Group #11729588 — subgroups

Subgroups — #11729588

Order 896.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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×#533	C₂	1	1	yes	maximal, characteristic
2	#89696	C₂	1	2	yes	maximal
3	#89342	C₂	1	4	yes	maximal, complemented
4	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
5	C₇×#125	C₄	1	2	yes	—
6	C₇×C₂²⋊Q₈	—	2	2	no	—
7	C₄×C₇×Q₈	—	2	2	no	—
8	C₇×#117	C₄	1	2	yes	—
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×#1136	—	2	2	no	—
11	C₄⋊(C₇⋊C₈)	—	2	2	no	—
12	C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
13	#1561	—	7	1	no	maximal, Sylow
14	C₂×C₇×Q₈	—	2	2	no	—
15	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₇	D₄	1	2	yes	—
17	C₇×C₄⋊C₄	—	2	4	no	—
18	C₇×C₄⋊C₄	—	2	2	no	—
19	C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
20	C₇×C₂²⋊C₄	—	2	2	no	—
21	C₄²×C₇	—	2	2	no	—
22	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₇	D₄	1	2	yes	—
24	C₄²×C₇	C₂³	1	1	yes	characteristic
25	#1136	—	4	4	no	—
26	C₂×C₇⋊C₈	—	2	4	no	—
27	#533	D₇	1	1	yes	characteristic
28	#423	—	7	2	no	—
29	#332	—	7	4	no	—
30	C₂³×C₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₇×Q₈	—	4	2	no	—
32	C₇×Q₈	—	4	2	no	—
33	C₂×C₄×C₇	—	2	2	no	—
34	C₂×C₄×C₇	—	2	2	no	—
35	C₂×C₄×C₇	—	2	2	no	—
36	C₂×C₄×C₇	—	2	4	no	—
37	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
41	C₇⋊C₈	—	4	4	no	—
42	C₂×M₁₆	—	14	2	no	—
43	C₄⋊Q₈	C₂×D₇	1	2	yes	—
44	#125	Dic₇	1	2	yes	—
45	C₂²⋊Q₈	—	2	2	no	—
46	C₄×Q₈	—	2	2	no	—
47	#117	Dic₇	1	2	yes	—
48	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
49	C₄⋊C₈	—	7	4	no	—
50	C₄⋊C₈	—	14	2	no	—
51	C₂²×C₇	—	2	1	no	—
52	C₂²×C₇	—	2	2	no	—
53	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₇	—	2	1	no	—
55	C₄×C₇	—	2	2	no	—
56	C₄×C₇	—	4	4	no	—
57	C₄×C₇	#101	1	2	yes	—
58	C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
59	C₄×C₇	#137	1	2	yes	—
60	C₂×Q₈	—	2	2	no	—
61	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
62	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
63	M₁₆	—	28	4	no	—
64	C₂×C₈	—	14	4	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	C₂×Dic₇	1	4	yes	—
68	C₂²⋊C₄	—	2	2	no	—
69	C₄²	—	2	2	no	—
70	C₄²	C₂×Dic₇	1	1	yes	characteristic
71	C₄²	C₂²⋊D₇	1	2	yes	—
72	C₄²	C₂²×D₇	1	1	yes	characteristic
73	C₂×C₇	—	4	1	no	—
74	C₂×C₇	C₂×#101	1	1	yes	characteristic
75	C₂×C₇	#417	1	2	yes	—
76	C₂³	C₂²⋊Dic₇	1	1	yes	characteristic
77	Q₈	—	4	2	no	—
78	Q₈	—	4	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	C₂²⋊Dic₇	1	2	yes	—
84	C₂×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
85	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
86	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
87	C₈	—	28	4	no	—
88	C₇	#1561	1	1	yes	characteristic, complemented, Sylow
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#6410	1	2	yes	—
96	C₄	#6512	1	2	yes	—
97	C₄	#6514	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#6410	1	1	yes	characteristic
100	C₂	#90004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#533

C₂

yes

maximal, characteristic

#89696

C₂

yes

maximal

#89342

C₂

yes

maximal, complemented

C₇×C₄⋊Q₈

C₂²

yes

—

C₇×#125

C₄

yes

—

C₇×C₂²⋊Q₈

—

C₄×C₇×Q₈

—

C₇×#117

C₄

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×#1136

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

C₂²

yes

—

#1561

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₇

D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

C₂×C₄

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

C₂×C₄

yes

characteristic

C₄²×C₇

D₄

yes

—

C₄²×C₇

C₂³

yes

characteristic

#1136

—

C₂×C₇⋊C₈

—

#533

D₇

yes

characteristic

#423

—

#332

—

C₂³×C₇

C₂²⋊C₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊C₄

yes

—

C₂×C₄×C₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

—

C₇⋊C₈

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₇

yes

—

#125

Dic₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₇

yes

—

C₂×C₄²

C₂×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#101

yes

—

C₄×C₇

AGL(1,ℤ/8)

yes

—

C₄×C₇

#137

yes

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₇

yes

characteristic

C₂²×C₄

C₂²⋊D₇

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₇

yes

characteristic

C₄²

C₂²⋊D₇

yes

—

C₄²

C₂²×D₇

yes

characteristic

C₂×C₇

—

C₂×C₇

C₂×#101

yes

characteristic

C₂×C₇

#417

yes

—

C₂³

C₂²⋊Dic₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₇

yes

—

C₂×C₄

C₂²⋊Dic₇

yes

characteristic

C₂×C₄

C₂²×Dic₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇

yes

—

C₈

—

C₇

#1561

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊Dic₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#6410

yes

—

C₄

#6512

yes

—

C₄

#6514

yes

—

C₂

—

C₂

C₂×#6410

yes

characteristic

100

C₂

#90004

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂²×C₄
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#533	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₇	C₂²⋊C₄
cosocle	C₄²×C₇	C₂³
hypercenter	#533	D₇
hypocenter	C₇	#1561
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇

commutator

C₂×C₄×C₇

C₂²×C₄

Frattini

C₄²

C₂²×D₇

Fitting

C₇×#533

C₂

socle

C₂²×C₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₇

C₂²⋊C₄

cosocle

C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #89342	C₂	semidirect
88. C₇	#1561	semidirect

H ≅ G/N

type

3. #89342

C₂

semidirect

88. C₇

#1561

semidirect

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

subgroup	index
C₇×#533	2
#89696	2
#89342	2
#1561	7

subgroup

index

Subgroups — #11729588

Conjugacy classes of subgroups

Complements

Maximal subgroups