Group #26033106 — subgroups

Subgroups — Q₈×D₇₁₂ #26033106

Order 11392.

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×D₃₅₆	C₂	1	2	yes	maximal, complemented
2	#19152003	C₂	1	3	yes	maximal, complemented
3	#19151879	C₂	1	6	yes	maximal, complemented
4	C₄×D₇₁₂	C₂	1	3	yes	maximal
5	C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₈₉	—	2	2	no	—
7	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
8	#15668012	C₂²	1	6	yes	—
9	#15667992	—	2	6	no	—
10	C₄×D₃₅₆	C₂²	1	6	yes	—
11	#15667946	C₂²	1	6	yes	—
12	#15667946	C₂²	1	3	yes	—
13	#15667934	C₂²	1	6	yes	—
14	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
16	C₄×C₈×C₈₉	C₂²	1	3	yes	—
17	C₂×Dic₁₇₈	—	2	6	no	—
18	Q₈×D₈₉	—	4	8	no	—
19	C₂×D₃₅₆	C₂³	1	2	yes	—
20	C₂×C₄×D₈₉	—	2	6	no	—
21	D₇₁₂	Q₈	1	4	yes	direct product factor
22	C₄⋊Dic₈₉	—	2	6	no	—
23	C₄⋊Dic₈₉	C₂³	1	6	yes	—
24	#13083551	—	2	6	no	—
25	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₈₉	C₂³	1	3	yes	—
27	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	D₄	1	3	yes	—
29	C₄²×C₈₉	C₂³	1	3	yes	—
30	C₂²×D₈₉	—	2	2	no	—
31	Q₈×C₈₉	D₈	1	4	yes	complemented
32	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₈₉	—	2	6	no	—
36	D₃₅₆	—	2	2	no	—
37	D₃₅₆	C₂×Q₈	1	4	yes	complemented
38	C₄×D₈₉	—	4	12	no	—
39	Dic₁₇₈	—	4	12	no	—
40	C₈×C₈₉	—	2	3	no	—
41	C₈×C₈₉	C₂×Q₈	1	2	yes	complemented
42	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
43	C₂×D₈₉	—	2	4	no	—
44	C₂×D₈₉	—	4	2	no	—
45	C₂×D₈₉	—	4	2	no	—
46	C₄×C₈₉	—	2	3	no	—
47	C₄×C₈₉	C₂×D₈	1	6	yes	—
48	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
49	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
50	Dic₈₉	—	4	6	no	—
51	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
52	C₂×C₈₉	C₂²×D₈	1	1	yes	characteristic
53	C₂×C₈₉	#578	1	1	yes	characteristic
54	D₈₉	—	4	4	no	—
55	Q₈×D₈	—	89	1	no	maximal, Sylow
56	C₈₉	Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
57	D₄×Q₈	—	89	2	no	—
58	#500	—	89	3	no	—
59	#474	—	89	6	no	—
60	C₈×Q₈	D₈₉	1	1	yes	characteristic, complemented
61	C₄×D₈	—	89	3	no	—
62	C₂²×Q₈	—	178	2	no	—
63	C₂×D₈	—	89	1	no	—
64	C₄⋊Q₈	—	89	6	no	—
65	C₂²⋊Q₈	—	178	6	no	—
66	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
67	C₄×D₄	—	89	6	no	—
68	#107	—	89	3	no	—
69	#107	—	89	6	no	—
70	C₄⋊C₈	C₂×D₈₉	1	3	yes	—
71	#102	—	89	6	no	—
72	C₄×C₈	C₂×D₈₉	1	3	yes	—
73	C₂×Q₈	—	356	8	no	—
74	C₂×Q₈	—	178	6	no	—
75	C₂×Q₈	D₃₅₆	1	1	yes	characteristic
76	C₂×D₄	—	89	2	no	—
77	C₂²×C₄	—	178	6	no	—
78	D₈	—	89	4	no	—
79	C₂×C₈	C₂²×D₈₉	1	3	yes	—
80	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
81	C₄⋊C₄	—	89	6	no	—
82	C₄⋊C₄	—	178	6	no	—
83	C₄⋊C₄	D₃₅₆	1	3	yes	—
84	C₂²⋊C₄	—	178	6	no	—
85	C₄²	C₂²×D₈₉	1	3	yes	—
86	C₂³	—	178	2	no	—
87	Q₈	—	356	12	no	—
88	Q₈	D₇₁₂	1	4	yes	direct product factor
89	D₄	—	89	4	no	—
90	D₄	—	178	2	no	—
91	C₂×C₄	—	356	12	no	—
92	C₂×C₄	—	178	6	no	—
93	C₂×C₄	C₂×D₃₅₆	1	3	yes	—
94	C₂×C₄	C₂×D₃₅₆	1	3	yes	—
95	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
96	C₈	—	2	3	no	—
97	C₈	Q₈×D₈₉	1	2	yes	complemented
98	C₂²	—	356	2	no	—
99	C₂²	—	356	2	no	—
100	C₂²	—	178	4	no	—
101	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
102	C₄	—	2	3	no	—
103	C₄	—	356	6	no	—
104	C₄	C₂×D₇₁₂	1	6	yes	—
105	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
106	C₄	#15668059	1	1	yes	characteristic
107	C₂	—	356	4	no	—
108	C₂	Q₈×D₃₅₆	1	1	yes	characteristic
109	C₂	C₂²×D₇₁₂	1	1	yes	characteristic
110	C₂	#19152415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₃₅₆

