Group #35213674 — subgroups

Subgroups — #35213674

Order 20864.

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

normal subgroups only

This group has 206 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	#23782395	C₂	1	1	yes	maximal, characteristic
2	#23781957	C₂	1	2	yes	maximal
3	#23781414	C₂	1	1	yes	maximal, characteristic
4	#23781397	C₂	1	2	yes	maximal
5	#494×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#19104734	—	2	2	no	—
7	#19104733	—	2	2	no	—
8	C₄×Dic₃₂₆	C₂²	1	2	yes	—
9	#19104700	—	2	2	no	—
10	#19104668	—	2	2	no	—
11	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
12	#19104658	—	2	2	no	—
13	C₂×Q₁₆×C₁₆₃	—	2	2	no	—
14	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₆₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₂₆	—	2	2	no	—
18	C₂×Dic₃₂₆	D₄	1	2	yes	—
19	C₂×C₁₆₃⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₆₃	D₄	1	2	yes	—
21	Dic₁₆₃⋊C₄	—	2	4	no	—
22	Dic₁₆₃⋊C₄	—	2	2	no	—
23	C₄×Dic₁₆₃	—	2	2	no	—
24	C₄×Dic₁₆₃	—	2	2	no	—
25	C₂×Q₈×C₁₆₃	—	2	1	no	—
26	C₂×Q₈×C₁₆₃	D₄	1	2	yes	—
27	Q₁₆×C₁₆₃	—	4	4	no	—
28	C₂×C₈×C₁₆₃	—	2	1	no	—
29	C₄⋊C₄×C₁₆₃	—	2	2	no	—
30	C₄⋊C₄×C₁₆₃	—	2	2	no	—
31	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
32	Dic₃₂₆	—	2	2	no	—
33	Dic₃₂₆	—	8	2	no	—
34	Dic₃₂₆	Q₁₆	1	4	yes	complemented
35	C₂×Dic₁₆₃	—	2	2	no	—
36	C₂×Dic₁₆₃	—	2	2	no	—
37	C₁₆₃⋊C₈	—	4	2	no	—
38	Q₈×C₁₆₃	—	4	2	no	—
39	Q₈×C₁₆₃	—	4	2	no	—
40	C₂×C₄×C₁₆₃	—	2	2	no	—
41	C₂×C₄×C₁₆₃	—	4	1	no	—
42	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₆₃	—	2	2	no	—
45	C₂²×C₁₆₃	#120	1	1	yes	characteristic
46	C₄×C₁₆₃	—	2	1	no	—
47	C₄×C₁₆₃	—	4	2	no	—
48	C₄×C₁₆₃	—	8	1	no	—
49	C₄×C₁₆₃	C₂×Q₁₆	1	2	yes	—
50	C₄×C₁₆₃	#137	1	2	yes	—
51	Dic₁₆₃	—	2	4	no	—
52	Dic₁₆₃	—	4	2	no	—
53	C₂×C₁₆₃	#451	1	2	yes	—
54	C₂×C₁₆₃	#453	1	1	yes	characteristic
55	C₁₆₃	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	163	1	no	maximal, Sylow
57	Q₈²	—	163	1	no	—
58	#494	D₁₆₃	1	1	yes	characteristic
59	#477	—	163	2	no	—
60	#333	—	163	1	no	—
61	Q₈⋊C₈	—	163	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	326	2	no	—
64	C₄⋊Q₈	—	326	2	no	—
65	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₁₆₃	1	1	yes	characteristic
67	C₄×Q₈	—	326	2	no	—
68	C₄×Q₈	—	163	2	no	—
69	#107	—	326	2	no	—
70	C₄⋊C₈	—	163	2	no	—
71	#103	—	326	2	no	—
72	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	326	2	no	—
75	C₂×Q₈	—	163	2	no	—
76	C₂×Q₈	C₂²⋊D₁₆₃	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	326	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	326	4	no	—
83	C₄⋊C₄	—	326	2	no	—
84	C₄⋊C₄	—	163	2	no	—
85	C₄²	—	326	2	no	—
86	C₄²	—	326	2	no	—
87	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	1304	2	no	—
91	Q₈	—	326	2	no	—
92	Q₈	—	163	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	326	2	no	—
96	C₂×C₄	—	326	2	no	—
97	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	652	2	no	—
101	C₂²	#19104703	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	326	4	no	—
106	C₄	—	652	2	no	—
107	C₄	Q₁₆×D₁₆₃	1	2	yes	—
108	C₄	#19104767	1	2	yes	—
109	C₂	#23781819	1	2	yes	—
110	C₂	#23781832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23782395

