Group #13274428 — subgroups

Subgroups — #13274428

Order 1792.

158 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₁₆×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	Dic₇⋊C₃₂	C₂	1	2	yes	maximal
4	#11725395	C₂	1	2	yes	maximal
5	C₄×C₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₄	C₄	1	1	yes	characteristic
7	C₂×C₇×C₃₂	C₂²	1	2	yes	—
8	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	Dic₇⋊C₁₆	C₄	1	2	yes	—
10	C₁₆×Dic₇	C₂²	1	2	yes	—
11	C₂×C₇⋊C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₃₂	C₂²	1	1	yes	characteristic
13	C₄⋊(C₇⋊C₁₆)	C₄	1	1	yes	characteristic
14	#13660	—	7	1	no	maximal, Sylow
15	C₄×Dic₁₄	C₈	1	1	yes	characteristic
16	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
18	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	Dic₇⋊C₈	C₈	1	2	yes	—
20	C₈×Dic₇	C₂×C₄	1	2	yes	—
21	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₄⋊(C₇⋊C₈)	C₈	1	1	yes	characteristic
23	C₇×C₃₂	—	2	2	no	—
24	C₇⋊C₃₂	—	2	1	no	—
25	C₇⋊C₃₂	Q₈	1	2	yes	complemented
26	Q₈×C₁₆	—	7	1	no	—
27	C₄⋊C₃₂	—	7	2	no	—
28	C₄⋊C₃₂	D₇	1	1	yes	characteristic
29	#1397	—	7	2	no	—
30	C₄×C₃₂	—	7	1	no	—
31	C₂×Dic₁₄	C₁₆	1	1	yes	characteristic
32	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
35	C₄⋊Dic₇	C₁₆	1	1	yes	characteristic
36	Dic₇⋊C₄	C₁₆	1	2	yes	—
37	C₄×Dic₇	C₂×C₈	1	2	yes	—
38	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
39	C₇×C₁₆	—	2	1	no	—
40	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
41	C₇×C₁₆	#41	1	1	yes	characteristic
42	C₇⋊C₁₆	—	2	2	no	—
43	C₈×Q₈	—	7	1	no	—
44	C₂×C₃₂	—	7	1	no	—
45	C₂×C₃₂	—	7	1	no	—
46	C₂×C₃₂	C₂×D₇	1	2	yes	—
47	C₄⋊C₁₆	—	7	2	no	—
48	C₄⋊C₁₆	—	7	1	no	—
49	C₄×C₁₆	—	7	2	no	—
50	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
51	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
54	C₂×Dic₇	C₂×C₁₆	1	2	yes	—
55	C₂×Dic₇	C₂×C₁₆	1	2	yes	—
56	Dic₁₄	—	2	2	no	—
57	C₇×C₈	—	2	1	no	—
58	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
59	C₇×C₈	#131	1	1	yes	characteristic
60	C₇⋊C₈	—	2	2	no	—
61	C₄×Q₈	—	7	1	no	—
62	C₂×C₁₆	—	7	2	no	—
63	C₂×C₁₆	C₄×D₇	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
65	C₄⋊C₈	—	7	2	no	—
66	C₄⋊C₈	—	7	1	no	—
67	C₄×C₈	—	7	2	no	—
68	C₄×C₈	C₄×D₇	1	1	yes	characteristic
69	C₃₂	—	2	2	no	—
70	C₃₂	—	7	2	no	—
71	C₃₂	—	14	1	no	—
72	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
73	C₄×C₇	—	2	1	no	—
74	C₄×C₇	M₆₄	1	2	yes	—
75	C₄×C₇	C₈×Q₈	1	1	yes	characteristic
76	C₄×C₇	#504	1	1	yes	characteristic
77	Dic₇	—	2	2	no	—
78	Dic₇	—	2	2	no	—
79	C₂×Q₈	—	7	1	no	—
80	C₂×C₈	—	7	2	no	—
81	C₂×C₈	C₈×D₇	1	1	yes	characteristic
82	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
83	C₄⋊C₄	—	7	2	no	—
84	C₄⋊C₄	—	7	1	no	—
85	C₄²	—	7	2	no	—
86	C₄²	C₈×D₇	1	1	yes	characteristic
87	C₁₆	—	2	1	no	—
88	C₁₆	—	14	2	no	—
89	C₁₆	#1159	1	1	yes	characteristic
90	C₁₆	Q₈×D₇	1	1	yes	characteristic
91	C₂×C₇	Q₈×C₁₆	1	1	yes	characteristic
92	C₂×C₇	C₂×M₆₄	1	1	yes	characteristic
93	C₂×C₇	#2258	1	1	yes	characteristic
94	Q₈	—	14	2	no	—
95	C₂×C₄	—	7	2	no	—
96	C₂×C₄	—	7	2	no	—
97	C₂×C₄	D₇×C₁₆	1	1	yes	characteristic
98	C₂×C₄	D₇×C₁₆	1	1	yes	characteristic
99	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	—	14	2	no	—
102	C₈	#6450	1	1	yes	characteristic
103	C₈	#6470	1	1	yes	characteristic
104	C₇	#13660	1	1	yes	characteristic, complemented, Sylow
105	C₂²	C₂×D₇×C₁₆	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	14	2	no	—
108	C₄	—	14	2	no	—
109	C₄	#89242	1	2	yes	—
110	C₄	#89602	1	1	yes	characteristic
111	C₄	#89679	1	1	yes	characteristic
112	C₂	#11727034	1	1	yes	characteristic
113	C₂	C₂×#89242	1	1	yes	characteristic
114	C₂	#11727126	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₄

