Group #46849938 — subgroups

Subgroups — Q₈²×D₂₇₁ #46849938

Order 34688.

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×Dic₅₄₂	C₂	1	6	yes	maximal, complemented
2	#31774627	C₂	1	9	yes	maximal, complemented
3	C₄⋊Q₈×D₂₇₁	C₂	1	9	yes	maximal
4	C₄×Q₈×D₂₇₁	C₂	1	6	yes	maximal
5	Q₈²×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₂₇₁	C₂²	1	6	yes	—
7	C₂×Q₈×D₂₇₁	C₂²	1	2	yes	—
8	#22576160	C₂²	1	9	yes	—
9	#22576159	C₂²	1	36	yes	—
10	#22576158	C₂²	1	18	yes	—
11	Q₈×Dic₂₇₁	C₂²	1	6	yes	—
12	C₄×Dic₅₄₂	C₂²	1	18	yes	—
13	#22576125	C₂²	1	18	yes	—
14	C₄⋊C₄×D₂₇₁	C₂²	1	18	yes	—
15	C₄²×D₂₇₁	C₂²	1	9	yes	—
16	C₄⋊Q₈×C₂₇₁	C₂²	1	9	yes	—
17	C₄×Q₈×C₂₇₁	C₂²	1	6	yes	—
18	C₂×Dic₅₄₂	C₂³	1	18	yes	—
19	C₂×Dic₅₄₂	C₂³	1	6	yes	—
20	Q₈×D₂₇₁	—	4	12	no	—
21	Q₈×D₂₇₁	Q₈	1	16	yes	direct product factor
22	C₂×C₄×D₂₇₁	C₂³	1	9	yes	—
23	C₂×C₄×D₂₇₁	C₂³	1	6	yes	—
24	C₄⋊Dic₂₇₁	C₂³	1	18	yes	—
25	Dic₂₇₁⋊C₄	C₂³	1	36	yes	—
26	C₄×Dic₂₇₁	C₂³	1	9	yes	—
27	C₄×Dic₂₇₁	C₂³	1	18	yes	—
28	C₂×Q₈×C₂₇₁	C₂³	1	6	yes	—
29	C₂×Q₈×C₂₇₁	C₂³	1	2	yes	—
30	C₄⋊C₄×C₂₇₁	C₂³	1	18	yes	—
31	C₄²×C₂₇₁	C₂³	1	9	yes	—
32	C₂²×D₂₇₁	C₂⁴	1	1	yes	characteristic
33	Dic₅₄₂	—	4	18	no	—
34	Dic₅₄₂	C₂×Q₈	1	24	yes	complemented
35	C₂×Dic₂₇₁	C₂⁴	1	9	yes	—
36	C₂×Dic₂₇₁	C₂⁴	1	6	yes	—
37	C₄×D₂₇₁	—	2	18	no	—
38	C₄×D₂₇₁	C₂×Q₈	1	24	yes	—
39	Q₈×C₂₇₁	—	4	6	no	—
40	Q₈×C₂₇₁	C₂×Q₈	1	8	yes	complemented
41	C₂×C₄×C₂₇₁	C₂⁴	1	9	yes	—
42	C₂×C₄×C₂₇₁	C₂⁴	1	6	yes	—
43	C₂²×C₂₇₁	C₂⁵	1	1	yes	characteristic
44	C₂×D₂₇₁	C₂²×Q₈	1	4	yes	—
45	C₂×D₂₇₁	H(5,𝔽₂)	1	2	yes	—
46	C₄×C₂₇₁	—	2	9	no	—
47	C₄×C₂₇₁	C₂²×Q₈	1	12	yes	—
48	Dic₂₇₁	—	2	9	no	—
49	Dic₂₇₁	C₂²×Q₈	1	12	yes	—
50	C₂×C₂₇₁	C₂³×Q₈	1	2	yes	—
51	C₂×C₂₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
52	D₂₇₁	Q₈²	1	4	yes	direct product factor
53	C₂₇₁	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
54	C₂×Q₈²	—	271	1	no	maximal, Sylow
55	Q₈²	—	271	9	no	—
56	Q₈²	—	271	6	no	—
57	Q₈²	D₂₇₁	1	1	yes	characteristic, direct product factor
58	C₂×C₄⋊Q₈	—	271	9	no	—
59	C₂×C₄×Q₈	—	271	6	no	—
60	C₂²×Q₈	—	271	2	no	—
61	C₂²×Q₈	—	271	6	no	—
62	C₄⋊Q₈	—	271	18	no	—
63	C₄⋊Q₈	—	271	36	no	—
64	C₄⋊Q₈	—	271	9	no	—
65	C₄⋊Q₈	C₂×D₂₇₁	1	9	yes	—
66	C₄×Q₈	—	271	6	no	—
67	C₄×Q₈	—	271	18	no	—
68	C₄×Q₈	—	271	18	no	—
69	C₄×Q₈	C₂×D₂₇₁	1	6	yes	—
70	C₂×C₄⋊C₄	—	271	18	no	—
71	C₂×C₄²	—	271	9	no	—
72	C₂×Q₈	—	271	16	no	—
73	C₂×Q₈	—	271	6	no	—
74	C₂×Q₈	—	271	18	no	—
75	C₂×Q₈	—	1084	12	no	—
76	C₂×Q₈	C₂²×D₂₇₁	1	6	yes	—
77	C₂×Q₈	C₂²×D₂₇₁	1	2	yes	—
78	C₂²×C₄	—	271	6	no	—
79	C₂²×C₄	—	271	9	no	—
80	C₄⋊C₄	—	271	36	no	—
81	C₄⋊C₄	—	271	18	no	—
82	C₄⋊C₄	C₂²×D₂₇₁	1	18	yes	—
83	C₄²	—	271	9	no	—
84	C₄²	—	271	18	no	—
85	C₄²	C₂²×D₂₇₁	1	9	yes	—
86	C₂³	—	271	1	no	—
87	Q₈	—	4	6	no	—
88	Q₈	—	271	24	no	—
89	Q₈	—	1084	18	no	—
90	Q₈	Q₈×D₂₇₁	1	8	yes	direct product factor
91	C₂×C₄	—	271	6	no	—
92	C₂×C₄	—	271	24	no	—
93	C₂×C₄	—	271	9	no	—
94	C₂×C₄	—	542	18	no	—
95	C₂×C₄	C₂³×D₂₇₁	1	9	yes	—
96	C₂×C₄	C₂³×D₂₇₁	1	6	yes	—
97	C₂²	—	271	4	no	—
98	C₂²	—	271	2	no	—
99	C₂²	C₂⁴×D₂₇₁	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	271	12	no	—
102	C₄	—	542	9	no	—
103	C₄	C₂×Q₈×D₂₇₁	1	12	yes	—
104	C₂	—	271	4	no	—
105	C₂	C₂²×Q₈×D₂₇₁	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₂₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₅₄₂

