Group #46782604 — subgroups

Subgroups — #46782604

Order 33664.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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×C₄⋊(C₂₆₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#22554381	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#30681693	C₂	1	2	yes	maximal
5	#376×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₆₃⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₂₆₃⋊C₈	C₂²	1	2	yes	—
8	C₄⋊(C₂₆₃⋊C₈)	C₄	1	8	yes	complemented
9	#22554381	—	2	2	no	—
10	C₄×C₂₆₃⋊C₈	—	2	2	no	—
11	C₂×C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
14	#95×C₂₆₃	C₄	1	4	yes	—
15	C₂×C₂₆₃⋊C₈	—	2	2	no	—
16	C₂×C₂₆₃⋊C₈	—	2	2	no	—
17	C₂×C₂₆₃⋊C₈	—	2	2	no	—
18	C₂×C₂₆₃⋊C₈	—	2	2	no	—
19	C₂×C₂₆₃⋊C₈	C₂×C₄	1	4	yes	—
20	C₂×C₂₆₃⋊C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₂₆₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₆₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₆₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
25	C₄²×C₂₆₃	—	2	2	no	—
26	C₄²×C₂₆₃	C₂×C₄	1	4	yes	—
27	C₄²×C₂₆₃	D₄	1	2	yes	—
28	C₄²×C₂₆₃	Q₈	1	2	yes	—
29	C₂₆₃⋊C₈	—	2	4	no	—
30	C₂₆₃⋊C₈	—	2	4	no	—
31	C₂³×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₆₃	—	2	4	no	—
33	C₂×C₄×C₂₆₃	—	2	4	no	—
34	C₂×C₄×C₂₆₃	—	2	2	no	—
35	C₂×C₄×C₂₆₃	—	2	4	no	—
36	C₂×C₄×C₂₆₃	C₄²	1	4	yes	—
37	C₂×C₄×C₂₆₃	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₂₆₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₆₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₆₃	#41	1	2	yes	—
42	C₂²×C₂₆₃	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₂₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₂₆₃	#117	1	1	yes	characteristic
45	C₂²×C₂₆₃	#131	1	4	yes	—
46	C₄×C₂₆₃	—	2	4	no	—
47	C₄×C₂₆₃	—	2	4	no	—
48	C₄×C₂₆₃	—	4	2	no	—
49	C₄×C₂₆₃	C₄×D₄	1	2	yes	—
50	C₄×C₂₆₃	C₄×Q₈	1	2	yes	—
51	C₂×C₂₆₃	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₂₆₃	#405	1	2	yes	—
53	C₂×C₂₆₃	#424	1	2	yes	—
54	C₂×C₂₆₃	#433	1	2	yes	—
55	C₂₆₃	#1768	1	1	yes	characteristic, complemented, Sylow
56	#1768	—	263	1	no	maximal, Sylow
57	C₂×C₄⋊C₈	—	263	2	no	—
58	C₂×#97	—	263	1	no	—
59	C₂×C₄×C₈	—	263	1	no	—
60	#376	D₂₆₃	1	1	yes	characteristic
61	#336	—	263	2	no	—
62	C₂²×C₈	—	263	2	no	—
63	C₂²×C₈	—	263	2	no	—
64	C₂×C₄²	C₂×D₂₆₃	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₂₆₃	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₂₆₃	1	1	yes	characteristic
67	C₄⋊C₈	—	263	8	no	—
68	#97	—	526	2	no	—
69	C₄×C₈	—	526	2	no	—
70	#95	Dic₂₆₃	1	4	yes	—
71	C₂²×C₄	C₂×Dic₂₆₃	1	2	yes	—
72	C₂²×C₄	C₂×Dic₂₆₃	1	2	yes	—
73	C₂²×C₄	C₂×Dic₂₆₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
75	C₂×C₈	—	263	4	no	—
76	C₂×C₈	—	263	4	no	—
77	C₂×C₈	—	526	2	no	—
78	C₂×C₈	—	526	2	no	—
79	C₂×C₈	—	526	2	no	—
80	C₂×C₈	—	526	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	C₄×D₂₆₃	1	4	yes	—
83	C₄²	D₁₀₅₂	1	2	yes	—
84	C₄²	Dic₅₂₆	1	2	yes	—
85	C₂³	C₂²×Dic₂₆₃	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×Dic₂₆₃	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₂₆₃	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₂₆₃	1	2	yes	—
93	C₂×C₄	C₂×D₁₀₅₂	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₅₂₆	1	1	yes	characteristic
95	C₂×C₄	#16862376	1	2	yes	—
96	C₈	—	526	4	no	—
97	C₈	—	526	4	no	—
98	C₂²	C₂×C₄×Dic₂₆₃	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₂₆₃	1	1	yes	characteristic
100	C₂²	#22554429	1	1	yes	characteristic
101	C₂²	#22554477	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₁₀₅₂	1	2	yes	—
106	C₄	C₄×Dic₅₂₆	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₂₆₃	1	1	yes	characteristic
108	C₂	#30681894	1	2	yes	—
109	C₂	#30681965	1	2	yes	—
110	C₂	#30681999	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₆₃⋊C₈)

C₂

yes

maximal

C₂×#22554381

C₂

yes

maximal, characteristic

C₂×C₄×C₂₆₃⋊C₈

C₂

yes

maximal, characteristic

#30681693

C₂

yes

maximal

#376×C₂₆₃

C₂

yes

maximal, characteristic

C₂²×C₂₆₃⋊C₈

C₂²

yes

—

C₂²×C₂₆₃⋊C₈

C₂²

yes

—

C₄⋊(C₂₆₃⋊C₈)