C₂

yes

maximal, characteristic

C₇×C₄⋊C₃₂

C₂

yes

maximal, characteristic

Dic₇⋊C₃₂

C₂

yes

maximal

#11725395

C₂

yes

maximal

C₄×C₇⋊C₃₂

C₂

yes

maximal, characteristic

C₈×Dic₁₄

C₄

yes

characteristic

C₂×C₇×C₃₂

C₂²

yes

—

C₄×C₇×C₁₆

C₂²

yes

characteristic

Dic₇⋊C₁₆

C₄

yes

—

C₁₆×Dic₇

C₂²

yes

—

C₂×C₇⋊C₃₂

C₂²

yes

characteristic

C₂×C₇⋊C₃₂

C₂²

yes

characteristic

C₄⋊(C₇⋊C₁₆)

C₄

yes

characteristic

#13660

—

maximal, Sylow

C₄×Dic₁₄

C₈

yes

characteristic

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₂×C₇×C₁₆

C₂³

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

Dic₇⋊C₈

C₈

yes

—

C₈×Dic₇

C₂×C₄

yes

—

C₂×C₇⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₇⋊C₈)

C₈

yes

characteristic

C₇×C₃₂

—

C₇⋊C₃₂

—

C₇⋊C₃₂

Q₈

yes

complemented

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

D₇

yes

characteristic

#1397

—

C₄×C₃₂

—

C₂×Dic₁₄

C₁₆

yes

characteristic

C₂×C₇×C₈

C₂×C₈

yes

characteristic

C₂×C₇×C₈

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×C₈

yes

characteristic

C₄⋊Dic₇

C₁₆

yes

characteristic

Dic₇⋊C₄

C₁₆

yes

—

C₄×Dic₇

C₂×C₈

yes

—

C₂×C₇⋊C₈

C₂×C₈

yes

—

C₇×C₁₆

—

C₇×C₁₆

C₂×Q₈

yes

characteristic

C₇×C₁₆

#41

yes

characteristic

C₇⋊C₁₆

—

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇

yes

characteristic

C₂×C₄×C₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇

C₂²×C₈

yes

characteristic

C₂×Dic₇

C₂×C₁₆

yes

—

C₂×Dic₇

C₂×C₁₆

yes

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

C₄×Q₈

yes

characteristic

C₇×C₈

#131

yes

characteristic

C₇⋊C₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇

yes

characteristic

C₂×C₁₆

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

C₂²×C₇

C₂²×C₁₆

yes

characteristic

C₄×C₇

—

C₄×C₇

M₆₄

yes

—

C₄×C₇

C₈×Q₈

yes

characteristic

C₄×C₇

#504

yes

characteristic

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₈×D₇

yes

characteristic

C₂×C₈

C₂×C₄×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₈×D₇

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#1159

yes

characteristic

C₁₆

Q₈×D₇

yes

characteristic

C₂×C₇

Q₈×C₁₆

yes

characteristic

C₂×C₇

C₂×M₆₄

yes

characteristic

C₂×C₇

#2258

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₇×C₁₆

yes

characteristic

C₂×C₄

D₇×C₁₆

yes

characteristic

C₂×C₄

C₂×C₈×D₇

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

#6450

yes

characteristic

103

C₈

#6470

yes

characteristic

104

C₇

#13660

yes

characteristic, complemented, Sylow

105

C₂²

C₂×D₇×C₁₆

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#89242

yes

—

110

C₄

#89602

yes

characteristic

111

C₄

#89679

yes

characteristic

112

C₂

#11727034

yes

characteristic

113

C₂

C₂×#89242

yes

characteristic

114

C₂

#11727126

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇
commutator, socle, prime radical	C₂²×C₇	C₂²×C₁₆
Fitting	C₇×C₄⋊C₃₂	C₂
cosocle	C₂×C₇×C₁₆	C₂³
hypercenter	C₄⋊C₃₂	D₇
hypocenter	C₇	#13660
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇

commutator, socle, prime radical

C₂²×C₇

C₂²×C₁₆

Fitting

C₇×C₄⋊C₃₂

C₂

cosocle

C₂×C₇×C₁₆

C₂³

hypercenter

C₄⋊C₃₂

D₇

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
25. C₇⋊C₃₂	Q₈	semidirect
104. C₇	#13660	semidirect

H ≅ G/N

type

25. C₇⋊C₃₂

Q₈

semidirect

104. C₇

#13660

semidirect

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

subgroup	index
C₁₆×Dic₁₄	2
C₇×C₄⋊C₃₂	2
Dic₇⋊C₃₂	2
#11725395	2
C₄×C₇⋊C₃₂	2
#13660	7

subgroup

index

C₁₆×Dic₁₄

C₇×C₄⋊C₃₂

Dic₇⋊C₃₂

#11725395

C₄×C₇⋊C₃₂

#13660

Subgroups — #13274428

Conjugacy classes of subgroups

Complements

Maximal subgroups