Group #13268534 — subgroups

Subgroups — #13268534

Order 1792.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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
2	#11729139	C₂	1	2	yes	maximal
3	#11729138	C₂	1	2	yes	maximal
4	#11725818	C₂	1	1	yes	maximal, characteristic
5	C₈×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₇×#500	C₂²	1	2	yes	—
7	C₇×#487	—	2	2	no	—
8	#89454	C₂²	1	2	yes	—
9	#89453	—	2	2	no	—
10	C₇×C₈²	C₂²	1	1	yes	characteristic
11	#89285	—	2	4	no	—
12	#89284	—	2	4	no	—
13	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	#12522	—	7	1	no	maximal, Sylow
16	C₇×C₄⋊Q₈	—	2	2	no	—
17	#6432	—	2	2	no	—
18	C₇×#107	—	2	2	no	—
19	C₇×#107	—	2	2	no	—
20	C₇×#103	—	4	4	no	—
21	C₄×C₇×C₈	D₄	1	2	yes	—
22	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
23	#6392	—	2	2	no	—
24	#6392	—	2	2	no	—
25	#6390	—	4	4	no	—
26	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
27	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
28	#2254	—	7	2	no	—
29	#2252	—	7	2	no	—
30	#1710	—	7	1	no	—
31	#1710	D₇	1	1	yes	characteristic
32	C₈×C₁₆	—	7	1	no	—
33	C₂×C₇×Q₈	—	4	2	no	—
34	C₂×Dic₁₄	—	4	2	no	—
35	C₂×C₇×C₈	D₈	1	4	yes	—
36	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
38	C₇×C₄⋊C₄	—	4	2	no	—
39	C₇×C₄⋊C₄	—	4	2	no	—
40	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₇	—	4	2	no	—
42	C₄⋊Dic₇	—	4	2	no	—
43	C₇⋊C₁₆	Q₁₆	1	4	yes	—
44	C₇⋊C₁₆	Q₁₆	1	4	yes	—
45	#500	—	7	2	no	—
46	#500	C₂×D₇	1	2	yes	—
47	#487	—	2	2	no	—
48	#487	—	14	2	no	—
49	#367	—	14	4	no	—
50	#366	—	14	4	no	—
51	C₄×C₁₆	—	7	1	no	—
52	C₄×C₁₆	—	7	1	no	—
53	C₈²	C₂×D₇	1	1	yes	characteristic
54	C₇×Q₈	—	8	4	no	—
55	C₂×C₄×C₇	—	8	2	no	—
56	C₂×C₄×C₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
58	C₂×Dic₇	—	8	2	no	—
59	Dic₁₄	—	8	4	no	—
60	C₇×C₈	—	2	4	no	—
61	C₇×C₈	C₂×Q₁₆	1	2	yes	—
62	C₇×C₈	C₂×Q₁₆	1	2	yes	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	14	2	no	—
65	C₂×C₁₆	—	7	2	no	—
66	C₂×C₁₆	—	7	2	no	—
67	#107	—	2	2	no	—
68	#107	—	2	2	no	—
69	#107	—	14	2	no	—
70	#107	—	14	2	no	—
71	#103	—	4	4	no	—
72	#103	—	28	4	no	—
73	C₄×C₈	C₂²⋊D₇	1	2	yes	—
74	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
75	C₂²×C₇	#493	1	1	yes	characteristic
76	C₄×C₇	—	16	2	no	—
77	C₄×C₇	#494	1	1	yes	characteristic
78	C₄×C₇	#494	1	1	yes	characteristic
79	C₄×C₇	#508	1	4	yes	—
80	Dic₇	—	16	2	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	28	2	no	—
83	C₂×C₈	#1141	1	4	yes	—
84	C₂×C₈	D₄×D₇	1	1	yes	characteristic
85	C₂×C₈	D₄×D₇	1	1	yes	characteristic
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	4	2	no	—
88	C₄⋊C₄	—	28	2	no	—
89	C₄⋊C₄	—	28	2	no	—
90	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
91	C₁₆	—	7	4	no	—
92	C₁₆	—	7	4	no	—
93	C₂×C₇	#1713	1	1	yes	characteristic
94	C₂×C₇	#2249	1	2	yes	—
95	Q₈	—	8	4	no	—
96	Q₈	—	56	4	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	56	2	no	—
99	C₂×C₄	C₂×#1141	1	2	yes	—
100	C₂×C₄	#6503	1	1	yes	characteristic
101	C₈	—	2	4	no	—
102	C₈	D₇×Q₁₆	1	2	yes	—
103	C₈	D₇×Q₁₆	1	2	yes	—
104	C₇	#12522	1	1	yes	characteristic, complemented, Sylow
105	C₂²	#89845	1	1	yes	characteristic
106	C₄	—	16	2	no	—
107	C₄	—	112	2	no	—
108	C₄	#89953	1	1	yes	characteristic
109	C₄	#89953	1	1	yes	characteristic
110	C₄	#89966	1	4	yes	—
111	C₂	#11729052	1	1	yes	characteristic
112	C₂	#11729074	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1710

C₂

yes

maximal, characteristic

#11729139

C₂

yes

maximal

#11729138

C₂

yes

maximal

#11725818

C₂

yes

maximal, characteristic

C₈×C₇⋊C₁₆

C₂

yes

maximal, characteristic

C₇×#500

C₂²

yes

—

C₇×#487

—

#89454

C₂²

yes

—

#89453

—

C₇×C₈²

C₂²

yes

characteristic

#89285

—

#89284

—

C₄×C₇⋊C₁₆

C₂²

yes

characteristic

C₄×C₇⋊C₁₆

C₂²

yes

characteristic

#12522

—

maximal, Sylow

C₇×C₄⋊Q₈

—

—

—

—

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

—

—

—

C₂×C₇⋊C₁₆

D₄

yes

—

C₂×C₇⋊C₁₆

D₄

yes

—

—

—

—

D₇

yes

characteristic

C₈×C₁₆

—

C₂×C₇×Q₈

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₇⋊C₁₆

Q₁₆

yes

—

C₇⋊C₁₆

Q₁₆

yes

—

#500

—

#500

C₂×D₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×D₇

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×Dic₇

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

C₂×Q₁₆

yes

—

C₇×C₈

C₂×Q₁₆

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄×C₈

C₂²⋊D₇

yes

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₄×C₇

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

#1141

yes

—

C₂×C₈

D₄×D₇

yes

characteristic

C₂×C₈

D₄×D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂×C₇

#1713

yes

characteristic

C₂×C₇

#2249

yes

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#1141

yes

—

100

C₂×C₄

#6503

yes

characteristic

101

C₈

—

102

C₈

D₇×Q₁₆

yes

—

103

C₈

D₇×Q₁₆

yes

—

104

C₇

#12522

yes

characteristic, complemented, Sylow

105

C₂²

#89845

yes

characteristic

106

C₄

—

107

C₄

—

112

—

108

C₄

#89953

yes

characteristic

109

C₄

#89953

yes

characteristic

110

C₄

#89966

yes

—

111

C₂

#11729052

yes

characteristic

112

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, prime radical	C₂²×C₇	#493
hypercenter	#1710	D₇
hypocenter	C₇	#12522
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

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
104. C₇	#12522	semidirect

H ≅ G/N

type

104. C₇

#12522

semidirect

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

subgroup	index
C₇×#1710	2
#11729139	2
#11729138	2
#11725818	2
C₈×C₇⋊C₁₆	2
#12522	7

subgroup

index

C₈×C₇⋊C₁₆

Subgroups — #13268534

Conjugacy classes of subgroups

Complements

Maximal subgroups