Group #30595281 — subgroups

Subgroups — #30595281

Order 15616.

186 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 22 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	#21460951	C₂	1	2	yes	maximal, complemented
2	#21445433	C₂	1	1	yes	maximal, characteristic
3	#16840667	—	2	2	no	—
4	#16840139	C₂²	1	1	yes	characteristic
5	#16840138	—	2	2	no	—
6	C₈×#92418	C₄	1	1	yes	characteristic, complemented
7	C₆₁×#493	C₄	1	1	yes	characteristic, complemented
8	#14519836	—	4	2	no	—
9	C₂×D₄₈₈	—	4	1	no	—
10	C₂×#11758199	—	4	4	no	—
11	#14519774	D₄	1	2	yes	—
12	C₄×#92418	—	2	1	no	—
13	C₂×D₈×C₆₁	—	4	1	no	—
14	#127×C₆₁	—	2	1	no	—
15	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×D₄×C₆₁	—	4	2	no	—
17	C₂×D₄×C₆₁	—	4	1	no	—
18	C₂×D₂₄₄	—	2	2	no	—
19	C₂×D₂₄₄	—	4	2	no	—
20	C₂²⋊#6923	—	8	2	no	—
21	D₈×C₆₁	—	8	2	no	—
22	C₂×C₈×C₆₁	—	2	1	no	—
23	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
24	#11758199	—	8	8	no	—
25	C₂×#92418	—	2	2	no	—
26	D₄₈₈	—	8	2	no	—
27	C₂³×C₆₁	—	8	1	no	—
28	C₂²×D₆₁	—	4	2	no	—
29	C₂×#6923	—	16	2	no	—
30	D₄×C₆₁	—	8	4	no	—
31	D₄×C₆₁	—	8	2	no	—
32	C₂×C₄×C₆₁	—	2	1	no	—
33	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
34	D₂₄₄	—	8	2	no	—
35	D₂₄₄	—	8	4	no	—
36	C₈×C₆₁	—	2	2	no	—
37	#92418	—	2	4	no	—
38	#7529	—	61	1	no	maximal, Sylow
39	C₂²×C₆₁	—	16	2	no	—
40	C₂²×C₆₁	—	16	1	no	—
41	C₂²×C₆₁	#353	1	1	yes	characteristic
42	C₂×D₆₁	—	8	2	no	—
43	C₂×D₆₁	—	16	2	no	—
44	#6923	—	32	2	no	—
45	C₄×C₆₁	—	2	2	no	—
46	C₄×C₆₁	—	2	1	no	—
47	#1712	—	61	1	no	—
48	#1408	—	61	2	no	—
49	C₂×C₆₁	—	2	1	no	—
50	C₂×C₆₁	—	32	1	no	—
51	C₂×C₆₁	#1408	1	1	yes	characteristic
52	D₆₁	—	16	2	no	—
53	#493	—	61	1	no	—
54	#493	—	122	2	no	—
55	#493	#6923	1	1	yes	characteristic, complemented
56	#353	—	122	2	no	—
57	C₈²	—	61	1	no	—
58	C₆₁	#7529	1	1	yes	characteristic, complemented, Sylow
59	C₂×D₈	—	4	1	no	—
60	C₂×D₈	—	244	1	no	—
61	C₂×D₈	—	244	4	no	—
62	#127	—	2	1	no	—
63	#127	—	61	2	no	—
64	#99	—	244	2	no	—
65	C₄×C₈	—	122	1	no	—
66	C₄×C₈	C₂×#6923	1	1	yes	characteristic
67	C₂×D₄	—	4	2	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	122	2	no	—
70	C₂×D₄	—	244	2	no	—
71	D₈	—	8	2	no	—
72	D₈	—	488	2	no	—
73	D₈	—	488	8	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	122	2	no	—
76	C₂²⋊C₄	—	488	2	no	—
77	C₄²	C₂²⋊#6923	1	1	yes	characteristic
78	C₂³	—	8	1	no	—
79	C₂³	—	244	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	488	4	no	—
83	D₄	—	488	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	976	2	no	—
86	C₂×C₄	#14519835	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	—	122	4	no	—
89	C₂²	—	16	2	no	—
90	C₂²	—	16	1	no	—
91	C₂²	—	976	2	no	—
92	C₂²	—	488	2	no	—
93	C₂²	#16840666	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	1952	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	32	1	no	—
99	C₂	—	976	2	no	—
100	C₂	#21460950	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21460951

