Group #13268492 — subgroups

Subgroups — #13268492

Order 1792.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
2	#11729136	C₂	1	2	yes	maximal, complemented
3	#11729135	C₂	1	2	yes	maximal
4	#11725814	C₂	1	1	yes	maximal, characteristic
5	C₈×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#500	C₂²	1	2	yes	—
7	C₇×#487	—	2	2	no	—
8	#89471	C₂²	1	1	yes	characteristic
9	#89467	C₂²	1	1	yes	characteristic
10	#89466	—	2	2	no	—
11	C₇×C₈²	C₂²	1	1	yes	characteristic
12	#89287	—	2	4	no	—
13	#89286	—	2	4	no	—
14	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
15	#12511	—	7	1	no	maximal, Sylow
16	C₇×C₄⋊Q₈	—	2	2	no	—
17	C₂×Dic₂₈	—	2	2	no	—
18	C₂×D₅₆	—	2	2	no	—
19	#6437	—	2	1	no	—
20	#6432	—	2	1	no	—
21	C₇×#107	—	2	2	no	—
22	C₇×#107	—	2	2	no	—
23	C₇×#103	—	4	4	no	—
24	C₄×C₇×C₈	D₄	1	2	yes	—
25	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
26	#6395	—	4	4	no	—
27	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
28	#2241	—	7	2	no	—
29	#2240	—	7	2	no	—
30	#1710	D₇	1	1	yes	characteristic, complemented
31	#1709	—	7	1	no	—
32	C₈×C₁₆	—	7	1	no	—
33	C₂×C₇×Q₈	—	4	2	no	—
34	C₂×D₂₈	—	4	1	no	—
35	C₂×D₂₈	—	4	2	no	—
36	C₂×Dic₁₄	—	4	1	no	—
37	C₂×C₇×C₈	D₈	1	4	yes	—
38	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
39	C₇×C₄⋊C₄	—	4	2	no	—
40	C₇×C₄⋊C₄	—	4	2	no	—
41	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₇	—	4	2	no	—
43	Dic₂₈	—	4	4	no	—
44	D₅₆	—	4	4	no	—
45	C₇⋊C₁₆	—	2	4	no	—
46	#500	C₂×D₇	1	2	yes	—
47	#494	—	7	1	no	—
48	#493	—	7	1	no	—
49	#487	—	2	2	no	—
50	#486	—	14	2	no	—
51	#358	—	14	4	no	—
52	#357	—	14	4	no	—
53	C₄×C₁₆	—	7	2	no	—
54	C₈²	C₂×D₇	1	1	yes	characteristic
55	C₂²×D₇	—	8	1	no	—
56	C₇×Q₈	—	8	4	no	—
57	C₂×C₄×C₇	—	8	2	no	—
58	C₂×C₄×C₇	#127	1	1	yes	characteristic
59	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
60	C₂×Dic₇	—	8	1	no	—
61	D₂₈	—	8	2	no	—
62	D₂₈	—	8	4	no	—
63	Dic₁₄	—	8	2	no	—
64	C₇×C₈	—	2	4	no	—
65	C₇×C₈	#135	1	4	yes	—
66	C₂×Q₁₆	—	14	2	no	—
67	C₂×D₈	—	14	2	no	—
68	C₄⋊Q₈	—	2	2	no	—
69	C₄⋊Q₈	—	14	1	no	—
70	#127	—	14	1	no	—
71	C₂×C₁₆	—	7	4	no	—
72	#107	—	2	2	no	—
73	#107	—	2	2	no	—
74	#103	—	4	4	no	—
75	#102	—	28	4	no	—
76	C₄×C₈	C₂²⋊D₇	1	2	yes	—
77	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
78	C₂²×C₇	#493	1	1	yes	characteristic
79	C₂×D₇	—	16	1	no	—
80	C₂×D₇	—	16	2	no	—
81	C₄×C₇	—	16	2	no	—
82	C₄×C₇	#495	1	2	yes	—
83	C₄×C₇	#508	1	4	yes	—
84	Dic₇	—	16	1	no	—
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	28	1	no	—
87	C₂×D₄	—	28	2	no	—
88	C₂×D₄	—	28	1	no	—
89	Q₁₆	—	28	4	no	—
90	D₈	—	28	4	no	—
91	C₂×C₈	#1141	1	4	yes	—
92	C₂×C₈	D₄×D₇	1	2	yes	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	28	2	no	—
96	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
97	C₁₆	—	14	4	no	—
98	C₂×C₇	#1709	1	1	yes	characteristic
99	C₂×C₇	#2249	1	2	yes	—
100	D₇	—	32	1	no	—
101	C₂³	—	56	1	no	—
102	Q₈	—	8	4	no	—
103	Q₈	—	56	2	no	—
104	D₄	—	56	2	no	—
105	D₄	—	56	4	no	—
106	C₂×C₄	—	8	2	no	—
107	C₂×C₄	—	56	1	no	—
108	C₂×C₄	C₂×#1141	1	2	yes	—
109	C₂×C₄	#6503	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₈	#6482	1	4	yes	—
112	C₇	#12511	1	1	yes	characteristic, complemented, Sylow
113	C₂²	—	112	1	no	—
114	C₂²	—	112	2	no	—
115	C₂²	#89845	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	112	1	no	—
118	C₄	#89963	1	2	yes	—
119	C₄	#89966	1	4	yes	—
120	C₂	—	224	1	no	—
121	C₂	#11729051	1	1	yes	characteristic
122	C₂	#11729074	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1710

