Group #24923516 — subgroups

Subgroups — #24923516

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×#14581939	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14581933	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128812	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128807	C₂	1	1	yes	maximal, characteristic
5	#19128755	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128752	C₂	1	1	yes	maximal, characteristic, complemented
7	#107×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₃₄	C₂²	1	1	yes	characteristic
9	C₂×D₆₆₈	—	2	1	no	—
10	C₂×D₆₆₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₆₇	C₂²	1	1	yes	characteristic
12	#14581939	—	2	2	no	—
13	#14581933	—	2	2	no	—
14	C₂×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₆₇	—	2	1	no	—
16	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
17	(C₂×D₁₆₇)⋊C₄	—	2	1	no	—
18	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₆₇	—	2	1	no	—
22	C₂²×D₁₆₇	D₄	1	1	yes	characteristic
23	Dic₃₃₄	—	2	1	no	—
24	Dic₃₃₄	—	2	1	no	—
25	D₆₆₈	—	2	1	no	—
26	D₆₆₈	—	2	1	no	—
27	D₆₆₈	—	4	2	no	—
28	C₂×Dic₁₆₇	—	2	1	no	—
29	C₂×Dic₁₆₇	D₄	1	1	yes	characteristic
30	C₄×D₁₆₇	—	2	1	no	—
31	C₄×D₁₆₇	—	2	1	no	—
32	C₁₆₇⋊C₈	—	2	1	no	—
33	C₂×C₄×C₁₆₇	—	2	1	no	—
34	C₂×C₄×C₁₆₇	—	2	1	no	—
35	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
36	C₈×C₁₆₇	D₄	1	2	yes	complemented
37	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×D₁₆₇	—	2	1	no	—
39	C₂×D₁₆₇	—	2	1	no	—
40	C₂×D₁₆₇	—	2	1	no	—
41	C₂×D₁₆₇	—	4	1	no	—
42	C₂×D₁₆₇	—	4	1	no	—
43	C₂×D₁₆₇	—	4	1	no	—
44	C₄×C₁₆₇	—	4	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	—	2	1	no	—
49	Dic₁₆₇	—	4	1	no	—
50	C₂×C₁₆₇	#121	1	1	yes	characteristic
51	C₂×C₁₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
52	C₂×C₁₆₇	#137	1	1	yes	characteristic
53	D₁₆₇	—	4	1	no	—
54	D₁₆₇	—	8	1	no	—
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	—	167	1	no	—
61	#107	D₁₆₇	1	1	yes	characteristic, complemented
62	#103	—	167	1	no	—
63	#102	—	167	1	no	—
64	C₂×Q₈	—	167	1	no	—
65	C₂×D₄	—	334	1	no	—
66	C₂×D₄	—	167	1	no	—
67	C₂²×C₄	—	167	1	no	—
68	SD₁₆	—	334	2	no	—
69	M₁₆	—	334	2	no	—
70	C₂×C₈	—	167	1	no	—
71	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
72	C₄⋊C₄	—	334	1	no	—
73	C₄⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
74	C₄⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
75	C₂²⋊C₄	—	334	1	no	—
76	C₂²⋊C₄	—	334	1	no	—
77	C₂³	—	334	1	no	—
78	C₂³	—	167	1	no	—
79	Q₈	—	334	1	no	—
80	Q₈	—	334	1	no	—
81	D₄	—	334	1	no	—
82	D₄	—	334	1	no	—
83	D₄	—	668	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	334	1	no	—
87	C₂×C₄	—	334	1	no	—
88	C₂×C₄	—	334	1	no	—
89	C₂×C₄	—	167	1	no	—
90	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
91	C₈	—	334	1	no	—
92	C₈	D₆₆₈	1	2	yes	complemented
93	C₂²	—	334	1	no	—
94	C₂²	—	334	1	no	—
95	C₂²	—	334	1	no	—
96	C₂²	—	668	1	no	—
97	C₂²	—	668	1	no	—
98	C₂²	—	668	1	no	—
99	C₂²	D₄×D₁₆₇	1	1	yes	characteristic
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	334	1	no	—
103	C₄	—	668	1	no	—
104	C₄	C₂×D₆₆₈	1	1	yes	characteristic
105	C₄	#14581951	1	1	yes	characteristic
106	C₂	—	1336	1	no	—
107	C₂	—	668	1	no	—
108	C₂	#19128807	1	1	yes	characteristic
109	C₂	#19128856	1	1	yes	characteristic
110	C₂	#19128864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581939

