Group #24923513 — subgroups

Subgroups — #24923513

Order 10688.

120 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 41 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	C₂×#14581938	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14581934	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128811	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128765	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128755	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128753	C₂	1	1	yes	maximal, characteristic, complemented
7	#121×C₁₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₃₄	C₂²	1	1	yes	characteristic
9	#14581938	—	2	2	no	—
10	#14581934	—	2	2	no	—
11	C₂×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₆₇	C₂²	1	1	yes	characteristic
14	Dic₁₆₇⋊C₄	—	2	1	no	—
15	C₂²⋊Dic₁₆₇	—	2	1	no	—
16	C₂×D₄×C₁₆₇	—	2	1	no	—
17	C₂×D₄×C₁₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₁₆₇	—	2	1	no	—
21	Dic₃₃₄	—	2	1	no	—
22	Dic₃₃₄	—	2	1	no	—
23	C₂×Dic₁₆₇	—	2	1	no	—
24	C₂×Dic₁₆₇	—	2	1	no	—
25	C₁₆₇⋊C₈	—	2	1	no	—
26	C₁₆₇⋊C₈	D₄	1	2	yes	complemented
27	C₂³×C₁₆₇	—	2	1	no	—
28	C₂³×C₁₆₇	D₄	1	1	yes	characteristic
29	D₄×C₁₆₇	—	2	1	no	—
30	D₄×C₁₆₇	—	2	1	no	—
31	D₄×C₁₆₇	—	4	2	no	—
32	C₂×C₄×C₁₆₇	—	2	1	no	—
33	C₂×C₄×C₁₆₇	—	2	1	no	—
34	C₂×C₄×C₁₆₇	—	2	1	no	—
35	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₁₆₇	—	2	1	no	—
38	C₂²×C₁₆₇	—	2	1	no	—
39	C₂²×C₁₆₇	—	2	1	no	—
40	C₂²×C₁₆₇	—	4	1	no	—
41	C₂²×C₁₆₇	—	4	1	no	—
42	C₂²×C₁₆₇	—	4	1	no	—
43	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
44	C₄×C₁₆₇	—	2	1	no	—
45	C₄×C₁₆₇	—	4	1	no	—
46	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₆₇	#41	1	1	yes	characteristic
48	Dic₁₆₇	—	4	1	no	—
49	Dic₁₆₇	—	4	1	no	—
50	C₂×C₁₆₇	—	4	1	no	—
51	C₂×C₁₆₇	—	8	1	no	—
52	C₂×C₁₆₇	#121	1	1	yes	characteristic
53	C₂×C₁₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×C₁₆₇	#137	1	1	yes	characteristic
55	C₁₆₇	#468	1	1	yes	characteristic, complemented, Sylow
56	#468	—	167	1	no	maximal, Sylow
57	C₂×SD₁₆	—	167	1	no	—
58	C₂×M₁₆	—	167	1	no	—
59	C₂²⋊Q₈	—	167	1	no	—
60	#121	D₁₆₇	1	1	yes	characteristic
61	#107	—	167	1	no	—
62	#103	—	167	1	no	—
63	#102	—	167	1	no	—
64	C₂×Q₈	—	167	1	no	—
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	C₂×D₁₆₇	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₁₆₇	1	1	yes	characteristic
68	SD₁₆	—	334	2	no	—
69	M₁₆	—	334	2	no	—
70	C₂×C₈	—	167	1	no	—
71	C₂×C₈	—	167	1	no	—
72	C₄⋊C₄	—	334	1	no	—
73	C₄⋊C₄	—	167	1	no	—
74	C₄⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
75	C₂²⋊C₄	—	2	1	no	—
76	C₂²⋊C₄	—	334	1	no	—
77	C₂³	—	2	1	no	—
78	C₂³	C₂²⋊D₁₆₇	1	1	yes	characteristic
79	Q₈	—	334	1	no	—
80	Q₈	—	334	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	4	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	334	1	no	—
88	C₂×C₄	—	334	1	no	—
89	C₂×C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
91	C₈	—	334	1	no	—
92	C₈	—	167	2	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	—	4	1	no	—
99	C₂²	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	668	1	no	—
103	C₄	—	668	1	no	—
104	C₄	D₄×D₁₆₇	1	1	yes	characteristic
105	C₄	#14581949	1	1	yes	characteristic
106	C₂	—	4	1	no	—
107	C₂	—	8	1	no	—
108	C₂	#19128804	1	1	yes	characteristic
109	C₂	#19128858	1	1	yes	characteristic
110	C₂	#19128862	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581938

