Group #24924105 — subgroups

Subgroups — #24924105

Order 10688.

810 conjugacy classes of subgroups; only the 443 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 443 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	#19128880	C₂	1	2	yes	maximal, complemented
2	#19128879	C₂	1	4	yes	maximal, complemented
3	#19128878	C₂	1	4	yes	maximal, complemented
4	#19128877	C₂	1	6	yes	maximal, complemented
5	C₂×#14581950	C₂	1	6	yes	maximal, complemented
6	#41×D₁₆₇	C₂	1	8	yes	maximal, complemented
7	C₂×#41×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581951	C₂²	1	8	yes	complemented
9	#14581950	C₂²	1	12	yes	complemented
10	#14581950	C₂²	1	4	yes	complemented
11	#14581950	C₂²	1	12	yes	complemented
12	#14581950	C₂²	1	12	yes	complemented
13	#14581949	C₂²	1	24	yes	complemented
14	C₂×Dic₃₃₄	C₂²	1	6	yes	complemented
15	Q₈×D₁₆₇	C₂²	1	8	yes	complemented
16	C₂×D₆₆₈	C₂²	1	6	yes	complemented
17	C₂×C₂²⋊D₁₆₇	C₂²	1	12	yes	complemented
18	D₄×D₁₆₇	C₂²	1	24	yes	complemented
19	C₂×C₄×D₁₆₇	C₂²	1	12	yes	complemented
20	#41×C₁₆₇	C₂²	1	4	yes	complemented
21	#41×C₁₆₇	C₂²	1	4	yes	complemented
22	C₂×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×D₄×C₁₆₇	C₂²	1	3	yes	complemented
24	C₂²×C₄×C₁₆₇	C₂²	1	3	yes	complemented
25	C₂²×D₁₆₇	C₂³	1	12	yes	—
26	Dic₃₃₄	C₂³	1	12	yes	—
27	Dic₃₃₄	C₂³	1	4	yes	—
28	D₆₆₈	C₂³	1	12	yes	—
29	D₆₆₈	C₂³	1	4	yes	—
30	C₂²⋊D₁₆₇	C₂³	1	24	yes	—
31	C₂²⋊D₁₆₇	C₂³	1	8	yes	—
32	C₂×Dic₁₆₇	C₂³	1	12	yes	—
33	C₄×D₁₆₇	C₂³	1	8	yes	—
34	C₄×D₁₆₇	C₂³	1	24	yes	—
35	C₂³×C₁₆₇	C₂³	1	3	yes	—
36	Q₈×C₁₆₇	C₂³	1	4	yes	—
37	D₄×C₁₆₇	C₂³	1	12	yes	—
38	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₆₇	C₂³	1	3	yes	—
40	C₂×C₄×C₁₆₇	C₂³	1	6	yes	—
41	C₂×C₄×C₁₆₇	C₂³	1	6	yes	—
42	C₂²×C₁₆₇	—	4	3	no	—
43	C₂²×C₁₆₇	C₂⁴	1	1	yes	characteristic
44	C₂²×C₁₆₇	C₂⁴	1	6	yes	—
45	C₂×D₁₆₇	—	4	12	no	—
46	C₂×D₁₆₇	C₂⁴	1	8	yes	—
