Group #46836031 — subgroups

Subgroups — #46836031

Order 34688.

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

normal subgroups only

This group has 242 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	#31774273	C₂	1	2	yes	maximal, complemented
2	#31774270	C₂	1	1	yes	maximal, characteristic, complemented
3	#31774243	C₂	1	2	yes	maximal, complemented
4	#31773618	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16864643	—	2	2	no	—
7	C₂×#16864639	—	2	4	no	—
8	#22576160	C₂²	1	1	yes	characteristic
9	#22576157	C₂²	1	1	yes	characteristic
10	#22576081	—	2	4	no	—
11	C₄×C₂₇₁⋊C₈	C₂²	1	2	yes	—
12	C₂×D₈×C₂₇₁	—	2	2	no	—
13	#127×C₂₇₁	C₂²	1	2	yes	—
14	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₄₂	—	2	1	no	—
16	C₂×D₁₀₈₄	—	2	2	no	—
17	C₂×D₁₀₈₄	—	2	1	no	—
18	#16864643	—	4	4	no	—
19	#16864639	—	4	8	no	—
20	C₂×C₂₇₁⋊C₈	D₄	1	4	yes	—
21	C₄⋊Dic₂₇₁	—	2	2	no	—
22	C₂×D₄×C₂₇₁	—	2	2	no	—
23	C₂×D₄×C₂₇₁	—	2	2	no	—
24	C₂×D₄×C₂₇₁	—	2	2	no	—
25	D₈×C₂₇₁	—	4	4	no	—
26	C₂×C₈×C₂₇₁	D₄	1	2	yes	—
27	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
28	C₂²×D₂₇₁	—	4	1	no	—
29	Dic₅₄₂	—	4	2	no	—
30	D₁₀₈₄	—	4	2	no	—
31	D₁₀₈₄	—	4	4	no	—
32	C₂×Dic₂₇₁	—	4	1	no	—
33	C₂₇₁⋊C₈	—	2	4	no	—
34	C₂³×C₂₇₁	—	4	2	no	—
35	D₄×C₂₇₁	—	4	4	no	—
36	D₄×C₂₇₁	—	4	4	no	—
37	D₄×C₂₇₁	—	4	4	no	—
38	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₇₁	D₈	1	4	yes	complemented
41	C₂²×C₂₇₁	—	8	2	no	—
42	C₂²×C₂₇₁	—	8	2	no	—
43	C₂²×C₂₇₁	—	8	2	no	—
44	C₂²×C₂₇₁	#127	1	1	yes	characteristic
45	C₂×D₂₇₁	—	8	1	no	—
46	C₂×D₂₇₁	—	8	2	no	—
47	C₄×C₂₇₁	C₂×D₈	1	2	yes	—
48	C₄×C₂₇₁	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₂₇₁	—	8	1	no	—
50	C₂×C₂₇₁	—	16	2	no	—
51	C₂×C₂₇₁	#493	1	1	yes	characteristic
52	C₂×C₂₇₁	#496	1	2	yes	—
53	D₂₇₁	—	16	1	no	—
54	C₂₇₁	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	271	1	no	maximal, Sylow
56	#493	—	271	2	no	—
57	#493	D₂₇₁	1	1	yes	characteristic, complemented
58	#492	—	271	1	no	—
59	#486	—	271	2	no	—
60	#322	—	271	1	no	—
61	C₂×SD₁₆	—	542	2	no	—
62	C₂×D₈	—	2	2	no	—
63	C₂×D₈	—	542	4	no	—
64	C₄⋊Q₈	—	271	1	no	—
65	#127	—	271	1	no	—
66	#127	C₂×D₂₇₁	1	2	yes	—
67	#102	—	542	4	no	—
68	C₄×C₈	—	271	2	no	—
69	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
70	C₂×Q₈	—	542	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	542	2	no	—
75	C₂×D₄	—	542	1	no	—
76	SD₁₆	—	1084	4	no	—
77	D₈	—	4	4	no	—
78	D₈	—	1084	8	no	—
79	C₂×C₈	—	271	4	no	—
80	C₂×C₈	C₂²⋊D₂₇₁	1	2	yes	—
81	C₄⋊C₄	—	542	2	no	—
82	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
83	C₂³	—	4	2	no	—
84	C₂³	—	1084	1	no	—
85	Q₈	—	1084	2	no	—
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	4	no	—
89	D₄	—	1084	2	no	—
90	D₄	—	1084	4	no	—
91	C₂×C₄	—	1084	1	no	—
92	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
94	C₈	—	542	4	no	—
95	C₈	#16864639	1	4	yes	complemented
96	C₂²	—	8	2	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	8	2	no	—
99	C₂²	—	2168	1	no	—
100	C₂²	—	2168	2	no	—
101	C₂²	#22576156	1	1	yes	characteristic
102	C₄	—	2168	1	no	—
103	C₄	C₂×#16864639	1	2	yes	—
104	C₄	#22576184	1	4	yes	—
105	C₂	—	16	2	no	—
106	C₂	—	4336	1	no	—
107	C₂	#31774273	1	1	yes	characteristic
108	C₂	#31774285	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31774273

