Group #46848168 — subgroups

Subgroups — #46848168

Order 34688.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#31774710	C₂	1	4	yes	maximal, complemented
2	#31774169	C₂	1	4	yes	maximal, complemented
3	#31774079	C₂	1	4	yes	maximal, complemented
4	#31774015	C₂	1	2	yes	maximal, complemented
5	#494×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22576208	C₂²	1	4	yes	—
7	#22576192	—	2	8	no	—
8	#22576191	C₂²	1	8	yes	—
9	#22576183	—	2	4	no	—
10	#22576182	C₂²	1	2	yes	—
11	Q₁₆×D₂₇₁	—	2	4	no	—
12	#22576166	C₂²	1	4	yes	—
13	#22576096	C₂²	1	2	yes	—
14	#22576088	C₂²	1	4	yes	—
15	#22576079	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₂₇₁	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₂₇₁	C₂²	1	2	yes	—
18	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
19	#16864655	—	2	8	no	—
20	#16864654	—	4	4	no	—
21	#16864654	C₂³	1	4	yes	—
22	Q₈×D₂₇₁	—	2	8	no	—
23	Dic₁₀₈₄	D₄	1	2	yes	—
24	#16864644	—	2	8	no	—
25	#16864643	D₄	1	4	yes	—
26	#16864641	—	2	8	no	—
27	D₂₁₆₈	D₄	1	2	yes	—
28	#16864638	C₂³	1	4	yes	—
29	#16864637	—	2	4	no	—
30	C₈×D₂₇₁	—	2	4	no	—
31	C₂×Q₈×C₂₇₁	C₂³	1	4	yes	—
32	Q₁₆×C₂₇₁	—	2	4	no	—
33	Q₁₆×C₂₇₁	—	2	4	no	—
34	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
35	C₄⋊C₄×C₂₇₁	—	4	2	no	—
36	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
37	Dic₅₄₂	—	4	4	no	—
38	Dic₅₄₂	C₂×D₄	1	4	yes	—
39	D₁₀₈₄	—	4	4	no	—
40	D₁₀₈₄	C₂×D₄	1	4	yes	—
41	C₂²⋊D₂₇₁	—	2	4	no	—
42	C₄×D₂₇₁	—	2	4	no	—
43	C₄×D₂₇₁	—	4	8	no	—
44	C₂₇₁⋊C₈	—	2	4	no	—
45	Q₈×C₂₇₁	—	2	8	no	—
46	C₂×C₄×C₂₇₁	—	2	1	no	—
47	C₂×C₄×C₂₇₁	—	2	4	no	—
48	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
49	C₈×C₂₇₁	C₂×D₄	1	2	yes	—
50	C₈×C₂₇₁	C₂×D₄	1	2	yes	—
51	C₂²×C₂₇₁	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₂₇₁	—	2	4	no	—
53	C₄×C₂₇₁	—	2	2	no	—
54	C₄×C₂₇₁	—	4	4	no	—
55	C₄×C₂₇₁	C₂²×D₄	1	2	yes	—
56	Dic₂₇₁	—	2	4	no	—
57	C₂×C₂₇₁	—	2	1	no	—
58	C₂×C₂₇₁	D₄²	1	1	yes	characteristic
59	D₂₇₁	—	4	4	no	—
60	C₂₇₁	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	271	1	no	maximal, Sylow
62	#578	—	271	4	no	—
63	#494	D₂₇₁	1	1	yes	characteristic, complemented
64	#473	—	271	4	no	—
65	#456	—	271	4	no	—
66	#443	—	271	2	no	—
67	2₋¹⁺⁴	—	271	4	no	—
68	#137	—	271	8	no	—
69	#137	—	542	8	no	—
70	#135	—	271	2	no	—
71	#135	—	542	4	no	—
72	C₂×Q₁₆	—	542	4	no	—
73	C₂×Q₁₆	C₂×D₂₇₁	1	4	yes	—
74	#131	—	271	4	no	—
75	C₄⋊Q₈	C₂×D₂₇₁	1	2	yes	—
76	#108	—	271	2	no	—
77	#104	—	271	4	no	—
78	#101	—	271	4	no	—
79	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
80	#41	—	271	4	no	—
81	#41	—	542	8	no	—
82	#41	—	1084	4	no	—
83	C₂×Q₈	—	542	8	no	—
84	C₂×Q₈	C₂²×D₂₇₁	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	271	2	no	—
88	Q₁₆	—	542	8	no	—
89	SD₁₆	—	271	4	no	—
90	SD₁₆	—	542	8	no	—
91	D₈	—	271	2	no	—
92	M₁₆	—	271	4	no	—
93	M₁₆	—	542	4	no	—
94	C₂×C₈	—	542	4	no	—
95	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	271	4	no	—
100	Q₈	—	1084	4	no	—
101	D₄	—	271	4	no	—
102	D₄	—	542	4	no	—
103	D₄	—	1084	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	542	4	no	—
107	C₂×C₄	—	1084	8	no	—
108	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
109	C₈	—	542	4	no	—
110	C₈	D₄×D₂₇₁	1	2	yes	—
111	C₈	D₄×D₂₇₁	1	2	yes	—
112	C₂²	—	542	4	no	—
113	C₂²	#22576205	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	542	4	no	—
117	C₄	C₂×D₄×D₂₇₁	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	1084	4	no	—
120	C₂	#31774532	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31774710

