Group #23776254 — subgroups

Subgroups — Q₈×D₆₃₂ #23776254

Order 10112.

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×D₃₁₆	C₂	1	2	yes	maximal, complemented
2	#19098968	C₂	1	3	yes	maximal, complemented
3	#19098844	C₂	1	6	yes	maximal, complemented
4	C₄×D₆₃₂	C₂	1	3	yes	maximal
5	C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₇₉	—	2	2	no	—
7	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
8	#14579161	C₂²	1	6	yes	—
9	#14579141	—	2	6	no	—
10	C₄×D₃₁₆	C₂²	1	6	yes	—
11	#14579095	C₂²	1	3	yes	—
12	#14579095	C₂²	1	6	yes	—
13	#14579083	C₂²	1	6	yes	—
14	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₇₉	C₂²	1	3	yes	—
16	C₄×C₈×C₇₉	C₂²	1	3	yes	—
17	C₂×Dic₁₅₈	—	2	6	no	—
18	Q₈×D₇₉	—	4	8	no	—
19	C₂×D₃₁₆	C₂³	1	2	yes	—
20	C₂×C₄×D₇₉	—	2	6	no	—
21	D₆₃₂	Q₈	1	4	yes	direct product factor
22	C₄⋊Dic₇₉	—	2	6	no	—
23	C₄⋊Dic₇₉	C₂³	1	6	yes	—
24	(C₂×D₇₉)⋊C₄	—	2	6	no	—
25	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₉	C₂³	1	3	yes	—
27	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₇₉	D₄	1	3	yes	—
29	C₄²×C₇₉	C₂³	1	3	yes	—
30	C₂²×D₇₉	—	2	2	no	—
31	Q₈×C₇₉	D₈	1	4	yes	complemented
32	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₇₉	—	2	6	no	—
36	D₃₁₆	—	2	2	no	—
37	D₃₁₆	C₂×Q₈	1	4	yes	complemented
38	C₄×D₇₉	—	4	12	no	—
39	Dic₁₅₈	—	4	12	no	—
40	C₈×C₇₉	—	2	3	no	—
41	C₈×C₇₉	C₂×Q₈	1	2	yes	complemented
42	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
43	C₂×D₇₉	—	2	4	no	—
44	C₂×D₇₉	—	4	2	no	—
45	C₂×D₇₉	—	4	2	no	—
46	C₄×C₇₉	—	2	3	no	—
47	C₄×C₇₉	C₂×D₈	1	6	yes	—
48	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
49	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
50	Dic₇₉	—	4	6	no	—
51	C₂×C₇₉	D₄×Q₈	1	1	yes	characteristic
52	C₂×C₇₉	C₂²×D₈	1	1	yes	characteristic
53	C₂×C₇₉	#578	1	1	yes	characteristic
54	D₇₉	—	4	4	no	—
55	Q₈×D₈	—	79	1	no	maximal, Sylow
56	C₇₉	Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
57	D₄×Q₈	—	79	2	no	—
58	#500	—	79	3	no	—
59	#474	—	79	6	no	—
60	C₈×Q₈	D₇₉	1	1	yes	characteristic, complemented
61	C₄×D₈	—	79	3	no	—
62	C₂²×Q₈	—	158	2	no	—
63	C₂×D₈	—	79	1	no	—
64	C₄⋊Q₈	—	79	6	no	—
65	C₂²⋊Q₈	—	158	6	no	—
66	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
67	C₄×D₄	—	79	6	no	—
68	#107	—	79	3	no	—
69	#107	—	79	6	no	—
70	C₄⋊C₈	C₂×D₇₉	1	3	yes	—
71	#102	—	79	6	no	—
72	C₄×C₈	C₂×D₇₉	1	3	yes	—
73	C₂×Q₈	—	316	8	no	—
74	C₂×Q₈	—	158	6	no	—
75	C₂×Q₈	D₃₁₆	1	1	yes	characteristic
76	C₂×D₄	—	79	2	no	—
77	C₂²×C₄	—	158	6	no	—
78	D₈	—	79	4	no	—
79	C₂×C₈	C₂²×D₇₉	1	3	yes	—
80	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
81	C₄⋊C₄	—	79	6	no	—
82	C₄⋊C₄	—	158	6	no	—
83	C₄⋊C₄	D₃₁₆	1	3	yes	—
84	C₂²⋊C₄	—	158	6	no	—
85	C₄²	C₂²×D₇₉	1	3	yes	—
86	C₂³	—	158	2	no	—
87	Q₈	—	316	12	no	—
88	Q₈	D₆₃₂	1	4	yes	direct product factor
89	D₄	—	79	4	no	—
90	D₄	—	158	2	no	—
91	C₂×C₄	—	316	12	no	—
92	C₂×C₄	—	158	6	no	—
93	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
94	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
95	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
96	C₈	—	2	3	no	—
97	C₈	Q₈×D₇₉	1	2	yes	complemented
98	C₂²	—	316	2	no	—
99	C₂²	—	316	2	no	—
100	C₂²	—	158	4	no	—
101	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
102	C₄	—	2	3	no	—
103	C₄	—	316	6	no	—
104	C₄	C₂×D₆₃₂	1	6	yes	—
105	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
106	C₄	#14579208	1	1	yes	characteristic
107	C₂	—	316	4	no	—
108	C₂	Q₈×D₃₁₆	1	1	yes	characteristic
109	C₂	C₂²×D₆₃₂	1	1	yes	characteristic
110	C₂	#19099380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₃₁₆

