Group #24924084 — subgroups

Subgroups — #24924084

Order 10688.

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

normal subgroups only

This group has 248 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	#19128863	C₂	1	3	yes	maximal, complemented
3	#19128862	C₂	1	3	yes	maximal, complemented
4	#19128852	C₂	1	3	yes	maximal, complemented
5	C₂×#14581940	C₂	1	3	yes	maximal, complemented
6	#19128839	C₂	1	1	yes	maximal, characteristic, complemented
7	2₋¹⁺⁴×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581950	C₂²	1	3	yes	—
9	#14581949	—	2	3	no	—
10	C₂×Dic₃₃₄	C₂²	1	3	yes	—
11	Q₈×D₁₆₇	—	2	1	no	—
12	#14581940	C₂²	1	3	yes	—
13	#14581940	C₂²	1	6	yes	—
14	#14581938	C₂²	1	3	yes	—
15	#14581937	C₂²	1	3	yes	—
16	#14581935	C₂²	1	1	yes	characteristic
17	#14581934	C₂²	1	3	yes	—
18	C₂×C₁₆₇⋊C₈	C₂²	1	3	yes	—
19	#41×C₁₆₇	—	2	3	no	—
20	#41×C₁₆₇	C₂²	1	3	yes	—
21	#41×C₁₆₇	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₁₆₇	—	2	1	no	—
23	C₂×Q₈×C₁₆₇	C₂²	1	3	yes	—
24	Dic₃₃₄	—	2	3	no	—
25	Dic₃₃₄	C₂³	1	3	yes	—
26	D₆₆₈	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₆₇	—	2	3	no	—
28	C₂×Dic₁₆₇	—	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	Q₈×C₁₆₇	—	2	3	no	—
33	Q₈×C₁₆₇	D₄	1	1	yes	characteristic
34	Q₈×C₁₆₇	C₂³	1	3	yes	—
35	D₄×C₁₆₇	—	2	3	no	—
36	D₄×C₁₆₇	D₄	1	3	yes	—
37	D₄×C₁₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₆₇	—	2	3	no	—
39	C₂×C₄×C₁₆₇	—	2	3	no	—
40	C₂×C₄×C₁₆₇	C₂³	1	3	yes	—
41	C₂²×C₁₆₇	—	2	1	no	—
42	C₂²×C₁₆₇	C₂×D₄	1	3	yes	—
43	C₂×D₁₆₇	—	2	1	no	—
44	C₄×C₁₆₇	—	2	3	no	—
45	C₄×C₁₆₇	C₂×D₄	1	3	yes	—
46	C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
47	Dic₁₆₇	—	2	3	no	—
48	C₂×C₁₆₇	—	2	3	no	—
49	C₂×C₁₆₇	—	4	1	no	—
50	C₂×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
51	D₁₆₇	—	4	1	no	—
52	C₁₆₇	#578	1	1	yes	characteristic, complemented, Sylow
53	#578	—	167	1	no	maximal, Sylow
54	2₋¹⁺⁴	—	167	1	no	—
55	2₋¹⁺⁴	D₁₆₇	1	1	yes	characteristic, complemented
56	#137	—	167	3	no	—
57	#137	—	167	3	no	—
58	#135	—	167	3	no	—
59	C₂×Q₁₆	—	167	3	no	—
60	#131	—	167	1	no	—
61	#41	—	2	3	no	—
62	#41	—	334	3	no	—
63	#41	—	167	3	no	—
64	#41	C₂×D₁₆₇	1	3	yes	—
65	#41	C₂×D₁₆₇	1	1	yes	characteristic
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	334	1	no	—
68	C₂×Q₈	—	167	3	no	—
69	C₂×Q₈	C₂×D₁₆₇	1	3	yes	—
70	Q₁₆	—	167	3	no	—
71	Q₁₆	—	167	6	no	—
72	SD₁₆	—	167	3	no	—
73	SD₁₆	—	167	3	no	—
74	D₈	—	167	1	no	—
75	M₁₆	—	167	3	no	—
76	C₂×C₈	—	167	3	no	—
77	Q₈	—	2	3	no	—
78	Q₈	—	334	3	no	—
79	Q₈	—	167	3	no	—
80	Q₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
81	Q₈	C₂²×D₁₆₇	1	3	yes	—
82	D₄	—	2	3	no	—
83	D₄	—	334	3	no	—
84	D₄	—	167	1	no	—
85	D₄	C₂²⋊D₁₆₇	1	3	yes	—
86	D₄	C₂²×D₁₆₇	1	1	yes	characteristic
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	334	3	no	—
90	C₂×C₄	—	334	3	no	—
91	C₂×C₄	C₂²×D₁₆₇	1	3	yes	—
92	C₈	—	167	1	no	—
93	C₈	—	167	3	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	334	1	no	—
96	C₂²	C₂×C₂²⋊D₁₆₇	1	3	yes	—
97	C₄	—	2	3	no	—
98	C₄	—	334	3	no	—
99	C₄	C₂×C₂²⋊D₁₆₇	1	3	yes	—
100	C₄	C₂³×D₁₆₇	1	1	yes	characteristic
101	C₂	—	2	3	no	—
102	C₂	—	4	1	no	—
103	C₂	—	668	1	no	—
104	C₂	C₂²×C₂²⋊D₁₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128879

