Group #24924083 — subgroups

Subgroups — #24924083

Order 10688.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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	2	yes	maximal, complemented
2	#19128862	C₂	1	2	yes	maximal, complemented
3	#19128861	C₂	1	4	yes	maximal, complemented
4	#19128851	C₂	1	2	yes	maximal, complemented
5	C₂×Dic₆₆₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×D₁₆₇	C₂	1	2	yes	maximal, complemented
7	#19128838	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14581950	—	2	2	no	—
10	#14581950	C₂²	1	1	yes	characteristic
11	#14581949	—	2	4	no	—
12	#14581949	C₂²	1	4	yes	—
13	C₂×Dic₃₃₄	—	2	2	no	—
14	C₂×Dic₃₃₄	C₂²	1	2	yes	—
15	Q₈×D₁₆₇	C₂²	1	4	yes	—
16	Dic₆₆₈	C₂²	1	2	yes	—
17	Dic₆₆₈	C₂²	1	2	yes	—
18	#14581940	C₂²	1	4	yes	—
19	#14581938	C₂²	1	4	yes	—
20	#14581934	C₂²	1	1	yes	characteristic
21	#14581933	C₂²	1	2	yes	—
22	C₈×D₁₆₇	C₂²	1	2	yes	—
23	#41×C₁₆₇	C₂²	1	2	yes	—
24	Q₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
25	SD₁₆×C₁₆₇	C₂²	1	2	yes	—
26	D₈×C₁₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
28	Dic₃₃₄	—	2	2	no	—
29	Dic₃₃₄	—	2	4	no	—
30	Dic₃₃₄	D₄	1	1	yes	characteristic
31	Dic₃₃₄	C₂³	1	4	yes	—
32	D₆₆₈	D₄	1	1	yes	characteristic
33	C₂²⋊D₁₆₇	—	2	4	no	—
34	C₂²⋊D₁₆₇	D₄	1	2	yes	—
35	C₂×Dic₁₆₇	—	2	2	no	—
36	C₂×Dic₁₆₇	—	2	4	no	—
37	C₄×D₁₆₇	—	2	4	no	—
38	C₄×D₁₆₇	C₂³	1	2	yes	—
39	C₁₆₇⋊C₈	C₂³	1	2	yes	—
40	Q₈×C₁₆₇	C₂³	1	2	yes	—
41	D₄×C₁₆₇	—	2	2	no	—
42	D₄×C₁₆₇	C₂³	1	2	yes	—
43	C₂×C₄×C₁₆₇	—	2	2	no	—
44	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
45	C₈×C₁₆₇	C₂³	1	1	yes	characteristic
46	C₈×C₁₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₁₆₇	—	2	2	no	—
48	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₆₇	C₂×D₄	1	2	yes	—
50	C₄×C₁₆₇	—	2	2	no	—
51	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
52	C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
53	Dic₁₆₇	—	2	4	no	—
54	Dic₁₆₇	C₂×D₄	1	2	yes	—
55	C₂×C₁₆₇	—	2	1	no	—
56	C₂×C₁₆₇	—	4	2	no	—
57	C₂×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
58	D₁₆₇	—	2	2	no	—
59	C₁₆₇	#578	1	1	yes	characteristic, complemented, Sylow
60	#578	—	167	1	no	maximal, Sylow
61	2₋¹⁺⁴	—	167	2	no	—
62	#137	—	167	2	no	—
63	#137	—	167	4	no	—
64	#135	—	167	2	no	—
65	#135	D₁₆₇	1	1	yes	characteristic, complemented
66	C₂×Q₁₆	—	167	2	no	—
67	C₂×Q₁₆	—	167	1	no	—
68	#131	—	167	1	no	—
69	#41	—	334	2	no	—
70	#41	—	334	4	no	—
71	#41	—	167	4	no	—
72	#41	—	167	1	no	—
73	#41	C₂×D₁₆₇	1	2	yes	—
74	C₂×Q₈	—	334	2	no	—
75	C₂×Q₈	—	167	4	no	—
76	C₂×Q₈	—	167	2	no	—
77	Q₁₆	—	167	4	no	—
78	Q₁₆	—	167	2	no	—
79	Q₁₆	—	167	2	no	—
80	Q₁₆	C₂×D₁₆₇	1	1	yes	characteristic
81	SD₁₆	—	167	4	no	—
82	SD₁₆	C₂×D₁₆₇	1	2	yes	—
83	D₈	C₂×D₁₆₇	1	1	yes	characteristic
84	M₁₆	—	167	1	no	—
85	M₁₆	—	167	2	no	—
86	C₂×C₈	—	167	2	no	—
87	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
88	Q₈	—	334	4	no	—
89	Q₈	—	334	2	no	—
90	Q₈	—	167	4	no	—
91	Q₈	—	167	1	no	—
92	Q₈	C₂²×D₁₆₇	1	2	yes	—
93	D₄	—	2	2	no	—
94	D₄	—	334	4	no	—
95	D₄	—	167	2	no	—
96	D₄	—	167	1	no	—
97	D₄	C₂²×D₁₆₇	1	2	yes	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	334	4	no	—
100	C₂×C₄	—	334	2	no	—
101	C₂×C₄	—	334	4	no	—
102	C₂×C₄	—	167	2	no	—
103	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
104	C₈	—	167	2	no	—
105	C₈	C₂²×D₁₆₇	1	1	yes	characteristic
106	C₈	C₂²×D₁₆₇	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	167	2	no	—
109	C₂²	D₄×D₁₆₇	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	334	4	no	—
112	C₄	—	167	2	no	—
113	C₄	D₄×D₁₆₇	1	1	yes	characteristic
114	C₄	C₂³×D₁₆₇	1	1	yes	characteristic
115	C₂	—	2	1	no	—
116	C₂	—	4	2	no	—
117	C₂	—	334	2	no	—
118	C₂	C₂×D₄×D₁₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128879

