Group #13057061 — subgroups

Subgroups — #13057061

Order 1344.

118 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 27 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	SL(2,𝔽₃)×Dic₇	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#859	C₂	1	1	yes	maximal, characteristic
3	C₇×#859	C₂	1	1	yes	maximal, characteristic
4	#89566	—	3	1	no	maximal, Hall
5	C₂×C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	Dic₇⋊Dic₃	—	4	1	no	maximal
7	Q₈×Dic₇	D₃	1	1	yes	characteristic
8	#6451	—	3	1	no	—
9	C₇×#103	—	3	1	no	—
10	#6409	—	3	1	no	—
11	#6392	—	3	1	no	—
12	Dic₇⋊C₈	—	3	1	no	—
13	#6381	—	3	1	no	—
14	#5391	—	7	1	no	maximal, Hall
15	C₂×Dic₂₁	—	4	1	no	—
16	C₂×C₇×Dic₃	—	4	1	no	—
17	C₂×C₃×Dic₇	—	4	1	no	—
18	C₇×SL(2,𝔽₃)	Q₈	1	1	yes	characteristic, complemented
19	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
20	C₂×Dic₁₄	—	6	1	no	—
21	C₂×C₇×C₈	—	3	1	no	—
22	C₇×C₄⋊C₄	—	3	1	no	—
23	C₄⋊Dic₇	—	3	1	no	—
24	C₄⋊Dic₇	—	3	1	no	—
25	Dic₇⋊C₄	—	6	1	no	—
26	C₄×Dic₇	—	3	1	no	—
27	C₂×C₇⋊C₈	—	3	1	no	—
28	C₄×SL(2,𝔽₃)	—	7	1	no	—
29	#859	—	7	1	no	—
30	#859	D₇	1	1	yes	characteristic
31	C₂²×C₃×C₇	—	4	1	no	—
32	Dic₂₁	—	8	1	no	—
33	C₃×Dic₇	—	4	2	no	—
34	C₇×Dic₃	—	8	1	no	—
35	#477	—	21	1	no	Sylow
36	C₇×Q₈	—	3	1	no	—
37	C₇×Q₈	Dic₆	1	1	yes	characteristic
38	C₂×C₄×C₇	—	3	1	no	—
39	C₂×C₄×C₇	—	6	1	no	—
40	C₂×Dic₇	—	3	1	no	—
41	C₂×Dic₇	—	6	1	no	—
42	C₂×Dic₇	S₄	1	1	yes	characteristic
43	Dic₁₄	—	12	2	no	—
44	C₇×C₈	—	6	1	no	—
45	C₇⋊C₈	—	6	1	no	—
46	C₂×SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic
47	C₄⋊Dic₃	—	28	1	no	—
48	C₂×C₃×C₇	—	4	1	no	—
49	C₂×C₃×C₇	—	4	1	no	—
50	C₂×C₃×C₇	—	4	1	no	—
51	C₄⋊Q₈	—	21	1	no	—
52	C₄×Q₈	—	7	1	no	—
53	#107	—	21	1	no	—
54	#107	—	21	1	no	—
55	C₄⋊C₈	—	21	1	no	—
56	#103	—	3	1	no	—
57	#103	—	21	1	no	—
58	C₂²×C₇	C₂×S₄	1	1	yes	characteristic
59	C₄×C₇	—	3	1	no	—
60	C₄×C₇	—	3	1	no	—
61	C₄×C₇	—	12	1	no	—
62	Dic₇	—	6	1	no	—
63	Dic₇	—	12	1	no	—
64	Dic₇	#226	1	2	yes	—
65	C₂×C₃×C₄	—	28	1	no	—
66	C₂×Dic₃	—	4	1	no	—
67	C₂×Dic₃	—	28	1	no	—
68	SL(2,𝔽₃)	Dic₁₄	1	1	yes	characteristic, complemented
69	C₃×C₇	—	4	1	no	Hall
70	C₂×Q₈	—	42	1	no	—
71	C₂×Q₈	D₃×D₇	1	1	yes	characteristic
72	C₂×C₈	—	3	1	no	—
73	C₂×C₈	—	21	1	no	—
74	C₄⋊C₄	—	3	1	no	—
75	C₄⋊C₄	—	21	1	no	—
76	C₄⋊C₄	—	21	1	no	—
77	C₄⋊C₄	—	42	1	no	—
78	C₄²	—	21	1	no	—
79	C₂×C₇	A₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₇	C₂×#226	1	1	yes	characteristic
81	C₂×C₇	#986	1	1	yes	characteristic
82	C₂²×C₃	—	4	1	no	—
83	C₃×C₄	—	28	2	no	—
84	Dic₃	—	8	1	no	—
85	Dic₃	—	56	1	no	—
86	Q₈	—	3	1	no	—
87	Q₈	—	84	2	no	—
88	Q₈	#4248	1	1	yes	characteristic
89	C₂×C₄	—	3	1	no	—
90	C₂×C₄	—	6	1	no	—
91	C₂×C₄	—	7	1	no	—
92	C₂×C₄	—	21	1	no	—
93	C₂×C₄	—	42	1	no	—
94	C₈	—	6	1	no	—
95	C₈	—	42	1	no	—
96	C₇	#5391	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₃	—	4	1	no	—
98	C₂×C₃	—	4	1	no	—
99	C₂×C₃	—	4	1	no	—
100	C₂²	D₇×S₄	1	1	yes	characteristic
101	C₄	—	3	1	no	—
102	C₄	—	3	1	no	—
103	C₄	—	7	2	no	—
104	C₄	—	12	1	no	—
105	C₄	—	42	1	no	—
106	C₄	—	84	1	no	—
107	C₃	—	4	1	no	Sylow
108	C₂	#10621472	1	1	yes	characteristic
109	C₂	D₇×#226	1	1	yes	characteristic
110	C₂	#10621477	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