C₂

yes

maximal, characteristic, complemented

C₂×#14581933

C₂

yes

maximal, characteristic, complemented

#19128812

C₂

yes

maximal, characteristic, complemented

#19128807

C₂

yes

maximal, characteristic

#19128755

C₂

yes

maximal, characteristic, complemented

#19128752

C₂

yes

maximal, characteristic, complemented

#107×C₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₃₄

C₂²

yes

characteristic

C₂×D₆₆₈

—

C₂×D₆₆₈

C₂²

yes

characteristic

C₂×C₄×D₁₆₇

C₂²

yes

characteristic

#14581939

—

#14581933

—

C₂×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₆₇

—

(C₂×D₁₆₇)⋊C₄

—

(C₂×D₁₆₇)⋊C₄

—

C₂×C₈×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₂²×D₁₆₇

—

C₂²×D₁₆₇

D₄

yes

characteristic

Dic₃₃₄

—

Dic₃₃₄

—

D₆₆₈

—

D₆₆₈

—

D₆₆₈

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

D₄

yes

characteristic

C₄×D₁₆₇

—

C₄×D₁₆₇

—

C₁₆₇⋊C₈

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₈×C₁₆₇

D₄

yes

complemented

C₂²×C₁₆₇

C₂×D₄

yes

characteristic

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₄×C₁₆₇

#41

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

D₁₆₇

—

D₁₆₇

—

C₁₆₇

#468

yes

characteristic, complemented, Sylow

#468

—

167

maximal, Sylow

C₂×SD₁₆

—

167

—

C₂×M₁₆

—

167

—

C₂²⋊Q₈

—

167

—

#121

—

167

—

#107

D₁₆₇

yes

characteristic, complemented

#103

—

167

—

#102

—

167

—

C₂×Q₈

—

167

—

C₂×D₄

—

334

—

C₂×D₄

—

167

—

C₂²×C₄

—

167

—

SD₁₆

—

334

—

M₁₆

—

334

—

C₂×C₈

—

167

—

C₂×C₈

C₂×D₁₆₇

yes

characteristic

C₄⋊C₄

—

334

—

C₄⋊C₄

C₂×D₁₆₇

yes

characteristic

C₄⋊C₄

C₂×D₁₆₇

yes

characteristic

C₂²⋊C₄

—

334

—

C₂²⋊C₄

—

334

—

C₂³

—

334

—

C₂³

—

167

—

Q₈

—

334

—

Q₈

—

334

—

D₄

—

334

—

D₄

—

334

—

D₄

—

668

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

—

167

—

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₈

—

334

—

C₈

D₆₆₈

yes

complemented

C₂²

—

334

—

C₂²

—

334

—

C₂²

—

334

—

C₂²

—

668

—

C₂²

—

668

—

C₂²

—

668

—

C₂²

D₄×D₁₆₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

334

—

103

C₄

—

668

—

104

C₄

C₂×D₆₆₈

yes

characteristic

105

C₄

#14581951

yes

characteristic

106

C₂

—

1336

—

107

C₂

—

668

—

108

C₂

#19128807

yes

characteristic

109

C₂

#19128856

yes

characteristic

110

C₂

#19128864

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₆₇
Fitting	#107×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₄
prime radical	#19128807	C₂
hypercenter	#107	D₁₆₇
hypocenter	C₁₆₇	#468
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×D₁₆₇

commutator, cosocle

C₂×C₄×C₁₆₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₆₇

Fitting

#107×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂×D₄

C₂

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
1. C₂×#14581939	C₂	semidirect
2. C₂×#14581933	C₂	semidirect
3. #19128812	C₂	semidirect
5. #19128755	C₂	semidirect
6. #19128752	C₂	semidirect
7. #107×C₁₆₇	C₂	semidirect
36. C₈×C₁₆₇	D₄	semidirect
55. C₁₆₇	#468	semidirect
61. #107	D₁₆₇	semidirect
92. C₈	D₆₆₈	semidirect

H ≅ G/N

type

1. C₂×#14581939

C₂

semidirect

2. C₂×#14581933

C₂

semidirect

3. #19128812

C₂

semidirect

5. #19128755

C₂

semidirect

6. #19128752

C₂

semidirect

7. #107×C₁₆₇

C₂

semidirect

36. C₈×C₁₆₇

D₄

semidirect

55. C₁₆₇

#468

semidirect

61. #107

D₁₆₇

semidirect

92. C₈

D₆₆₈

semidirect

Subgroups — #24923516

Conjugacy classes of subgroups

Complements