Group #46781826 — subgroups

Subgroups — #46781826

Order 33664.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#30682668	C₂	1	1	yes	maximal, characteristic, complemented
2	#30682312	C₂	1	1	yes	maximal, characteristic
3	#30682139	C₂	1	2	yes	maximal
4	#30681668	C₂	1	2	yes	maximal, complemented
5	#333×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16862365	—	2	2	no	—
7	C₂×#16862363	—	2	2	no	—
8	#22554469	—	2	2	no	—
9	#22554468	—	2	2	no	—
10	#22554467	C₂²	1	1	yes	characteristic
11	C₄×Dic₅₂₆	C₂²	1	2	yes	—
12	#22554435	—	2	2	no	—
13	#22554392	—	2	2	no	—
14	C₄×C₂₆₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₆₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₆₃	C₂²	1	2	yes	—
17	C₂×Dic₅₂₆	—	2	2	no	—
18	C₂×Dic₅₂₆	D₄	1	2	yes	—
19	C₂×D₁₀₅₂	—	2	2	no	—
20	C₂×D₁₀₅₂	—	2	1	no	—
21	#16862365	—	4	4	no	—
22	#16862363	—	4	4	no	—
23	C₂×C₂₆₃⋊C₈	—	2	1	no	—
24	C₄⋊Dic₂₆₃	D₄	1	2	yes	—
25	Dic₂₆₃⋊C₄	—	2	4	no	—
26	Dic₂₆₃⋊C₄	—	2	2	no	—
27	C₄×Dic₂₆₃	—	2	2	no	—
28	C₄×Dic₂₆₃	—	2	2	no	—
29	C₂×Q₈×C₂₆₃	D₄	1	2	yes	—
30	C₂×C₈×C₂₆₃	—	2	2	no	—
31	C₄⋊C₄×C₂₆₃	—	2	2	no	—
32	C₄²×C₂₆₃	C₂³	1	1	yes	characteristic
33	C₂²×D₂₆₃	—	4	1	no	—
34	Dic₅₂₆	—	2	2	no	—
35	Dic₅₂₆	—	8	2	no	—
36	Dic₅₂₆	SD₁₆	1	4	yes	complemented
37	D₁₀₅₂	—	4	2	no	—
38	D₁₀₅₂	—	4	2	no	—
39	D₁₀₅₂	—	4	2	no	—
40	C₂×Dic₂₆₃	—	2	2	no	—
41	C₂×Dic₂₆₃	—	2	2	no	—
42	C₂₆₃⋊C₈	—	2	2	no	—
43	Q₈×C₂₆₃	—	4	2	no	—
44	C₂×C₄×C₂₆₃	—	2	2	no	—
45	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
47	C₈×C₂₆₃	—	4	2	no	—
48	C₂²×C₂₆₃	#120	1	1	yes	characteristic
49	C₂×D₂₆₃	—	8	2	no	—
50	C₂×D₂₆₃	—	8	1	no	—
51	C₄×C₂₆₃	—	2	1	no	—
52	C₄×C₂₆₃	—	4	2	no	—
53	C₄×C₂₆₃	C₂×SD₁₆	1	2	yes	—
54	C₄×C₂₆₃	#137	1	2	yes	—
55	Dic₂₆₃	—	2	4	no	—
56	Dic₂₆₃	—	4	2	no	—
