Group #28417176 — subgroups

Subgroups — #28417176

Order 14912.

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	#21418532	C₂	1	1	yes	maximal, characteristic, complemented
2	#21418529	C₂	1	1	yes	maximal, characteristic, complemented
3	#21418520	C₂	1	3	yes	maximal, complemented
4	#21418510	C₂	1	3	yes	maximal, complemented
5	#21418508	C₂	1	3	yes	maximal, complemented
6	#21418505	C₂	1	3	yes	maximal, complemented
7	#126×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₂₃₃	C₂²	1	1	yes	characteristic
9	C₂×D₉₃₂	C₂²	1	3	yes	—
10	C₂×D₉₃₂	C₂²	1	3	yes	—
11	C₂×D₉₃₂	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₂₃₃	C₂²	1	3	yes	—
13	C₂×C₄×D₂₃₃	C₂²	1	3	yes	complemented
14	#16836573	C₂²	1	3	yes	complemented
15	#16836571	C₂²	1	3	yes	—
16	#16836571	C₂²	1	3	yes	—
17	#16836571	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₂₃₃	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₂₃₃	C₂²	1	3	yes	—
20	C₄²×C₂₃₃	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₂₃₃	—	4	3	no	—
22	C₂²×D₂₃₃	C₂³	1	1	yes	characteristic
23	C₂²×D₂₃₃	C₂³	1	1	yes	characteristic
24	C₂²×D₂₃₃	C₂³	1	3	yes	—
25	D₉₃₂	—	4	3	no	—
26	D₉₃₂	—	4	3	no	—
27	D₉₃₂	—	4	3	no	—
28	C₂²⋊D₂₃₃	—	4	3	no	—
29	C₂×Dic₂₃₃	C₂³	1	3	yes	—
30	C₄×D₂₃₃	—	4	3	no	—
31	C₂³×C₂₃₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₂₃₃	C₂³	1	3	yes	—
33	C₂×C₄×C₂₃₃	C₂³	1	3	yes	—
34	C₂²×C₂₃₃	—	2	3	no	—
35	C₂²×C₂₃₃	C₂⁴	1	1	yes	characteristic
36	C₂×D₂₃₃	—	2	3	no	—
37	C₂×D₂₃₃	—	2	3	no	—
38	C₂×D₂₃₃	—	2	3	no	—
39	C₂×D₂₃₃	—	2	3	no	—
40	C₂×D₂₃₃	—	2	3	no	—
41	C₂×D₂₃₃	—	4	4	no	—
42	C₄×C₂₃₃	—	2	3	no	—
43	C₄×C₂₃₃	—	2	3	no	—
44	Dic₂₃₃	—	2	3	no	—
45	C₂×C₂₃₃	—	4	1	no	—
46	C₂×C₂₃₃	H(5,𝔽₂)	1	3	yes	—
47	D₂₃₃	—	4	3	no	—
48	D₂₃₃	—	4	1	no	—
49	D₂₃₃	—	4	1	no	—
50	C₂₃₃	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	233	1	no	maximal, Sylow
52	#127	—	233	1	no	—
53	#126	—	233	1	no	—
54	#126	D₂₃₃	1	1	yes	characteristic, complemented
55	#123	—	233	3	no	—
56	#121	—	233	3	no	—
57	#121	—	233	3	no	—
58	#120	—	233	3	no	—
59	C₂⁴	—	233	1	no	—
60	C₂×D₄	—	233	3	no	—
61	C₂×D₄	—	233	3	no	—
62	C₂×D₄	—	233	3	no	—
63	C₂×D₄	—	233	3	no	—
64	C₂²×C₄	—	233	3	no	—
65	C₄⋊C₄	—	233	3	no	—
66	C₄⋊C₄	C₂×D₂₃₃	1	3	yes	complemented
67	C₂²⋊C₄	—	233	3	no	—
68	C₂²⋊C₄	—	233	3	no	—
69	C₂²⋊C₄	—	233	3	no	—
70	C₂²⋊C₄	C₂×D₂₃₃	1	3	yes	—
71	C₄²	C₂×D₂₃₃	1	1	yes	characteristic, complemented
72	C₂³	—	932	3	no	—
73	C₂³	—	233	1	no	—
74	C₂³	—	233	1	no	—
75	C₂³	—	233	3	no	—
76	C₂³	C₂²×D₂₃₃	1	1	yes	characteristic
77	D₄	—	932	3	no	—
78	D₄	—	932	3	no	—
79	D₄	—	932	3	no	—
80	D₄	—	932	3	no	—
81	C₂×C₄	—	932	3	no	—
82	C₂×C₄	—	233	3	no	—
83	C₂×C₄	C₂²×D₂₃₃	1	3	yes	—
84	C₂×C₄	C₂²×D₂₃₃	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	932	4	no	—
87	C₂²	—	466	3	no	—
88	C₂²	—	466	3	no	—
89	C₂²	—	466	3	no	—
90	C₂²	—	466	3	no	—
91	C₂²	—	466	3	no	—
92	C₂²	C₂³×D₂₃₃	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	466	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	932	1	no	—
98	C₂	—	932	1	no	—
99	C₂	—	932	3	no	—
100	C₂	#21418581	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21418532

