Group #22569311 — subgroups

Subgroups — #22569311

Order 8576.

330 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 205 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	#16863527	C₂	1	4	yes	maximal, complemented
2	Q₈×C₆₇⋊C₈	C₂	1	4	yes	maximal, complemented
3	#16863457	C₂	1	4	yes	maximal
4	#16863449	C₂	1	2	yes	maximal
5	#533×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
8	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
10	#14523954	C₂²	1	4	yes	—
11	C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
12	C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
13	C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
14	#125×C₆₇	C₄	1	2	yes	—
15	C₂²⋊Q₈×C₆₇	C₄	1	4	yes	—
16	C₄×Q₈×C₆₇	C₂²	1	4	yes	—
17	#117×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
20	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
21	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
28	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₆₇	C₂³	1	2	yes	—
31	C₄²×C₆₇	C₂³	1	4	yes	—
32	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	Q₈×C₆₇	—	2	8	no	—
34	C₂×C₄×C₆₇	—	2	4	no	—
35	C₂×C₄×C₆₇	—	2	2	no	—
36	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₆₇	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
41	C₆₇⋊C₈	—	2	4	no	—
42	C₆₇⋊C₈	—	2	4	no	—
43	C₂²×C₆₇	—	2	1	no	—
44	C₂²×C₆₇	—	2	2	no	—
45	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
46	C₄×C₆₇	—	2	1	no	—
47	C₄×C₆₇	—	2	2	no	—
48	C₄×C₆₇	—	2	8	no	—
49	C₄×C₆₇	#131	1	4	yes	—
50	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
51	C₂×C₆₇	—	4	1	no	—
52	C₂×C₆₇	#519	1	1	yes	characteristic
53	C₂×C₆₇	C₂×#131	1	2	yes	—
54	#2986	—	67	1	no	maximal, Sylow
55	C₆₇	#2986	1	1	yes	characteristic, complemented, Sylow
56	#533	D₆₇	1	1	yes	characteristic
57	#446	—	67	4	no	—
58	C₈×Q₈	—	67	4	no	—
59	#433	—	67	4	no	—
60	#431	—	67	2	no	—
61	C₄⋊Q₈	Dic₆₇	1	2	yes	—
62	#125	Dic₆₇	1	2	yes	—
63	C₂²⋊Q₈	Dic₆₇	1	4	yes	—
64	C₄×Q₈	C₂×D₆₇	1	4	yes	—
65	#117	C₂×D₆₇	1	2	yes	—
66	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
67	C₄⋊C₈	—	67	2	no	—
68	C₄⋊C₈	—	67	2	no	—
69	C₄⋊C₈	—	67	8	no	—
70	C₂²⋊C₈	—	67	4	no	—
71	#97	—	67	4	no	—
72	C₄×C₈	—	67	4	no	—
73	C₄×C₈	—	67	4	no	—
74	C₂×Q₈	C₂×Dic₆₇	1	4	yes	—
75	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
76	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
77	C₂×C₈	—	67	4	no	—
78	C₂×C₈	—	67	4	no	—
79	C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
80	C₄⋊C₄	C₂×Dic₆₇	1	8	yes	—
81	C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
82	C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
83	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
84	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
85	C₄²	C₂²×D₆₇	1	2	yes	—
86	C₄²	C₂²×D₆₇	1	4	yes	—
87	C₂³	C₂²×Dic₆₇	1	1	yes	characteristic
88	Q₈	—	2	8	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
93	C₂×C₄	C₂²×Dic₆₇	1	8	yes	—
94	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
95	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
96	C₈	—	134	4	no	—
97	C₈	—	134	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	2	no	—
103	C₄	—	2	8	no	—
104	C₄	#14524050	1	4	yes	—
105	C₄	#14524091	1	2	yes	—
106	C₂	—	4	1	no	—
107	C₂	#16863874	1	1	yes	characteristic
108	C₂	C₂×#14524050	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863527

C₂

yes

maximal, complemented

Q₈×C₆₇⋊C₈

C₂

yes

maximal, complemented

#16863457

C₂

yes

maximal

#16863449

C₂

yes

maximal

#533×C₆₇

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523954

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₄

yes

—

#125×C₆₇

C₄

yes

—

C₂²⋊Q₈×C₆₇

C₄

yes

—

C₄×Q₈×C₆₇

C₂²

yes

—

#117×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#131

yes

—

C₄×C₆₇

2₋¹⁺⁴

yes

—

C₂×C₆₇

—

C₂×C₆₇

#519

yes

characteristic

C₂×C₆₇

C₂×#131

yes

—

#2986

—

maximal, Sylow

C₆₇

#2986

yes

characteristic, complemented, Sylow

#533

D₆₇

yes

characteristic

#446

—

C₈×Q₈

—

#433

—

#431

—

C₄⋊Q₈

Dic₆₇

yes

—

#125

Dic₆₇

yes

—

C₂²⋊Q₈

Dic₆₇

yes

—

C₄×Q₈

C₂×D₆₇

yes

—

#117

C₂×D₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₂²⋊C₄

C₂×Dic₆₇

yes

—

C₄²

C₂×Dic₆₇

yes

characteristic

C₄²

C₂×Dic₆₇

yes

characteristic

C₄²

C₂²×D₆₇

yes

—

C₄²

C₂²×D₆₇

yes

—

C₂³

C₂²×Dic₆₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₄

C₂²×Dic₆₇

yes

—

C₂×C₄

C₂²×Dic₆₇

yes

—

C₂×C₄

C₂²×Dic₆₇

yes

—

C₂×C₄

C₂³×D₆₇

yes

characteristic

C₈

—

134

—

C₈

—

134

—

C₂²

—

C₂²

—

100

C₂²

C₂³×Dic₆₇

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#14524050

yes

—

105

C₄

#14524091

yes

—

106

C₂

—

107

C₂

#16863874

yes

characteristic

108

C₂

C₂×#14524050

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#533×C₆₇	C₂
hypercenter	#533	D₆₇
hypocenter	C₆₇	#2986
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₇

commutator, socle

C₂²×C₆₇

C₂³×C₄

Fitting

#533×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #16863527	C₂	semidirect
2. Q₈×C₆₇⋊C₈	C₂	semidirect
55. C₆₇	#2986	semidirect

H ≅ G/N

type

1. #16863527

C₂

semidirect

2. Q₈×C₆₇⋊C₈

C₂

semidirect

55. C₆₇

#2986

semidirect

Subgroups — #22569311

Conjugacy classes of subgroups

Complements