Group #24923498 — subgroups

Subgroups — #24923498

Order 10688.

124 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 47 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	#19128811	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128806	C₂	1	1	yes	maximal, characteristic
4	#19128764	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128757	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128754	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₃₄	C₂²	1	1	yes	characteristic
9	C₂×D₆₆₈	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₆₇	—	2	1	no	—
11	#14581939	—	2	2	no	—
12	C₄⋊Dic₁₆₇	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×D₁₆₇)⋊C₄	—	2	1	no	—
17	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₆₇	—	2	2	no	—
19	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₆₇	—	2	1	no	—
22	Dic₃₃₄	—	2	1	no	—
23	Dic₃₃₄	—	2	1	no	—
24	D₆₆₈	—	2	1	no	—
25	D₆₆₈	—	2	1	no	—
26	C₂²⋊D₁₆₇	—	4	2	no	—
27	C₂×Dic₁₆₇	—	2	1	no	—
28	C₂×Dic₁₆₇	—	2	1	no	—
29	C₂×Dic₁₆₇	—	2	1	no	—
30	C₂³×C₁₆₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₆₇	—	2	1	no	—
32	C₂×C₄×C₁₆₇	—	2	1	no	—
33	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
35	C₈×C₁₆₇	—	2	1	no	—
36	C₈×C₁₆₇	D₄	1	2	yes	complemented
37	C₂²×C₁₆₇	—	2	1	no	—
38	C₂²×C₁₆₇	—	2	1	no	—
39	C₂²×C₁₆₇	SD₁₆	1	2	yes	complemented
40	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₆₇	—	4	1	no	—
42	C₂×D₁₆₇	—	4	1	no	—
43	C₂×D₁₆₇	—	4	1	no	—
44	C₄×C₁₆₇	—	2	1	no	—
45	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
46	C₄×C₁₆₇	#41	1	1	yes	characteristic
47	Dic₁₆₇	—	4	1	no	—
48	Dic₁₆₇	—	4	1	no	—
49	Dic₁₆₇	—	4	1	no	—
50	C₂×C₁₆₇	—	2	2	no	—
51	C₂×C₁₆₇	#121	1	1	yes	characteristic
52	C₂×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂×C₁₆₇	#135	1	1	yes	characteristic
54	D₁₆₇	—	8	1	no	—
55	C₁₆₇	#465	1	1	yes	characteristic, complemented, Sylow
56	#465	—	167	1	no	maximal, Sylow
57	C₂×SD₁₆	—	167	1	no	—
58	C₂²×C₈	D₁₆₇	1	1	yes	characteristic, complemented
59	C₂²⋊Q₈	—	167	1	no	—
60	#121	—	167	1	no	—
61	#106	—	167	1	no	—
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	C₂×D₁₆₇	1	1	yes	characteristic
68	SD₁₆	—	334	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
71	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
72	C₄⋊C₄	—	334	1	no	—
73	C₄⋊C₄	—	167	1	no	—
74	C₄⋊C₄	—	167	1	no	—
75	C₂²⋊C₄	—	334	1	no	—
76	C₂²⋊C₄	—	334	1	no	—
77	C₂³	—	334	1	no	—
78	C₂³	D₆₆₈	1	1	yes	characteristic
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	D₆₆₈	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
91	C₈	—	2	1	no	—
92	C₈	C₂²⋊D₁₆₇	1	2	yes	complemented
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	668	1	no	—
96	C₂²	—	668	1	no	—
97	C₂²	—	668	1	no	—
98	C₂²	#14581939	1	2	yes	complemented
99	C₂²	C₂×D₆₆₈	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	668	1	no	—
102	C₄	—	668	1	no	—
103	C₄	—	668	1	no	—
104	C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
105	C₄	#14581950	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	1336	1	no	—
108	C₂	#19128806	1	1	yes	characteristic
109	C₂	C₂×#14581939	1	1	yes	characteristic
110	C₂	#19128854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581939

C₂

yes

maximal, characteristic, complemented

#19128811

C₂

yes

maximal, characteristic, complemented

#19128806

C₂

yes

maximal, characteristic

#19128764

C₂

yes

maximal, characteristic, complemented

#19128757

C₂

yes

maximal, characteristic, complemented

#19128754

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×C₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₃₄

