Group #27133474 — subgroups

Subgroups — #27133474

Order 12224.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20259369	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259352	C₂	1	3	yes	maximal, complemented
3	#20259348	C₂	1	3	yes	maximal, complemented
4	#20259342	C₂	1	3	yes	maximal, complemented
5	C₂×#15690275	C₂	1	3	yes	maximal, complemented
6	#20259329	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#15690287	C₂²	1	3	yes	complemented
9	#15690286	—	2	3	no	—
10	C₂×Dic₃₈₂	C₂²	1	3	yes	complemented
11	Q₈×D₁₉₁	—	2	1	no	—
12	#15690277	C₂²	1	3	yes	complemented
13	#15690275	C₂²	1	3	yes	complemented
14	#15690275	C₂²	1	6	yes	complemented
15	#15690274	C₂²	1	1	yes	characteristic, complemented
16	#15690272	C₂²	1	3	yes	complemented
17	#15690271	C₂²	1	3	yes	complemented
18	C₂×C₁₉₁⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₁₉₁	—	2	1	no	—
20	#41×C₁₉₁	C₂²	1	3	yes	—
21	#41×C₁₉₁	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₁₉₁	—	2	3	no	—
23	C₂×D₄×C₁₉₁	C₂²	1	3	yes	—
24	Dic₃₈₂	—	2	3	no	—
25	Dic₃₈₂	C₂³	1	3	yes	—
26	D₇₆₄	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₉₁	—	2	3	no	—
28	C₂×Dic₁₉₁	—	2	3	no	—
29	C₄×D₁₉₁	—	2	3	no	—
30	C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
31	C₁₉₁⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₁₉₁	—	2	3	no	—
33	Q₈×C₁₉₁	D₄	1	1	yes	characteristic
34	Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
35	D₄×C₁₉₁	—	2	3	no	—
36	D₄×C₁₉₁	—	2	3	no	—
37	D₄×C₁₉₁	D₄	1	3	yes	—
38	D₄×C₁₉₁	C₂³	1	3	yes	—
39	C₂×C₄×C₁₉₁	—	2	3	no	—
40	C₂×C₄×C₁₉₁	C₂³	1	3	yes	—
41	C₂²×C₁₉₁	—	2	3	no	—
42	C₂²×C₁₉₁	—	8	3	no	—
43	C₂²×C₁₉₁	C₂×D₄	1	3	yes	—
44	C₂×D₁₉₁	—	2	1	no	—
45	C₄×C₁₉₁	—	2	1	no	—
46	C₄×C₁₉₁	C₂×D₄	1	3	yes	—
47	C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
48	Dic₁₉₁	—	2	3	no	—
49	C₂×C₁₉₁	—	2	3	no	—
50	C₂×C₁₉₁	—	4	3	no	—
51	C₂×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
52	D₁₉₁	—	4	1	no	—
53	C₁₉₁	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	191	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	191	1	no	—
56	H(5,𝔽₂)	D₁₉₁	1	1	yes	characteristic, complemented
57	#137	—	191	3	no	—
58	AGL(1,ℤ/8)	—	191	3	no	—
59	#135	—	191	3	no	—
60	C₂×SD₁₆	—	191	3	no	—
61	#131	—	191	1	no	—
62	#41	—	2	1	no	—
63	#41	—	382	3	no	—
64	#41	—	191	3	no	—
65	#41	C₂×D₁₉₁	1	3	yes	—
66	#41	C₂×D₁₉₁	1	1	yes	characteristic
67	C₂×Q₈	—	382	1	no	—
68	C₂×Q₈	—	191	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₁₉₁	1	3	yes	—
71	Q₁₆	—	191	3	no	—
72	SD₁₆	—	191	1	no	—
73	SD₁₆	—	191	3	no	—
74	SD₁₆	—	191	6	no	—
75	D₈	—	191	3	no	—
76	M₁₆	—	191	3	no	—
77	C₂×C₈	—	191	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	382	3	no	—
80	Q₈	—	191	3	no	—
81	Q₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
82	Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	382	3	no	—
86	D₄	—	191	1	no	—
87	D₄	C₂²⋊D₁₉₁	1	3	yes	—
88	D₄	C₂²×D₁₉₁	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	382	3	no	—
91	C₂×C₄	—	382	3	no	—
92	C₂×C₄	C₂²×D₁₉₁	1	3	yes	—
93	C₈	—	191	1	no	—
94	C₈	—	191	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	382	1	no	—
98	C₂²	C₂×C₂²⋊D₁₉₁	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	382	3	no	—
101	C₄	C₂×C₂²⋊D₁₉₁	1	3	yes	—
102	C₄	C₂³×D₁₉₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	764	1	no	—
106	C₂	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20259369