C₂

yes

maximal, complemented

#21445433

C₂

yes

maximal, characteristic

#16840667

—

#16840139

C₂²

yes

characteristic

#16840138

—

C₈×#92418

C₄

yes

characteristic, complemented

C₆₁×#493

C₄

yes

characteristic, complemented

#14519836

—

C₂×D₄₈₈

—

C₂×#11758199

—

#14519774

D₄

yes

—

C₄×#92418

—

C₂×D₈×C₆₁

—

#127×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂²⋊#6923

—

D₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

#11758199

—

C₂×#92418

—

D₄₈₈

—

C₂³×C₆₁

—

C₂²×D₆₁

—

C₂×#6923

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#99

yes

characteristic

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

#92418

—

#7529

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#353

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

—

#1712

—

#1408

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1408

yes

characteristic

D₆₁

—

#493

—

#493

—

122

—

#493

#6923

yes

characteristic, complemented

#353

—

122

—

C₈²

—

C₆₁

#7529

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

244

—

C₂×D₈

—

244

—

#127

—

#127

—

#99

—

244

—

C₄×C₈

—

122

—

C₄×C₈

C₂×#6923

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

122

—

C₂×D₄

—

244

—

D₈

—

D₈

—

488

—

D₈

—

488

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂²⋊C₄

—

488

—

C₄²

C₂²⋊#6923

yes

characteristic

C₂³

—

C₂³

—

244

—

D₄

—

D₄

—

D₄

—

488

—

D₄

—

488

—

C₂×C₄

—

C₂×C₄

—

976

—

C₂×C₄

#14519835

yes

characteristic

C₈

—

C₈

—

122

—

C₂²

—

C₂²

—

C₂²

—

976

—

C₂²

—

488

—

C₂²

#16840666

yes

characteristic

C₄

—

C₄

—

C₄

—

1952

—

C₂

—

C₂

—

C₂

—

976

—

100

C₂

#21460950

yes

characteristic

	subgroup	G/H
center	C₂	#21460950
commutator	C₄×C₈×C₆₁	C₂×C₄
Frattini	C₄×C₈	C₂×#6923
Fitting	C₆₁×#493	C₄
socle	C₂×C₆₁	#1408
prime radical	#21445433	C₂
cosocle	#16840139	C₂²
hypercenter	#493	#6923
hypocenter	C₆₁	#7529
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#21460950

commutator

C₄×C₈×C₆₁

C₂×C₄

Frattini

C₄×C₈

C₂×#6923

Fitting

C₆₁×#493

C₄

C₂×C₆₁

C₂

C₂²

C₆₁

G ▷ C₄×C₈×C₆₁ ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #21460951	C₂	semidirect
6. C₈×#92418	C₄	semidirect
7. C₆₁×#493	C₄	semidirect
55. #493	#6923	semidirect
58. C₆₁	#7529	semidirect

H ≅ G/N

type

1. #21460951

C₂

semidirect

6. C₈×#92418

C₄

semidirect

7. C₆₁×#493

C₄

semidirect

55. #493

#6923

semidirect

58. C₆₁

#7529

semidirect

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

subgroup	index
#21460951	2
#21445433	2
#7529	61

subgroup

index

#21460951

#21445433

#7529

Subgroups — #30595281

Conjugacy classes of subgroups

Complements

Maximal subgroups