Group #46838441 — subgroups

Subgroups — #46838441

Order 34688.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#31774335	C₂	1	2	yes	maximal, complemented
2	#31774274	C₂	1	1	yes	maximal, characteristic, complemented
3	#31774015	C₂	1	1	yes	maximal, characteristic, complemented
4	#31773711	C₂	1	2	yes	maximal, complemented
5	#364×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22576182	C₂²	1	1	yes	characteristic
7	C₂×D₂₁₆₈	—	2	1	no	—
8	C₂×D₂₁₆₈	C₂²	1	2	yes	—
9	#22576166	—	2	1	no	—
10	#22576157	—	2	1	no	—
11	#22576105	—	2	2	no	—
12	D₄₃₃₆	—	2	2	no	—
13	#22576096	C₂²	1	1	yes	characteristic
14	#22576088	—	2	1	no	—
15	M₃₂×C₂₇₁	C₂²	1	2	yes	—
16	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
17	#16864654	—	2	1	no	—
18	C₂×D₁₀₈₄	—	2	2	no	—
19	C₂×D₁₀₈₄	—	4	2	no	—
20	Dic₁₀₈₄	D₄	1	1	yes	characteristic, complemented
21	#16864643	—	2	1	no	—
22	D₂₁₆₈	—	2	2	no	—
23	D₂₁₆₈	—	2	2	no	—
24	D₂₁₆₈	—	4	1	no	—
25	D₂₁₆₈	—	4	1	no	—
26	D₂₁₆₈	D₄	1	1	yes	characteristic, complemented
27	#16864638	—	2	1	no	—
28	#16864637	—	4	1	no	—
29	C₈×D₂₇₁	—	4	1	no	—
30	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
32	C₄²×C₂₇₁	D₄	1	1	yes	characteristic
33	C₁₆×C₂₇₁	—	2	2	no	—
34	C₂²×D₂₇₁	—	4	2	no	—
35	Dic₅₄₂	—	2	1	no	—
36	D₁₀₈₄	—	2	1	no	—
37	D₁₀₈₄	—	4	2	no	—
38	D₁₀₈₄	—	4	2	no	—
39	D₁₀₈₄	—	8	4	no	—
40	C₂²⋊D₂₇₁	—	4	1	no	—
41	C₄×D₂₇₁	—	4	1	no	—
42	C₂₇₁⋊C₈	—	4	1	no	—
43	C₂×C₄×C₂₇₁	—	2	1	no	—
44	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
45	C₈×C₂₇₁	D₈	1	1	yes	characteristic
46	C₈×C₂₇₁	D₈	1	1	yes	characteristic
47	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
48	C₈×C₂₇₁	#41	1	1	yes	characteristic
49	C₂²×C₂₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₂₇₁	—	4	1	no	—
51	C₂×D₂₇₁	—	8	4	no	—
52	C₂×D₂₇₁	—	8	2	no	—
53	C₄×C₂₇₁	—	2	2	no	—
54	C₄×C₂₇₁	#121	1	1	yes	characteristic
55	C₄×C₂₇₁	C₂×D₈	1	1	yes	characteristic
56	Dic₂₇₁	—	4	1	no	—
57	C₂×C₂₇₁	—	2	1	no	—
58	C₂×C₂₇₁	#459	1	1	yes	characteristic
59	D₂₇₁	—	8	1	no	—
60	D₂₇₁	—	16	2	no	—
61	C₂₇₁	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	271	1	no	maximal, Sylow
63	#509	—	271	2	no	—
64	#493	—	271	1	no	—
65	#443	—	271	1	no	—
66	#364	D₂₇₁	1	1	yes	characteristic, complemented
67	#361	—	271	2	no	—
68	#135	—	271	1	no	—
69	C₂×D₈	—	271	2	no	—
70	C₂×D₈	—	542	1	no	—
71	#131	—	542	1	no	—
72	#127	—	542	1	no	—
73	SD₃₂	—	542	2	no	—
74	D₁₆	—	542	2	no	—
75	M₃₂	C₂×D₂₇₁	1	2	yes	—
76	#108	—	271	1	no	—
77	#104	—	542	1	no	—
78	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
79	#41	—	542	1	no	—
80	C₂×D₄	—	542	2	no	—
81	C₂×D₄	—	1084	2	no	—
82	Q₁₆	—	271	1	no	—
83	SD₁₆	—	542	1	no	—
84	D₈	—	271	1	no	—
85	D₈	—	542	2	no	—
86	D₈	—	542	2	no	—
87	D₈	—	1084	1	no	—
88	D₈	—	1084	1	no	—
89	M₁₆	—	542	1	no	—
90	M₁₆	—	1084	1	no	—
91	C₂×C₈	—	1084	1	no	—
92	C₂×C₈	D₁₀₈₄	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
94	C₄²	D₁₀₈₄	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	1084	2	no	—
97	Q₈	—	542	1	no	—
98	D₄	—	542	1	no	—
99	D₄	—	1084	1	no	—
100	D₄	—	1084	2	no	—
101	D₄	—	1084	2	no	—
102	D₄	—	2168	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	1084	1	no	—
105	C₂×C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
106	C₈	—	1084	1	no	—
107	C₈	D₂₁₆₈	1	1	yes	characteristic
108	C₈	D₂₁₆₈	1	1	yes	characteristic
109	C₈	D₄×D₂₇₁	1	1	yes	characteristic
110	C₈	#16864655	1	1	yes	characteristic
111	C₂²	—	1084	1	no	—
112	C₂²	—	2168	4	no	—
113	C₂²	—	2168	2	no	—
114	C₂²	#22576188	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	1084	1	no	—
117	C₄	#22576135	1	1	yes	characteristic
118	C₄	C₂×D₂₁₆₈	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2168	1	no	—
121	C₂	—	4336	2	no	—
122	C₂	#31774093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31774335

