Group #22573063 — subgroups

Subgroups — Q₈×D₅₃₆ #22573063

Order 8576.

404 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 149 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×D₂₆₈	C₂	1	2	yes	maximal, complemented
2	#16863801	C₂	1	3	yes	maximal, complemented
3	#16863677	C₂	1	6	yes	maximal, complemented
4	C₄×D₅₃₆	C₂	1	3	yes	maximal
5	C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₆₇	—	2	2	no	—
7	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
8	#14524043	C₂²	1	6	yes	—
9	#14524023	—	2	6	no	—
10	C₄×D₂₆₈	C₂²	1	6	yes	—
11	#14523977	C₂²	1	3	yes	—
12	#14523977	C₂²	1	6	yes	—
13	#14523965	C₂²	1	6	yes	—
14	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
16	C₄×C₈×C₆₇	C₂²	1	3	yes	—
17	C₂×Dic₁₃₄	—	2	6	no	—
18	Q₈×D₆₇	—	4	8	no	—
19	C₂×D₂₆₈	C₂³	1	2	yes	—
20	C₂×C₄×D₆₇	—	2	6	no	—
21	D₅₃₆	Q₈	1	4	yes	direct product factor
22	C₄⋊Dic₆₇	—	2	6	no	—
23	C₄⋊Dic₆₇	C₂³	1	6	yes	—
24	(C₂×D₆₇)⋊C₄	—	2	6	no	—
25	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	C₂³	1	3	yes	—
27	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₆₇	D₄	1	3	yes	—
29	C₄²×C₆₇	C₂³	1	3	yes	—
30	C₂²×D₆₇	—	2	2	no	—
31	Q₈×C₆₇	D₈	1	4	yes	complemented
32	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₆₇	—	2	6	no	—
36	D₂₆₈	—	2	2	no	—
37	D₂₆₈	C₂×Q₈	1	4	yes	complemented
38	C₄×D₆₇	—	4	12	no	—
39	Dic₁₃₄	—	4	12	no	—
40	C₈×C₆₇	—	2	3	no	—
41	C₈×C₆₇	C₂×Q₈	1	2	yes	complemented
42	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
43	C₂×D₆₇	—	2	4	no	—
44	C₂×D₆₇	—	4	2	no	—
45	C₂×D₆₇	—	4	2	no	—
46	C₄×C₆₇	—	2	3	no	—
47	C₄×C₆₇	C₂×D₈	1	6	yes	—
48	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
49	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
50	Dic₆₇	—	4	6	no	—
51	C₂×C₆₇	D₄×Q₈	1	1	yes	characteristic
52	C₂×C₆₇	C₂²×D₈	1	1	yes	characteristic
53	C₂×C₆₇	#578	1	1	yes	characteristic
54	D₆₇	—	4	4	no	—
55	Q₈×D₈	—	67	1	no	maximal, Sylow
56	C₆₇	Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
57	D₄×Q₈	—	67	2	no	—
58	#500	—	67	3	no	—
59	#474	—	67	6	no	—
60	C₈×Q₈	D₆₇	1	1	yes	characteristic, complemented
61	C₄×D₈	—	67	3	no	—
62	C₂²×Q₈	—	134	2	no	—
63	C₂×D₈	—	67	1	no	—
64	C₄⋊Q₈	—	67	6	no	—
65	C₂²⋊Q₈	—	134	6	no	—
66	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
67	C₄×D₄	—	67	6	no	—
68	#107	—	67	3	no	—
69	#107	—	67	6	no	—
70	C₄⋊C₈	C₂×D₆₇	1	3	yes	—
71	#102	—	67	6	no	—
72	C₄×C₈	C₂×D₆₇	1	3	yes	—
73	C₂×Q₈	—	268	8	no	—
74	C₂×Q₈	—	134	6	no	—
75	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
76	C₂×D₄	—	67	2	no	—
77	C₂²×C₄	—	134	6	no	—
78	D₈	—	67	4	no	—
79	C₂×C₈	C₂²×D₆₇	1	3	yes	—
80	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
81	C₄⋊C₄	—	67	6	no	—
82	C₄⋊C₄	—	134	6	no	—
83	C₄⋊C₄	D₂₆₈	1	3	yes	—
84	C₂²⋊C₄	—	134	6	no	—
85	C₄²	C₂²×D₆₇	1	3	yes	—
86	C₂³	—	134	2	no	—
87	Q₈	—	268	12	no	—
88	Q₈	D₅₃₆	1	4	yes	direct product factor
89	D₄	—	67	4	no	—
90	D₄	—	134	2	no	—
91	C₂×C₄	—	268	12	no	—
92	C₂×C₄	—	134	6	no	—
93	C₂×C₄	C₂×D₂₆₈	1	3	yes	—
94	C₂×C₄	C₂×D₂₆₈	1	3	yes	—
95	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
96	C₈	—	2	3	no	—
97	C₈	Q₈×D₆₇	1	2	yes	complemented
98	C₂²	—	268	2	no	—
99	C₂²	—	268	2	no	—
100	C₂²	—	134	4	no	—
101	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
102	C₄	—	2	3	no	—
103	C₄	—	268	6	no	—
104	C₄	C₂×D₅₃₆	1	6	yes	—
105	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
106	C₄	#14524090	1	1	yes	characteristic
107	C₂	—	268	4	no	—
108	C₂	Q₈×D₂₆₈	1	1	yes	characteristic
109	C₂	C₂²×D₅₃₆	1	1	yes	characteristic
110	C₂	#16864213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₆₈

