Group #28412222 — subgroups

Subgroups — #28412222

Order 14656.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20300985	C₂	1	1	yes	maximal, characteristic, complemented
2	#20300982	C₂	1	1	yes	maximal, characteristic, complemented
3	#20300973	C₂	1	3	yes	maximal, complemented
4	#20300963	C₂	1	3	yes	maximal, complemented
5	#20300961	C₂	1	3	yes	maximal, complemented
6	#20300958	C₂	1	3	yes	maximal, complemented
7	#126×C₂₂₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₂₂₉	C₂²	1	1	yes	characteristic
9	C₂×D₉₁₆	C₂²	1	3	yes	—
10	C₂×D₉₁₆	C₂²	1	3	yes	—
11	C₂×D₉₁₆	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₂₂₉	C₂²	1	3	yes	—
13	C₂×C₄×D₂₂₉	C₂²	1	3	yes	complemented
14	#16814645	C₂²	1	3	yes	complemented
15	#16814643	C₂²	1	3	yes	—
16	#16814643	C₂²	1	3	yes	—
17	#16814643	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₂₂₉	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₂₂₉	C₂²	1	3	yes	—
20	C₄²×C₂₂₉	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₂₂₉	—	4	3	no	—
22	C₂²×D₂₂₉	C₂³	1	1	yes	characteristic
23	C₂²×D₂₂₉	C₂³	1	1	yes	characteristic
24	C₂²×D₂₂₉	C₂³	1	3	yes	—
25	D₉₁₆	—	4	3	no	—
26	D₉₁₆	—	4	3	no	—
27	D₉₁₆	—	4	3	no	—
28	C₂²⋊D₂₂₉	—	4	3	no	—
29	C₂×Dic₂₂₉	C₂³	1	3	yes	—
30	C₄×D₂₂₉	—	4	3	no	—
31	C₂³×C₂₂₉	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₂₂₉	C₂³	1	3	yes	—
33	C₂×C₄×C₂₂₉	C₂³	1	3	yes	—
34	C₂²×C₂₂₉	—	2	3	no	—
35	C₂²×C₂₂₉	C₂⁴	1	1	yes	characteristic
36	C₂×D₂₂₉	—	2	3	no	—
37	C₂×D₂₂₉	—	2	3	no	—
38	C₂×D₂₂₉	—	2	3	no	—
39	C₂×D₂₂₉	—	2	3	no	—
40	C₂×D₂₂₉	—	2	3	no	—
41	C₂×D₂₂₉	—	4	4	no	—
42	C₄×C₂₂₉	—	2	3	no	—
43	C₄×C₂₂₉	—	2	3	no	—
44	Dic₂₂₉	—	2	3	no	—
45	C₂×C₂₂₉	—	4	1	no	—
46	C₂×C₂₂₉	H(5,𝔽₂)	1	3	yes	—
47	D₂₂₉	—	4	3	no	—
48	D₂₂₉	—	4	1	no	—
49	D₂₂₉	—	4	1	no	—
50	C₂₂₉	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	229	1	no	maximal, Sylow
52	#127	—	229	1	no	—
53	#126	—	229	1	no	—
54	#126	D₂₂₉	1	1	yes	characteristic, complemented
55	#123	—	229	3	no	—
56	#121	—	229	3	no	—
57	#121	—	229	3	no	—
58	#120	—	229	3	no	—
59	C₂⁴	—	229	1	no	—
60	C₂×D₄	—	229	3	no	—
61	C₂×D₄	—	229	3	no	—
62	C₂×D₄	—	229	3	no	—
63	C₂×D₄	—	229	3	no	—
64	C₂²×C₄	—	229	3	no	—
65	C₄⋊C₄	—	229	3	no	—
66	C₄⋊C₄	C₂×D₂₂₉	1	3	yes	complemented
67	C₂²⋊C₄	—	229	3	no	—
68	C₂²⋊C₄	—	229	3	no	—
69	C₂²⋊C₄	—	229	3	no	—
70	C₂²⋊C₄	C₂×D₂₂₉	1	3	yes	—
71	C₄²	C₂×D₂₂₉	1	1	yes	characteristic, complemented
72	C₂³	—	916	3	no	—
73	C₂³	—	229	1	no	—
74	C₂³	—	229	1	no	—
75	C₂³	—	229	3	no	—
76	C₂³	C₂²×D₂₂₉	1	1	yes	characteristic
77	D₄	—	916	3	no	—
78	D₄	—	916	3	no	—
79	D₄	—	916	3	no	—
80	D₄	—	916	3	no	—
81	C₂×C₄	—	916	3	no	—
82	C₂×C₄	—	229	3	no	—
83	C₂×C₄	C₂²×D₂₂₉	1	3	yes	—
84	C₂×C₄	C₂²×D₂₂₉	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	916	4	no	—
87	C₂²	—	458	3	no	—
88	C₂²	—	458	3	no	—
89	C₂²	—	458	3	no	—
90	C₂²	—	458	3	no	—
91	C₂²	—	458	3	no	—
92	C₂²	C₂³×D₂₂₉	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	458	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	916	1	no	—
98	C₂	—	916	1	no	—
99	C₂	—	916	3	no	—
100	C₂	#20301034	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20300985