C₄

yes

complemented

#22554381

—

C₄×C₂₆₃⋊C₈

—

C₂×C₄²×C₂₆₃

C₂²

yes

characteristic

C₂×C₄²×C₂₆₃

C₂²

yes

characteristic

C₂×C₄²×C₂₆₃

C₂²

yes

characteristic

#95×C₂₆₃

C₄

yes

—

C₂×C₂₆₃⋊C₈

—

C₂×C₂₆₃⋊C₈

—

C₂×C₂₆₃⋊C₈

—

C₂×C₂₆₃⋊C₈

—

C₂×C₂₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₆₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₂₆₃

C₂×C₄

yes

—

C₂²×C₄×C₂₆₃

C₂×C₄

yes

—

C₂²×C₄×C₂₆₃

C₂×C₄

yes

—

C₂²×C₄×C₂₆₃

C₂³

yes

characteristic

C₄²×C₂₆₃

—

C₄²×C₂₆₃

C₂×C₄

yes

—

C₄²×C₂₆₃

D₄

yes

—

C₄²×C₂₆₃

Q₈

yes

—

C₂₆₃⋊C₈

—

C₂₆₃⋊C₈

—

C₂³×C₂₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

C₄²

yes

—

C₂×C₄×C₂₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₆₃

C₂²×C₄

yes

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₃

#41

yes

—

C₂²×C₂₆₃

C₂×C₄²

yes

characteristic

C₂²×C₂₆₃

C₂×C₄⋊C₄

yes

C₂²×C₂₆₃

yes

C₂²×C₂₆₃

yes

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

C₄×D₄

yes

—

C₄×C₂₆₃

C₄×Q₈

yes

—

C₂×C₂₆₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₆₃

#405

yes

—

C₂×C₂₆₃

#424

yes

—

C₂×C₂₆₃

#433

yes

—

C₂₆₃

#1768

yes

characteristic, complemented, Sylow

#1768

—

263

maximal, Sylow

C₂×C₄⋊C₈

—

263

—

C₂×#97

—

263

—

C₂×C₄×C₈

—

263

—

#376

D₂₆₃

yes

characteristic

#336

—

263

—

C₂²×C₈

—

263

—

C₂²×C₈

—

263

—

C₂×C₄²

C₂×D₂₆₃

yes

characteristic

C₂×C₄²

C₂×D₂₆₃

yes

characteristic

C₂×C₄²

C₂×D₂₆₃

yes

characteristic

C₄⋊C₈

—

263

—

#97

—

526

—

C₄×C₈

—

526

—

#95

Dic₂₆₃

yes

—

C₂²×C₄

C₂×Dic₂₆₃

yes

—

C₂²×C₄

C₂×Dic₂₆₃

yes

—

C₂²×C₄

C₂×Dic₂₆₃

yes

—

C₂²×C₄

C₂²×D₂₆₃

yes

characteristic

C₂×C₈

—

263

—

C₂×C₈

—

263

—

C₂×C₈

—

526

—

C₂×C₈

—

526

—

C₂×C₈

—

526

—

C₂×C₈

—

526

—

C₄²

—

C₄²

C₄×D₂₆₃

yes

—

C₄²

D₁₀₅₂

yes

—

C₄²

Dic₅₂₆

yes

—

C₂³

C₂²×Dic₂₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₆₃

yes

—

C₂×C₄

C₄⋊Dic₂₆₃

yes

—

C₂×C₄

C₂×C₄×D₂₆₃

yes

—

C₂×C₄

C₂×D₁₀₅₂

yes

characteristic

C₂×C₄

C₂×Dic₅₂₆

yes

characteristic

C₂×C₄

#16862376

yes

—

C₈

—

526

—

C₈

—

526

—

C₂²

C₂×C₄×Dic₂₆₃

yes

characteristic

C₂²

C₂×C₄⋊Dic₂₆₃

yes

characteristic

100

C₂²

#22554429

yes

characteristic

101

C₂²

#22554477

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₁₀₅₂

yes

—

106

C₄

C₄×Dic₅₂₆

yes

—

107

C₂

C₄×C₄⋊Dic₂₆₃

yes

characteristic

108

C₂

#30681894

yes

—

109

C₂

#30681965

yes

—

110

C₂

#30681999

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₆₃
commutator	C₂²×C₂₆₃	C₂×C₄²
Fitting	#376×C₂₆₃	C₂
socle, prime radical	C₂³×C₂₆₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₆₃	C₂³
hypercenter	#376	D₂₆₃
hypocenter	C₂₆₃	#1768
LCS	G ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₆₃

commutator

C₂²×C₂₆₃

C₂×C₄²

Fitting

#376×C₂₆₃

C₂

socle, prime radical

C₂³×C₂₆₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₆₃

C₂³

D₂₆₃

C₂₆₃

G ▷ C₂²×C₂₆₃ ▷ C₂₆₃

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
8. C₄⋊(C₂₆₃⋊C₈)	C₄	semidirect
55. C₂₆₃	#1768	semidirect

H ≅ G/N

type

8. C₄⋊(C₂₆₃⋊C₈)

C₄

semidirect

55. C₂₆₃

#1768

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₆₃⋊C₈)	2
C₂×#22554381	2
C₂×C₄×C₂₆₃⋊C₈	2
#30681693	2
#376×C₂₆₃	2
#1768	263

subgroup

index

C₂×C₄⋊(C₂₆₃⋊C₈)

C₂×#22554381

C₂×C₄×C₂₆₃⋊C₈

#30681693

#376×C₂₆₃

#1768

263

Subgroups — #46782604

Conjugacy classes of subgroups

Complements

Maximal subgroups