Group #27133475 — subgroups

Subgroups — #27133475

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	#20259368	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259355	C₂	1	3	yes	maximal, complemented
4	#20259350	C₂	1	3	yes	maximal, complemented
5	#20259344	C₂	1	3	yes	maximal, complemented
6	C₂×#15690276	C₂	1	3	yes	maximal, complemented
7	#131×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#15690288	—	2	1	no	—
9	#15690287	C₂²	1	3	yes	—
10	#15690287	C₂²	1	3	yes	complemented
11	#15690286	—	2	3	no	—
12	C₂×Dic₃₈₂	C₂²	1	3	yes	complemented
13	Q₈×D₁₉₁	—	2	1	no	—
14	C₂×D₇₆₄	C₂²	1	3	yes	—
15	D₄×D₁₉₁	—	2	3	no	—
16	Dic₇₆₄	C₂²	1	3	yes	complemented
17	#15690276	C₂²	1	3	yes	complemented
18	#15690276	C₂²	1	6	yes	complemented
19	#15690276	C₂²	1	1	yes	characteristic, complemented
20	D₁₅₂₈	C₂²	1	3	yes	complemented
21	#41×C₁₉₁	C₂²	1	1	yes	characteristic
22	M₁₆×C₁₉₁	C₂²	1	3	yes	complemented
23	C₂×C₈×C₁₉₁	C₂²	1	3	yes	complemented
24	C₂²×D₁₉₁	—	2	3	no	—
25	Dic₃₈₂	—	2	3	no	—
26	Dic₃₈₂	C₂³	1	3	yes	—
27	Dic₃₈₂	C₂³	1	1	yes	characteristic
28	D₇₆₄	—	2	3	no	—
29	D₇₆₄	C₂³	1	1	yes	characteristic
30	D₇₆₄	C₂³	1	3	yes	—
31	C₂²⋊D₁₉₁	—	2	3	no	—
32	C₂²⋊D₁₉₁	—	2	3	no	—
33	C₂×Dic₁₉₁	—	2	3	no	—
34	C₄×D₁₉₁	—	2	3	no	—
35	C₄×D₁₉₁	—	2	3	no	—
36	Q₈×C₁₉₁	D₄	1	1	yes	characteristic
37	D₄×C₁₉₁	D₄	1	3	yes	—
38	C₂×C₄×C₁₉₁	C₂³	1	3	yes	—
39	C₈×C₁₉₁	C₂³	1	1	yes	characteristic
40	C₈×C₁₉₁	C₂³	1	3	yes	—
41	C₂²×C₁₉₁	C₂×D₄	1	3	yes	—
42	C₂×D₁₉₁	—	2	1	no	—
43	C₂×D₁₉₁	—	2	3	no	—
44	C₂×D₁₉₁	—	8	3	no	—
45	C₄×C₁₉₁	C₂×D₄	1	3	yes	—
46	C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
47	Dic₁₉₁	—	2	1	no	—
48	Dic₁₉₁	—	2	3	no	—
49	C₂×C₁₉₁	—	2	3	no	—
50	C₂×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
51	D₁₉₁	—	4	1	no	—
52	D₁₉₁	—	4	3	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)	—	191	1	no	—
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	D₁₉₁	1	1	yes	characteristic, complemented
62	#41	—	382	1	no	—
63	#41	—	382	3	no	—
64	#41	—	191	3	no	—
65	#41	—	191	3	no	—
66	#41	C₂×D₁₉₁	1	1	yes	characteristic
67	C₂×Q₈	—	382	1	no	—
68	C₂×Q₈	—	191	3	no	—
69	C₂×D₄	—	382	3	no	—
70	C₂×D₄	—	191	3	no	—
71	Q₁₆	—	191	3	no	—
72	SD₁₆	—	191	3	no	—
73	SD₁₆	—	191	6	no	—
74	SD₁₆	—	191	1	no	—
75	D₈	—	191	3	no	—
76	M₁₆	C₂×D₁₉₁	1	3	yes	complemented
77	C₂×C₈	C₂×D₁₉₁	1	3	yes	complemented
78	C₂³	—	382	3	no	—
79	Q₈	—	382	3	no	—
80	Q₈	—	191	3	no	—
81	Q₈	—	191	1	no	—
82	Q₈	D₇₆₄	1	1	yes	characteristic
83	D₄	—	382	3	no	—
84	D₄	—	382	3	no	—
85	D₄	—	382	3	no	—
86	D₄	—	191	1	no	—
87	D₄	—	191	3	no	—
88	D₄	D₇₆₄	1	3	yes	—
89	C₂×C₄	—	382	3	no	—
90	C₂×C₄	—	382	3	no	—
91	C₂×C₄	—	382	3	no	—
92	C₂×C₄	C₂²×D₁₉₁	1	3	yes	—
93	C₈	C₂²×D₁₉₁	1	1	yes	characteristic
94	C₈	C₂²×D₁₉₁	1	3	yes	—
95	C₂²	—	382	1	no	—
96	C₂²	—	382	3	no	—
97	C₂²	—	1528	3	no	—
98	C₂²	C₂×D₇₆₄	1	3	yes	—
99	C₄	—	382	1	no	—
100	C₄	—	382	3	no	—
101	C₄	C₂×D₇₆₄	1	3	yes	—
102	C₄	C₂³×D₁₉₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	764	3	no	—
105	C₂	—	764	1	no	—
106	C₂	C₂²×D₇₆₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20259369

