

—

#89294

—

C₂×C₇⋊C₃₂

C₂²

yes

—

#13628

—

maximal, Sylow

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

#6489

C₈

yes

characteristic

C₂×#1136

C₈

yes

characteristic

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₂×C₇×C₁₆

D₄

yes

—

C₂×C₇×C₁₆

Q₈

yes

—

C₂×C₇×C₁₆

C₂³

yes

characteristic

#6389

C₈

yes

—

C₈×Dic₇

C₂×C₄

yes

—

#6386

—

C₂×C₇⋊C₁₆

C₂×C₄

yes

—

C₇×C₃₂

—

C₇⋊C₃₂

—

C₂×M₆₄

—

C₂²×C₃₂

D₇

yes

characteristic

#2108

—

C₄⋊C₃₂

—

C₂²×C₄×C₇

C₂×C₈

yes

characteristic

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₂×C₇×C₈

C₂×C₈

yes

characteristic

C₂×C₇×C₈

M₁₆

yes

—

C₂×C₇×C₈

C₂²×C₄

yes

characteristic

C₂²⋊Dic₇

C₁₆

yes

—

C₄⋊Dic₇

C₁₆

yes

—

C₄×Dic₇

C₂×C₈

yes

—

#1136

—

C₂×C₇⋊C₈

C₂×C₈

yes

—

C₇×C₁₆

C₄⋊C₄

yes

—

C₇×C₁₆

C₂×D₄

yes

characteristic

C₇×C₁₆

C₂×Q₈

yes

characteristic

C₇⋊C₁₆

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₇

yes

characteristic

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇

yes

—

#346

—

C₄×C₁₆

—

C₂³×C₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇

C₄⋊C₈

yes

—

C₂×C₄×C₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇

M₃₂

yes

—

C₂×C₄×C₇

C₂²×C₈

yes

characteristic

C₂×Dic₇

C₂×C₁₆

yes

—

C₇×C₈

C₄⋊C₈

yes

—

C₇×C₈

C₂×C₄⋊C₄

yes

characteristic

C₇×C₈

C₂×M₁₆

yes

characteristic

C₇⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

C₄×D₇

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

Dic₁₄

yes

—

C₂×C₁₆

C₄×D₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₇

yes

—

C₂×C₁₆

C₂²×D₇

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂²×C₇

—

C₂²×C₇

C₄⋊C₁₆

yes

—

C₂²×C₇

C₂²×C₁₆

yes

characteristic

C₄×C₇

C₄⋊C₁₆

yes

—

C₄×C₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇

C₂×M₃₂

yes

characteristic

Dic₇

—

C₂²×C₄

C₈×D₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₈×D₇

yes

characteristic

C₂×C₈

#1131

yes

—

C₂×C₈

Dic₇⋊C₄

yes

—

C₂×C₈

C₂×C₄×D₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

Dic₇⋊C₄

yes

—

C₁₆

C₂×Dic₁₄

yes

characteristic

C₁₆

C₂×C₂²⋊D₇

yes

characteristic

C₂×C₇

—

100

C₂×C₇

C₂×C₄⋊C₁₆

yes

characteristic

101

C₂×C₇

#2258

yes

—

102

C₂³

D₇×C₁₆

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

D₇×C₁₆

yes

characteristic

105

C₂×C₄

#6372

yes

—

106

C₂×C₄

Dic₇⋊C₈

yes

—

107

C₂×C₄

C₂×C₈×D₇

yes

characteristic

108

C₈

—

109

C₈

Dic₇⋊C₈

yes

—

110

C₈

C₂×#1131

yes

characteristic

111

C₈

C₂×Dic₇⋊C₄

yes

characteristic

112

C₇

#13628

yes

characteristic, complemented, Sylow

113

C₂²

—

114

C₂²

Dic₇⋊C₁₆

yes

—

115

C₂²

C₂×D₇×C₁₆

yes

characteristic

116

C₄

—

117

C₄

Dic₇⋊C₁₆

yes

—

118

C₄

C₂×#6372

yes

characteristic

119

C₄

C₂×Dic₇⋊C₈

yes

characteristic

120

C₂

—

121

C₂

#11727120

yes

—

122

C₂

C₂×Dic₇⋊C₁₆

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇
commutator, socle	C₂²×C₇	C₂²×C₁₆
Fitting	C₂²×C₇×C₃₂	C₂
prime radical	C₂³×C₇	C₂×C₁₆
cosocle	C₂×C₇×C₁₆	C₂³
hypercenter	C₂²×C₃₂	D₇
hypocenter	C₇	#13628
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇

commutator, socle

C₂²×C₇

C₂²×C₁₆

Fitting

C₂²×C₇×C₃₂

C₂

prime radical

C₂³×C₇

C₂×C₁₆

cosocle

C₂×C₇×C₁₆

C₂³

hypercenter

C₂²×C₃₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
4. Dic₇⋊C₃₂	C₂	semidirect
112. C₇	#13628	semidirect

H ≅ G/N

type

4. Dic₇⋊C₃₂

C₂

semidirect

112. C₇

#13628

semidirect

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

subgroup	index
C₂²×C₇×C₃₂	2
#11729153	2
C₂×#89294	2
Dic₇⋊C₃₂	2
#13628	7

subgroup

index

C₂²×C₇×C₃₂

#11729153

C₂×#89294

Dic₇⋊C₃₂

#13628

Subgroups — #13274281

Conjugacy classes of subgroups

Complements

Maximal subgroups