C₂

yes

maximal, complemented

#31774274

C₂

yes

maximal, characteristic, complemented

#31774015

C₂

yes

maximal, characteristic, complemented

#31773711

C₂

yes

maximal, complemented

#364×C₂₇₁

C₂

yes

maximal, characteristic, complemented

#22576182

C₂²

yes

characteristic

C₂×D₂₁₆₈

—

C₂×D₂₁₆₈

C₂²

yes

—

#22576166

—

#22576157

—

#22576105

—

D₄₃₃₆

—

#22576096

C₂²

yes

characteristic

#22576088

—

M₃₂×C₂₇₁

C₂²

yes

—

C₄×C₈×C₂₇₁

C₂²

yes

characteristic

#16864654

—

C₂×D₁₀₈₄

—

C₂×D₁₀₈₄

—

Dic₁₀₈₄

D₄

yes

characteristic, complemented

#16864643

—

D₂₁₆₈

—

D₂₁₆₈

—

D₂₁₆₈

—

D₂₁₆₈

—

D₂₁₆₈

D₄

yes

characteristic, complemented

#16864638

—

#16864637

—

C₈×D₂₇₁

—

C₂×C₈×C₂₇₁

D₄

yes

characteristic

C₂×C₈×C₂₇₁

C₂³

yes

characteristic

C₄²×C₂₇₁

D₄

yes

characteristic

C₁₆×C₂₇₁

—

C₂²×D₂₇₁

—

Dic₅₄₂

—

D₁₀₈₄

—

D₁₀₈₄

—

D₁₀₈₄

—

D₁₀₈₄

—

C₂²⋊D₂₇₁

—

C₄×D₂₇₁

—

C₂₇₁⋊C₈

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

D₈

yes

characteristic

C₈×C₂₇₁

D₈

yes

characteristic

C₈×C₂₇₁

C₂×D₄

yes

C₈×C₂₇₁

yes

C₂²×C₂₇₁

yes

C₂×D₂₇₁

—

C₂×D₂₇₁

—

C₂×D₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

#121

yes

characteristic

C₄×C₂₇₁

C₂×D₈

yes

characteristic

Dic₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

#459

yes

characteristic

D₂₇₁

—

D₂₇₁

—

C₂₇₁

#2213

yes

characteristic, complemented, Sylow

#2213

—

271

maximal, Sylow

#509

—

271

—

#493

—

271

—

#443

—

271

—

#364

D₂₇₁

yes

characteristic, complemented

#361

—

271

—

#135

—

271

—

C₂×D₈

—

271

—

C₂×D₈

—

542

—

#131

—

542

—

#127

—

542

—

SD₃₂

—

542

—

D₁₆

—

542

—

M₃₂

C₂×D₂₇₁

yes

—

#108

—

271

—

#104

—

542

—

C₄×C₈

C₂×D₂₇₁

yes

characteristic

#41

—

542

—

C₂×D₄

—

542

—

C₂×D₄

—

1084

—

Q₁₆

—

271

—

SD₁₆

—

542

—

D₈

—

271

—

D₈

—

542

—

D₈

—

542

—

D₈

—

1084

—

D₈

—

1084

—

M₁₆

—

542

—

M₁₆

—

1084

—

C₂×C₈

—

1084

—

C₂×C₈

D₁₀₈₄

yes

characteristic