C₂

yes

maximal, characteristic, complemented

#21418529

C₂

yes

maximal, characteristic, complemented

#21418520

C₂

yes

maximal, complemented

#21418510

C₂

yes

maximal, complemented

#21418508

C₂

yes

maximal, complemented

#21418505

C₂

yes

maximal, complemented

#126×C₂₃₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₂₃₃

C₂²

yes

characteristic

C₂×D₉₃₂

C₂²

yes

—

C₂×D₉₃₂

C₂²

yes

—

C₂×D₉₃₂

C₂²

yes

complemented

C₂×C₂²⋊D₂₃₃

C₂²

yes

—

C₂×C₄×D₂₃₃

C₂²

yes

complemented

#16836573

C₂²

yes

complemented

#16836571

C₂²

yes

—

#16836571

C₂²

yes

—

#16836571

C₂²

yes

complemented

C₄⋊C₄×C₂₃₃

C₂²

yes

complemented

C₂²⋊C₄×C₂₃₃

C₂²

yes

—

C₄²×C₂₃₃

C₂²

yes

characteristic, complemented

C₂²×D₂₃₃

—

C₂²×D₂₃₃

C₂³

yes

characteristic

C₂²×D₂₃₃

C₂³

yes

characteristic

C₂²×D₂₃₃

C₂³

yes

—

D₉₃₂

—

D₉₃₂

—

D₉₃₂

—

C₂²⋊D₂₃₃

—

C₂×Dic₂₃₃

C₂³

yes

—

C₄×D₂₃₃

—

C₂³×C₂₃₃

C₂³

yes

characteristic

C₂×C₄×C₂₃₃

C₂³

yes

—

C₂×C₄×C₂₃₃

C₂³

yes

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

C₂⁴

yes

characteristic

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

Dic₂₃₃

—

C₂×C₂₃₃

—

C₂×C₂₃₃

H(5,𝔽₂)