C₂

yes

maximal, characteristic, complemented

#20259352

C₂

yes

maximal, complemented

#20259348

C₂

yes

maximal, complemented

#20259342

C₂

yes

maximal, complemented

C₂×#15690275

C₂

yes

maximal, complemented

#20259329

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₉₁

C₂

yes

maximal, characteristic, complemented

#15690287

C₂²

yes

complemented

#15690286

—

C₂×Dic₃₈₂

C₂²

yes

complemented

Q₈×D₁₉₁

—

#15690277

C₂²

yes

complemented

#15690275

C₂²

yes

complemented

#15690275

C₂²

yes

complemented

#15690274

C₂²

yes

characteristic, complemented

#15690272

C₂²

yes

complemented

#15690271

C₂²

yes

complemented

C₂×C₁₉₁⋊C₈

C₂²

yes

complemented

#41×C₁₉₁

—

#41×C₁₉₁

C₂²

yes

—

#41×C₁₉₁

C₂²

yes

characteristic

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

C₂²

yes

—

Dic₃₈₂

—

Dic₃₈₂

C₂³

yes

—

D₇₆₄

C₂³

yes

characteristic

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

C₂³

yes

characteristic

C₁₉₁⋊C₈

C₂³

yes

—

C₂³×C₁₉₁

—

Q₈×C₁₉₁

D₄

yes

characteristic

Q₈×C₁₉₁

C₂³

yes

characteristic

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

D₄

yes

—

D₄×C₁₉₁

C₂³

yes

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×D₄

yes

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×D₄

yes

—

C₄×C₁₉₁

C₂⁴

yes

characteristic

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂²×D₄

yes

characteristic

D₁₉₁

—

C₁₉₁

#577

yes

characteristic, complemented, Sylow

#577

—

191

maximal, Sylow

2₋¹⁺⁴

—

191

—

H(5,𝔽₂)

D₁₉₁

yes

characteristic, complemented

#137

—

191

—

AGL(1,ℤ/8)

—

191

—

#135

—

191

—

C₂×SD₁₆

—

191

—

#131

—

191

—

#41

—

#41

—

382

—

#41

—

191

—

#41

C₂×D₁₉₁

yes

—

#41

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂×D₄

—

C₂×D₄

C₂×D₁₉₁

yes

—

Q₁₆

—

191

—

SD₁₆

—

191

—

SD₁₆

—

191

—

SD₁₆

—

191

—

D₈

—

191

—

M₁₆

—

191

—

C₂×C₈

—

191

—

C₂³

—

Q₈

—

382

—

Q₈

—

191

—

Q₈

C₂²⋊D₁₉₁

yes

characteristic

Q₈

C₂²×D₁₉₁

yes

characteristic

D₄

—

D₄

—

D₄

—

382

—

D₄

—

191

—

D₄

C₂²⋊D₁₉₁

yes

—

D₄

C₂²×D₁₉₁

yes

—

C₂×C₄

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

C₂²×D₁₉₁

yes

—

C₈

—

191

—

C₈

—

191

—

C₂²

—

C₂²

—

C₂²

—

382

—

C₂²

C₂×C₂²⋊D₁₉₁

yes

—

C₄

—

100

C₄

—

382

—

101

C₄

C₂×C₂²⋊D₁₉₁

yes

—

102

C₄

C₂³×D₁₉₁

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

764

—

106

C₂

C₂²×C₂²⋊D₁₉₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₉₁
commutator, cosocle	C₄×C₁₉₁	C₂⁴
Frattini	C₄	C₂³×D₁₉₁
Fitting	H(5,𝔽₂)×C₁₉₁	C₂
socle	C₂×C₁₉₁	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₉₁
hypocenter	C₁₉₁	#577
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₉₁

commutator, cosocle

C₄×C₁₉₁

C₂⁴

Frattini

C₄

C₂³×D₁₉₁

Fitting

H(5,𝔽₂)×C₁₉₁

C₂

socle

C₂×C₁₉₁

C₂²×D₄

D₁₉₁

C₁₉₁

G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #20259369	C₂	semidirect
2. #20259352	C₂	semidirect
3. #20259348	C₂	semidirect
4. #20259342	C₂	semidirect
5. C₂×#15690275	C₂	semidirect
6. #20259329	C₂	semidirect
7. H(5,𝔽₂)×C₁₉₁	C₂	semidirect
8. #15690287	C₂²	semidirect
10. C₂×Dic₃₈₂	C₂²	semidirect
12. #15690277	C₂²	semidirect
13. #15690275	C₂²	semidirect
14. #15690275	C₂²	semidirect
15. #15690274	C₂²	semidirect
16. #15690272	C₂²	semidirect
17. #15690271	C₂²	semidirect
18. C₂×C₁₉₁⋊C₈	C₂²	semidirect
53. C₁₉₁	#577	semidirect
56. H(5,𝔽₂)	D₁₉₁	semidirect

H ≅ G/N

type

1. #20259369

C₂

semidirect

2. #20259352

C₂

semidirect

3. #20259348

C₂

semidirect

4. #20259342

C₂

semidirect

5. C₂×#15690275

C₂

semidirect

6. #20259329

C₂

semidirect

7. H(5,𝔽₂)×C₁₉₁

C₂

semidirect

8. #15690287

C₂²

semidirect

10. C₂×Dic₃₈₂

C₂²

semidirect

12. #15690277

C₂²

semidirect

13. #15690275

C₂²

semidirect

14. #15690275

C₂²

semidirect

15. #15690274

C₂²

semidirect

16. #15690272

C₂²

semidirect

17. #15690271

C₂²

semidirect

18. C₂×C₁₉₁⋊C₈

C₂²

semidirect

53. C₁₉₁

#577

semidirect

56. H(5,𝔽₂)

D₁₉₁

semidirect

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

subgroup	index
#20259369	2
#20259352	2
#20259348	2
#20259342	2
C₂×#15690275	2
#20259329	2
H(5,𝔽₂)×C₁₉₁	2
#577	191

subgroup

index

H(5,𝔽₂)×C₁₉₁

191

Subgroups — #27133474

Conjugacy classes of subgroups

Complements

Maximal subgroups