C₂

yes

maximal, characteristic, complemented

C₂×#14581934

C₂

yes

maximal, characteristic, complemented

#19128811

C₂

yes

maximal, characteristic, complemented

#19128765

C₂

yes

maximal, characteristic, complemented

#19128755

C₂

yes

maximal, characteristic, complemented

#19128753

C₂

yes

maximal, characteristic, complemented

#121×C₁₆₇

C₂

yes

maximal, characteristic

C₂×Dic₃₃₄

C₂²

yes

characteristic

#14581938

—

#14581934

—

C₂×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₆₇

C₂²

yes

characteristic

Dic₁₆₇⋊C₄

—

C₂²⋊Dic₁₆₇

—

C₂×D₄×C₁₆₇

—

C₂×D₄×C₁₆₇

C₂²

yes

characteristic

C₂²×C₄×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₆₇

—

Dic₃₃₄

—

Dic₃₃₄

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

D₄

yes

complemented

C₂³×C₁₆₇

—

C₂³×C₁₆₇

D₄

yes

characteristic

D₄×C₁₆₇

—

D₄×C₁₆₇

—

D₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂×D₄

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₄×C₁₆₇

#41

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#468

—

167

maximal, Sylow

C₂×SD₁₆

—

167

—

C₂×M₁₆

—

167

—

C₂²⋊Q₈

—

167

—

#121

D₁₆₇

yes

characteristic

#107

—

167

—

#103

—

167

—

#102

—

167

—

C₂×Q₈

—

167

—

C₂×D₄

—

C₂×D₄

C₂×D₁₆₇

yes

characteristic

C₂²×C₄

C₂×D₁₆₇

yes

characteristic

SD₁₆

—

334

—

M₁₆

—

334

—

C₂×C₈

—

167

—

C₂×C₈

—

167

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

167

—

C₄⋊C₄

C₂×D₁₆₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

334

—

C₂³

—

C₂³

C₂²⋊D₁₆₇

yes

characteristic

Q₈

—

334

—

Q₈

—

334

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

C₂²⋊D₁₆₇

yes

characteristic

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₈

—

334

—

C₈

—

167

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₁₆₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

668

—

103

C₄

—

668

—

104

C₄

D₄×D₁₆₇

yes

characteristic

105

C₄

#14581949

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#19128804

yes

characteristic

109

C₂

#19128858

yes

characteristic

110

C₂

#19128862

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₆₇
Fitting, prime radical	#121×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₄
hypercenter	#121	D₁₆₇
hypocenter	C₁₆₇	#468
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₆₇

commutator, cosocle

C₂×C₄×C₁₆₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₆₇

Fitting, prime radical

#121×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂×D₄

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

N	H ≅ G/N	type
1. C₂×#14581938	C₂	semidirect
2. C₂×#14581934	C₂	semidirect
3. #19128811	C₂	semidirect
4. #19128765	C₂	semidirect
5. #19128755	C₂	semidirect
6. #19128753	C₂	semidirect
26. C₁₆₇⋊C₈	D₄	semidirect
55. C₁₆₇	#468	semidirect

H ≅ G/N

type

1. C₂×#14581938

C₂

semidirect

2. C₂×#14581934

C₂

semidirect

3. #19128811

C₂

semidirect

4. #19128765

C₂

semidirect

5. #19128755

C₂

semidirect

6. #19128753

C₂

semidirect

26. C₁₆₇⋊C₈

D₄

semidirect

55. C₁₆₇

#468

semidirect

Subgroups — #24923513

Conjugacy classes of subgroups

Complements