Group #24924081 — subgroups

Subgroups — #24924081

Order 10688.

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	#19128879	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128878	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128865	C₂	1	3	yes	maximal, complemented
4	#19128860	C₂	1	3	yes	maximal, complemented
5	#19128854	C₂	1	3	yes	maximal, complemented
6	C₂×#14581939	C₂	1	3	yes	maximal, complemented
7	#131×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581951	—	2	1	no	—
9	#14581950	C₂²	1	3	yes	—
10	#14581950	C₂²	1	3	yes	complemented
11	#14581949	—	2	3	no	—
12	C₂×Dic₃₃₄	C₂²	1	3	yes	complemented
13	Q₈×D₁₆₇	—	2	1	no	—
14	C₂×D₆₆₈	C₂²	1	3	yes	—
15	D₄×D₁₆₇	—	2	3	no	—
16	Dic₆₆₈	C₂²	1	3	yes	complemented
17	#14581939	C₂²	1	3	yes	complemented
18	#14581939	C₂²	1	6	yes	complemented
19	#14581939	C₂²	1	1	yes	characteristic, complemented
20	D₁₃₃₆	C₂²	1	3	yes	complemented
21	#41×C₁₆₇	C₂²	1	1	yes	characteristic
22	M₁₆×C₁₆₇	C₂²	1	3	yes	complemented
23	C₂×C₈×C₁₆₇	C₂²	1	3	yes	complemented
24	C₂²×D₁₆₇	—	2	3	no	—
25	Dic₃₃₄	—	2	3	no	—
26	Dic₃₃₄	C₂³	1	3	yes	—
27	Dic₃₃₄	C₂³	1	1	yes	characteristic
28	D₆₆₈	—	2	3	no	—
29	D₆₆₈	C₂³	1	1	yes	characteristic
30	D₆₆₈	C₂³	1	3	yes	—
31	C₂²⋊D₁₆₇	—	2	3	no	—
32	C₂²⋊D₁₆₇	—	2	3	no	—
33	C₂×Dic₁₆₇	—	2	3	no	—
34	C₄×D₁₆₇	—	2	3	no	—
35	C₄×D₁₆₇	—	2	3	no	—
36	Q₈×C₁₆₇	D₄	1	1	yes	characteristic
37	D₄×C₁₆₇	D₄	1	3	yes	—
38	C₂×C₄×C₁₆₇	C₂³	1	3	yes	—
39	C₈×C₁₆₇	C₂³	1	1	yes	characteristic
40	C₈×C₁₆₇	C₂³	1	3	yes	—
41	C₂²×C₁₆₇	C₂×D₄	1	3	yes	—
42	C₂×D₁₆₇	—	2	1	no	—
43	C₂×D₁₆₇	—	2	3	no	—
44	C₂×D₁₆₇	—	8	3	no	—
45	C₄×C₁₆₇	C₂×D₄	1	3	yes	—
46	C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
47	Dic₁₆₇	—	2	1	no	—
48	Dic₁₆₇	—	2	3	no	—
49	C₂×C₁₆₇	—	2	3	no	—
50	C₂×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
51	D₁₆₇	—	4	1	no	—
52	D₁₆₇	—	4	3	no	—
53	C₁₆₇	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	167	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	167	1	no	—
56	H(5,𝔽₂)	—	167	1	no	—
57	#137	—	167	3	no	—
58	AGL(1,ℤ/8)	—	167	3	no	—
59	#135	—	167	3	no	—
60	C₂×SD₁₆	—	167	3	no	—
61	#131	D₁₆₇	1	1	yes	characteristic, complemented
62	#41	—	334	1	no	—
63	#41	—	334	3	no	—
64	#41	—	167	3	no	—
65	#41	—	167	3	no	—
66	#41	C₂×D₁₆₇	1	1	yes	characteristic
67	C₂×Q₈	—	334	1	no	—
68	C₂×Q₈	—	167	3	no	—
69	C₂×D₄	—	334	3	no	—
70	C₂×D₄	—	167	3	no	—
71	Q₁₆	—	167	3	no	—
72	SD₁₆	—	167	3	no	—
73	SD₁₆	—	167	6	no	—
74	SD₁₆	—	167	1	no	—
75	D₈	—	167	3	no	—
76	M₁₆	C₂×D₁₆₇	1	3	yes	complemented
77	C₂×C₈	C₂×D₁₆₇	1	3	yes	complemented
78	C₂³	—	334	3	no	—
79	Q₈	—	334	3	no	—
80	Q₈	—	167	3	no	—
81	Q₈	—	167	1	no	—
82	Q₈	D₆₆₈	1	1	yes	characteristic
83	D₄	—	334	3	no	—
84	D₄	—	334	3	no	—
85	D₄	—	334	3	no	—
86	D₄	—	167	1	no	—
87	D₄	—	167	3	no	—
88	D₄	D₆₆₈	1	3	yes	—
89	C₂×C₄	—	334	3	no	—
90	C₂×C₄	—	334	3	no	—
91	C₂×C₄	—	334	3	no	—
92	C₂×C₄	C₂²×D₁₆₇	1	3	yes	—
93	C₈	C₂²×D₁₆₇	1	1	yes	characteristic
94	C₈	C₂²×D₁₆₇	1	3	yes	—
95	C₂²	—	1336	3	no	—
96	C₂²	—	334	1	no	—
97	C₂²	—	334	3	no	—
98	C₂²	C₂×D₆₆₈	1	3	yes	—
99	C₄	—	334	1	no	—
100	C₄	—	334	3	no	—
101	C₄	C₂×D₆₆₈	1	3	yes	—
102	C₄	C₂³×D₁₆₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	668	3	no	—
105	C₂	—	668	1	no	—
106	C₂	C₂²×D₆₆₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128879