C₂

yes

maximal, complemented

#19152003

C₂

yes

maximal, complemented

#19151879

C₂

yes

maximal, complemented

C₄×D₇₁₂

C₂

yes

maximal

C₈×Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₈₉

—

C₂×D₇₁₂

C₂²

yes

characteristic

#15668012

C₂²

yes

—

#15667992

—

C₄×D₃₅₆

C₂²

yes

—

#15667946

C₂²

yes

—

#15667946

C₂²

yes

—

#15667934

C₂²

yes

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

Q₈×D₈₉

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₄×D₈₉

—

D₇₁₂

Q₈

yes

direct product factor

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083551

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂²×D₈₉

—

Q₈×C₈₉

D₈

yes

complemented

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

C₂×Q₈

yes

complemented

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₈

yes

complemented

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×D₈

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

C₂×C₈₉

D₄×Q₈

yes

characteristic

C₂×C₈₉

C₂²×D₈

yes

characteristic

C₂×C₈₉

#578

yes

characteristic

D₈₉

—

Q₈×D₈

—

maximal, Sylow

C₈₉

Q₈×D₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

#500

—

#474

—

C₈×Q₈

D₈₉

yes

characteristic, complemented

C₄×D₈

—

C₂²×Q₈

—

178

—

C₂×D₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

C₂×D₈₉

yes

—

#102

—

C₄×C₈

C₂×D₈₉

yes

—

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂×Q₈

D₃₅₆

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

178

—

D₈

—

C₂×C₈

C₂²×D₈₉

yes

—

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

D₃₅₆

yes

—

C₂²⋊C₄

—

178

—

C₄²

C₂²×D₈₉

yes

—

C₂³

—

178

—

Q₈

—

356

—

Q₈

D₇₁₂

yes

direct product factor

D₄

—

D₄

—

178

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂×C₄

C₂³×D₈₉

yes

characteristic

C₈

—

C₈

Q₈×D₈₉

yes

complemented

C₂²

—

356

—

C₂²

—

356

—

100

C₂²

—

178

—

101

C₂²

C₂²×D₃₅₆

yes

characteristic

102

C₄

—

103

C₄

—

356

—

104

C₄

C₂×D₇₁₂

yes

—

105

C₄

C₂×Q₈×D₈₉

yes

characteristic

106

C₄

#15668059

yes

characteristic

107

C₂

—

356

—

108

C₂

Q₈×D₃₅₆

yes

characteristic

109

C₂

C₂²×D₇₁₂

yes

characteristic

110

C₂

#19152415

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₈×Q₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₇₁₂	C₂²
hypercenter	C₈×Q₈	D₈₉
hypocenter	C₈₉	Q₈×D₈
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₈×Q₈×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂×D₇₁₂

C₂²

hypercenter

C₈×Q₈

D₈₉

hypocenter

C₈₉

Q₈×D₈

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×D₃₅₆	C₂	semidirect
2. #19152003	C₂	semidirect
3. #19151879	C₂	semidirect
5. C₈×Q₈×C₈₉	C₂	semidirect
21. D₇₁₂	Q₈	direct (N × H)
31. Q₈×C₈₉	D₈	semidirect
37. D₃₅₆	C₂×Q₈	semidirect
41. C₈×C₈₉	C₂×Q₈	semidirect
56. C₈₉	Q₈×D₈	semidirect
60. C₈×Q₈	D₈₉	semidirect
88. Q₈	D₇₁₂	direct (N × H)
97. C₈	Q₈×D₈₉	semidirect

H ≅ G/N

type

1. Q₈×D₃₅₆

C₂

semidirect

2. #19152003

C₂

semidirect

3. #19151879

C₂

semidirect

5. C₈×Q₈×C₈₉

C₂

semidirect

21. D₇₁₂

Q₈

direct (N × H)

31. Q₈×C₈₉

D₈

semidirect

37. D₃₅₆

C₂×Q₈

semidirect

41. C₈×C₈₉

C₂×Q₈

semidirect

56. C₈₉

Q₈×D₈

semidirect

60. C₈×Q₈

D₈₉

semidirect

88. Q₈

D₇₁₂

direct (N × H)

97. C₈

Q₈×D₈₉

semidirect

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

subgroup	index
Q₈×D₃₅₆	2
#19152003	2
#19151879	2
C₄×D₇₁₂	2
C₈×Q₈×C₈₉	2
Q₈×D₈	89

subgroup

index

Q₈×D₃₅₆

#19152003

#19151879

C₄×D₇₁₂

C₈×Q₈×C₈₉

Q₈×D₈

Subgroups — Q₈×D₇₁₂ #26033106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×D712 #26033106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₇₁₂ #26033106