yes

—

D₂₃₃

—

D₂₃₃

—

D₂₃₃

—

C₂₃₃

#560

yes

characteristic, complemented, Sylow

#560

—

233

maximal, Sylow

#127

—

233

—

#126

—

233

—

#126

D₂₃₃

yes

characteristic, complemented

#123

—

233

—

#121

—

233

—

#121

—

233

—

#120

—

233

—

C₂⁴

—

233

—

C₂×D₄

—

233

—

C₂×D₄

—

233

—

C₂×D₄

—

233

—

C₂×D₄

—

233

—

C₂²×C₄

—

233

—

C₄⋊C₄

—

233

—

C₄⋊C₄

C₂×D₂₃₃

yes

complemented

C₂²⋊C₄

—

233

—

C₂²⋊C₄

—

233

—

C₂²⋊C₄

—

233

—

C₂²⋊C₄

C₂×D₂₃₃

yes

—

C₄²

C₂×D₂₃₃

yes

characteristic, complemented

C₂³

—

932

—

C₂³

—

233

—

C₂³

—

233

—

C₂³

—

233

—

C₂³

C₂²×D₂₃₃

yes

characteristic

D₄

—

932

—

D₄

—

932

—

D₄

—

932

—

D₄

—

932

—

C₂×C₄

—

932

—

C₂×C₄

—

233

—

C₂×C₄

C₂²×D₂₃₃

yes

—

C₂×C₄

C₂²×D₂₃₃

yes

—

C₂²

—

C₂²

—

932

—

C₂²

—

466

—

C₂²

—

466

—

C₂²

—

466

—

C₂²

—

466

—

C₂²

—

466

—

C₂²

C₂³×D₂₃₃

yes

characteristic

C₄

—

C₄

—

C₄

—

466

—

C₂

—

C₂

—

932

—

C₂

—

932

—

C₂

—

932

—

100

C₂

#21418581

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃₃
commutator, socle, cosocle	C₂²×C₂₃₃	C₂⁴
Fitting	#126×C₂₃₃	C₂
hypercenter	#126	D₂₃₃
hypocenter	C₂₃₃	#560
LCS	G ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃₃

commutator, socle, cosocle

C₂²×C₂₃₃

C₂⁴

Fitting

#126×C₂₃₃

C₂

D₂₃₃

C₂₃₃

G ▷ C₂²×C₂₃₃ ▷ C₂₃₃

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #21418532	C₂	semidirect
2. #21418529	C₂	semidirect
3. #21418520	C₂	semidirect
4. #21418510	C₂	semidirect
5. #21418508	C₂	semidirect
6. #21418505	C₂	semidirect
7. #126×C₂₃₃	C₂	semidirect
11. C₂×D₉₃₂	C₂²	semidirect
13. C₂×C₄×D₂₃₃	C₂²	semidirect
14. #16836573	C₂²	semidirect
17. #16836571	C₂²	semidirect
18. C₄⋊C₄×C₂₃₃	C₂²	semidirect
20. C₄²×C₂₃₃	C₂²	semidirect
50. C₂₃₃	#560	semidirect
54. #126	D₂₃₃	semidirect
66. C₄⋊C₄	C₂×D₂₃₃	semidirect
71. C₄²	C₂×D₂₃₃	semidirect

H ≅ G/N

type

1. #21418532

C₂

semidirect

2. #21418529

C₂

semidirect

3. #21418520

C₂

semidirect

4. #21418510

C₂

semidirect

5. #21418508

C₂

semidirect

6. #21418505

C₂

semidirect

7. #126×C₂₃₃

C₂

semidirect

11. C₂×D₉₃₂

C₂²

semidirect

13. C₂×C₄×D₂₃₃

C₂²

semidirect

14. #16836573

C₂²

semidirect

17. #16836571

C₂²

semidirect

18. C₄⋊C₄×C₂₃₃

C₂²

semidirect

20. C₄²×C₂₃₃

C₂²

semidirect

50. C₂₃₃

#560

semidirect

54. #126

D₂₃₃

semidirect

66. C₄⋊C₄

C₂×D₂₃₃

semidirect

71. C₄²

C₂×D₂₃₃

semidirect

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

subgroup	index
#21418532	2
#21418529	2
#21418520	2
#21418510	2
#21418508	2
#21418505	2
#126×C₂₃₃	2
#560	233

subgroup

index

#126×C₂₃₃

233

Subgroups — #28417176

Conjugacy classes of subgroups

Complements

Maximal subgroups