Group #20261546 — subgroups

Subgroups — #20261546

Order 6208.

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	#15690857	C₂	1	1	yes	maximal, characteristic, complemented
2	#15690840	C₂	1	3	yes	maximal, complemented
3	#15690836	C₂	1	3	yes	maximal, complemented
4	#15690830	C₂	1	3	yes	maximal, complemented
5	C₂×#13095499	C₂	1	3	yes	maximal, complemented
6	#15690817	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#13095511	C₂²	1	3	yes	complemented
9	#13095510	—	2	3	no	—
10	C₂×Dic₁₉₄	C₂²	1	3	yes	complemented
11	Q₈×D₉₇	—	2	1	no	—
12	#13095501	C₂²	1	3	yes	complemented
13	#13095499	C₂²	1	3	yes	complemented
14	#13095499	C₂²	1	6	yes	complemented
15	#13095498	C₂²	1	1	yes	characteristic, complemented
16	#13095496	C₂²	1	3	yes	complemented
17	#13095495	C₂²	1	3	yes	complemented
18	C₂×#11715895	C₂²	1	3	yes	complemented
19	#41×C₉₇	—	2	1	no	—
20	#41×C₉₇	C₂²	1	3	yes	—
21	#41×C₉₇	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₉₇	—	2	3	no	—
23	C₂×D₄×C₉₇	C₂²	1	3	yes	—
24	C₂³×C₉₇	—	2	3	no	—
25	Q₈×C₉₇	D₄	1	1	yes	characteristic
26	Q₈×C₉₇	C₂³	1	1	yes	characteristic
27	D₄×C₉₇	—	2	3	no	—
28	D₄×C₉₇	—	2	3	no	—
29	D₄×C₉₇	D₄	1	3	yes	—
30	D₄×C₉₇	C₂³	1	3	yes	—
31	C₂×C₄×C₉₇	—	2	3	no	—
32	C₂×C₄×C₉₇	C₂³	1	3	yes	—
33	C₂²⋊D₉₇	—	2	3	no	—
34	C₂×Dic₉₇	—	2	3	no	—
35	D₃₈₈	C₂³	1	1	yes	characteristic
36	C₄×D₉₇	—	2	3	no	—
37	Dic₁₉₄	—	2	3	no	—
38	Dic₁₉₄	C₂³	1	3	yes	—
39	#11715895	C₂³	1	1	yes	characteristic
40	#11715895	C₂³	1	3	yes	—
41	C₂²×C₉₇	—	2	3	no	—
42	C₂²×C₉₇	—	8	3	no	—
43	C₂²×C₉₇	C₂×D₄	1	3	yes	—
44	C₂×D₉₇	—	2	1	no	—
45	C₄×C₉₇	—	2	1	no	—
46	C₄×C₉₇	C₂×D₄	1	3	yes	—
47	C₄×C₉₇	C₂⁴	1	1	yes	characteristic
48	Dic₉₇	—	2	3	no	—
49	C₂×C₉₇	—	2	3	no	—
50	C₂×C₉₇	—	4	3	no	—
51	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
52	D₉₇	—	4	1	no	—
53	C₉₇	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	97	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	97	1	no	—
56	H(5,𝔽₂)	D₉₇	1	1	yes	characteristic, complemented
57	#137	—	97	3	no	—
58	AGL(1,ℤ/8)	—	97	3	no	—
59	#135	—	97	3	no	—
60	C₂×SD₁₆	—	97	3	no	—
61	#131	—	97	1	no	—
62	#41	—	2	1	no	—
63	#41	—	97	3	no	—
64	#41	—	194	3	no	—
65	#41	C₂×D₉₇	1	3	yes	—
66	#41	C₂×D₉₇	1	1	yes	characteristic
67	C₂×Q₈	—	97	3	no	—
68	C₂×Q₈	—	194	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₉₇	1	3	yes	—
71	Q₁₆	—	97	3	no	—
72	SD₁₆	—	97	1	no	—
73	SD₁₆	—	97	3	no	—
74	SD₁₆	—	97	6	no	—
75	D₈	—	97	3	no	—
76	M₁₆	—	97	3	no	—
77	C₂×C₈	—	97	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	97	3	no	—
80	Q₈	—	194	3	no	—
81	Q₈	C₂²⋊D₉₇	1	1	yes	characteristic
82	Q₈	C₂²×D₉₇	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	97	1	no	—
86	D₄	—	194	3	no	—
87	D₄	C₂²⋊D₉₇	1	3	yes	—
88	D₄	C₂²×D₉₇	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	194	3	no	—
91	C₂×C₄	—	194	3	no	—
92	C₂×C₄	C₂²×D₉₇	1	3	yes	—
93	C₈	—	97	1	no	—
94	C₈	—	97	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	194	1	no	—
98	C₂²	C₂×C₂²⋊D₉₇	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	194	3	no	—
101	C₄	C₂×C₂²⋊D₉₇	1	3	yes	—
102	C₄	C₂³×D₉₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	388	1	no	—
106	C₂	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15690857