C₂

yes

maximal, characteristic, complemented

#19128878

C₂

yes

maximal, characteristic, complemented

#19128865

C₂

yes

maximal, complemented

#19128860

C₂

yes

maximal, complemented

#19128854

C₂

yes

maximal, complemented

C₂×#14581939

C₂

yes

maximal, complemented

#131×C₁₆₇

C₂

yes

maximal, characteristic, complemented

#14581951

—

#14581950

C₂²

yes

—

#14581950

C₂²

yes

complemented

#14581949

—

C₂×Dic₃₃₄

C₂²

yes

complemented

Q₈×D₁₆₇

—

C₂×D₆₆₈

C₂²

yes

—

D₄×D₁₆₇

—

Dic₆₆₈

C₂²

yes

complemented

#14581939

C₂²

yes

complemented

#14581939

C₂²

yes

complemented

#14581939

C₂²

yes

characteristic, complemented

D₁₃₃₆

C₂²

yes

complemented

#41×C₁₆₇

C₂²

yes

characteristic

M₁₆×C₁₆₇

C₂²

yes

complemented

C₂×C₈×C₁₆₇

C₂²

yes

complemented

C₂²×D₁₆₇

—

Dic₃₃₄

—

Dic₃₃₄

C₂³

yes

—

Dic₃₃₄

C₂³

yes

characteristic

D₆₆₈

—

D₆₆₈

C₂³

yes

characteristic

D₆₆₈

C₂³

yes

—

C₂²⋊D₁₆₇

—

C₂²⋊D₁₆₇

—

C₂×Dic₁₆₇

—

C₄×D₁₆₇

—

C₄×D₁₆₇

—

Q₈×C₁₆₇

D₄

yes

characteristic

D₄×C₁₆₇

D₄

yes

—

C₂×C₄×C₁₆₇

C₂³

yes

—

C₈×C₁₆₇

C₂³

yes

characteristic

C₈×C₁₆₇

C₂³

yes

—

C₂²×C₁₆₇

C₂×D₄

yes

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₄×C₁₆₇

C₂×D₄

yes

—

C₄×C₁₆₇

C₂⁴

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

C₂²×D₄

yes

characteristic

D₁₆₇

—

D₁₆₇

—

C₁₆₇

#577

yes

characteristic, complemented, Sylow

#577

—

167

maximal, Sylow

2₋¹⁺⁴

—

167

—

H(5,𝔽₂)

—

167

—

#137

—

167

—

AGL(1,ℤ/8)

