Group #22569131 — subgroups

Subgroups — C₂×Q₈×C₆₇⋊C₈ #22569131

Order 8576.

440 conjugacy classes of subgroups; only the 333 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 333 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	Q₈×C₆₇⋊C₈	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₆₇⋊C₈)	C₂	1	3	yes	maximal
3	C₂×C₄×C₆₇⋊C₈	C₂	1	3	yes	maximal
4	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₈	C₂²	1	3	yes	—
6	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊(C₆₇⋊C₈)	C₂²	1	12	yes	—
8	C₄×C₆₇⋊C₈	C₂²	1	12	yes	—
9	C₂²×Q₈×C₆₇	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₆₇	C₄	1	4	yes	—
11	C₄×Q₈×C₆₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
13	C₂×C₄²×C₆₇	C₂²	1	3	yes	—
14	C₂×C₆₇⋊C₈	—	2	6	no	—
15	C₂×C₆₇⋊C₈	Q₈	1	4	yes	direct product factor
16	C₂×C₆₇⋊C₈	C₂³	1	6	yes	—
17	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
18	C₂×Q₈×C₆₇	C₈	1	8	yes	complemented
19	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₆₇	C₂×C₄	1	3	yes	—
21	C₂²×C₄×C₆₇	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₆₇	C₂×C₄	1	12	yes	—
24	C₄²×C₆₇	C₂×C₄	1	6	yes	—
25	C₄²×C₆₇	C₂³	1	6	yes	—
26	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
27	Q₈×C₆₇	C₂×C₈	1	16	yes	complemented
28	C₂×C₄×C₆₇	—	2	6	no	—
29	C₂×C₄×C₆₇	C₂×C₈	1	12	yes	—
30	C₂×C₄×C₆₇	C₂²×C₄	1	6	yes	—
31	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₂²×C₄	1	6	yes	—
33	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₆₇	#41	1	2	yes	—
35	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
36	C₆₇⋊C₈	—	2	6	no	—
37	C₆₇⋊C₈	C₂×Q₈	1	4	yes	direct product factor
38	C₂²×C₆₇	C₄×Q₈	1	2	yes	—
39	C₂²×C₆₇	C₂²×C₈	1	2	yes	—
40	C₂²×C₆₇	#131	1	2	yes	—
41	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
42	C₄×C₆₇	—	2	6	no	—
43	C₄×C₆₇	C₄×Q₈	1	2	yes	—
44	C₄×C₆₇	C₂²×C₈	1	12	yes	—
45	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
46	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
47	C₂×C₆₇	C₈×Q₈	1	4	yes	complemented
48	C₂×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
49	C₂×C₆₇	C₂³×C₈	1	1	yes	characteristic
50	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
51	C₂×C₈×Q₈	—	67	1	no	maximal, Sylow
52	C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic
54	C₈×Q₈	—	67	8	no	—
55	C₂×C₄⋊C₈	—	67	3	no	—
56	C₂×C₄×C₈	—	67	3	no	—
57	C₂²×Q₈	Dic₆₇	1	1	yes	characteristic
58	C₂²×C₈	—	67	1	no	—
59	C₂²×C₈	—	67	3	no	—
60	C₄×Q₈	Dic₆₇	1	4	yes	—
61	C₄×Q₈	C₂×D₆₇	1	4	yes	—
62	C₂×C₄⋊C₄	Dic₆₇	1	3	yes	—
63	C₂×C₄²	C₂×D₆₇	1	3	yes	—
64	C₄⋊C₈	—	67	12	no	—
65	C₄×C₈	—	67	12	no	—
66	C₂×Q₈	C₆₇⋊C₈	1	8	yes	direct product factor
67	C₂×Q₈	C₂×Dic₆₇	1	4	yes	—
68	C₂²×C₄	C₂×Dic₆₇	1	3	yes	—
69	C₂²×C₄	C₂×Dic₆₇	1	3	yes	—
70	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
71	C₂×C₈	—	67	4	no	—
72	C₂×C₈	—	67	2	no	—
73	C₂×C₈	—	67	6	no	—
74	C₂×C₈	—	134	6	no	—
75	C₄⋊C₄	C₂×Dic₆₇	1	12	yes	—
76	C₄²	C₂×Dic₆₇	1	6	yes	—
77	C₄²	C₂²×D₆₇	1	6	yes	—
78	C₂³	C₂²×Dic₆₇	1	1	yes	characteristic
79	Q₈	C₂×C₆₇⋊C₈	1	16	yes	direct product factor
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	C₂×C₆₇⋊C₈	1	12	yes	—
82	C₂×C₄	C₂²×Dic₆₇	1	6	yes	—
83	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
84	C₂×C₄	C₂²×Dic₆₇	1	6	yes	—
85	C₂×C₄	Q₈×D₆₇	1	2	yes	—
86	C₂×C₄	#11761629	1	2	yes	—
87	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
88	C₈	—	67	4	no	—
89	C₈	—	134	6	no	—
90	C₂²	Q₈×Dic₆₇	1	2	yes	—
91	C₂²	C₂²×C₆₇⋊C₈	1	2	yes	—
92	C₂²	#14524050	1	2	yes	—
93	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
94	C₄	—	2	6	no	—
95	C₄	Q₈×Dic₆₇	1	2	yes	—
96	C₄	C₂²×C₆₇⋊C₈	1	12	yes	—
97	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
98	C₄	C₂×#11761629	1	1	yes	characteristic
99	C₂	Q₈×C₆₇⋊C₈	1	4	yes	direct product factor
100	C₂	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
101	C₂	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
102	C₂	C₂×#14524050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₆₇⋊C₈