C₂

yes

maximal, complemented

#16863801

C₂

yes

maximal, complemented

#16863677

C₂

yes

maximal, complemented

C₄×D₅₃₆

C₂

yes

maximal

C₈×Q₈×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₇

—

C₂×D₅₃₆

C₂²

yes

characteristic

#14524043

C₂²

yes

—

#14524023

—

C₄×D₂₆₈

C₂²

yes

—

#14523977

C₂²

yes

—

#14523977

C₂²

yes

—

#14523965

C₂²

yes

—

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

Q₈×D₆₇

—

C₂×D₂₆₈

C₂³

yes

—

C₂×C₄×D₆₇

—

D₅₃₆

Q₈

yes

direct product factor

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

(C₂×D₆₇)⋊C₄

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂²×D₆₇

—

Q₈×C₆₇

D₈

yes

complemented

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

C₂×Q₈

yes

complemented

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₈

yes

complemented

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

Dic₆₇

—

C₂×C₆₇

D₄×Q₈

yes

characteristic

C₂×C₆₇

C₂²×D₈

yes

characteristic

C₂×C₆₇

#578

yes

characteristic

D₆₇

—

Q₈×D₈

—

maximal, Sylow

C₆₇

Q₈×D₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

#500

—

#474

—

C₈×Q₈

D₆₇

yes

characteristic, complemented

C₄×D₈

—

C₂²×Q₈

—

134

—

C₂×D₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

C₂×D₆₇

yes

—

#102

—

C₄×C₈

C₂×D₆₇

yes

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂×Q₈

D₂₆₈

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

134

—

D₈

—

C₂×C₈

C₂²×D₆₇

yes

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

D₂₆₈

yes

—

C₂²⋊C₄

—

134

—

C₄²

C₂²×D₆₇

yes

—

C₂³

—

134

—

Q₈

—

268

—

Q₈

D₅₃₆

yes

direct product factor

D₄

—

D₄

—

134

—

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

C₂×D₂₆₈

yes

—

C₂×C₄

C₂×D₂₆₈

yes

—

C₂×C₄

C₂³×D₆₇

yes

characteristic

C₈

—

C₈

Q₈×D₆₇

yes

complemented

C₂²

—

268

—

C₂²

—

268

—

100

C₂²

—

134

—

101

C₂²

C₂²×D₂₆₈

yes

characteristic

102

C₄

—

103

C₄

—

268

—

104

C₄

C₂×D₅₃₆

yes

—

105

C₄

C₂×Q₈×D₆₇

yes

characteristic

106

C₄

#14524090

yes

characteristic

107

C₂

—

268

—

108

C₂

Q₈×D₂₆₈

yes

characteristic

109

C₂

C₂²×D₅₃₆

yes

characteristic

110

C₂

#16864213

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₈×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₈×Q₈	D₆₇
hypocenter	C₆₇	Q₈×D₈
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₈×Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₈×Q₈

D₆₇

hypocenter

C₆₇

Q₈×D₈

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×D₂₆₈	C₂	semidirect
2. #16863801	C₂	semidirect
3. #16863677	C₂	semidirect
5. C₈×Q₈×C₆₇	C₂	semidirect
21. D₅₃₆	Q₈	direct (N × H)
31. Q₈×C₆₇	D₈	semidirect
37. D₂₆₈	C₂×Q₈	semidirect
41. C₈×C₆₇	C₂×Q₈	semidirect
56. C₆₇	Q₈×D₈	semidirect
60. C₈×Q₈	D₆₇	semidirect
88. Q₈	D₅₃₆	direct (N × H)
97. C₈	Q₈×D₆₇	semidirect

H ≅ G/N

type

1. Q₈×D₂₆₈

C₂

semidirect

2. #16863801

C₂

semidirect

3. #16863677

C₂

semidirect

5. C₈×Q₈×C₆₇

C₂

semidirect

21. D₅₃₆

Q₈

direct (N × H)

31. Q₈×C₆₇

D₈

semidirect

37. D₂₆₈

C₂×Q₈

semidirect

41. C₈×C₆₇

C₂×Q₈

semidirect

56. C₆₇

Q₈×D₈

semidirect

60. C₈×Q₈

D₆₇

semidirect

88. Q₈

D₅₃₆

direct (N × H)

97. C₈

Q₈×D₆₇

semidirect

Subgroups — Q₈×D₅₃₆ #22573063

Conjugacy classes of subgroups

Complements

Subgroups — Q8×D536 #22573063

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×D₅₃₆ #22573063