47	C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
48	C₄×C₁₆₇	C₂⁴	1	6	yes	—
49	C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
50	Dic₁₆₇	C₂⁴	1	8	yes	—
51	C₂×C₁₆₇	—	2	6	no	—
52	C₂×C₁₆₇	—	2	1	no	—
53	C₂×C₁₆₇	C₂⁵	1	1	yes	characteristic
54	D₁₆₇	—	2	8	no	—
55	C₁₆₇	#585	1	1	yes	characteristic, complemented, Sylow
56	#585	—	167	1	no	maximal, Sylow
57	2₋¹⁺⁴	—	167	4	no	—
58	2₋¹⁺⁴	—	167	2	no	—
59	H(5,𝔽₂)	—	167	4	no	—
60	H(5,𝔽₂)	—	167	6	no	—
61	C₂×#41	—	167	8	no	—
62	C₂×#41	—	167	6	no	—
63	C₂×#41	D₁₆₇	1	1	yes	characteristic, complemented
64	#41	—	167	8	no	—
65	#41	—	167	24	no	—
66	#41	—	167	12	no	—
67	#41	—	167	4	no	—
68	#41	—	167	12	no	—
69	#41	—	167	12	no	—
70	#41	C₂×D₁₆₇	1	4	yes	complemented
71	#41	C₂×D₁₆₇	1	4	yes	complemented
72	C₂×Q₈	—	167	8	no	—
73	C₂×Q₈	—	167	6	no	—
74	C₂×Q₈	C₂×D₁₆₇	1	1	yes	characteristic, complemented
75	C₂×D₄	—	167	12	no	—
76	C₂×D₄	—	167	24	no	—
77	C₂×D₄	—	167	6	no	—
78	C₂×D₄	C₂×D₁₆₇	1	3	yes	complemented
79	C₂²×C₄	—	167	12	no	—
80	C₂²×C₄	C₂×D₁₆₇	1	3	yes	complemented
81	C₂³	—	167	12	no	—
82	C₂³	C₂²×D₁₆₇	1	3	yes	—
83	Q₈	—	167	12	no	—
84	Q₈	—	167	4	no	—
85	Q₈	C₂²×D₁₆₇	1	4	yes	—
86	D₄	—	167	24	no	—
87	D₄	—	167	8	no	—
88	D₄	—	167	12	no	—
89	D₄	—	167	4	no	—
90	D₄	C₂²×D₁₆₇	1	12	yes	—
91	C₂×C₄	—	167	12	no	—
92	C₂×C₄	—	167	24	no	—
93	C₂×C₄	—	167	8	no	—
94	C₂×C₄	C₂²×D₁₆₇	1	6	yes	—
95	C₂×C₄	C₂²×D₁₆₇	1	3	yes	—
96	C₂×C₄	C₂²×D₁₆₇	1	6	yes	—
97	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
98	C₂²	—	4	3	no	—
99	C₂²	—	668	12	no	—
100	C₂²	—	167	8	no	—
101	C₂²	C₂³×D₁₆₇	1	6	yes	—
102	C₂²	C₂³×D₁₆₇	1	1	yes	characteristic
103	C₄	—	167	8	no	—
104	C₄	C₂³×D₁₆₇	1	6	yes	—
105	C₄	C₂³×D₁₆₇	1	1	yes	characteristic
106	C₄	C₂³×D₁₆₇	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	6	no	—
109	C₂	—	334	8	no	—
110	C₂	C₂⁴×D₁₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128880

