Group #35320816 — subgroups

Subgroups — #35320816

Order 21248.

414 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 275 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⋊(C₈₃⋊C₁₆)	C₂	1	6	yes	maximal
2	C₄²×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈₃⋊C₁₆	C₂²	1	4	yes	—
4	C₄⋊(C₈₃⋊C₁₆)	—	2	12	no	—
5	C₂×C₄×C₈×C₈₃	C₄	1	3	yes	—
6	C₂×C₄×C₈×C₈₃	C₂²	1	3	yes	—
7	C₄³×C₈₃	C₄	1	1	yes	characteristic
8	C₂×C₈₃⋊C₁₆	—	2	12	no	—
9	C₂²×C₈×C₈₃	C₂×C₄	1	3	yes	—
10	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₈₃	C₈	1	3	yes	—
12	C₂×C₄²×C₈₃	C₈	1	1	yes	characteristic
13	C₂×C₄²×C₈₃	C₂×C₄	1	3	yes	—
14	C₄×C₈×C₈₃	—	2	6	no	—
15	C₄×C₈×C₈₃	D₄	1	6	yes	—
16	C₄×C₈×C₈₃	Q₈	1	6	yes	—
17	C₈₃⋊C₁₆	—	4	4	no	—
18	C₂²×C₄×C₈₃	C₂×C₈	1	3	yes	—
19	C₂²×C₄×C₈₃	C₂×C₈	1	3	yes	—
20	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₈×C₈₃	—	2	6	no	—
22	C₂×C₈×C₈₃	C₄⋊C₄	1	6	yes	—
23	C₂×C₈×C₈₃	C₂×D₄	1	3	yes	—
24	C₂×C₈×C₈₃	C₂×Q₈	1	3	yes	—
25	C₄²×C₈₃	—	2	6	no	—
26	C₄²×C₈₃	C₁₆	1	4	yes	complemented
27	C₄²×C₈₃	C₄⋊C₄	1	6	yes	—
28	C₄²×C₈₃	M₁₆	1	6	yes	—
29	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
30	C₂×C₄×C₈₃	—	2	6	no	—
31	C₂×C₄×C₈₃	C₄⋊C₈	1	6	yes	—
32	C₂×C₄×C₈₃	C₄⋊C₈	1	6	yes	—
33	C₂×C₄×C₈₃	C₂×C₁₆	1	6	yes	—
34	C₂×C₄×C₈₃	M₃₂	1	6	yes	—
35	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
36	C₂×C₄×C₈₃	C₂×M₁₆	1	3	yes	—
37	C₈×C₈₃	—	2	6	no	—
38	C₈×C₈₃	#127	1	1	yes	characteristic
39	C₈×C₈₃	C₄⋊Q₈	1	3	yes	—
40	C₂²×C₈₃	C₂×C₄⋊C₈	1	3	yes	—
41	C₂²×C₈₃	C₂²×C₁₆	1	1	yes	characteristic
42	C₂²×C₈₃	C₂×M₃₂	1	3	yes	—
43	C₄×C₈₃	—	2	6	no	—
44	C₄×C₈₃	C₄⋊C₁₆	1	12	yes	complemented
45	C₄×C₈₃	#384	1	1	yes	characteristic
46	C₄×C₈₃	#423	1	3	yes	—
47	#12898	—	83	1	no	maximal, Sylow
48	C₂×C₈₃	#1842	1	1	yes	characteristic
49	C₂×C₈₃	C₂×C₄⋊C₁₆	1	3	yes	—
50	C₂×C₈₃	#2150	1	3	yes	—
51	C₂×C₄⋊C₁₆	—	83	6	no	—
52	C₄²×C₈	D₈₃	1	1	yes	characteristic
53	C₈₃	#12898	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₁₆	—	83	4	no	—
55	C₂×C₄×C₈	Dic₈₃	1	3	yes	—
56	C₂×C₄×C₈	C₂×D₈₃	1	3	yes	—
57	C₄³	Dic₈₃	1	1	yes	characteristic
58	C₄⋊C₁₆	—	166	12	no	—
59	C₂²×C₈	C₂×Dic₈₃	1	3	yes	—
60	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
61	C₂×C₄²	C₈₃⋊C₈	1	3	yes	—
62	C₂×C₄²	C₈₃⋊C₈	1	1	yes	characteristic
63	C₂×C₄²	C₂×Dic₈₃	1	3	yes	—
64	C₂×C₁₆	—	166	12	no	—
65	C₄×C₈	—	2	6	no	—
66	C₄×C₈	Dic₁₆₆	1	6	yes	—
67	C₄×C₈	D₃₃₂	1	6	yes	—
68	C₂²×C₄	C₂×C₈₃⋊C₈	1	3	yes	—
69	C₂²×C₄	C₂×C₈₃⋊C₈	1	3	yes	—
70	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	C₄⋊Dic₈₃	1	6	yes	—
73	C₂×C₈	C₂×D₃₃₂	1	3	yes	—
74	C₂×C₈	C₂×Dic₁₆₆	1	3	yes	—
75	C₄²	—	2	6	no	—
76	C₄²	C₈₃⋊C₁₆	1	4	yes	complemented
77	C₄²	C₄⋊Dic₈₃	1	6	yes	—
78	C₄²	#13050574	1	6	yes	—
79	C₁₆	—	332	4	no	—
80	C₂³	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	6	yes	—
83	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	6	yes	—
84	C₂×C₄	C₂×C₈₃⋊C₁₆	1	6	yes	—
85	C₂×C₄	#14581732	1	6	yes	—
86	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
87	C₂×C₄	C₂×#13050574	1	3	yes	—
88	C₈	—	2	6	no	—
89	C₈	#14581789	1	1	yes	characteristic
90	C₈	#14581792	1	3	yes	—
91	C₂²	C₂×C₄⋊(C₈₃⋊C₈)	1	3	yes	—
92	C₂²	C₂²×C₈₃⋊C₁₆	1	1	yes	characteristic
93	C₂²	C₂×#14581732	1	3	yes	—
94	C₄	—	2	6	no	—
95	C₄	C₄⋊(C₈₃⋊C₁₆)	1	12	yes	complemented
96	C₄	#19127571	1	1	yes	characteristic
97	C₄	#19127725	1	3	yes	—
98	C₂	#24907615	1	1	yes	characteristic
99	C₂	C₂×C₄⋊(C₈₃⋊C₁₆)	1	3	yes	—
100	C₂	#24909219	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₈₃⋊C₁₆)