C₂

yes

maximal, characteristic

#23781957

C₂

yes

maximal

#23781414

C₂

yes

maximal, characteristic

#23781397

C₂

yes

maximal

#494×C₁₆₃

C₂

yes

maximal, characteristic

#19104734

—

#19104733

—

C₄×Dic₃₂₆

C₂²

yes

—

#19104700

—

#19104668

—

C₄⋊(C₁₆₃⋊C₈)

C₂²

yes

—

#19104658

—

C₂×Q₁₆×C₁₆₃

—

C₄⋊Q₈×C₁₆₃

C₂²

yes

characteristic

C₄⋊Q₈×C₁₆₃

C₂²

yes

characteristic

C₄×C₈×C₁₆₃

C₂²

yes

characteristic

C₂×Dic₃₂₆

—

C₂×Dic₃₂₆

D₄

yes

—

C₂×C₁₆₃⋊C₈

—

C₄⋊Dic₁₆₃

D₄

yes

—

Dic₁₆₃⋊C₄

—

Dic₁₆₃⋊C₄

—

C₄×Dic₁₆₃

—

C₄×Dic₁₆₃

—

C₂×Q₈×C₁₆₃

—

C₂×Q₈×C₁₆₃

D₄

yes

—

Q₁₆×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂³

yes

characteristic

Dic₃₂₆

—

Dic₃₂₆

—

Dic₃₂₆

Q₁₆

yes

complemented

C₂×Dic₁₆₃

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

Q₈×C₁₆₃

—

Q₈×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₈×C₁₆₃

—

C₂²×C₁₆₃

#120

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂×Q₁₆

yes

—

C₄×C₁₆₃

#137

yes

—

Dic₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

#451

yes

—

C₂×C₁₆₃

#453

yes

characteristic

C₁₆₃

#1633

yes

characteristic, complemented, Sylow

#1633

—

163

maximal, Sylow

Q₈²

—

163

—

#494

D₁₆₃

yes

characteristic

#477

—

163

—

#333

—

163

—

Q₈⋊C₈

—

163

—

C₂×Q₁₆

—

C₄⋊Q₈

—

326

—

C₄⋊Q₈

—

326

—

C₄⋊Q₈

C₂×D₁₆₃

yes

characteristic

C₄⋊Q₈

C₂×D₁₆₃

yes

characteristic

C₄×Q₈

—

326

—

C₄×Q₈

—

163

—

#107

—

326

—

C₄⋊C₈

—

163

—

#103

—

326

—

C₄×C₈

C₂×D₁₆₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

326

—

C₂×Q₈

—

163

—

C₂×Q₈

C₂²⋊D₁₆₃

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

326

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

326

—

C₄⋊C₄

—

163

—

C₄²

—

326

—

C₄²

—

326

—

C₄²

C₂²×D₁₆₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

1304

—

Q₈

—

326

—

Q₈

—

163

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

326

—

C₂×C₄

—

326

—

C₂×C₄

D₄×D₁₆₃

yes

—

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

C₈

—

100

C₈

—

652

—

101

C₂²

#19104703

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

326

—

106

C₄

—

652

—

107

C₄

Q₁₆×D₁₆₃

yes

—

108

C₄

#19104767

yes

—

109

C₂

#23781819

yes

—

110

C₂

#23781832

yes

characteristic

	subgroup	G/H
center	C₂²	#19104703
commutator, cosocle	C₄²×C₁₆₃	C₂³
Frattini	C₄²	C₂²×D₁₆₃
Fitting	#494×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	#120
hypercenter	#494	D₁₆₃
hypocenter	C₁₆₃	#1633
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#19104703

commutator, cosocle

C₄²×C₁₆₃

C₂³

Frattini

C₄²

C₂²×D₁₆₃

Fitting

#494×C₁₆₃

C₂

C₂²×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₃₂₆	Q₁₆	semidirect
55. C₁₆₃	#1633	semidirect

H ≅ G/N

type

34. Dic₃₂₆

Q₁₆

semidirect

55. C₁₆₃

#1633

semidirect

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

subgroup	index
#23782395	2
#23781957	2
#23781414	2
#23781397	2
#494×C₁₆₃	2
#1633	163

subgroup

index

#494×C₁₆₃

163

Subgroups — #35213674

Conjugacy classes of subgroups

Complements

Maximal subgroups