C₂

yes

maximal, characteristic, complemented

#20259368

C₂

yes

maximal, characteristic, complemented

#20259355

C₂

yes

maximal, complemented

#20259350

C₂

yes

maximal, complemented

#20259344

C₂

yes

maximal, complemented

C₂×#15690276

C₂

yes

maximal, complemented

#131×C₁₉₁

C₂

yes

maximal, characteristic, complemented

#15690288

—

#15690287

C₂²

yes

—

#15690287

C₂²

yes

complemented

#15690286

—

C₂×Dic₃₈₂

C₂²

yes

complemented

Q₈×D₁₉₁

—

C₂×D₇₆₄

C₂²

yes

—

D₄×D₁₉₁

—

Dic₇₆₄

C₂²

yes

complemented

#15690276

C₂²

yes

complemented

#15690276

C₂²

yes

complemented

#15690276

C₂²

yes

characteristic, complemented

D₁₅₂₈

C₂²

yes

complemented

#41×C₁₉₁

C₂²

yes

characteristic

M₁₆×C₁₉₁

C₂²

yes

complemented

C₂×C₈×C₁₉₁

C₂²

yes

complemented

C₂²×D₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

C₂³

yes

—

Dic₃₈₂

C₂³

yes

characteristic

D₇₆₄

—

D₇₆₄

C₂³

yes

characteristic

D₇₆₄

C₂³

yes

—

C₂²⋊D₁₉₁

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

Q₈×C₁₉₁

D₄

yes

characteristic

D₄×C₁₉₁

D₄

yes

—

C₂×C₄×C₁₉₁

C₂³

yes

—

C₈×C₁₉₁

C₂³

yes

characteristic

C₈×C₁₉₁

C₂³

yes

—

C₂²×C₁₉₁

C₂×D₄

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

C₂×D₄

yes

—

C₄×C₁₉₁

C₂⁴

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂²×D₄

yes

characteristic

D₁₉₁

—

D₁₉₁

—

C₁₉₁

#577

yes

characteristic, complemented, Sylow

#577

—

191

maximal, Sylow

2₋¹⁺⁴

—

191

—

H(5,𝔽₂)

—

191

—

#137

—

191

—

AGL(1,ℤ/8)