C₂

yes

maximal

C₄²×C₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₁₆

C₂²

yes

—

C₄⋊(C₈₃⋊C₁₆)

—

C₂×C₄×C₈×C₈₃

C₄

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

C₄³×C₈₃

C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

—

C₂²×C₈×C₈₃

C₂×C₄

yes

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄²×C₈₃

C₈

yes

—

C₂×C₄²×C₈₃

C₈

yes

characteristic

C₂×C₄²×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

C₂×C₈

yes

—

C₂²×C₄×C₈₃

C₂×C₈

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×Q₈

yes

—

C₄²×C₈₃

—

C₄²×C₈₃

C₁₆

yes

complemented

C₄²×C₈₃

C₄⋊C₄

yes

—

C₄²×C₈₃

M₁₆

yes

—

C₂³×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₈

yes

—

C₂×C₄×C₈₃

C₄⋊C₈

yes

—

C₂×C₄×C₈₃

C₂×C₁₆

yes

—

C₂×C₄×C₈₃

M₃₂

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×M₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

#127

yes

characteristic

C₈×C₈₃

C₄⋊Q₈

yes

—

C₂²×C₈₃

C₂×C₄⋊C₈

yes

—

C₂²×C₈₃

C₂²×C₁₆

yes

characteristic

C₂²×C₈₃

C₂×M₃₂

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₄⋊C₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

