Group #24924072 — subgroups

Subgroups — #24924072

Order 10688.

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

normal subgroups only

This group has 268 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	#19128878	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128858	C₂	1	3	yes	maximal, complemented
3	#19128857	C₂	1	3	yes	maximal, complemented
4	#19128852	C₂	1	3	yes	maximal, complemented
5	C₂×#14581935	C₂	1	3	yes	maximal, complemented
6	#19128839	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581951	—	2	1	no	—
9	#14581950	C₂²	1	3	yes	complemented
10	C₂×D₆₆₈	C₂²	1	3	yes	complemented
11	D₄×D₁₆₇	—	2	3	no	—
12	#14581940	C₂²	1	1	yes	characteristic, complemented
13	#14581938	C₂²	1	3	yes	complemented
14	#14581937	C₂²	1	3	yes	complemented
15	#14581935	C₂²	1	3	yes	complemented
16	#14581935	C₂²	1	6	yes	complemented
17	#14581934	C₂²	1	3	yes	complemented
18	C₂×C₁₆₇⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₁₆₇	—	2	1	no	—
20	#41×C₁₆₇	C₂²	1	1	yes	characteristic
21	#41×C₁₆₇	C₂²	1	3	yes	complemented
22	C₂×D₄×C₁₆₇	—	2	3	no	—
23	C₂×D₄×C₁₆₇	C₂²	1	3	yes	complemented
24	C₂²×D₁₆₇	—	2	3	no	—
25	Dic₃₃₄	C₂³	1	1	yes	characteristic
26	D₆₆₈	—	2	3	no	—
27	D₆₆₈	C₂³	1	3	yes	—
28	C₂²⋊D₁₆₇	—	2	3	no	—
29	C₄×D₁₆₇	—	2	3	no	—
30	C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₁₆₇⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₁₆₇	—	2	3	no	—
33	Q₈×C₁₆₇	D₄	1	1	yes	characteristic
34	Q₈×C₁₆₇	C₂³	1	1	yes	characteristic
35	D₄×C₁₆₇	—	2	3	no	—
36	D₄×C₁₆₇	—	2	3	no	—
37	D₄×C₁₆₇	D₄	1	3	yes	—
38	D₄×C₁₆₇	C₂³	1	3	yes	—
39	C₂×C₄×C₁₆₇	—	2	3	no	—
40	C₂×C₄×C₁₆₇	C₂³	1	3	yes	—
41	C₂²×C₁₆₇	—	2	3	no	—
42	C₂²×C₁₆₇	—	8	3	no	—
43	C₂²×C₁₆₇	C₂×D₄	1	3	yes	—
44	C₂×D₁₆₇	—	2	3	no	—
45	C₂×D₁₆₇	—	8	3	no	—
46	C₄×C₁₆₇	—	2	1	no	—
47	C₄×C₁₆₇	C₂×D₄	1	3	yes	—
48	C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
49	Dic₁₆₇	—	2	1	no	—
50	C₂×C₁₆₇	—	2	3	no	—
51	C₂×C₁₆₇	—	4	3	no	—
52	C₂×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
53	D₁₆₇	—	4	3	no	—
54	C₁₆₇	#576	1	1	yes	characteristic, complemented, Sylow
55	#576	—	167	1	no	maximal, Sylow
56	H(5,𝔽₂)	—	167	1	no	—
57	H(5,𝔽₂)	D₁₆₇	1	1	yes	characteristic, complemented
58	AGL(1,ℤ/8)	—	167	3	no	—
59	AGL(1,ℤ/8)	—	167	3	no	—
60	#135	—	167	3	no	—
61	C₂×D₈	—	167	3	no	—
62	#131	—	167	1	no	—
63	#41	—	2	1	no	—
64	#41	—	334	1	no	—
65	#41	—	167	3	no	—
66	#41	C₂×D₁₆₇	1	1	yes	characteristic
67	#41	C₂×D₁₆₇	1	3	yes	complemented
68	C₂×D₄	—	2	3	no	—
69	C₂×D₄	—	334	3	no	—
70	C₂×D₄	—	167	3	no	—
71	C₂×D₄	C₂×D₁₆₇	1	3	yes	complemented
72	Q₁₆	—	167	1	no	—
73	SD₁₆	—	167	3	no	—
74	SD₁₆	—	167	3	no	—
75	D₈	—	167	3	no	—
76	D₈	—	167	6	no	—
77	M₁₆	—	167	3	no	—
78	C₂×C₈	—	167	3	no	—
79	C₂³	—	2	3	no	—
80	C₂³	—	334	3	no	—
81	Q₈	—	167	1	no	—
82	Q₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
83	Q₈	C₂²×D₁₆₇	1	1	yes	characteristic
84	D₄	—	2	3	no	—
85	D₄	—	2	3	no	—
86	D₄	—	334	3	no	—
87	D₄	—	334	3	no	—
88	D₄	—	167	3	no	—
89	D₄	C₂²⋊D₁₆₇	1	3	yes	—
90	D₄	C₂²×D₁₆₇	1	3	yes	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	334	3	no	—
93	C₂×C₄	C₂²×D₁₆₇	1	3	yes	—
94	C₈	—	167	1	no	—
95	C₈	—	167	3	no	—
96	C₂²	—	2	3	no	—
97	C₂²	—	8	3	no	—
98	C₂²	—	1336	3	no	—
99	C₂²	—	334	3	no	—
100	C₂²	C₂×C₂²⋊D₁₆₇	1	3	yes	—
101	C₄	—	2	1	no	—
102	C₄	—	334	1	no	—
103	C₄	C₂×C₂²⋊D₁₆₇	1	3	yes	—
104	C₄	C₂³×D₁₆₇	1	1	yes	characteristic
105	C₂	—	2	3	no	—
106	C₂	—	4	3	no	—
107	C₂	—	668	3	no	—
108	C₂	C₂²×C₂²⋊D₁₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128878