C₂

yes

maximal, complemented

#31774627

C₂

yes

maximal, complemented

C₄⋊Q₈×D₂₇₁

C₂

yes

maximal

C₄×Q₈×D₂₇₁

C₂

yes

maximal

Q₈²×C₂₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₂₇₁

C₂²

yes

—

C₂×Q₈×D₂₇₁

C₂²

yes

—

#22576160

C₂²

yes

—

#22576159

C₂²

yes

—

#22576158

C₂²

yes

—

Q₈×Dic₂₇₁

C₂²

yes

—

C₄×Dic₅₄₂

C₂²

yes

—

#22576125

C₂²

yes

—

C₄⋊C₄×D₂₇₁

C₂²

yes

—

C₄²×D₂₇₁

C₂²

yes

—

C₄⋊Q₈×C₂₇₁

C₂²

yes

—

C₄×Q₈×C₂₇₁

C₂²

yes

—

C₂×Dic₅₄₂

C₂³

yes

—

C₂×Dic₅₄₂

C₂³

yes

—

Q₈×D₂₇₁

—

Q₈×D₂₇₁

Q₈

yes

direct product factor

C₂×C₄×D₂₇₁

C₂³

yes

—

C₂×C₄×D₂₇₁

C₂³

yes

—

C₄⋊Dic₂₇₁

C₂³

yes

—

Dic₂₇₁⋊C₄

C₂³

yes

—

C₄×Dic₂₇₁

C₂³

yes

—

C₄×Dic₂₇₁

C₂³

yes

—

C₂×Q₈×C₂₇₁

C₂³

yes

—

C₂×Q₈×C₂₇₁

C₂³

yes

—

C₄⋊C₄×C₂₇₁

C₂³

yes

—

C₄²×C₂₇₁

C₂³

yes

—

C₂²×D₂₇₁

C₂⁴

yes

characteristic

Dic₅₄₂

—

Dic₅₄₂

C₂×Q₈

yes

complemented

C₂×Dic₂₇₁

C₂⁴

yes

—

C₂×Dic₂₇₁

C₂⁴

yes

—

C₄×D₂₇₁

—

C₄×D₂₇₁

C₂×Q₈

yes

—

Q₈×C₂₇₁

—

Q₈×C₂₇₁

C₂×Q₈

yes

complemented

C₂×C₄×C₂₇₁

C₂⁴

yes

—

C₂×C₄×C₂₇₁

C₂⁴

yes

—

C₂²×C₂₇₁

C₂⁵

yes

characteristic

C₂×D₂₇₁

C₂²×Q₈

yes

—

C₂×D₂₇₁

H(5,𝔽₂)

yes

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂²×Q₈

yes

—

Dic₂₇₁

—

Dic₂₇₁

C₂²×Q₈

yes

—

C₂×C₂₇₁

C₂³×Q₈

yes

—

C₂×C₂₇₁

C₂×H(5,𝔽₂)

