Group #46835504 — subgroups

Subgroups — #46835504

Order 34688.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#31774276	C₂	1	1	yes	maximal, characteristic
2	#31774192	C₂	1	2	yes	maximal
3	#31773646	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₂₇₁⋊C₈)	C₂	1	2	yes	maximal
5	Q₈²×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16864644	—	2	2	no	—
7	#22576160	C₂²	1	1	yes	characteristic
8	#22576093	—	2	2	no	—
9	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
10	#22576084	—	2	2	no	—
11	C₄×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₂₇₁	—	2	2	no	—
13	C₄⋊Q₈×C₂₇₁	—	2	2	no	—
14	C₄⋊Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₂₇₁	—	2	2	no	—
16	C₄×Q₈×C₂₇₁	C₂²	1	2	yes	—
17	C₂×Dic₅₄₂	—	2	1	no	—
18	#16864644	—	4	4	no	—
19	C₂×C₂₇₁⋊C₈	—	2	1	no	—
20	C₂×C₂₇₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₂₇₁	—	2	2	no	—
22	C₂×Q₈×C₂₇₁	—	2	2	no	—
23	C₂×Q₈×C₂₇₁	D₄	1	2	yes	—
24	C₂×Q₈×C₂₇₁	D₄	1	2	yes	—
25	C₄⋊C₄×C₂₇₁	—	2	4	no	—
26	C₄⋊C₄×C₂₇₁	—	2	2	no	—
27	C₄⋊C₄×C₂₇₁	—	2	2	no	—
28	C₄⋊C₄×C₂₇₁	D₄	1	2	yes	—
29	C₄²×C₂₇₁	—	2	2	no	—
30	C₄²×C₂₇₁	—	2	2	no	—
31	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
32	Dic₅₄₂	—	4	2	no	—
33	C₂×Dic₂₇₁	—	4	1	no	—
34	C₂₇₁⋊C₈	—	2	2	no	—
35	C₂₇₁⋊C₈	—	4	2	no	—
36	Q₈×C₂₇₁	—	2	2	no	—
37	Q₈×C₂₇₁	—	4	2	no	—
38	Q₈×C₂₇₁	—	8	2	no	—
39	Q₈×C₂₇₁	Q₁₆	1	4	yes	complemented
40	C₂×C₄×C₂₇₁	—	2	2	no	—
41	C₂×C₄×C₂₇₁	—	2	2	no	—
42	C₂×C₄×C₂₇₁	—	2	2	no	—
43	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
45	C₂²×C₂₇₁	#120	1	1	yes	characteristic
46	C₄×C₂₇₁	—	2	4	no	—
47	C₄×C₂₇₁	—	2	1	no	—
48	C₄×C₂₇₁	—	4	2	no	—
49	C₄×C₂₇₁	—	4	2	no	—
50	C₄×C₂₇₁	C₂×Q₁₆	1	2	yes	—
51	C₄×C₂₇₁	#137	1	2	yes	—
52	Dic₂₇₁	—	8	1	no	—
53	C₂×C₂₇₁	#451	1	2	yes	—
54	C₂×C₂₇₁	#453	1	1	yes	characteristic
55	C₂₇₁	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	271	1	no	maximal, Sylow
57	Q₈²	D₂₇₁	1	1	yes	characteristic
58	#494	—	271	1	no	—
59	#477	—	271	2	no	—
60	#333	—	271	1	no	—
61	Q₈⋊C₈	—	271	2	no	—
62	C₂×Q₁₆	—	542	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	271	1	no	—
66	C₄⋊Q₈	C₂×D₂₇₁	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₂₇₁	1	2	yes	—
69	#107	—	542	2	no	—
70	C₄⋊C₈	—	271	2	no	—
71	#103	—	542	2	no	—
72	C₄×C₈	—	271	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	542	1	no	—
75	C₂×Q₈	C₂²⋊D₂₇₁	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₂₇₁	1	2	yes	—
77	Q₁₆	—	1084	4	no	—
78	C₂×C₈	—	542	1	no	—
79	C₂×C₈	—	542	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	542	2	no	—
84	C₄⋊C₄	C₂²⋊D₂₇₁	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	1084	2	no	—
92	Q₈	#16864644	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	1084	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	2	yes	—
99	C₈	—	542	2	no	—
100	C₈	—	1084	2	no	—
101	C₂²	#22576129	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	2168	1	no	—
107	C₄	C₂×#16864644	1	2	yes	—
108	C₄	#22576191	1	2	yes	—
109	C₂	#31774053	1	2	yes	—
110	C₂	#31774065	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31774276