C₂

yes

maximal, characteristic, complemented

#19128858

C₂

yes

maximal, complemented

#19128857

C₂

yes

maximal, complemented

#19128852

C₂

yes

maximal, complemented

C₂×#14581935

C₂

yes

maximal, complemented

#19128839

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₆₇

C₂

yes

maximal, characteristic, complemented

#14581951

—

#14581950

C₂²

yes

complemented

C₂×D₆₆₈

C₂²

yes

complemented

D₄×D₁₆₇

—

#14581940

C₂²

yes

characteristic, complemented

#14581938

C₂²

yes

complemented

#14581937

C₂²

yes

complemented

#14581935

C₂²

yes

complemented

#14581935

C₂²

yes

complemented

#14581934

C₂²

yes

complemented

C₂×C₁₆₇⋊C₈

C₂²

yes

complemented

#41×C₁₆₇

—

#41×C₁₆₇

C₂²

yes

characteristic

#41×C₁₆₇

C₂²

yes

complemented

C₂×D₄×C₁₆₇

—

C₂×D₄×C₁₆₇

C₂²

yes

complemented

C₂²×D₁₆₇

—

Dic₃₃₄

C₂³

yes

characteristic

D₆₆₈

—

D₆₆₈

C₂³

yes

—

C₂²⋊D₁₆₇

—

C₄×D₁₆₇

—

C₁₆₇⋊C₈

C₂³

yes

characteristic

C₁₆₇⋊C₈

C₂³

yes

—

C₂³×C₁₆₇

—

Q₈×C₁₆₇

D₄

yes

characteristic

Q₈×C₁₆₇

C₂³

yes

characteristic

D₄×C₁₆₇

—

D₄×C₁₆₇

—

D₄×C₁₆₇

D₄

yes

—

D₄×C₁₆₇

C₂³

yes

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂³

yes

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂×D₄

yes

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×D₄

yes

—

C₄×C₁₆₇

C₂⁴

yes

characteristic

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

C₂²×D₄

yes

characteristic

D₁₆₇

—

C₁₆₇

#576

yes

characteristic, complemented, Sylow

#576

—

167

maximal, Sylow

H(5,𝔽₂)

—

167

—

H(5,𝔽₂)

D₁₆₇

yes

characteristic, complemented

AGL(1,ℤ/8)

—

167

—

AGL(1,ℤ/8)

