Group #46846868 — subgroups

Subgroups — C₂×#31774299 #46846868

Order 34688.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×#22576152	C₂	1	2	yes	maximal
2	#31774299	C₂	1	8	yes	maximal, direct product factor
3	C₂×#22576094	C₂	1	2	yes	maximal
4	C₂×#22576092	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	#22576152	—	2	4	no	—
7	#22576152	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₂₇₁	C₂²	1	4	yes	—
9	C₂×Dic₂₇₁⋊C₄	—	2	4	no	—
10	#22576094	C₂²	1	8	yes	—
11	#22576092	C₂²	1	8	yes	—
12	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₇₁	C₂²	1	4	yes	—
16	C₂²×Dic₂₇₁	—	2	4	no	—
17	C₄⋊Dic₂₇₁	—	2	4	no	—
18	C₄⋊Dic₂₇₁	C₂³	1	8	yes	—
19	Dic₂₇₁⋊C₄	—	2	16	no	—
20	C₂²×C₄×C₂₇₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₇₁	Q₈	1	4	yes	—
23	C₂×C₈×C₂₇₁	Q₈	1	4	yes	—
24	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
25	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
26	C₄²×C₂₇₁	D₄	1	2	yes	—
27	C₄²×C₂₇₁	C₂³	1	2	yes	—
28	C₂×Dic₂₇₁	—	2	8	no	—
29	C₂×Dic₂₇₁	—	4	8	no	—
30	C₂³×C₂₇₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₇₁	—	2	4	no	—
32	C₂×C₄×C₂₇₁	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
37	C₈×C₂₇₁	C₂×Q₈	1	4	yes	—
38	C₈×C₂₇₁	C₂×Q₈	1	4	yes	—
39	C₂²×C₂₇₁	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₂₇₁	#135	1	4	yes	—
41	C₂²×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₂₇₁	—	2	4	no	—
43	C₄×C₂₇₁	C₄⋊Q₈	1	2	yes	—
44	C₄×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
46	Dic₂₇₁	—	4	8	no	—
47	C₂×C₂₇₁	#499	1	4	yes	complemented
48	C₂×C₂₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₂₇₁	C₂×#135	1	2	yes	—
50	C₂₇₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
51	C₂×#499	—	271	1	no	maximal, Sylow
52	C₂×#125	—	271	2	no	—
53	#499	—	271	8	no	—
54	C₂×#107	—	271	2	no	—
55	C₂×#106	—	271	2	no	—
56	C₂×C₄×C₈	D₂₇₁	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
59	#125	—	271	8	no	—
60	#125	—	542	4	no	—
61	C₂×C₄⋊C₄	—	271	4	no	—
62	C₂×C₄⋊C₄	—	542	4	no	—
63	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
64	#107	—	271	8	no	—
65	#106	—	271	8	no	—
66	C₄×C₈	C₂×D₂₇₁	1	4	yes	—
67	C₂²×C₄	—	542	4	no	—
68	C₂²×C₄	D₁₀₈₄	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
70	C₂×C₈	Dic₅₄₂	1	4	yes	—
71	C₂×C₈	Dic₅₄₂	1	4	yes	—
72	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
73	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
74	C₄⋊C₄	—	271	8	no	—
75	C₄⋊C₄	—	542	4	no	—
76	C₄⋊C₄	—	542	16	no	—
77	C₄²	D₁₀₈₄	1	2	yes	—
78	C₄²	C₂²×D₂₇₁	1	2	yes	—
79	C₂³	C₂×D₁₀₈₄	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	542	8	no	—
82	C₂×C₄	—	1084	8	no	—
83	C₂×C₄	C₂×D₁₀₈₄	1	4	yes	—
84	C₂×C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₅₄₂	1	2	yes	—
86	C₂×C₄	C₂×Dic₅₄₂	1	2	yes	—
87	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
88	C₈	C₂×Dic₅₄₂	1	4	yes	—
89	C₈	C₂×Dic₅₄₂	1	4	yes	—
90	C₂²	#22576160	1	2	yes	—
91	C₂²	#22576182	1	4	yes	—
92	C₂²	C₂²×D₁₀₈₄	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	1084	8	no	—
95	C₄	#22576160	1	2	yes	—
96	C₄	C₂²×Dic₅₄₂	1	1	yes	characteristic
97	C₄	C₂²×Dic₅₄₂	1	1	yes	characteristic
98	C₂	#31774299	1	4	yes	direct product factor
99	C₂	C₂×#22576160	1	1	yes	characteristic
100	C₂	C₂×#22576182	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22576152