57	C₂×C₂₆₃	#448	1	2	yes	—
58	C₂×C₂₆₃	#453	1	1	yes	characteristic
59	D₂₆₃	—	16	1	no	—
60	C₂₆₃	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	263	1	no	maximal, Sylow
62	Q₈²	—	263	1	no	—
63	#492	—	263	1	no	—
64	#460	—	263	2	no	—
65	#333	D₂₆₃	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	—	263	2	no	—
67	C₂×SD₁₆	—	526	2	no	—
68	C₂×SD₁₆	—	526	2	no	—
69	C₄⋊Q₈	—	526	2	no	—
70	C₄⋊Q₈	—	526	2	no	—
71	C₄⋊Q₈	C₂×D₂₆₃	1	1	yes	characteristic
72	#127	—	263	1	no	—
73	C₄×Q₈	—	263	2	no	—
74	C₄×Q₈	—	526	2	no	—
75	C₄⋊C₈	C₂×D₂₆₃	1	2	yes	—
76	#102	—	526	2	no	—
77	C₄×C₈	—	263	1	no	—
78	C₂×Q₈	—	263	2	no	—
79	C₂×Q₈	—	526	2	no	—
80	C₂×Q₈	D₁₀₅₂	1	2	yes	—
81	C₂×D₄	—	526	2	no	—
82	C₂×D₄	—	526	1	no	—
83	SD₁₆	—	1052	4	no	—
84	SD₁₆	—	1052	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	526	1	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	263	2	no	—
89	C₄⋊C₄	—	526	4	no	—
90	C₄⋊C₄	—	526	2	no	—
91	C₄²	—	526	2	no	—
92	C₄²	—	526	2	no	—
93	C₄²	C₂²×D₂₆₃	1	1	yes	characteristic
94	C₂³	—	1052	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	263	4	no	—
97	Q₈	—	526	2	no	—
98	Q₈	—	2104	2	no	—
99	D₄	—	1052	2	no	—
100	D₄	—	1052	2	no	—
101	D₄	—	1052	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	526	2	no	—
104	C₂×C₄	—	526	2	no	—
105	C₂×C₄	D₄×D₂₆₃	1	2	yes	—
106	C₂×C₄	C₂×D₁₀₅₂	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	526	2	no	—
109	C₂²	—	2104	2	no	—
110	C₂²	—	2104	1	no	—
111	C₂²	#22554440	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	526	4	no	—
115	C₄	—	1052	2	no	—
116	C₄	SD₁₆×D₂₆₃	1	2	yes	—
117	C₄	#22554502	1	2	yes	—
118	C₂	—	4208	1	no	—
119	C₂	#30682078	1	2	yes	—
120	C₂	#30682108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30682668