C₂×C₈

C₂²×D₂₇₁

yes

characteristic

C₄²

D₁₀₈₄

yes

characteristic

C₁₆

—

C₂³

—

1084

—

Q₈

—

542

—

D₄

—

542

—

D₄

—

1084

—

100

D₄

—

1084

—

101

D₄

—

1084

—

102

D₄

—

2168

—

103

C₂×C₄

—

104

C₂×C₄

—

1084

—

105

C₂×C₄

C₂×D₁₀₈₄

yes

characteristic

106

C₈

—

1084

—

107

C₈

D₂₁₆₈

yes

characteristic

108

C₈

D₂₁₆₈

yes

characteristic

109

C₈

D₄×D₂₇₁

yes

characteristic

110

C₈

#16864655

yes

characteristic

111

C₂²

—

1084

—

112

C₂²

—

2168

—

113

C₂²

—

2168

—

114

C₂²

#22576188

yes

characteristic

115

C₄

—

116

C₄

—

1084

—

117

C₄

#22576135

yes

characteristic

118

C₄

C₂×D₂₁₆₈

yes

characteristic

119

C₂

—

120

C₂

—

2168

—

121

C₂

—

4336

—

122

C₂

#31774093

yes

characteristic

	subgroup	G/H
center	C₂	#31774093
commutator, cosocle	C₂×C₈×C₂₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₇₁
Fitting	#364×C₂₇₁	C₂
socle	C₂×C₂₇₁	#459
hypercenter	#364	D₂₇₁
hypocenter	C₂₇₁	#2213
LCS	G ▷ C₂×C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#31774093

commutator, cosocle

C₂×C₈×C₂₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₂₇₁

Fitting

#364×C₂₇₁

C₂

C₂×C₂₇₁

D₂₇₁

C₂₇₁

G ▷ C₂×C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #31774335	C₂	semidirect
2. #31774274	C₂	semidirect
3. #31774015	C₂	semidirect
4. #31773711	C₂	semidirect
5. #364×C₂₇₁	C₂	semidirect
20. Dic₁₀₈₄	D₄	semidirect
26. D₂₁₆₈	D₄	semidirect
61. C₂₇₁	#2213	semidirect
66. #364	D₂₇₁	semidirect

H ≅ G/N

type

1. #31774335

C₂

semidirect

2. #31774274

C₂

semidirect

3. #31774015

C₂

semidirect

4. #31773711

C₂

semidirect

5. #364×C₂₇₁

C₂

semidirect

20. Dic₁₀₈₄

D₄

semidirect

26. D₂₁₆₈

D₄

semidirect

61. C₂₇₁

#2213

semidirect

66. #364

D₂₇₁

semidirect

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

subgroup	index
#31774335	2
#31774274	2
#31774015	2
#31773711	2
#364×C₂₇₁	2
#2213	271

subgroup

index

#364×C₂₇₁

271

Subgroups — #46838441

Conjugacy classes of subgroups

Complements

Maximal subgroups