yes

characteristic

D₂₇₁

Q₈²

yes

direct product factor

C₂₇₁

C₂×Q₈²

yes

characteristic, complemented, Sylow

C₂×Q₈²

—

271

maximal, Sylow

Q₈²

—

271

—

Q₈²

—

271

—

Q₈²

D₂₇₁

yes

characteristic, direct product factor

C₂×C₄⋊Q₈

—

271

—

C₂×C₄×Q₈

—

271

—

C₂²×Q₈

—

271

—

C₂²×Q₈

—

271

—

C₄⋊Q₈

—

271

—

C₄⋊Q₈

—

271

—

C₄⋊Q₈

—

271

—

C₄⋊Q₈

C₂×D₂₇₁

yes

—

C₄×Q₈

—

271

—

C₄×Q₈

—

271

—

C₄×Q₈

—

271

—

C₄×Q₈

C₂×D₂₇₁

yes

—

C₂×C₄⋊C₄

—

271

—

C₂×C₄²

—

271

—

C₂×Q₈

—

271

—

C₂×Q₈

—

271

—

C₂×Q₈

—

271

—

C₂×Q₈

—

1084

—

C₂×Q₈

C₂²×D₂₇₁

yes

—

C₂×Q₈

C₂²×D₂₇₁

yes

—

C₂²×C₄

—

271

—

C₂²×C₄

—

271

—

C₄⋊C₄

—

271

—

C₄⋊C₄

—

271

—

C₄⋊C₄

C₂²×D₂₇₁

yes

—

C₄²

—

271

—

C₄²

—

271

—

C₄²

C₂²×D₂₇₁

yes

—

C₂³

—

271

—

Q₈

—

Q₈

—

271

—

Q₈

—

1084

—

Q₈

Q₈×D₂₇₁

yes

direct product factor

C₂×C₄

—

271

—

C₂×C₄

—

271

—

C₂×C₄

—

271

—

C₂×C₄

—

542

—

C₂×C₄

C₂³×D₂₇₁

yes

—

C₂×C₄

C₂³×D₂₇₁

yes

—

C₂²

—

271

—

C₂²

—

271

—

C₂²

C₂⁴×D₂₇₁

yes

characteristic

100

C₄

—

101

C₄

—

271

—

102

C₄

—

542

—

103

C₄

C₂×Q₈×D₂₇₁

yes

—

104

C₂

—

271

—

105

C₂

C₂²×Q₈×D₂₇₁

yes

—

106

C₂

H(5,𝔽₂)×D₂₇₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₇₁
commutator, socle, cosocle	C₂²×C₂₇₁	C₂⁵
Fitting	Q₈²×C₂₇₁	C₂
prime radical	C₂²×D₂₇₁	C₂⁴
hypercenter	Q₈²	D₂₇₁
hypocenter	C₂₇₁	C₂×Q₈²
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₂₇₁

commutator, socle, cosocle

C₂²×C₂₇₁

C₂⁵

Fitting

Q₈²×C₂₇₁

C₂

prime radical

C₂²×D₂₇₁

C₂⁴

hypercenter

Q₈²

D₂₇₁

hypocenter

C₂₇₁

C₂×Q₈²

LCS

G ▷ C₂²×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. Q₈×Dic₅₄₂	C₂	semidirect
2. #31774627	C₂	semidirect
5. Q₈²×C₂₇₁	C₂	semidirect
21. Q₈×D₂₇₁	Q₈	direct (N × H)
34. Dic₅₄₂	C₂×Q₈	semidirect
40. Q₈×C₂₇₁	C₂×Q₈	semidirect
52. D₂₇₁	Q₈²	direct (N × H)
53. C₂₇₁	C₂×Q₈²	semidirect
57. Q₈²	D₂₇₁	direct (N × H)
90. Q₈	Q₈×D₂₇₁	direct (N × H)

H ≅ G/N

type

1. Q₈×Dic₅₄₂

C₂

semidirect

2. #31774627

C₂

semidirect

5. Q₈²×C₂₇₁

C₂

semidirect

21. Q₈×D₂₇₁

Q₈

direct (N × H)

34. Dic₅₄₂

C₂×Q₈

semidirect

40. Q₈×C₂₇₁

C₂×Q₈

semidirect

52. D₂₇₁

Q₈²

direct (N × H)

53. C₂₇₁

C₂×Q₈²

semidirect

57. Q₈²

D₂₇₁

direct (N × H)

90. Q₈

Q₈×D₂₇₁

direct (N × H)

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

subgroup	index
Q₈×Dic₅₄₂	2
#31774627	2
C₄⋊Q₈×D₂₇₁	2
C₄×Q₈×D₂₇₁	2
Q₈²×C₂₇₁	2
C₂×Q₈²	271

subgroup

index

Q₈×Dic₅₄₂

#31774627

C₄⋊Q₈×D₂₇₁

C₄×Q₈×D₂₇₁

Q₈²×C₂₇₁

C₂×Q₈²

271

Subgroups — Q₈²×D₂₇₁ #46849938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q82×D271 #46849938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈²×D₂₇₁ #46849938