C₂²

yes

characteristic

C₂×D₆₆₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₆₇

—

#14581939

—

C₄⋊Dic₁₆₇

C₂²

yes

characteristic

C₄⋊Dic₁₆₇

C₂²

yes

characteristic

Dic₁₆₇⋊C₄

—

C₂²⋊Dic₁₆₇

—

(C₂×D₁₆₇)⋊C₄

—

C₂²×C₄×C₁₆₇

C₂²

yes

characteristic

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×C₈×C₁₆₇

C₂²

yes

characteristic

C₂²×D₁₆₇

—

Dic₃₃₄

—

Dic₃₃₄

—

D₆₆₈

—

D₆₆₈

—

C₂²⋊D₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂³×C₁₆₇

D₄

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₈×C₁₆₇

—

C₈×C₁₆₇

D₄

yes

complemented

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

SD₁₆

yes

complemented

C₂²×C₁₆₇

C₂×D₄

yes

characteristic

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₄×C₁₆₇

#41

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

#121

yes

characteristic

C₂×C₁₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₆₇

#135

yes

characteristic

D₁₆₇

—

C₁₆₇

#465

yes

characteristic, complemented, Sylow

#465

—

167

maximal, Sylow

C₂×SD₁₆

—

167

—

C₂²×C₈

D₁₆₇

yes

characteristic, complemented

C₂²⋊Q₈

—

167

—

#121

—

167

—

#106

—

167

—

#103

—

167

—

#102

—

167

—

C₂×Q₈

—

167

—

C₂×D₄

—

334

—

C₂×D₄

—

167

—

C₂²×C₄

C₂×D₁₆₇

yes

characteristic

SD₁₆

—

334

—

C₂×C₈

—

C₂×C₈

C₂×D₁₆₇

yes

characteristic

C₂×C₈

C₂×D₁₆₇

yes

characteristic

C₄⋊C₄

—

334

—

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₂²⋊C₄

—

334

—

C₂²⋊C₄

—

334

—

C₂³

—

334

—

C₂³

D₆₆₈

yes

characteristic

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

D₆₆₈

yes

characteristic

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₈

—

C₈

C₂²⋊D₁₆₇

yes

complemented

C₂²

—

C₂²

—

C₂²

—

668

—

C₂²

—

668

—

C₂²

—

668

—

C₂²

#14581939

yes

complemented

C₂²

C₂×D₆₆₈

yes

characteristic

100

C₄

—

101

C₄

—

668

—

102

C₄

—

668

—

103

C₄

—

668

—

104

C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

105

C₄

#14581950

yes

characteristic

106

C₂

—

107

C₂

—

1336

—

108

C₂

#19128806

yes

characteristic

109

C₂

C₂×#14581939

yes

characteristic

110

C₂

#19128854

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₆₆₈
commutator, cosocle	C₂×C₄×C₁₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₆₇
Fitting	C₂²×C₈×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₄
prime radical	#19128806	C₂
hypercenter	C₂²×C₈	D₁₆₇
hypocenter	C₁₆₇	#465
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂²

C₂×D₆₆₈

commutator, cosocle

C₂×C₄×C₁₆₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₆₇

Fitting

C₂²×C₈×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂×D₄

prime radical

#19128806

C₂

hypercenter

C₂²×C₈

D₁₆₇

hypocenter

C₁₆₇

#465

LCS

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
1. C₂×#14581939	C₂	semidirect
2. #19128811	C₂	semidirect
4. #19128764	C₂	semidirect
5. #19128757	C₂	semidirect
6. #19128754	C₂	semidirect
7. C₂²×C₈×C₁₆₇	C₂	semidirect
36. C₈×C₁₆₇	D₄	semidirect
39. C₂²×C₁₆₇	SD₁₆	semidirect
55. C₁₆₇	#465	semidirect
58. C₂²×C₈	D₁₆₇	semidirect
92. C₈	C₂²⋊D₁₆₇	semidirect
98. C₂²	#14581939	semidirect

H ≅ G/N

type

1. C₂×#14581939

C₂

semidirect

2. #19128811

C₂

semidirect

4. #19128764

C₂

semidirect

5. #19128757

C₂

semidirect

6. #19128754

C₂

semidirect

7. C₂²×C₈×C₁₆₇

C₂

semidirect

36. C₈×C₁₆₇

D₄

semidirect

39. C₂²×C₁₆₇

SD₁₆

semidirect

55. C₁₆₇

#465

semidirect

58. C₂²×C₈

D₁₆₇

semidirect

92. C₈

C₂²⋊D₁₆₇

semidirect

98. C₂²

#14581939

semidirect

Subgroups — #24923498

Conjugacy classes of subgroups

Complements