—

C₂×C₈₃

yes

C₂×C₈₃

C₂×C₄⋊C₁₆

yes

—

C₂×C₈₃

#2150

yes

—

C₂×C₄⋊C₁₆

—

C₄²×C₈

D₈₃

yes

characteristic

C₈₃

#12898

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₈₃

yes

—

C₂×C₄×C₈

C₂×D₈₃

yes

—

C₄³

Dic₈₃

yes

characteristic

C₄⋊C₁₆

—

166

—

C₂²×C₈

C₂×Dic₈₃

yes

—

C₂²×C₈

C₂²×D₈₃

yes

characteristic

C₂×C₄²

C₈₃⋊C₈

yes

—

C₂×C₄²

C₈₃⋊C₈

yes

characteristic

C₂×C₄²

C₂×Dic₈₃

yes

—

C₂×C₁₆

—

166

—

C₄×C₈

—

C₄×C₈

Dic₁₆₆

yes

—

C₄×C₈

D₃₃₂

yes

—

C₂²×C₄

C₂×C₈₃⋊C₈

yes

—

C₂²×C₄

C₂×C₈₃⋊C₈

yes

—

C₂²×C₄

C₂²×Dic₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

C₂×D₃₃₂

yes

—

C₂×C₈

C₂×Dic₁₆₆

yes

—

C₄²

—

C₄²

C₈₃⋊C₁₆

yes

complemented

C₄²

C₄⋊Dic₈₃

yes

—

C₄²

#13050574

yes

—

C₁₆

—

332

—

C₂³

C₂²×C₈₃⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊(C₈₃⋊C₈)

yes

—

C₂×C₄

C₄⋊(C₈₃⋊C₈)

yes

—

C₂×C₄

C₂×C₈₃⋊C₁₆

yes

—

C₂×C₄

#14581732

yes

—

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

—

C₂×C₄

C₂×#13050574

yes

—

C₈

—

C₈

#14581789

yes

characteristic

C₈

#14581792

yes

—

C₂²

C₂×C₄⋊(C₈₃⋊C₈)

yes

—

C₂²

C₂²×C₈₃⋊C₁₆

yes

characteristic

C₂²

C₂×#14581732

yes

—

C₄

—

C₄

C₄⋊(C₈₃⋊C₁₆)

yes

C₄

yes

C₄

yes

—

C₂

#24907615

yes

characteristic

C₂

C₂×C₄⋊(C₈₃⋊C₁₆)

yes

—

100

C₂

#24909219

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂²×C₁₆
Fitting	C₄²×C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₄²×C₈	D₈₃
hypocenter	C₈₃	#12898
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₈₃

commutator

C₂²×C₈₃

C₂²×C₁₆

Fitting

C₄²×C₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂²×C₈

cosocle

C₂²×C₈×C₈₃

C₂³

hypercenter

C₄²×C₈

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
26. C₄²×C₈₃	C₁₆	semidirect
44. C₄×C₈₃	C₄⋊C₁₆	semidirect
53. C₈₃	#12898	semidirect
76. C₄²	C₈₃⋊C₁₆	semidirect
95. C₄	C₄⋊(C₈₃⋊C₁₆)	semidirect

H ≅ G/N

type

26. C₄²×C₈₃

C₁₆

semidirect

44. C₄×C₈₃

C₄⋊C₁₆

semidirect

53. C₈₃

#12898

semidirect

76. C₄²

C₈₃⋊C₁₆

semidirect

95. C₄

C₄⋊(C₈₃⋊C₁₆)

semidirect

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

subgroup	index
C₂×C₄⋊(C₈₃⋊C₁₆)	2
C₄²×C₈×C₈₃	2
#12898	83

subgroup

index

C₂×C₄⋊(C₈₃⋊C₁₆)

C₄²×C₈×C₈₃

#12898

Subgroups — #35320816

Conjugacy classes of subgroups

Complements

Maximal subgroups