—

167

—

#135

—

167

—

C₂×D₈

—

167

—

#131

—

167

—

#41

—

#41

—

334

—

#41

—

167

—

#41

C₂×D₁₆₇

yes

characteristic

#41

C₂×D₁₆₇

yes

complemented

C₂×D₄

—

C₂×D₄

—

334

—

C₂×D₄

—

167

—

C₂×D₄

C₂×D₁₆₇

yes

complemented

Q₁₆

—

167

—

SD₁₆

—

167

—

SD₁₆

—

167

—

D₈

—

167

—

D₈

—

167

—

M₁₆

—

167

—

C₂×C₈

—

167

—

C₂³

—

C₂³

—

334

—

Q₈

—

167

—

Q₈

C₂²⋊D₁₆₇

yes

characteristic

Q₈

C₂²×D₁₆₇

yes

characteristic

D₄

—

D₄

—

D₄

—

334

—

D₄

—

334

—

D₄

—

167

—

D₄

C₂²⋊D₁₆₇

yes

—

D₄

C₂²×D₁₆₇

yes

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

C₂²×D₁₆₇

yes

—

C₈

—

167

—

C₈

—

167

—

C₂²

—

C₂²

—

C₂²

—

1336

—

C₂²

—

334

—

100

C₂²

C₂×C₂²⋊D₁₆₇

yes

—

101

C₄

—

102

C₄

—

334

—

103

C₄

C₂×C₂²⋊D₁₆₇

yes

—

104

C₄

C₂³×D₁₆₇

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

668

—

108

C₂

C₂²×C₂²⋊D₁₆₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₆₇
commutator, cosocle	C₄×C₁₆₇	C₂⁴
Frattini	C₄	C₂³×D₁₆₇
Fitting	H(5,𝔽₂)×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₆₇
hypocenter	C₁₆₇	#576
LCS	G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₆₇

commutator, cosocle

C₄×C₁₆₇

C₂⁴

Frattini

C₄

C₂³×D₁₆₇

Fitting

H(5,𝔽₂)×C₁₆₇

C₂

socle

C₂×C₁₆₇

C₂²×D₄

D₁₆₇

C₁₆₇

G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #19128878	C₂	semidirect
2. #19128858	C₂	semidirect
3. #19128857	C₂	semidirect
4. #19128852	C₂	semidirect
5. C₂×#14581935	C₂	semidirect
6. #19128839	C₂	semidirect
7. H(5,𝔽₂)×C₁₆₇	C₂	semidirect
9. #14581950	C₂²	semidirect
10. C₂×D₆₆₈	C₂²	semidirect
12. #14581940	C₂²	semidirect
13. #14581938	C₂²	semidirect
14. #14581937	C₂²	semidirect
15. #14581935	C₂²	semidirect
16. #14581935	C₂²	semidirect
17. #14581934	C₂²	semidirect
18. C₂×C₁₆₇⋊C₈	C₂²	semidirect
21. #41×C₁₆₇	C₂²	semidirect
23. C₂×D₄×C₁₆₇	C₂²	semidirect
54. C₁₆₇	#576	semidirect
57. H(5,𝔽₂)	D₁₆₇	semidirect
67. #41	C₂×D₁₆₇	semidirect
71. C₂×D₄	C₂×D₁₆₇	semidirect

H ≅ G/N

type

1. #19128878

C₂

semidirect

2. #19128858

C₂

semidirect

3. #19128857

C₂

semidirect

4. #19128852

C₂

semidirect

5. C₂×#14581935

C₂

semidirect

6. #19128839

C₂

semidirect

7. H(5,𝔽₂)×C₁₆₇

C₂

semidirect

9. #14581950

C₂²

semidirect

10. C₂×D₆₆₈

C₂²

semidirect

12. #14581940

C₂²

semidirect

13. #14581938

C₂²

semidirect

14. #14581937

C₂²

semidirect

15. #14581935

C₂²

semidirect

16. #14581935

C₂²

semidirect

17. #14581934

C₂²

semidirect

18. C₂×C₁₆₇⋊C₈

C₂²

semidirect

21. #41×C₁₆₇

C₂²

semidirect

23. C₂×D₄×C₁₆₇

C₂²

semidirect

54. C₁₆₇

#576

semidirect

57. H(5,𝔽₂)

D₁₆₇

semidirect

67. #41

C₂×D₁₆₇

semidirect

71. C₂×D₄

C₂×D₁₆₇

semidirect

Subgroups — #24924072

Conjugacy classes of subgroups

Complements