C₂

yes

maximal, complemented

#31774270

C₂

yes

maximal, characteristic, complemented

#31774243

C₂

yes

maximal, complemented

#31773618

C₂

yes

maximal, characteristic, complemented

#493×C₂₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#16864643

—

C₂×#16864639

—

#22576160

C₂²

yes

characteristic

#22576157

C₂²

yes

characteristic

#22576081

—

C₄×C₂₇₁⋊C₈

C₂²

yes

—

C₂×D₈×C₂₇₁

—

#127×C₂₇₁

C₂²

yes

—

C₄×C₈×C₂₇₁

C₂²

yes

characteristic

C₂×Dic₅₄₂

—

C₂×D₁₀₈₄

—

C₂×D₁₀₈₄

—

#16864643

—

#16864639

—

C₂×C₂₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₇₁

—

C₂×D₄×C₂₇₁

—

C₂×D₄×C₂₇₁

—

C₂×D₄×C₂₇₁

—

D₈×C₂₇₁

—

C₂×C₈×C₂₇₁

D₄

yes

—

C₄²×C₂₇₁

C₂³

yes

characteristic

C₂²×D₂₇₁

—

Dic₅₄₂

—

D₁₀₈₄

—

D₁₀₈₄

—

C₂×Dic₂₇₁

—

C₂₇₁⋊C₈

—

C₂³×C₂₇₁

—

D₄×C₂₇₁

—

D₄×C₂₇₁

—

D₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

D₈

yes

complemented

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

#127

yes

characteristic

C₂×D₂₇₁

—

C₂×D₂₇₁

—

C₄×C₂₇₁

C₂×D₈

yes

—

C₄×C₂₇₁

AGL(1,ℤ/8)

yes

—

Dic₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

#493

yes

characteristic

C₂×C₂₇₁

#496

yes

—

D₂₇₁

—

C₂₇₁

#1721

yes

characteristic, complemented, Sylow

#1721

—

271

maximal, Sylow

#493

—

271

—

#493

D₂₇₁

yes

characteristic, complemented

#492

—

271

—

#486

—

271

—

#322

—

271

—

C₂×SD₁₆

—

542

—

C₂×D₈

—

C₂×D₈

—

542

—

C₄⋊Q₈

—

271

—

#127

—

271

—

#127

C₂×D₂₇₁

yes

—

#102

—

542

—

C₄×C₈

—

271

—

C₄×C₈

C₂×D₂₇₁

yes

characteristic

C₂×Q₈

—

542

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

542

—

C₂×D₄

—

542

—

SD₁₆

—

1084

—

D₈

—

D₈

—

1084

—

C₂×C₈

—

271

—

C₂×C₈

C₂²⋊D₂₇₁

yes

—

C₄⋊C₄

—

542

—

C₄²

C₂²×D₂₇₁

yes

characteristic

C₂³

—

C₂³

—

1084

—

Q₈

—

1084

—

D₄

—

D₄

—

D₄

—

D₄

—

1084

—

D₄

—

1084

—

C₂×C₄

—

1084

—

C₂×C₄

D₄×D₂₇₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

characteristic

C₈

—

542

—

C₈

#16864639

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

2168

—

100

C₂²

—

2168

—

101

C₂²

#22576156

yes

characteristic

102

C₄

—

2168

—

103

C₄

C₂×#16864639

yes

—

104

C₄

#22576184

yes

—

105

C₂

—

106

C₂

—

4336

—

107

C₂

#31774273

yes

characteristic

108

C₂

#31774285

yes

—

	subgroup	G/H
center	C₂²	#22576156
commutator, cosocle	C₄²×C₂₇₁	C₂³
Frattini	C₄²	C₂²×D₂₇₁
Fitting	#493×C₂₇₁	C₂
socle	C₂²×C₂₇₁	#127
hypercenter	#493	D₂₇₁
hypocenter	C₂₇₁	#1721
LCS	G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#22576156

commutator, cosocle

C₄²×C₂₇₁

C₂³

Frattini

C₄²

C₂²×D₂₇₁

Fitting

#493×C₂₇₁

C₂

C₂²×C₂₇₁

D₂₇₁

C₂₇₁

G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #31774273	C₂	semidirect
2. #31774270	C₂	semidirect
3. #31774243	C₂	semidirect
4. #31773618	C₂	semidirect
5. #493×C₂₇₁	C₂	semidirect
40. C₈×C₂₇₁	D₈	semidirect
54. C₂₇₁	#1721	semidirect
57. #493	D₂₇₁	semidirect
95. C₈	#16864639	semidirect

H ≅ G/N

type

1. #31774273

C₂

semidirect

2. #31774270

C₂

semidirect

3. #31774243

C₂

semidirect

4. #31773618

C₂

semidirect

5. #493×C₂₇₁

C₂

semidirect

40. C₈×C₂₇₁

D₈

semidirect

54. C₂₇₁

#1721

semidirect

57. #493

D₂₇₁

semidirect

95. C₈

#16864639

semidirect

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

subgroup	index
#31774273	2
#31774270	2
#31774243	2
#31773618	2
#493×C₂₇₁	2
#1721	271

subgroup

index

#493×C₂₇₁

271

Subgroups — #46836031

Conjugacy classes of subgroups

Complements

Maximal subgroups