C₂

yes

maximal, complemented

#19128862

C₂

yes

maximal, complemented

#19128861

C₂

yes

maximal, complemented

#19128851

C₂

yes

maximal, complemented

C₂×Dic₆₆₈

C₂

yes

maximal, characteristic, complemented

Q₁₆×D₁₆₇

C₂

yes

maximal, complemented

#19128838

C₂

yes

maximal, characteristic, complemented

#135×C₁₆₇

C₂

yes

maximal, characteristic, complemented

#14581950

—

#14581950

C₂²

yes

characteristic

#14581949

—

#14581949

C₂²

yes

—

C₂×Dic₃₃₄

—

C₂×Dic₃₃₄

C₂²

yes

—

Q₈×D₁₆₇

C₂²

yes

—

Dic₆₆₈

C₂²

yes

—

Dic₆₆₈

C₂²

yes

—

#14581940

C₂²

yes

—

#14581938

C₂²

yes

—

#14581934

C₂²

yes

characteristic

#14581933

C₂²

yes

—

C₈×D₁₆₇

C₂²

yes

—

#41×C₁₆₇

C₂²

yes

—

Q₁₆×C₁₆₇

C₂²

yes

characteristic

SD₁₆×C₁₆₇

C₂²

yes

—

D₈×C₁₆₇

C₂²

yes

characteristic

C₂×C₈×C₁₆₇

C₂²

yes

characteristic

Dic₃₃₄

—

Dic₃₃₄

—

Dic₃₃₄

D₄

yes

characteristic

Dic₃₃₄

C₂³

yes

—

D₆₆₈

D₄

yes

characteristic

C₂²⋊D₁₆₇

—

C₂²⋊D₁₆₇

D₄

yes

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₄×D₁₆₇

—

C₄×D₁₆₇

C₂³

yes

—

C₁₆₇⋊C₈

C₂³

yes

—

Q₈×C₁₆₇

C₂³

yes

—

D₄×C₁₆₇

—

D₄×C₁₆₇

C₂³

yes

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₈×C₁₆₇

C₂³

yes

characteristic

C₈×C₁₆₇

C₂³

yes

characteristic

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂×D₄

yes

characteristic

C₂×D₁₆₇

C₂×D₄

yes

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₄×C₁₆₇

C₂⁴

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

C₂×D₄

yes

—

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

—

#137

—

167

—

#137

—

167

—

#135

—

167

—

#135

D₁₆₇

yes

characteristic, complemented

C₂×Q₁₆

—

167

—

C₂×Q₁₆

—

167

—

#131

—

167

—

#41

—

334

—

#41

—

334

—

#41

—

167

—

#41

—

167

—

#41

C₂×D₁₆₇

yes

—

C₂×Q₈

—

334

—

C₂×Q₈

—

167

—

C₂×Q₈

—

167

—

Q₁₆

—

167

—

Q₁₆

—

167

—

Q₁₆

—

167

—

Q₁₆

C₂×D₁₆₇

yes