C₂

yes

maximal

#31774299

C₂

yes

maximal, direct product factor

C₂×#22576094

C₂

yes

maximal

C₂×#22576092

C₂

yes

maximal

C₂×C₄×C₈×C₂₇₁

C₂

yes

maximal, characteristic

#22576152

—

#22576152

C₂²

yes

—

C₂×C₄⋊Dic₂₇₁

C₂²

yes

—

C₂×Dic₂₇₁⋊C₄

—

#22576094

C₂²

yes

—

#22576092

C₂²

yes

—

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

C₂×C₄²×C₂₇₁

C₂²

yes

characteristic

C₄×C₈×C₂₇₁

C₂²

yes

—

C₂²×Dic₂₇₁

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

C₂³

yes

—

Dic₂₇₁⋊C₄

—

C₂²×C₄×C₂₇₁

D₄

yes

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

C₂×C₈×C₂₇₁

Q₈

yes

—

C₂×C₈×C₂₇₁

Q₈

yes

—

C₂×C₈×C₂₇₁

C₂³

yes

—

C₂×C₈×C₂₇₁

C₂³

yes

—

C₄²×C₂₇₁

D₄

yes

—

C₄²×C₂₇₁

C₂³

yes

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂³×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×Q₈

yes

—

C₂×C₄×C₂₇₁

C₂×Q₈

yes

—

C₂×C₄×C₂₇₁

C₂⁴

yes

characteristic

C₈×C₂₇₁

C₂×Q₈

yes

—

C₈×C₂₇₁

C₂×Q₈

yes

—

C₂²×C₂₇₁

C₄⋊Q₈

yes

—

C₂²×C₂₇₁

#135

yes

—

C₂²×C₂₇₁

C₂²×D₄

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

C₄⋊Q₈

yes

—

C₄×C₂₇₁

C₂²×Q₈

yes

characteristic

C₄×C₂₇₁

C₂²×Q₈

yes

characteristic

Dic₂₇₁

—

C₂×C₂₇₁

#499

yes

complemented

C₂×C₂₇₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₇₁

C₂×#135

yes

—

C₂₇₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

271

maximal, Sylow

C₂×#125

—

271

—

#499

—

271

—

C₂×#107

—

271

—

C₂×#106

—

271

—

C₂×C₄×C₈

D₂₇₁

yes

characteristic

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

#125

—

271

—

#125

—

542

—

C₂×C₄⋊C₄

—

271

—

C₂×C₄⋊C₄

—

542

—

C₂×C₄²

C₂×D₂₇₁

yes

characteristic

#107

—

271

—

#106

—

271

—

C₄×C₈

C₂×D₂₇₁

yes

—

C₂²×C₄

—

542

—

C₂²×C₄

D₁₀₈₄

yes

—

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

C₂×C₈

Dic₅₄₂

yes

—

C₂×C₈

Dic₅₄₂

yes

—

C₂×C₈

C₂²×D₂₇₁

yes

—

C₂×C₈

C₂²×D₂₇₁

yes

—

C₄⋊C₄

—

271

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄²

D₁₀₈₄

yes

—

C₄²

C₂²×D₂₇₁

yes

—

C₂³

C₂×D₁₀₈₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

542

—

C₂×C₄

—

1084

—

C₂×C₄

C₂×D₁₀₈₄

yes

—

C₂×C₄

C₂×D₁₀₈₄

yes

characteristic

C₂×C₄

C₂×Dic₅₄₂

yes

—

C₂×C₄

C₂×Dic₅₄₂

yes

—

C₂×C₄

C₂³×D₂₇₁

yes

characteristic

C₈

C₂×Dic₅₄₂

yes

—

C₈

C₂×Dic₅₄₂

yes

—

C₂²

#22576160

yes

—

C₂²

#22576182

yes

—

C₂²

C₂²×D₁₀₈₄

yes

characteristic

C₄

—

C₄

—

1084

—

C₄

#22576160

yes

—

C₄

C₂²×Dic₅₄₂

yes

characteristic

C₄

C₂²×Dic₅₄₂

yes

characteristic

C₂

#31774299

yes

direct product factor

C₂

C₂×#22576160

yes

characteristic

100

C₂

C₂×#22576182

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₀₈₄
commutator, cosocle	C₂×C₄×C₂₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	C₂×C₄×C₈×C₂₇₁	C₂
socle, prime radical	C₂³×C₂₇₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₂₇₁
hypocenter	C₂₇₁	C₂×#499
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₀₈₄

commutator, cosocle

C₂×C₄×C₂₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₇₁

Fitting

C₂×C₄×C₈×C₂₇₁

C₂

socle, prime radical

C₂³×C₂₇₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₂₇₁

hypocenter

C₂₇₁

C₂×#499

LCS

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #31774299	C₂	direct (N × H)
47. C₂×C₂₇₁	#499	semidirect
50. C₂₇₁	C₂×#499	semidirect
98. C₂	#31774299	direct (N × H)

H ≅ G/N

type

2. #31774299

C₂

direct (N × H)

47. C₂×C₂₇₁

#499

semidirect

50. C₂₇₁

C₂×#499

semidirect

98. C₂

#31774299

direct (N × H)

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

subgroup	index
C₂×#22576152	2
#31774299	2
C₂×#22576094	2
C₂×#22576092	2
C₂×C₄×C₈×C₂₇₁	2
C₂×#499	271

subgroup

index

C₂×C₄×C₈×C₂₇₁

271

Subgroups — C₂×#31774299 #46846868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#31774299 #46846868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31774299 #46846868