—

191

—

#135

—

191

—

C₂×SD₁₆

—

191

—

#131

D₁₉₁

yes

characteristic, complemented

#41

—

382

—

#41

—

382

—

#41

—

191

—

#41

—

191

—

#41

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂×D₄

—

382

—

C₂×D₄

—

191

—

Q₁₆

—

191

—

SD₁₆

—

191

—

SD₁₆

—

191

—

SD₁₆

—

191

—

D₈

—

191

—

M₁₆

C₂×D₁₉₁

yes

complemented

C₂×C₈

C₂×D₁₉₁

yes

complemented

C₂³

—

382

—

Q₈

—

382

—

Q₈

—

191

—

Q₈

—

191

—

Q₈

D₇₆₄

yes

characteristic

D₄

—

382

—

D₄

—

382

—

D₄

—

382

—

D₄

—

191

—

D₄

—

191

—

D₄

D₇₆₄

yes

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

C₂²×D₁₉₁

yes

—

C₈

C₂²×D₁₉₁

yes

characteristic

C₈

C₂²×D₁₉₁

yes

—

C₂²

—

382

—

C₂²

—

382

—

C₂²

—

1528

—

C₂²

C₂×D₇₆₄

yes

—

C₄

—

382

—

100

C₄

—

382

—

101

C₄

C₂×D₇₆₄

yes

—

102

C₄

C₂³×D₁₉₁

yes

characteristic

103

C₂

—

104

C₂

—

764

—

105

C₂

—

764

—

106

C₂

C₂²×D₇₆₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₇₆₄
commutator, cosocle	C₄×C₁₉₁	C₂⁴
Frattini	C₄	C₂³×D₁₉₁
Fitting	#131×C₁₉₁	C₂
socle	C₂×C₁₉₁	C₂²×D₄
hypercenter	#131	D₁₉₁
hypocenter	C₁₉₁	#577
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₇₆₄

commutator, cosocle

C₄×C₁₉₁

C₂⁴

Frattini

C₄

C₂³×D₁₉₁

Fitting

#131×C₁₉₁

C₂

socle

C₂×C₁₉₁

C₂²×D₄

D₁₉₁

C₁₉₁

G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #20259369	C₂	semidirect
2. #20259368	C₂	semidirect
3. #20259355	C₂	semidirect
4. #20259350	C₂	semidirect
5. #20259344	C₂	semidirect
6. C₂×#15690276	C₂	semidirect
7. #131×C₁₉₁	C₂	semidirect
10. #15690287	C₂²	semidirect
12. C₂×Dic₃₈₂	C₂²	semidirect
16. Dic₇₆₄	C₂²	semidirect
17. #15690276	C₂²	semidirect
18. #15690276	C₂²	semidirect
19. #15690276	C₂²	semidirect
20. D₁₅₂₈	C₂²	semidirect
22. M₁₆×C₁₉₁	C₂²	semidirect
23. C₂×C₈×C₁₉₁	C₂²	semidirect
53. C₁₉₁	#577	semidirect
61. #131	D₁₉₁	semidirect
76. M₁₆	C₂×D₁₉₁	semidirect
77. C₂×C₈	C₂×D₁₉₁	semidirect

H ≅ G/N

type

1. #20259369

C₂

semidirect

2. #20259368

C₂

semidirect

3. #20259355

C₂

semidirect

4. #20259350

C₂

semidirect

5. #20259344

C₂

semidirect

6. C₂×#15690276

C₂

semidirect

7. #131×C₁₉₁

C₂

semidirect

10. #15690287

C₂²

semidirect

12. C₂×Dic₃₈₂

C₂²

semidirect

16. Dic₇₆₄

C₂²

semidirect

17. #15690276

C₂²

semidirect

18. #15690276

C₂²

semidirect

19. #15690276

C₂²

semidirect

20. D₁₅₂₈

C₂²

semidirect

22. M₁₆×C₁₉₁

C₂²

semidirect

23. C₂×C₈×C₁₉₁

C₂²

semidirect

53. C₁₉₁

#577

semidirect

61. #131

D₁₉₁

semidirect

76. M₁₆

C₂×D₁₉₁

semidirect

77. C₂×C₈

C₂×D₁₉₁

semidirect

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

subgroup	index
#20259369	2
#20259368	2
#20259355	2
#20259350	2
#20259344	2
C₂×#15690276	2
#131×C₁₉₁	2
#577	191

subgroup

index

#131×C₁₉₁

191

Subgroups — #27133475

Conjugacy classes of subgroups

Complements

Maximal subgroups