C₂

yes

maximal, characteristic, complemented

#20300982

C₂

yes

maximal, characteristic, complemented

#20300973

C₂

yes

maximal, complemented

#20300963

C₂

yes

maximal, complemented

#20300961

C₂

yes

maximal, complemented

#20300958

C₂

yes

maximal, complemented

#126×C₂₂₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₂₂₉

C₂²

yes

characteristic

C₂×D₉₁₆

C₂²

yes

—

C₂×D₉₁₆

C₂²

yes

—

C₂×D₉₁₆

C₂²

yes

complemented

C₂×C₂²⋊D₂₂₉

C₂²

yes

—

C₂×C₄×D₂₂₉

C₂²

yes

complemented

#16814645

C₂²

yes

complemented

#16814643

C₂²

yes

—

#16814643

C₂²

yes

—

#16814643

C₂²

yes

complemented

C₄⋊C₄×C₂₂₉

C₂²

yes

complemented

C₂²⋊C₄×C₂₂₉

C₂²

yes

—

C₄²×C₂₂₉

C₂²

yes

characteristic, complemented

C₂²×D₂₂₉

—

C₂²×D₂₂₉

C₂³

yes

characteristic

C₂²×D₂₂₉

C₂³

yes

characteristic

C₂²×D₂₂₉

C₂³

yes

—

D₉₁₆

—

D₉₁₆

—

D₉₁₆

—

C₂²⋊D₂₂₉

—

C₂×Dic₂₂₉

C₂³

yes

—

C₄×D₂₂₉

—

C₂³×C₂₂₉

C₂³

yes

characteristic

C₂×C₄×C₂₂₉

C₂³

yes

—

C₂×C₄×C₂₂₉

C₂³

yes

—

C₂²×C₂₂₉

—

C₂²×C₂₂₉

C₂⁴

yes

characteristic

C₂×D₂₂₉

—

C₂×D₂₂₉

—

C₂×D₂₂₉

—

C₂×D₂₂₉

—

C₂×D₂₂₉

—

C₂×D₂₂₉

—

C₄×C₂₂₉

—

C₄×C₂₂₉

—

Dic₂₂₉

—

C₂×C₂₂₉

—

C₂×C₂₂₉

H(5,𝔽₂)