C₂

yes

maximal, complemented

#19128879

C₂

yes

maximal, complemented

#19128878

C₂

yes

maximal, complemented

#19128877

C₂

yes

maximal, complemented

C₂×#14581950

C₂

yes

maximal, complemented

#41×D₁₆₇

C₂

yes

maximal, complemented

C₂×#41×C₁₆₇

C₂

yes

maximal, characteristic, complemented

#14581951

C₂²

yes

complemented

#14581950

C₂²

yes

complemented

#14581950

C₂²

yes

complemented

#14581950

C₂²

yes

complemented

#14581950

C₂²

yes

complemented

#14581949

C₂²

yes

complemented

C₂×Dic₃₃₄

C₂²

yes

complemented

Q₈×D₁₆₇

C₂²

yes

complemented

C₂×D₆₆₈

C₂²

yes

complemented

C₂×C₂²⋊D₁₆₇

C₂²

yes

complemented

D₄×D₁₆₇

C₂²

yes

complemented

C₂×C₄×D₁₆₇

C₂²

yes

complemented

#41×C₁₆₇

C₂²

yes

complemented

#41×C₁₆₇

C₂²

yes

complemented

C₂×Q₈×C₁₆₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₁₆₇

C₂²

yes

complemented

C₂²×C₄×C₁₆₇

C₂²

yes

complemented

C₂²×D₁₆₇

C₂³

yes

—

Dic₃₃₄

C₂³

yes

—

Dic₃₃₄

C₂³

yes

—

D₆₆₈

C₂³

yes

—

D₆₆₈

C₂³

yes

—

C₂²⋊D₁₆₇

C₂³

yes

—

C₂²⋊D₁₆₇

C₂³

yes

—

C₂×Dic₁₆₇

C₂³

yes

—

C₄×D₁₆₇

C₂³

yes

—

C₄×D₁₆₇

C₂³

yes

—

C₂³×C₁₆₇

C₂³

yes

—

Q₈×C₁₆₇

C₂³

yes

—

D₄×C₁₆₇

C₂³

yes

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

—

C₂×C₄×C₁₆₇

C₂³

yes

—

C₂×C₄×C₁₆₇

C₂³

yes

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂⁴

yes

characteristic

C₂²×C₁₆₇

C₂⁴

yes

—

C₂×D₁₆₇

—

C₂×D₁₆₇

C₂⁴

yes

—

C₄×C₁₆₇

C₂⁴

yes

characteristic

C₄×C₁₆₇

C₂⁴

yes

—

C₄×C₁₆₇

C₂⁴

yes

characteristic

Dic₁₆₇

C₂⁴

yes

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

C₂⁵

yes

characteristic

D₁₆₇

—

C₁₆₇

#585

yes

characteristic, complemented, Sylow

#585

—

167

maximal, Sylow

2₋¹⁺⁴

—

167

—

2₋¹⁺⁴

—

167

—

H(5,𝔽₂)

—

167

—

H(5,𝔽₂)

—

167

—

C₂×#41

—

167

—

C₂×#41

—

167

—

C₂×#41

D₁₆₇

yes

characteristic, complemented

#41

—

167

—

#41

—

167

—

#41

—

167

—

#41

—

167

—

#41

—

167

—

#41

—

167

—

#41

C₂×D₁₆₇

yes

complemented

#41

C₂×D₁₆₇

yes

complemented

C₂×Q₈

—

167

—

C₂×Q₈

—

167

—

C₂×Q₈

C₂×D₁₆₇

yes

characteristic, complemented

C₂×D₄

—

167

—

C₂×D₄

—

167

—

C₂×D₄

—

167

—

C₂×D₄

C₂×D₁₆₇

yes

complemented

C₂²×C₄

—

167

—

C₂²×C₄

C₂×D₁₆₇

yes

complemented

C₂³

—

167

—

C₂³

C₂²×D₁₆₇

yes

—

Q₈

—

167

—

Q₈

—

167

—

Q₈

C₂²×D₁₆₇

yes

—

D₄

—

167

—

D₄

—

167

—

D₄

—

167

—

D₄

—

167

—

D₄

C₂²×D₁₆₇

yes

—

C₂×C₄

—

167

—

C₂×C₄

—

167

—

C₂×C₄

—

167

—

C₂×C₄

C₂²×D₁₆₇

yes

—

C₂×C₄

C₂²×D₁₆₇

yes

—

C₂×C₄

C₂²×D₁₆₇

yes

—

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₂²

—

C₂²

—

668

—

100

C₂²

—

167

—

101

C₂²

C₂³×D₁₆₇

yes

—

102

C₂²

C₂³×D₁₆₇

yes

characteristic

103

C₄

—

167

—

104

C₄

C₂³×D₁₆₇

yes

—

105

C₄

C₂³×D₁₆₇

yes

characteristic

106

C₄

C₂³×D₁₆₇

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

334

—

110

C₂

C₂⁴×D₁₆₇

yes

characteristic

	subgroup	G/H
center	C₄	C₂³×D₁₆₇
commutator, socle, cosocle	C₂×C₁₆₇	C₂⁵
Frattini	C₂	C₂⁴×D₁₆₇
Fitting	C₂×#41×C₁₆₇	C₂
hypercenter	C₂×#41	D₁₆₇
hypocenter	C₁₆₇	#585
LCS	G ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₄ ◁ C₂×#41