C₂

yes

maximal, characteristic, complemented

#11729136

C₂

yes

maximal, complemented

#11729135

C₂

yes

maximal

#11725814

C₂

yes

maximal, characteristic

C₈×C₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₇×#500

C₂²

yes

—

C₇×#487

—

#89471

C₂²

yes

characteristic

#89467

C₂²

yes

characteristic

#89466

—

C₇×C₈²

C₂²

yes

characteristic

#89287

—

#89286

—

C₄×C₇⋊C₁₆

C₂²

yes

—

#12511

—

maximal, Sylow

C₇×C₄⋊Q₈

—

C₂×Dic₂₈

—

C₂×D₅₆

—

—

—

—

—

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

characteristic

#6395

—

C₂×C₇⋊C₁₆

D₄

yes

—

#2241

—

#2240

—

#1710

D₇

yes

characteristic, complemented

#1709

—

C₈×C₁₆

—

C₂×C₇×Q₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

Dic₂₈

—

D₅₆

—

C₇⋊C₁₆

—

#500

C₂×D₇

yes

—

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

#135

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

C₂²⋊D₇

yes

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

#495

yes

—

C₄×C₇

#508

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#1141

yes

—

C₂×C₈

D₄×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₇

yes

characteristic

C₁₆

—

C₂×C₇

#1709

yes

characteristic

C₂×C₇

#2249

yes

—

100

D₇

—

101

C₂³

—

102

Q₈

—

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×#1141

yes

—

109

C₂×C₄

#6503

yes

characteristic

110

C₈

—

111

C₈

#6482

yes

—

112

C₇

#12511

yes

characteristic, complemented, Sylow

113

C₂²

—

112

—

114

C₂²

—

112

—

115

C₂²

#89845

yes

characteristic

116

C₄

—

117

C₄

—

112

—

118

C₄

#89963

yes

—

119

C₄

#89966

yes

—

120

C₂

—

224

—

121

C₂

#11729051

yes

characteristic

122

C₂

#11729074

yes

—

	subgroup	G/H
center	C₂²	#89845
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1710	C₂
socle	C₂²×C₇	#493
prime radical	#89467	C₂²
hypercenter	#1710	D₇
hypocenter	C₇	#12511
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#89845

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

C₂²

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. C₇×#1710	C₂	semidirect
2. #11729136	C₂	semidirect
5. C₈×C₇⋊C₁₆	C₂	semidirect
30. #1710	D₇	semidirect
112. C₇	#12511	semidirect

H ≅ G/N

type

1. C₇×#1710

C₂

semidirect

2. #11729136

C₂

semidirect

5. C₈×C₇⋊C₁₆

C₂

semidirect

30. #1710

D₇

semidirect

112. C₇

#12511

semidirect

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

subgroup	index
C₇×#1710	2
#11729136	2
#11729135	2
#11725814	2
C₈×C₇⋊C₁₆	2
#12511	7

subgroup

index

C₈×C₇⋊C₁₆

Subgroups — #13268492

Conjugacy classes of subgroups

Complements

Maximal subgroups