C₂

yes

maximal, complemented

#19098968

C₂

yes

maximal, complemented

#19098844

C₂

yes

maximal, complemented

C₄×D₆₃₂

C₂

yes

maximal

C₈×Q₈×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₉

—

C₂×D₆₃₂

C₂²

yes

characteristic

#14579161

C₂²

yes

—

#14579141

—

C₄×D₃₁₆

C₂²

yes

—

#14579095

C₂²

yes

—

#14579095

C₂²

yes

—

#14579083

C₂²

yes

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

Q₈×D₇₉

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₄×D₇₉

—

D₆₃₂

Q₈

yes

direct product factor

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂²×D₇₉

—

Q₈×C₇₉

D₈

yes

complemented

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

C₂×Q₈

yes

complemented

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×Q₈

yes

complemented

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

C₂×C₇₉

D₄×Q₈

yes

characteristic

C₂×C₇₉

C₂²×D₈

yes

characteristic

C₂×C₇₉

#578

yes

characteristic

D₇₉

—

Q₈×D₈

—

maximal, Sylow

C₇₉

Q₈×D₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

#500

—

#474

—

C₈×Q₈

D₇₉

yes

characteristic, complemented

C₄×D₈

—

C₂²×Q₈

—

158

—

C₂×D₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

C₂×D₇₉

yes

characteristic

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

C₂×D₇₉

yes

—

#102

—

C₄×C₈

C₂×D₇₉

yes

—

C₂×Q₈

—

316

—

C₂×Q₈

—

158

—

C₂×Q₈

D₃₁₆

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

158

—

D₈

—

C₂×C₈

C₂²×D₇₉

yes

—

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄⋊C₄

D₃₁₆

yes

—

C₂²⋊C₄

—

158

—

C₄²

C₂²×D₇₉

yes

—

C₂³

—

158

—

Q₈

—

316

—

Q₈

D₆₃₂

yes

direct product factor

D₄

—

D₄

—

158

—

C₂×C₄

—

316

—

C₂×C₄

—

158

—

C₂×C₄

C₂×D₃₁₆

yes

—

C₂×C₄

C₂×D₃₁₆

yes

—

C₂×C₄

C₂³×D₇₉

yes

characteristic

C₈

—

C₈

Q₈×D₇₉

yes

complemented

C₂²

—

316

—

C₂²

—

316

—

100

C₂²

—

158

—

101

C₂²

C₂²×D₃₁₆

yes

characteristic

102

C₄

—

103

C₄

—

316

—

104

C₄

C₂×D₆₃₂

yes

—

105

C₄

C₂×Q₈×D₇₉

yes

characteristic

106

C₄

#14579208

yes

characteristic

107

C₂

—

316

—

108

C₂

Q₈×D₃₁₆

yes

characteristic

109

C₂

C₂²×D₆₃₂

yes

characteristic

110

C₂

#19099380

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₈×Q₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×D₆₃₂	C₂²
hypercenter	C₈×Q₈	D₇₉
hypocenter	C₇₉	Q₈×D₈
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₈×Q₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂×D₆₃₂

C₂²

hypercenter

C₈×Q₈

D₇₉

hypocenter

C₇₉

Q₈×D₈

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×D₃₁₆	C₂	semidirect
2. #19098968	C₂	semidirect
3. #19098844	C₂	semidirect
5. C₈×Q₈×C₇₉	C₂	semidirect
21. D₆₃₂	Q₈	direct (N × H)
31. Q₈×C₇₉	D₈	semidirect
37. D₃₁₆	C₂×Q₈	semidirect
41. C₈×C₇₉	C₂×Q₈	semidirect
56. C₇₉	Q₈×D₈	semidirect
60. C₈×Q₈	D₇₉	semidirect
88. Q₈	D₆₃₂	direct (N × H)
97. C₈	Q₈×D₇₉	semidirect

H ≅ G/N

type

1. Q₈×D₃₁₆

C₂

semidirect

2. #19098968

C₂

semidirect

3. #19098844

C₂

semidirect

5. C₈×Q₈×C₇₉

C₂

semidirect

21. D₆₃₂

Q₈

direct (N × H)

31. Q₈×C₇₉

D₈

semidirect

37. D₃₁₆

C₂×Q₈

semidirect

41. C₈×C₇₉

C₂×Q₈

semidirect

56. C₇₉

Q₈×D₈

semidirect

60. C₈×Q₈

D₇₉

semidirect

88. Q₈

D₆₃₂

direct (N × H)

97. C₈

Q₈×D₇₉

semidirect

Subgroups — Q₈×D₆₃₂ #23776254

Conjugacy classes of subgroups

Complements

Subgroups — Q8×D632 #23776254

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×D₆₃₂ #23776254