characteristic

SD₁₆

—

167

—

SD₁₆

C₂×D₁₆₇

yes

—

D₈

C₂×D₁₆₇

yes

characteristic

M₁₆

—

167

—

M₁₆

—

167

—

C₂×C₈

—

167

—

C₂×C₈

C₂×D₁₆₇

yes

characteristic

Q₈

—

334

—

Q₈

—

334

—

Q₈

—

167

—

Q₈

—

167

—

Q₈

C₂²×D₁₆₇

yes

—

D₄

—

D₄

—

334

—

D₄

—

167

—

D₄

—

167

—

D₄

C₂²×D₁₆₇

yes

—

C₂×C₄

—

C₂×C₄

—

334

—

100

C₂×C₄

—

334

—

101

C₂×C₄

—

334

—

102

C₂×C₄

—

167

—

103

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

104

C₈

—

167

—

105

C₈

C₂²×D₁₆₇

yes

characteristic

106

C₈

C₂²×D₁₆₇

yes

characteristic

107

C₂²

—

108

C₂²

—

167

—

109

C₂²

D₄×D₁₆₇

yes

characteristic

110

C₄

—

111

C₄

—

334

—

112

C₄

—

167

—

113

C₄

D₄×D₁₆₇

yes

characteristic

114

C₄

C₂³×D₁₆₇

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

334

—

118

C₂

C₂×D₄×D₁₆₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₁₆₇
commutator, cosocle	C₄×C₁₆₇	C₂⁴
Frattini	C₄	C₂³×D₁₆₇
Fitting	#135×C₁₆₇	C₂
socle	C₂×C₁₆₇	C₂²×D₄
hypercenter	#135	D₁₆₇
hypocenter	C₁₆₇	#578
LCS	G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₂

C₂×D₄×D₁₆₇

commutator, cosocle

C₄×C₁₆₇

C₂⁴

Frattini

C₄

C₂³×D₁₆₇

Fitting

#135×C₁₆₇

C₂

socle

C₂×C₁₆₇

C₂²×D₄

D₁₆₇

C₁₆₇

G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #19128879	C₂	semidirect
2. #19128862	C₂	semidirect
3. #19128861	C₂	semidirect
4. #19128851	C₂	semidirect
5. C₂×Dic₆₆₈	C₂	semidirect
6. Q₁₆×D₁₆₇	C₂	semidirect
7. #19128838	C₂	semidirect
8. #135×C₁₆₇	C₂	semidirect
59. C₁₆₇	#578	semidirect
65. #135	D₁₆₇	semidirect

H ≅ G/N

type

1. #19128879

C₂

semidirect

2. #19128862

C₂

semidirect

3. #19128861

C₂

semidirect

4. #19128851

C₂

semidirect

5. C₂×Dic₆₆₈

C₂

semidirect

6. Q₁₆×D₁₆₇

C₂

semidirect

7. #19128838

C₂

semidirect

8. #135×C₁₆₇

C₂

semidirect

59. C₁₆₇

#578

semidirect

65. #135

D₁₆₇

semidirect

Subgroups — #24924083

Conjugacy classes of subgroups

Complements