Group #22601914 — subgroups

Subgroups — Q₈²×D₇₁ #22601914

Order 9088.

1112 conjugacy classes of subgroups; only the 575 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 575 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×Dic₁₄₂	C₂	1	6	yes	maximal, complemented
2	#17986729	C₂	1	9	yes	maximal, complemented
3	C₄⋊Q₈×D₇₁	C₂	1	9	yes	maximal
4	C₄×Q₈×D₇₁	C₂	1	6	yes	maximal
5	Q₈²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₇₁	C₂²	1	6	yes	—
7	C₂×Q₈×D₇₁	C₂²	1	2	yes	—
8	#14556128	C₂²	1	9	yes	—
9	#14556127	C₂²	1	36	yes	—
10	#14556126	C₂²	1	18	yes	—
11	Q₈×Dic₇₁	C₂²	1	6	yes	—
12	C₄×Dic₁₄₂	C₂²	1	18	yes	—
13	#14556093	C₂²	1	18	yes	—
14	C₄⋊C₄×D₇₁	C₂²	1	18	yes	—
15	C₄²×D₇₁	C₂²	1	9	yes	—
16	C₄⋊Q₈×C₇₁	C₂²	1	9	yes	—
17	C₄×Q₈×C₇₁	C₂²	1	6	yes	—
18	C₂×Dic₁₄₂	C₂³	1	18	yes	—
19	C₂×Dic₁₄₂	C₂³	1	6	yes	—
20	Q₈×D₇₁	—	4	12	no	—
21	Q₈×D₇₁	Q₈	1	16	yes	direct product factor
22	C₂×C₄×D₇₁	C₂³	1	9	yes	—
23	C₂×C₄×D₇₁	C₂³	1	6	yes	—
24	C₄⋊Dic₇₁	C₂³	1	18	yes	—
25	Dic₇₁⋊C₄	C₂³	1	36	yes	—
26	C₄×Dic₇₁	C₂³	1	9	yes	—
27	C₄×Dic₇₁	C₂³	1	18	yes	—
28	C₂×Q₈×C₇₁	C₂³	1	6	yes	—
29	C₂×Q₈×C₇₁	C₂³	1	2	yes	—
30	C₄⋊C₄×C₇₁	C₂³	1	18	yes	—
31	C₄²×C₇₁	C₂³	1	9	yes	—
32	C₂²×D₇₁	C₂⁴	1	1	yes	characteristic
33	Q₈×C₇₁	—	4	6	no	—
34	Q₈×C₇₁	C₂×Q₈	1	8	yes	complemented
35	C₂×C₄×C₇₁	C₂⁴	1	9	yes	—
36	C₂×C₄×C₇₁	C₂⁴	1	6	yes	—
37	C₂×Dic₇₁	C₂⁴	1	9	yes	—
38	C₂×Dic₇₁	C₂⁴	1	6	yes	—
39	C₄×D₇₁	—	2	18	no	—
40	C₄×D₇₁	C₂×Q₈	1	24	yes	—
41	Dic₁₄₂	—	4	18	no	—
42	Dic₁₄₂	C₂×Q₈	1	24	yes	complemented
43	C₂²×C₇₁	C₂⁵	1	1	yes	characteristic
44	C₂×D₇₁	C₂²×Q₈	1	4	yes	—
45	C₂×D₇₁	H(5,𝔽₂)	1	2	yes	—
46	C₄×C₇₁	—	2	9	no	—
47	C₄×C₇₁	C₂²×Q₈	1	12	yes	—
48	Dic₇₁	—	2	9	no	—
49	Dic₇₁	C₂²×Q₈	1	12	yes	—
50	C₂×C₇₁	C₂³×Q₈	1	2	yes	—
51	C₂×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
52	D₇₁	Q₈²	1	4	yes	direct product factor
53	C₂×Q₈²	—	71	1	no	maximal, Sylow
54	C₇₁	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
55	Q₈²	—	71	9	no	—
56	Q₈²	—	71	6	no	—
57	Q₈²	D₇₁	1	1	yes	characteristic, direct product factor
58	C₂×C₄⋊Q₈	—	71	9	no	—
59	C₂×C₄×Q₈	—	71	6	no	—
60	C₂²×Q₈	—	71	6	no	—
61	C₂²×Q₈	—	71	2	no	—
62	C₄⋊Q₈	—	71	18	no	—
63	C₄⋊Q₈	—	71	36	no	—
64	C₄⋊Q₈	—	71	9	no	—
65	C₄⋊Q₈	C₂×D₇₁	1	9	yes	—
66	C₄×Q₈	—	71	6	no	—
67	C₄×Q₈	—	71	18	no	—
68	C₄×Q₈	—	71	18	no	—
69	C₄×Q₈	C₂×D₇₁	1	6	yes	—
70	C₂×C₄⋊C₄	—	71	18	no	—
71	C₂×C₄²	—	71	9	no	—
72	C₂×Q₈	—	284	12	no	—
73	C₂×Q₈	—	71	16	no	—
74	C₂×Q₈	—	71	18	no	—
75	C₂×Q₈	—	71	6	no	—
76	C₂×Q₈	C₂²×D₇₁	1	6	yes	—
77	C₂×Q₈	C₂²×D₇₁	1	2	yes	—
78	C₂²×C₄	—	71	6	no	—
79	C₂²×C₄	—	71	9	no	—
80	C₄⋊C₄	—	71	36	no	—
81	C₄⋊C₄	—	71	18	no	—
82	C₄⋊C₄	C₂²×D₇₁	1	18	yes	—
83	C₄²	—	71	9	no	—
84	C₄²	—	71	18	no	—
85	C₄²	C₂²×D₇₁	1	9	yes	—
86	C₂³	—	71	1	no	—
87	Q₈	—	4	6	no	—
88	Q₈	—	284	18	no	—
89	Q₈	—	71	24	no	—
90	Q₈	Q₈×D₇₁	1	8	yes	direct product factor
91	C₂×C₄	—	71	6	no	—
92	C₂×C₄	—	71	24	no	—
93	C₂×C₄	—	71	9	no	—
94	C₂×C₄	—	142	18	no	—
95	C₂×C₄	C₂³×D₇₁	1	9	yes	—
96	C₂×C₄	C₂³×D₇₁	1	6	yes	—
97	C₂²	—	71	4	no	—
98	C₂²	—	71	2	no	—
99	C₂²	C₂⁴×D₇₁	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	71	12	no	—
102	C₄	—	142	9	no	—
103	C₄	C₂×Q₈×D₇₁	1	12	yes	—
104	C₂	—	71	4	no	—
105	C₂	C₂²×Q₈×D₇₁	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₄₂