subgroup

G/H

center

C₄

C₂³×D₁₆₇

commutator, socle, cosocle

C₂×C₁₆₇

C₂⁵

Frattini

C₂

C₂⁴×D₁₆₇

Fitting

C₂×#41×C₁₆₇

C₂

hypercenter

C₂×#41

D₁₆₇

C₁₆₇

G ▷ C₂×C₁₆₇ ▷ C₁₆₇

C₁ ◁ C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. #19128880	C₂	semidirect
2. #19128879	C₂	semidirect
3. #19128878	C₂	semidirect
4. #19128877	C₂	semidirect
5. C₂×#14581950	C₂	semidirect
6. #41×D₁₆₇	C₂	semidirect
7. C₂×#41×C₁₆₇	C₂	semidirect
8. #14581951	C₂²	semidirect
9. #14581950	C₂²	semidirect
10. #14581950	C₂²	semidirect
11. #14581950	C₂²	semidirect
12. #14581950	C₂²	semidirect
13. #14581949	C₂²	semidirect
14. C₂×Dic₃₃₄	C₂²	semidirect
15. Q₈×D₁₆₇	C₂²	semidirect
16. C₂×D₆₆₈	C₂²	semidirect
17. C₂×C₂²⋊D₁₆₇	C₂²	semidirect
18. D₄×D₁₆₇	C₂²	semidirect
19. C₂×C₄×D₁₆₇	C₂²	semidirect
20. #41×C₁₆₇	C₂²	semidirect
21. #41×C₁₆₇	C₂²	semidirect
22. C₂×Q₈×C₁₆₇	C₂²	semidirect
23. C₂×D₄×C₁₆₇	C₂²	semidirect
24. C₂²×C₄×C₁₆₇	C₂²	semidirect
55. C₁₆₇	#585	semidirect
63. C₂×#41	D₁₆₇	semidirect
70. #41	C₂×D₁₆₇	semidirect
71. #41	C₂×D₁₆₇	semidirect
74. C₂×Q₈	C₂×D₁₆₇	semidirect
78. C₂×D₄	C₂×D₁₆₇	semidirect
80. C₂²×C₄	C₂×D₁₆₇	semidirect

H ≅ G/N

type

1. #19128880

C₂

semidirect

2. #19128879

C₂

semidirect

3. #19128878

C₂

semidirect

4. #19128877

C₂

semidirect

5. C₂×#14581950

C₂

semidirect

6. #41×D₁₆₇

C₂

semidirect

7. C₂×#41×C₁₆₇

C₂

semidirect

8. #14581951

C₂²

semidirect

9. #14581950

C₂²

semidirect

10. #14581950

C₂²

semidirect

11. #14581950

C₂²

semidirect

12. #14581950

C₂²

semidirect

13. #14581949

C₂²

semidirect

14. C₂×Dic₃₃₄

C₂²

semidirect

15. Q₈×D₁₆₇

C₂²

semidirect

16. C₂×D₆₆₈

C₂²

semidirect

17. C₂×C₂²⋊D₁₆₇

C₂²

semidirect

18. D₄×D₁₆₇

C₂²

semidirect

19. C₂×C₄×D₁₆₇

C₂²

semidirect

20. #41×C₁₆₇

C₂²

semidirect

21. #41×C₁₆₇

C₂²

semidirect

22. C₂×Q₈×C₁₆₇

C₂²

semidirect

23. C₂×D₄×C₁₆₇

C₂²

semidirect

24. C₂²×C₄×C₁₆₇

C₂²

semidirect

55. C₁₆₇

#585

semidirect

63. C₂×#41

D₁₆₇

semidirect

70. #41

C₂×D₁₆₇

semidirect

71. #41

C₂×D₁₆₇

semidirect

74. C₂×Q₈

C₂×D₁₆₇

semidirect

78. C₂×D₄

C₂×D₁₆₇

semidirect

80. C₂²×C₄

C₂×D₁₆₇

semidirect

Subgroups — #24924105

Conjugacy classes of subgroups

Complements