SL(2,𝔽₃)×Dic₇

C₂

yes

maximal, characteristic

C₇⋊#859

C₂

yes

maximal, characteristic

C₇×#859

C₂

yes

maximal, characteristic

#89566

—

maximal, Hall

C₂×C₇×SL(2,𝔽₃)

C₂²

yes

characteristic

Dic₇⋊Dic₃

—

maximal

Q₈×Dic₇

D₃

yes

—

—

—

—

Dic₇⋊C₈

—

#6381

—

#5391

—

maximal, Hall

C₂×Dic₂₁

—

C₂×C₇×Dic₃

—

C₂×C₃×Dic₇

—

C₇×SL(2,𝔽₃)

Q₈

yes

characteristic, complemented

C₂×C₇×Q₈

C₂×D₃

yes

characteristic

C₂×Dic₁₄

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

C₄×SL(2,𝔽₃)

—

#859

—

#859

D₇

yes

characteristic

C₂²×C₃×C₇

—

Dic₂₁

—

C₃×Dic₇

—

C₇×Dic₃

—

#477

—

Sylow

C₇×Q₈

—

C₇×Q₈

Dic₆

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

S₄

yes

characteristic

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

C₂×SL(2,𝔽₃)

C₂×D₇

yes

characteristic

C₄⋊Dic₃

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

#103

—

#103

—

C₂²×C₇

C₂×S₄

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

Dic₇

—

Dic₇

#226

yes

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

SL(2,𝔽₃)

Dic₁₄

yes

characteristic, complemented

C₃×C₇

—

Hall

C₂×Q₈

—

C₂×Q₈

D₃×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂×C₇

A₄⋊Q₈

yes

C₂×C₇

yes

C₂×C₇

yes

C₂²×C₃

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Q₈

—

Q₈

—

Q₈

#4248

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

—

C₇

#5391

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

100

C₂²

D₇×S₄

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₃

—

Sylow

108

C₂

#10621472

yes

characteristic

109

C₂

D₇×#226

yes

characteristic

110

C₂

#10621477

yes

characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₇×S₄
commutator, prime radical, cosocle	C₂×C₇×SL(2,𝔽₃)	C₂²
Fitting	C₂×C₇×Q₈	C₂×D₃
socle	C₂²×C₇	C₂×S₄
hypocenter	C₇×SL(2,𝔽₃)	Q₈
LCS	G ▷ C₂×C₇×SL(2,𝔽₃) ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂²

subgroup

G/H

center, Frattini, hypercenter

C₂²

D₇×S₄

commutator, prime radical, cosocle

C₂×C₇×SL(2,𝔽₃)

C₂²

Fitting

C₂×C₇×Q₈

C₂×D₃

socle

C₂²×C₇

C₂×S₄

hypocenter

C₇×SL(2,𝔽₃)

Q₈

LCS

G ▷ C₂×C₇×SL(2,𝔽₃) ▷ C₇×SL(2,𝔽₃)

UCS

C₁ ◁ C₂²

N	H ≅ G/N	type
18. C₇×SL(2,𝔽₃)	Q₈	semidirect
68. SL(2,𝔽₃)	Dic₁₄	semidirect
96. C₇	#5391	semidirect

H ≅ G/N

type

18. C₇×SL(2,𝔽₃)

Q₈

semidirect

68. SL(2,𝔽₃)

Dic₁₄

semidirect

96. C₇

#5391

semidirect

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

subgroup	index
SL(2,𝔽₃)×Dic₇	2
C₇⋊#859	2
C₇×#859	2
#89566	3
Dic₇⋊Dic₃	4
#5391	7

subgroup

index

Dic₇⋊Dic₃

Subgroups — #13057061

Conjugacy classes of subgroups

Complements

Maximal subgroups