C₂

yes

maximal, complemented

#17986729

C₂

yes

maximal, complemented

C₄⋊Q₈×D₇₁

C₂

yes

maximal

C₄×Q₈×D₇₁

C₂

yes

maximal

Q₈²×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₇₁

C₂²

yes

—

C₂×Q₈×D₇₁

C₂²

yes

—

#14556128

C₂²

yes

—

#14556127

C₂²

yes

—

#14556126

C₂²

yes

—

Q₈×Dic₇₁

C₂²

yes

—

C₄×Dic₁₄₂

C₂²

yes

—

#14556093

C₂²

yes

—

C₄⋊C₄×D₇₁

C₂²

yes

—

C₄²×D₇₁

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₄×Q₈×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

C₂³

yes

—

C₂×Dic₁₄₂

C₂³

yes

—

Q₈×D₇₁

—

Q₈×D₇₁

Q₈

yes

direct product factor

C₂×C₄×D₇₁

C₂³

yes

—

C₂×C₄×D₇₁

C₂³

yes

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

C₂³

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂³

yes

—

C₂²×D₇₁

C₂⁴

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

C₂×Q₈

yes

complemented

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂×Dic₇₁

C₂⁴

yes

—

C₂×Dic₇₁

C₂⁴

yes

—

C₄×D₇₁

—

C₄×D₇₁

C₂×Q₈

yes

—

Dic₁₄₂

—

Dic₁₄₂

C₂×Q₈

yes

complemented

C₂²×C₇₁

C₂⁵

yes

characteristic

C₂×D₇₁

C₂²×Q₈

yes

—

C₂×D₇₁

H(5,𝔽₂)

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂²×Q₈

yes

—

Dic₇₁

—

Dic₇₁

C₂²×Q₈

yes

—

C₂×C₇₁

C₂³×Q₈

yes

—

C₂×C₇₁

C₂×H(5,𝔽₂)

yes

characteristic

D₇₁

Q₈²

yes

direct product factor

C₂×Q₈²

—

maximal, Sylow

C₇₁

C₂×Q₈²

yes

characteristic, complemented, Sylow

Q₈²

—

Q₈²

—

Q₈²

D₇₁

yes

characteristic, direct product factor

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×Q₈

—

284

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₇₁

yes

—

C₂×Q₈

C₂²×D₇₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₇₁

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₇₁

yes

—

C₂³

—

Q₈

—

Q₈

—

284

—

Q₈

—

Q₈

Q₈×D₇₁

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

C₂³×D₇₁

yes

—

C₂×C₄

C₂³×D₇₁

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×D₇₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

142

—

103

C₄

C₂×Q₈×D₇₁

yes

—

104

C₂

—

105

C₂

C₂²×Q₈×D₇₁

yes

—

106

C₂

H(5,𝔽₂)×D₇₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₁
commutator, socle	C₂²×C₇₁	C₂⁵
Fitting	Q₈²×C₇₁	C₂
hypercenter	Q₈²	D₇₁
hypocenter	C₇₁	C₂×Q₈²
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₇₁

commutator, socle

C₂²×C₇₁

C₂⁵

Fitting

Q₈²×C₇₁

C₂

hypercenter

Q₈²

D₇₁

hypocenter

C₇₁

C₂×Q₈²

LCS

G ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. Q₈×Dic₁₄₂	C₂	semidirect
2. #17986729	C₂	semidirect
5. Q₈²×C₇₁	C₂	semidirect
21. Q₈×D₇₁	Q₈	direct (N × H)
34. Q₈×C₇₁	C₂×Q₈	semidirect
42. Dic₁₄₂	C₂×Q₈	semidirect
52. D₇₁	Q₈²	direct (N × H)
54. C₇₁	C₂×Q₈²	semidirect
57. Q₈²	D₇₁	direct (N × H)
90. Q₈	Q₈×D₇₁	direct (N × H)

H ≅ G/N

type

1. Q₈×Dic₁₄₂

C₂

semidirect

2. #17986729

C₂

semidirect

5. Q₈²×C₇₁

C₂

semidirect

21. Q₈×D₇₁

Q₈

direct (N × H)

34. Q₈×C₇₁

C₂×Q₈

semidirect

42. Dic₁₄₂

C₂×Q₈

semidirect

52. D₇₁

Q₈²

direct (N × H)

54. C₇₁

C₂×Q₈²

semidirect

57. Q₈²

D₇₁

direct (N × H)

90. Q₈

Q₈×D₇₁

direct (N × H)

Subgroups — Q₈²×D₇₁ #22601914

Conjugacy classes of subgroups

Complements

Subgroups — Q82×D71 #22601914

Conjugacy classes of subgroups

Complements

Subgroups — Q₈²×D₇₁ #22601914