—

167

—

#135

—

167

—

C₂×SD₁₆

—

167

—

#131

D₁₆₇

yes

characteristic, complemented

#41

—

334

—

#41

—

334

—

#41

—

167

—

#41

—

167

—

#41

C₂×D₁₆₇

yes

characteristic

C₂×Q₈

—

334

—

C₂×Q₈

—

167

—

C₂×D₄

—

334

—

C₂×D₄

—

167

—

Q₁₆

—

167

—

SD₁₆

—

167

—

SD₁₆

—

167

—

SD₁₆

—

167

—

D₈

—

167

—

M₁₆

C₂×D₁₆₇

yes

complemented

C₂×C₈

C₂×D₁₆₇

yes

complemented

C₂³

—

334

—

Q₈

—

334

—

Q₈

—

167

—

Q₈

—

167

—

Q₈

D₆₆₈

yes

characteristic

D₄

—

334

—

D₄

—

334

—

D₄

—

334

—

D₄

—

167

—

D₄

—

167

—

D₄

D₆₆₈

yes

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

C₂²×D₁₆₇

yes

—

C₈

C₂²×D₁₆₇

yes

characteristic

C₈

C₂²×D₁₆₇

yes

—

C₂²

—

1336

—

C₂²

—

334

—

C₂²

—

334

—

C₂²

C₂×D₆₆₈

yes

—

C₄

—

334

—

100

C₄

—

334

—

101

C₄

C₂×D₆₆₈

yes

—

102

C₄

C₂³×D₁₆₇

yes

characteristic

103

C₂

—

104

C₂

—

668

—

105

C₂

—

668

—

106

C₂

C₂²×D₆₆₈

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₆₆₈
commutator, cosocle	C₄×C₁₆₇	C₂⁴
Frattini	C₄	C₂³×D₁₆₇
Fitting	#131×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₂²×D₄
hypercenter	#131	D₁₆₇
hypocenter	C₁₆₇	#577
LCS	G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₆₆₈

commutator, cosocle

C₄×C₁₆₇

C₂⁴

Frattini

C₄

C₂³×D₁₆₇

Fitting

#131×C₁₆₇

C₂

socle

C₂×C₁₆₇

C₂²×D₄

D₁₆₇

C₁₆₇

G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #19128879	C₂	semidirect
2. #19128878	C₂	semidirect
3. #19128865	C₂	semidirect
4. #19128860	C₂	semidirect
5. #19128854	C₂	semidirect
6. C₂×#14581939	C₂	semidirect
7. #131×C₁₆₇	C₂	semidirect
10. #14581950	C₂²	semidirect
12. C₂×Dic₃₃₄	C₂²	semidirect
16. Dic₆₆₈	C₂²	semidirect
17. #14581939	C₂²	semidirect
18. #14581939	C₂²	semidirect
19. #14581939	C₂²	semidirect
20. D₁₃₃₆	C₂²	semidirect
22. M₁₆×C₁₆₇	C₂²	semidirect
23. C₂×C₈×C₁₆₇	C₂²	semidirect
53. C₁₆₇	#577	semidirect
61. #131	D₁₆₇	semidirect
76. M₁₆	C₂×D₁₆₇	semidirect
77. C₂×C₈	C₂×D₁₆₇	semidirect

H ≅ G/N

type

1. #19128879

C₂

semidirect

2. #19128878

C₂

semidirect

3. #19128865

C₂

semidirect

4. #19128860

C₂

semidirect

5. #19128854

C₂

semidirect

6. C₂×#14581939

C₂

semidirect

7. #131×C₁₆₇

C₂

semidirect

10. #14581950

C₂²

semidirect

12. C₂×Dic₃₃₄

C₂²

semidirect

16. Dic₆₆₈

C₂²

semidirect

17. #14581939

C₂²

semidirect

18. #14581939

C₂²

semidirect

19. #14581939

C₂²

semidirect

20. D₁₃₃₆

C₂²

semidirect

22. M₁₆×C₁₆₇

C₂²

semidirect

23. C₂×C₈×C₁₆₇

C₂²

semidirect

53. C₁₆₇

#577

semidirect

61. #131

D₁₆₇

semidirect

76. M₁₆

C₂×D₁₆₇

semidirect

77. C₂×C₈

C₂×D₁₆₇

semidirect

Subgroups — #24924081

Conjugacy classes of subgroups

Complements