Group #13247506 — subgroups

Subgroups — #13247506

Order 1792.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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×#1450	C₂	1	1	yes	maximal, characteristic
2	#11727486	C₂	1	2	yes	maximal
3	#11725553	C₂	1	4	yes	maximal
4	C₇×#431	C₂²	1	2	yes	—
5	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
6	#89698	C₂²	1	4	yes	—
7	C₇×#322	C₄	1	2	yes	—
8	C₇×C₈²	C₄	1	2	yes	—
9	#89253	—	2	2	no	—
10	#89252	—	2	2	no	—
11	#8797	—	7	1	no	maximal, Sylow
12	C₂²×C₇×C₈	Q₈	1	2	yes	—
13	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
14	C₇×C₄⋊C₈	—	2	2	no	—
15	C₇×C₂²⋊C₈	—	2	2	no	—
16	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
17	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
18	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₄×C₇×C₈	D₄	1	2	yes	—
20	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
21	C₄⋊(C₇⋊C₈)	—	2	4	no	—
22	#6377	—	2	4	no	—
23	#1568	—	7	2	no	—
24	#1450	D₇	1	1	yes	characteristic
25	#1287	—	7	4	no	—
26	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₇	#41	1	2	yes	—
28	C₂×C₇×C₈	—	2	4	no	—
29	C₂×C₇×C₈	—	2	4	no	—
30	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
31	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
32	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₄²×C₇	#41	1	2	yes	—
35	C₂×C₇⋊C₈	—	4	4	no	—
36	#432	—	7	4	no	—
37	#431	C₂×D₇	1	2	yes	—
38	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
39	#335	—	14	2	no	—
40	#334	—	14	2	no	—
41	#322	Dic₇	1	2	yes	—
42	C₈²	Dic₇	1	2	yes	—
43	C₂³×C₇	#125	1	1	yes	characteristic
44	C₂×C₄×C₇	—	2	2	no	—
45	C₂×C₄×C₇	—	4	2	no	—
46	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇	#117	1	2	yes	—
48	C₂×C₄×C₇	#124	1	2	yes	—
49	C₂×C₄×C₇	#125	1	1	yes	characteristic
50	C₇×C₈	—	2	2	no	—
51	C₇×C₈	—	4	4	no	—
52	C₇×C₈	#108	1	4	yes	—
53	C₇⋊C₈	—	8	4	no	—
54	C₂²×C₈	Dic₁₄	1	2	yes	—
55	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	—	14	4	no	—
58	C₂²⋊C₈	—	2	2	no	—
59	C₂²⋊C₈	—	14	4	no	—
60	#97	—	14	4	no	—
61	C₄×C₈	D₂₈	1	2	yes	—
62	C₄×C₈	C₂×Dic₇	1	4	yes	—
63	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
64	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
65	C₂²×C₇	—	2	2	no	—
66	C₂²×C₇	—	2	1	no	—
67	C₂²×C₇	#382	1	1	yes	characteristic
68	C₄×C₇	—	2	2	no	—
69	C₄×C₇	—	2	2	no	—
70	C₄×C₇	—	2	1	no	—
71	C₄×C₇	C₂×#108	1	2	yes	—
72	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
73	C₂²×C₄	#1157	1	2	yes	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	28	4	no	—
77	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
78	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
79	C₄²	C₂×D₂₈	1	1	yes	characteristic
80	C₄²	#1157	1	2	yes	—
81	C₄²	C₂²×Dic₇	1	1	yes	characteristic
82	C₂×C₇	—	4	1	no	—
83	C₂×C₇	#1833	1	2	yes	—
84	C₂×C₇	#1837	1	1	yes	characteristic
85	C₂³	#6433	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	#6433	1	1	yes	characteristic
89	C₂×C₄	#6438	1	2	yes	—
90	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
91	C₂×C₄	#6489	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	4	4	no	—
94	C₈	—	56	4	no	—
95	C₈	#6393	1	4	yes	—
96	C₇	#8797	1	1	yes	characteristic, complemented, Sylow
97	C₂²	—	2	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#89708	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	2	2	no	—
102	C₄	—	2	1	no	—
103	C₄	C₂×#6393	1	2	yes	—
104	C₂	—	4	1	no	—
105	C₂	#11727545	1	1	yes	characteristic
106	C₂	#11727650	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1450

C₂

yes

maximal, characteristic

#11727486

C₂

yes

maximal

#11725553

C₂

yes

maximal

C₇×#431

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

#89698

C₂²

yes

—

C₇×#322

C₄

yes

—

C₇×C₈²

C₄

yes

—

—

—

—

C₂²×C₇×C₈

Q₈

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#6377

—

#1568

—

#1450

D₇

yes

characteristic

#1287

—

C₂²×C₄×C₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

#41

yes

—

C₂×C₇⋊C₈

—

#432

—

#431

C₂×D₇

yes

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

Dic₇

yes

—

C₈²

Dic₇

yes

—

C₂³×C₇

#125

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#124

yes

—

C₂×C₄×C₇

#125

yes

characteristic

C₇×C₈

—

C₇×C₈

—

C₇×C₈

#108

yes

—

C₇⋊C₈

—

C₂²×C₈

Dic₁₄

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

D₂₈

yes

—

C₄×C₈

C₂×Dic₇

yes

—

C₄×C₈

C₂×Dic₇

yes

characteristic

C₄×C₈

C₂×Dic₇

yes

characteristic

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#382

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×#108

yes

—

C₂²×C₄

C₂×Dic₁₄

yes

characteristic

C₂²×C₄

#1157

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₇

yes

—

C₂×C₈

C₄⋊Dic₇

yes

—

C₄²

C₂×D₂₈

yes

characteristic

C₄²

#1157

yes

—

C₄²

C₂²×Dic₇

yes

characteristic

C₂×C₇

—

C₂×C₇

#1833

yes

—

C₂×C₇

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6433

yes

characteristic

C₂×C₄

#6438

yes

—

C₂×C₄

C₂×C₄⋊Dic₇

yes

characteristic

C₂×C₄

#6489

yes

—

C₈

—

C₈

—

C₈

—

C₈

#6393

yes

—

C₇

#8797

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

#89708

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

C₂×#6393

yes

—

104

C₂

—

105

C₂

#11727545

yes

characteristic

106

C₂

#11727650

yes

—

	subgroup	G/H
center	C₂²	#89708
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1450	C₂
socle	C₂²×C₇	#382
prime radical	C₂³×C₇	#125
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1450	D₇
hypocenter	C₇	#8797
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#89708

commutator

C₄²×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
96. C₇	#8797	semidirect

H ≅ G/N

type

96. C₇

#8797

semidirect

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

subgroup	index
C₇×#1450	2
#11727486	2
#11725553	2
#8797	7

subgroup

index

Subgroups — #13247506

Conjugacy classes of subgroups

Complements

Maximal subgroups