C₂

yes

maximal, characteristic, complemented

#15690840

C₂

yes

maximal, complemented

#15690836

C₂

yes

maximal, complemented

#15690830

C₂

yes

maximal, complemented

C₂×#13095499

C₂

yes

maximal, complemented

#15690817

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₉₇

C₂

yes

maximal, characteristic, complemented

#13095511

C₂²

yes

complemented

#13095510

—

C₂×Dic₁₉₄

C₂²

yes

complemented

Q₈×D₉₇

—

#13095501

C₂²

yes

complemented

#13095499

C₂²

yes

complemented

#13095499

C₂²

yes

complemented

#13095498

C₂²

yes

characteristic, complemented

#13095496

C₂²

yes

complemented

#13095495

C₂²

yes

complemented

C₂×#11715895

C₂²

yes

complemented

#41×C₉₇

—

#41×C₉₇

C₂²

yes

—

#41×C₉₇

C₂²

yes

characteristic

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂²

yes

—

C₂³×C₉₇

—

Q₈×C₉₇

D₄

yes

characteristic

Q₈×C₉₇

C₂³

yes

characteristic

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

D₄

yes

—

D₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

D₃₈₈

C₂³

yes

characteristic

C₄×D₉₇

—

Dic₁₉₄

—

Dic₁₉₄

C₂³

yes

—

#11715895

C₂³

yes

characteristic

#11715895

C₂³

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×D₄

yes

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×D₄

yes

—

C₄×C₉₇

C₂⁴

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂²×D₄

yes

characteristic

D₉₇

—

C₉₇

#577

yes

characteristic, complemented, Sylow

#577

—

maximal, Sylow

2₋¹⁺⁴

—

H(5,𝔽₂)

D₉₇

yes

characteristic, complemented

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×SD₁₆

—

—

—

—

—

194

—

#41

C₂×D₉₇

yes

—

#41

C₂×D₉₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×D₄

—

C₂×D₄

C₂×D₉₇

yes

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂³

—

Q₈

—

Q₈

—

194

—

Q₈

C₂²⋊D₉₇

yes

characteristic

Q₈

C₂²×D₉₇

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

194

—

D₄

C₂²⋊D₉₇

yes

—

D₄

C₂²×D₉₇

yes

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₂²×D₉₇

yes

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

194

—

C₂²

C₂×C₂²⋊D₉₇

yes

—

C₄

—

100

C₄

—

194

—

101

C₄

C₂×C₂²⋊D₉₇

yes

—

102

C₄

C₂³×D₉₇

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

388

—

106

C₂

C₂²×C₂²⋊D₉₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₉₇
commutator	C₄×C₉₇	C₂⁴
Frattini	C₄	C₂³×D₉₇
Fitting	H(5,𝔽₂)×C₉₇	C₂
socle	C₂×C₉₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₉₇
hypocenter	C₉₇	#577
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₉₇

commutator

C₄×C₉₇

C₂⁴

Frattini

C₄

C₂³×D₉₇

Fitting

H(5,𝔽₂)×C₉₇

C₂

socle

C₂×C₉₇

C₂²×D₄

D₉₇

C₉₇

G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #15690857	C₂	semidirect
2. #15690840	C₂	semidirect
3. #15690836	C₂	semidirect
4. #15690830	C₂	semidirect
5. C₂×#13095499	C₂	semidirect
6. #15690817	C₂	semidirect
7. H(5,𝔽₂)×C₉₇	C₂	semidirect
8. #13095511	C₂²	semidirect
10. C₂×Dic₁₉₄	C₂²	semidirect
12. #13095501	C₂²	semidirect
13. #13095499	C₂²	semidirect
14. #13095499	C₂²	semidirect
15. #13095498	C₂²	semidirect
16. #13095496	C₂²	semidirect
17. #13095495	C₂²	semidirect
18. C₂×#11715895	C₂²	semidirect
53. C₉₇	#577	semidirect
56. H(5,𝔽₂)	D₉₇	semidirect

H ≅ G/N

type

1. #15690857

C₂

semidirect

2. #15690840

C₂

semidirect

3. #15690836

C₂

semidirect

4. #15690830

C₂

semidirect

5. C₂×#13095499

C₂

semidirect

6. #15690817

C₂

semidirect

7. H(5,𝔽₂)×C₉₇

C₂

semidirect

8. #13095511

C₂²

semidirect

10. C₂×Dic₁₉₄

C₂²

semidirect

12. #13095501

C₂²

semidirect

13. #13095499

C₂²

semidirect

14. #13095499

C₂²

semidirect

15. #13095498

C₂²

semidirect

16. #13095496

C₂²

semidirect

17. #13095495

C₂²

semidirect

18. C₂×#11715895

C₂²

semidirect

53. C₉₇

#577

semidirect

56. H(5,𝔽₂)

D₉₇

semidirect

Subgroups — #20261546

Conjugacy classes of subgroups

Complements