yes

—

D₂₂₉

—

D₂₂₉

—

D₂₂₉

—

C₂₂₉

#560

yes

characteristic, complemented, Sylow

#560

—

229

maximal, Sylow

#127

—

229

—

#126

—

229

—

#126

D₂₂₉

yes

characteristic, complemented

#123

—

229

—

#121

—

229

—

#121

—

229

—

#120

—

229

—

C₂⁴

—

229

—

C₂×D₄

—

229

—

C₂×D₄

—

229

—

C₂×D₄

—

229

—

C₂×D₄

—

229

—

C₂²×C₄

—

229

—

C₄⋊C₄

—

229

—

C₄⋊C₄

C₂×D₂₂₉

yes

complemented

C₂²⋊C₄

—

229

—

C₂²⋊C₄

—

229

—

C₂²⋊C₄

—

229

—

C₂²⋊C₄

C₂×D₂₂₉

yes

—

C₄²

C₂×D₂₂₉

yes

characteristic, complemented

C₂³

—

916

—

C₂³

—

229

—

C₂³

—

229

—

C₂³

—

229

—

C₂³

C₂²×D₂₂₉

yes

characteristic

D₄

—

916

—

D₄

—

916

—

D₄

—

916

—

D₄

—

916

—

C₂×C₄

—

916

—

C₂×C₄

—

229

—

C₂×C₄

C₂²×D₂₂₉

yes

—

C₂×C₄

C₂²×D₂₂₉

yes

—

C₂²

—

C₂²

—

916

—

C₂²

—

458

—

C₂²

—

458

—

C₂²

—

458

—

C₂²

—

458

—

C₂²

—

458

—

C₂²

C₂³×D₂₂₉

yes

characteristic

C₄

—

C₄

—

C₄

—

458

—

C₂

—

C₂

—

916

—

C₂

—

916

—

C₂

—

916

—

100

C₂

#20301034

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₉
commutator, socle, cosocle	C₂²×C₂₂₉	C₂⁴
Fitting	#126×C₂₂₉	C₂
hypercenter	#126	D₂₂₉
hypocenter	C₂₂₉	#560
LCS	G ▷ C₂²×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₂₉

commutator, socle, cosocle

C₂²×C₂₂₉

C₂⁴

Fitting

#126×C₂₂₉

C₂

D₂₂₉

C₂₂₉

G ▷ C₂²×C₂₂₉ ▷ C₂₂₉

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #20300985	C₂	semidirect
2. #20300982	C₂	semidirect
3. #20300973	C₂	semidirect
4. #20300963	C₂	semidirect
5. #20300961	C₂	semidirect
6. #20300958	C₂	semidirect
7. #126×C₂₂₉	C₂	semidirect
11. C₂×D₉₁₆	C₂²	semidirect
13. C₂×C₄×D₂₂₉	C₂²	semidirect
14. #16814645	C₂²	semidirect
17. #16814643	C₂²	semidirect
18. C₄⋊C₄×C₂₂₉	C₂²	semidirect
20. C₄²×C₂₂₉	C₂²	semidirect
50. C₂₂₉	#560	semidirect
54. #126	D₂₂₉	semidirect
66. C₄⋊C₄	C₂×D₂₂₉	semidirect
71. C₄²	C₂×D₂₂₉	semidirect

H ≅ G/N

type

1. #20300985

C₂

semidirect

2. #20300982

C₂

semidirect

3. #20300973

C₂

semidirect

4. #20300963

C₂

semidirect

5. #20300961

C₂

semidirect

6. #20300958

C₂

semidirect

7. #126×C₂₂₉

C₂

semidirect

11. C₂×D₉₁₆

C₂²

semidirect

13. C₂×C₄×D₂₂₉

C₂²

semidirect

14. #16814645

C₂²

semidirect

17. #16814643

C₂²

semidirect

18. C₄⋊C₄×C₂₂₉

C₂²

semidirect

20. C₄²×C₂₂₉

C₂²

semidirect

50. C₂₂₉

#560

semidirect

54. #126

D₂₂₉

semidirect

66. C₄⋊C₄

C₂×D₂₂₉

semidirect

71. C₄²

C₂×D₂₂₉

semidirect

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

subgroup	index
#20300985	2
#20300982	2
#20300973	2
#20300963	2
#20300961	2
#20300958	2
#126×C₂₂₉	2
#560	229

subgroup

index

#126×C₂₂₉

229

Subgroups — #28412222

Conjugacy classes of subgroups

Complements

Maximal subgroups