C₂

yes

maximal, characteristic, complemented

#19128863

C₂

yes

maximal, complemented

#19128862

C₂

yes

maximal, complemented

#19128852

C₂

yes

maximal, complemented

C₂×#14581940

C₂

yes

maximal, complemented

#19128839

C₂

yes

maximal, characteristic, complemented

2₋¹⁺⁴×C₁₆₇

C₂

yes

maximal, characteristic, complemented

#14581950

C₂²

yes

—

#14581949

—

C₂×Dic₃₃₄

C₂²

yes

—

Q₈×D₁₆₇

—

#14581940

C₂²

yes

—

#14581940

C₂²

yes

—

#14581938

C₂²

yes

—

#14581937

C₂²

yes

—

#14581935

C₂²

yes

characteristic

#14581934

C₂²

yes

—

C₂×C₁₆₇⋊C₈

C₂²

yes

—

#41×C₁₆₇

—

#41×C₁₆₇

C₂²

yes

—

#41×C₁₆₇

C₂²

yes

characteristic

C₂×Q₈×C₁₆₇

—

C₂×Q₈×C₁₆₇

C₂²

yes

—

Dic₃₃₄

—

Dic₃₃₄

C₂³

yes

—

D₆₆₈

C₂³

yes

characteristic

C₂²⋊D₁₆₇

—

C₂×Dic₁₆₇

—

C₄×D₁₆₇

—

C₁₆₇⋊C₈

C₂³

yes

characteristic

C₁₆₇⋊C₈

C₂³

yes

—

Q₈×C₁₆₇

—

Q₈×C₁₆₇

D₄

yes

characteristic

Q₈×C₁₆₇

C₂³

yes

—

D₄×C₁₆₇

—

D₄×C₁₆₇

D₄

yes

—

D₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂³

yes

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂×D₄

yes

—

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

#578

yes

characteristic, complemented, Sylow

#578

—

167

maximal, Sylow

2₋¹⁺⁴

—

167

—

2₋¹⁺⁴

D₁₆₇

yes

characteristic, complemented

#137

—

167

—

#137

—

167

—

#135

—

167

—

C₂×Q₁₆

—

167

—

#131

—

167

—

#41

—

#41

—

334

—

#41

—

167

—

#41

C₂×D₁₆₇

yes

—

#41

C₂×D₁₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

334

—

C₂×Q₈

—

167

—

C₂×Q₈

C₂×D₁₆₇

yes

—

Q₁₆

—

167

—

Q₁₆

—

167

—

SD₁₆

—

167

—

SD₁₆

—

167

—

D₈

—

167

—

M₁₆

—

167

—

C₂×C₈

—

167

—

Q₈

—

Q₈

—

334

—

Q₈

—

167

—

Q₈

C₂²⋊D₁₆₇

yes

characteristic

Q₈

C₂²×D₁₆₇

yes

—

D₄

—

D₄

—

334

—

D₄

—

167

—

D₄

C₂²⋊D₁₆₇

yes

—

D₄

C₂²×D₁₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

C₂²×D₁₆₇

yes

—

C₈

—

167

—

C₈

—

167

—

C₂²

—

C₂²

—

334

—

C₂²

C₂×C₂²⋊D₁₆₇

yes

—

C₄

—

C₄

—

334

—

C₄

C₂×C₂²⋊D₁₆₇

yes

—

100

C₄

C₂³×D₁₆₇

yes

characteristic

101

C₂

—

102

C₂

—

103

C₂

—

668

—

104

C₂

C₂²×C₂²⋊D₁₆₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₆₇
commutator, cosocle	C₄×C₁₆₇	C₂⁴
Frattini	C₄	C₂³×D₁₆₇
Fitting	2₋¹⁺⁴×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₂²×D₄
hypercenter	2₋¹⁺⁴	D₁₆₇
hypocenter	C₁₆₇	#578
LCS	G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ 2₋¹⁺⁴

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₆₇

commutator, cosocle

C₄×C₁₆₇

C₂⁴

Frattini

C₄

C₂³×D₁₆₇

Fitting

2₋¹⁺⁴×C₁₆₇

C₂

socle

C₂×C₁₆₇

C₂²×D₄

hypercenter

2₋¹⁺⁴

D₁₆₇

hypocenter

C₁₆₇

#578

LCS

G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ 2₋¹⁺⁴

N	H ≅ G/N	type
1. #19128879	C₂	semidirect
2. #19128863	C₂	semidirect
3. #19128862	C₂	semidirect
4. #19128852	C₂	semidirect
5. C₂×#14581940	C₂	semidirect
6. #19128839	C₂	semidirect
7. 2₋¹⁺⁴×C₁₆₇	C₂	semidirect
52. C₁₆₇	#578	semidirect
55. 2₋¹⁺⁴	D₁₆₇	semidirect

H ≅ G/N

type

1. #19128879

C₂

semidirect

2. #19128863

C₂

semidirect

3. #19128862

C₂

semidirect

4. #19128852

C₂

semidirect

5. C₂×#14581940

C₂

semidirect

6. #19128839

C₂

semidirect

7. 2₋¹⁺⁴×C₁₆₇

C₂

semidirect

52. C₁₆₇

#578

semidirect

55. 2₋¹⁺⁴

D₁₆₇

semidirect

Subgroups — #24924084

Conjugacy classes of subgroups

Complements