C₂

yes

maximal, complemented

#31774169

C₂

yes

maximal, complemented

#31774079

C₂

yes

maximal, complemented

#31774015

C₂

yes

maximal, complemented

#494×C₂₇₁

C₂

yes

maximal, characteristic, complemented

#22576208

C₂²

yes

—

#22576192

—

#22576191

C₂²

yes

—

#22576183

—

#22576182

C₂²

yes

—

Q₁₆×D₂₇₁

—

#22576166

C₂²

yes

—

#22576096

C₂²

yes

—

#22576088

C₂²

yes

—

#22576079

C₂²

yes

—

C₂×Q₁₆×C₂₇₁

C₂²

yes

—

C₄⋊Q₈×C₂₇₁

C₂²

yes

—

C₄×C₈×C₂₇₁

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₂₇₁

—

Dic₁₀₈₄

D₄

yes

—

#16864644

—

#16864643

D₄

yes

—

#16864641

—

D₂₁₆₈

D₄

yes

—

#16864638

C₂³

yes

—

#16864637

—

C₈×D₂₇₁

—

C₂×Q₈×C₂₇₁

C₂³

yes

—

Q₁₆×C₂₇₁

—

Q₁₆×C₂₇₁

—

C₂×C₈×C₂₇₁

C₂³

yes

—

C₄⋊C₄×C₂₇₁

—

C₄²×C₂₇₁

C₂³

yes

characteristic

Dic₅₄₂

—

Dic₅₄₂

C₂×D₄

yes

—

D₁₀₈₄

—

D₁₀₈₄

C₂×D₄

yes

—

C₂²⋊D₂₇₁

—

C₄×D₂₇₁

—

C₄×D₂₇₁

—

C₂₇₁⋊C₈

—

Q₈×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂⁴

yes

characteristic

C₈×C₂₇₁

C₂×D₄

yes

—

C₈×C₂₇₁

C₂×D₄

yes

—

C₂²×C₂₇₁

H(5,𝔽₂)

yes

characteristic

C₂×D₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂²×D₄

yes

—

Dic₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

D₄²

yes

characteristic

D₂₇₁

—

C₂₇₁

#3293

yes

characteristic, complemented, Sylow

#3293

—

271

maximal, Sylow

#578

—

271

—

#494

D₂₇₁

yes

characteristic, complemented

#473

—

271

—

#456

—

271

—

#443

—

271

—

2₋¹⁺⁴

—

271

—

#137

—

271

—

#137

—

542

—

#135

—

271

—

#135

—

542

—

C₂×Q₁₆

—

542

—

C₂×Q₁₆

C₂×D₂₇₁

yes

—

#131

—

271

—

C₄⋊Q₈

C₂×D₂₇₁

yes

—

#108

—

271

—

#104

—

271

—

#101

—

271

—

C₄×C₈

C₂×D₂₇₁

yes

characteristic

#41

—

271

—

#41

—

542

—

#41

—

1084

—

C₂×Q₈

—

542

—

C₂×Q₈

C₂²×D₂₇₁

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

271

—

Q₁₆

—

542

—

SD₁₆

—

271

—

SD₁₆

—

542

—

D₈

—

271

—

M₁₆

—

271

—

M₁₆

—

542

—

C₂×C₈

—

542

—

C₂×C₈

C₂²×D₂₇₁

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₇₁

yes

characteristic

Q₈

—

Q₈

—

271

—

100

Q₈

—

1084

—

101

D₄

—

271

—

102

D₄

—

542

—

103

D₄

—

1084

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

542

—

107

C₂×C₄

—

1084

—

108

C₂×C₄

C₂³×D₂₇₁

yes

characteristic

109

C₈

—

542

—

110

C₈

D₄×D₂₇₁

yes

—

111

C₈

D₄×D₂₇₁

yes

—

112

C₂²

—

542

—

113

C₂²

#22576205

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

542

—

117

C₄

C₂×D₄×D₂₇₁

yes

—

118

C₂

—

119

C₂

—

1084

—

120

C₂

#31774532

yes

characteristic

	subgroup	G/H
center	C₂	#31774532
commutator, cosocle	C₂×C₄×C₂₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	#494×C₂₇₁	C₂
socle	C₂×C₂₇₁	D₄²
hypercenter	#494	D₂₇₁
hypocenter	C₂₇₁	#3293
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#31774532

commutator, cosocle

C₂×C₄×C₂₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₇₁

Fitting

#494×C₂₇₁

C₂

socle

C₂×C₂₇₁

D₄²

D₂₇₁

C₂₇₁

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #31774710	C₂	semidirect
2. #31774169	C₂	semidirect
3. #31774079	C₂	semidirect
4. #31774015	C₂	semidirect
5. #494×C₂₇₁	C₂	semidirect
60. C₂₇₁	#3293	semidirect
63. #494	D₂₇₁	semidirect

H ≅ G/N

type

1. #31774710

C₂

semidirect

2. #31774169

C₂

semidirect

3. #31774079

C₂

semidirect

4. #31774015

C₂

semidirect

5. #494×C₂₇₁

C₂

semidirect

60. C₂₇₁

#3293

semidirect

63. #494

D₂₇₁

semidirect

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

subgroup	index
#31774710	2
#31774169	2
#31774079	2
#31774015	2
#494×C₂₇₁	2
#3293	271

subgroup

index

#494×C₂₇₁

271

Subgroups — #46848168

Conjugacy classes of subgroups

Complements

Maximal subgroups