C₂

yes

maximal, direct product factor

C₂×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal

C₂×C₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₆₇

C₄

yes

characteristic

C₄×Q₈×C₆₇

C₄

yes

—

C₄×Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₄

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₈

yes

direct product factor

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×Q₈×C₆₇

C₈

yes

complemented

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

Q₈×C₆₇

C₂×C₈

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂×Q₈

yes

direct product factor

C₂²×C₆₇

C₄×Q₈

yes

—

C₂²×C₆₇

C₂²×C₈

yes

—

C₂²×C₆₇

#131

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄×Q₈

yes

—

C₄×C₆₇

C₂²×C₈

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

C₂×C₆₇

C₈×Q₈

yes

complemented

C₂×C₆₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₇

C₂³×C₈

yes

characteristic

C₂×C₆₇

C₂×#131

yes

characteristic

C₂×C₈×Q₈

—

maximal, Sylow

C₆₇

C₂×C₈×Q₈

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₆₇

yes

characteristic

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂²×Q₈

Dic₆₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

Dic₆₇

yes

—

C₄×Q₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

Dic₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

—

C₄⋊C₈

—

C₄×C₈

—

C₂×Q₈

C₆₇⋊C₈

yes

direct product factor

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄²

C₂×Dic₆₇

yes

—

C₄²

C₂²×D₆₇

yes

—

C₂³

C₂²×Dic₆₇

yes

characteristic

Q₈

C₂×C₆₇⋊C₈

yes

direct product factor

C₂×C₄

—

C₂×C₄

C₂×C₆₇⋊C₈

yes

—

C₂×C₄

C₂²×Dic₆₇

yes

—

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₄

C₂²×Dic₆₇

yes

—

C₂×C₄

Q₈×D₆₇

yes

—

C₂×C₄

#11761629

yes

—

C₂×C₄

C₂³×D₆₇

yes

characteristic

C₈

—

C₈

—

134

—

C₂²

Q₈×Dic₆₇

yes

—

C₂²

C₂²×C₆₇⋊C₈

yes

—

C₂²

#14524050

yes

—

C₂²

C₂³×Dic₆₇

yes

characteristic

C₄

—

C₄

Q₈×Dic₆₇

yes

—

C₄

C₂²×C₆₇⋊C₈

yes

—

C₄

C₂×Q₈×D₆₇

yes

characteristic

C₄

C₂×#11761629

yes

characteristic

C₂

Q₈×C₆₇⋊C₈

yes

direct product factor

100

C₂

C₂×Q₈×Dic₆₇

yes

characteristic

101

C₂

C₂³×C₆₇⋊C₈

yes

characteristic

102

C₂

C₂×#14524050

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₆₇
commutator	C₂×C₆₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₄
hypercenter	C₂×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂×C₈×Q₈
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₆₇

commutator

C₂×C₆₇

C₂³×C₈

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄×Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂²×C₄

hypercenter

C₂×C₄×Q₈

D₆₇

hypocenter

C₆₇

C₂×C₈×Q₈

LCS

G ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. Q₈×C₆₇⋊C₈	C₂	direct (N × H)
15. C₂×C₆₇⋊C₈	Q₈	direct (N × H)
18. C₂×Q₈×C₆₇	C₈	semidirect
27. Q₈×C₆₇	C₂×C₈	semidirect
37. C₆₇⋊C₈	C₂×Q₈	direct (N × H)
47. C₂×C₆₇	C₈×Q₈	semidirect
52. C₆₇	C₂×C₈×Q₈	semidirect
66. C₂×Q₈	C₆₇⋊C₈	direct (N × H)
79. Q₈	C₂×C₆₇⋊C₈	direct (N × H)
99. C₂	Q₈×C₆₇⋊C₈	direct (N × H)

H ≅ G/N

type

1. Q₈×C₆₇⋊C₈

C₂

direct (N × H)

15. C₂×C₆₇⋊C₈

Q₈

direct (N × H)

18. C₂×Q₈×C₆₇

C₈

semidirect

27. Q₈×C₆₇

C₂×C₈

semidirect

37. C₆₇⋊C₈

C₂×Q₈

direct (N × H)

47. C₂×C₆₇

C₈×Q₈

semidirect

52. C₆₇

C₂×C₈×Q₈

semidirect

66. C₂×Q₈

C₆₇⋊C₈

direct (N × H)

79. Q₈

C₂×C₆₇⋊C₈

direct (N × H)

99. C₂

Q₈×C₆₇⋊C₈

direct (N × H)

Subgroups — C₂×Q₈×C₆₇⋊C₈ #22569131

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q8×C67⋊C8 #22569131

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₈×C₆₇⋊C₈ #22569131