C₂

yes

maximal, characteristic, complemented

#30682312

C₂

yes

maximal, characteristic

#30682139

C₂

yes

maximal

#30681668

C₂

yes

maximal, complemented

#333×C₂₆₃

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₅₂₆

C₂²

yes

—

#22554435

—

#22554392

—

C₄×C₂₆₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₆₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₃

C₂²

yes

—

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

D₄

yes

—

C₂×D₁₀₅₂

—

C₂×D₁₀₅₂

—

#16862365

—

#16862363

—

C₂×C₂₆₃⋊C₈

—

C₄⋊Dic₂₆₃

D₄

yes

—

Dic₂₆₃⋊C₄

—

Dic₂₆₃⋊C₄

—

C₄×Dic₂₆₃

—

C₄×Dic₂₆₃

—

C₂×Q₈×C₂₆₃

D₄

yes

—

C₂×C₈×C₂₆₃

—

C₄⋊C₄×C₂₆₃

—

C₄²×C₂₆₃

C₂³

yes

characteristic

C₂²×D₂₆₃

—

Dic₅₂₆

—

Dic₅₂₆

—

Dic₅₂₆

SD₁₆

yes

complemented

D₁₀₅₂

—

D₁₀₅₂

—

D₁₀₅₂

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂₆₃⋊C₈

—

Q₈×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

characteristic

C₈×C₂₆₃

—

C₂²×C₂₆₃

#120

yes

characteristic

C₂×D₂₆₃

—

C₂×D₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

C₂×SD₁₆

yes

—

C₄×C₂₆₃

#137

yes

—

Dic₂₆₃

—

Dic₂₆₃

—

C₂×C₂₆₃

#448

yes

—

C₂×C₂₆₃

#453

yes

characteristic

D₂₆₃

—

C₂₆₃

#1623

yes

characteristic, complemented, Sylow

#1623

—

263

maximal, Sylow

Q₈²

—

263

—

#492

—

263

—

#460

—

263

—

#333

D₂₆₃

yes

characteristic, complemented

Q₈⋊C₈

—

263

—

C₂×SD₁₆

—

526

—

C₂×SD₁₆

—

526

—

C₄⋊Q₈

—

526

—

C₄⋊Q₈

—

526

—

C₄⋊Q₈

C₂×D₂₆₃

yes

characteristic

#127

—

263

—

C₄×Q₈

—

263

—

C₄×Q₈

—

526

—

C₄⋊C₈

C₂×D₂₆₃

yes

—

#102

—

526

—

C₄×C₈

—

263

—

C₂×Q₈

—

263

—

C₂×Q₈

—

526

—

C₂×Q₈

D₁₀₅₂

yes

—

C₂×D₄

—

526

—

C₂×D₄

—

526

—

SD₁₆

—

1052

—

SD₁₆

—

1052

—

C₂×C₈

—

C₂×C₈

—

526

—

C₄⋊C₄

—

C₄⋊C₄

—

263

—

C₄⋊C₄

—

526

—

C₄⋊C₄

—

526

—

C₄²

—

526

—

C₄²

—

526

—

C₄²

C₂²×D₂₆₃

yes

characteristic

C₂³

—

1052

—

Q₈

—

Q₈

—

263

—

Q₈

—

526

—

Q₈

—

2104

—

D₄

—

1052

—

100

D₄

—

1052

—

101

D₄

—

1052

—

102

C₂×C₄

—

103

C₂×C₄

—

526

—

104

C₂×C₄

—

526

—

105

C₂×C₄

D₄×D₂₆₃

yes

—

106

C₂×C₄

C₂×D₁₀₅₂

yes

characteristic

107

C₈

—

108

C₈

—

526

—

109

C₂²

—

2104

—

110

C₂²

—

2104

—

111

C₂²

#22554440

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

526

—

115

C₄

—

1052

—

116

C₄

SD₁₆×D₂₆₃

yes

—

117

C₄

#22554502

yes

—

118

C₂

—

4208

—

119

C₂

#30682078

yes

—

120

C₂

#30682108

yes

characteristic

	subgroup	G/H
center	C₂²	#22554440
commutator, cosocle	C₄²×C₂₆₃	C₂³
Frattini	C₄²	C₂²×D₂₆₃
Fitting	#333×C₂₆₃	C₂
socle	C₂²×C₂₆₃	#120
prime radical	#22554467	C₂²
hypercenter	#333	D₂₆₃
hypocenter	C₂₆₃	#1623
LCS	G ▷ C₄²×C₂₆₃ ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#22554440

commutator, cosocle

C₄²×C₂₆₃

C₂³

Frattini

C₄²

C₂²×D₂₆₃

Fitting

#333×C₂₆₃

C₂

C₂²×C₂₆₃

C₂²

D₂₆₃

C₂₆₃

G ▷ C₄²×C₂₆₃ ▷ C₂²×C₂₆₃ ▷ C₂₆₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #30682668	C₂	semidirect
4. #30681668	C₂	semidirect
5. #333×C₂₆₃	C₂	semidirect
36. Dic₅₂₆	SD₁₆	semidirect
60. C₂₆₃	#1623	semidirect
65. #333	D₂₆₃	semidirect

H ≅ G/N

type

1. #30682668

C₂

semidirect

4. #30681668

C₂

semidirect

5. #333×C₂₆₃

C₂

semidirect

36. Dic₅₂₆

SD₁₆

semidirect

60. C₂₆₃

#1623

semidirect

65. #333

D₂₆₃

semidirect

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

subgroup	index
#30682668	2
#30682312	2
#30682139	2
#30681668	2
#333×C₂₆₃	2
#1623	263

subgroup

index

#333×C₂₆₃

263

Subgroups — #46781826

Conjugacy classes of subgroups

Complements

Maximal subgroups