C₂

yes

maximal, characteristic

#31774192

C₂

yes

maximal

#31773646

C₂

yes

maximal, characteristic

Q₈⋊(C₂₇₁⋊C₈)

C₂

yes

maximal

Q₈²×C₂₇₁

C₂

yes

maximal, characteristic

C₂×#16864644

—

#22576160

C₂²

yes

characteristic

#22576093

—

C₄⋊(C₂₇₁⋊C₈)

C₂²

yes

—

#22576084

—

C₄×C₂₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₇₁

—

C₄⋊Q₈×C₂₇₁

—

C₄⋊Q₈×C₂₇₁

C₂²

yes

characteristic

C₄×Q₈×C₂₇₁

—

C₄×Q₈×C₂₇₁

C₂²

yes

—

C₂×Dic₅₄₂

—

#16864644

—

C₂×C₂₇₁⋊C₈

—

C₂×C₂₇₁⋊C₈

—

C₄⋊Dic₂₇₁

—

C₂×Q₈×C₂₇₁

—

C₂×Q₈×C₂₇₁

D₄

yes

—

C₂×Q₈×C₂₇₁

D₄

yes

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

D₄

yes

—

C₄²×C₂₇₁

—

C₄²×C₂₇₁

—

C₄²×C₂₇₁

C₂³

yes

characteristic

Dic₅₄₂

—

C₂×Dic₂₇₁

—

C₂₇₁⋊C₈

—

C₂₇₁⋊C₈

—

Q₈×C₂₇₁

—

Q₈×C₂₇₁

—

Q₈×C₂₇₁

—

Q₈×C₂₇₁

Q₁₆

yes

complemented

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂²×C₂₇₁

#120

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂×Q₁₆

yes

—

C₄×C₂₇₁

#137

yes

—

Dic₂₇₁

—

C₂×C₂₇₁

#451

yes

—

C₂×C₂₇₁

#453

yes

characteristic

C₂₇₁

#1633

yes

characteristic, complemented, Sylow

#1633

—

271

maximal, Sylow

Q₈²

D₂₇₁

yes

characteristic

#494

—

271

—

#477

—

271

—

#333

—

271

—

Q₈⋊C₈

—

271

—

C₂×Q₁₆

—

542

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

271

—

C₄⋊Q₈

C₂×D₂₇₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₂₇₁

yes

—

#107

—

542

—

C₄⋊C₈

—

271

—

#103

—

542

—

C₄×C₈

—

271

—

C₂×Q₈

—

C₂×Q₈

—

542

—

C₂×Q₈

C₂²⋊D₂₇₁

yes

—

C₂×Q₈

C₂²⋊D₂₇₁

yes

—

Q₁₆

—

1084

—

C₂×C₈

—

542

—

C₂×C₈

—

542

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

542

—

C₄⋊C₄

C₂²⋊D₂₇₁

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₇₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

1084

—

Q₈

#16864644

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1084

—

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

—

C₈

—

542

—

100

C₈

—

1084

—

101

C₂²

#22576129

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

2168

—

107

C₄

C₂×#16864644

yes

—

108

C₄

#22576191

yes

—

109

C₂

#31774053

yes

—

110

C₂

#31774065

yes

characteristic

	subgroup	G/H
center	C₂²	#22576129
commutator, cosocle	C₄²×C₂₇₁	C₂³
Frattini	C₄²	C₂²×D₂₇₁
Fitting	Q₈²×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	#120
hypercenter	Q₈²	D₂₇₁
hypocenter	C₂₇₁	#1633
LCS	G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#22576129

commutator, cosocle

C₄²×C₂₇₁

C₂³

Frattini

C₄²

C₂²×D₂₇₁

Fitting

Q₈²×C₂₇₁

C₂

socle, prime radical

C₂²×C₂₇₁

#120

hypercenter

Q₈²

D₂₇₁

hypocenter

C₂₇₁

#1633

LCS

G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
39. Q₈×C₂₇₁	Q₁₆	semidirect
55. C₂₇₁	#1633	semidirect
92. Q₈	#16864644	semidirect

H ≅ G/N

type

39. Q₈×C₂₇₁

Q₁₆

semidirect

55. C₂₇₁

#1633

semidirect

92. Q₈

#16864644

semidirect

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

subgroup	index
#31774276	2
#31774192	2
#31773646	2
Q₈⋊(C₂₇₁⋊C₈)	2
Q₈²×C₂₇₁	2
#1633	271

subgroup

index

#31774276

#31774192

#31773646

Q₈⋊(C₂₇₁⋊C₈)

Q₈²×C₂₇₁

#1633

271

Subgroups — #46835504

Conjugacy classes of subgroups

Complements

Maximal subgroups