Group #24923344 — subgroups

Subgroups — #24923344

Order 10688.

122 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14581934	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14581933	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂²⋊D₁₆₇	C₂	1	1	yes	maximal, characteristic
4	Dic₁₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₁₆₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊D₁₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₆₇	C₂²	1	1	yes	characteristic
10	#14581934	—	2	2	no	—
11	#14581933	—	2	2	no	—
12	C₂×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	Dic₁₆₇⋊C₄	C₄	1	1	yes	characteristic
15	C₂²⋊Dic₁₆₇	C₄	1	1	yes	characteristic
16	(C₂×D₁₆₇)⋊C₄	C₄	1	1	yes	characteristic
17	C₄×Dic₁₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²⋊D₁₆₇	—	2	2	no	—
23	C₂×Dic₁₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₁₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₆₇	—	2	1	no	—
27	C₄×D₁₆₇	—	2	1	no	—
28	C₁₆₇⋊C₈	—	2	1	no	—
29	C₁₆₇⋊C₈	D₄	1	2	yes	complemented
30	C₂³×C₁₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₆₇	—	2	1	no	—
32	C₂×C₄×C₁₆₇	—	2	1	no	—
33	C₂×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
35	C₈×C₁₆₇	—	2	1	no	—
36	C₈×C₁₆₇	—	2	1	no	—
37	C₂²×C₁₆₇	—	2	1	no	—
38	C₂²×C₁₆₇	—	2	1	no	—
39	C₂²×C₁₆₇	—	2	1	no	—
40	C₂²×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×D₁₆₇	—	2	1	no	—
42	C₂×D₁₆₇	—	2	1	no	—
43	C₂×D₁₆₇	—	2	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	—	2	1	no	—
48	Dic₁₆₇	—	2	1	no	—
49	Dic₁₆₇	M₁₆	1	2	yes	complemented
50	C₂×C₁₆₇	—	4	1	no	—
51	C₂×C₁₆₇	C₄×D₄	1	1	yes	characteristic
52	C₂×C₁₆₇	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₁₆₇	#131	1	1	yes	characteristic
54	D₁₆₇	—	4	1	no	—
55	C₁₆₇	#436	1	1	yes	characteristic, complemented, Sylow
56	#436	—	167	1	no	maximal, Sylow
57	C₂×M₁₆	—	167	1	no	—
58	C₂×M₁₆	—	167	1	no	—
59	C₄×D₄	—	167	1	no	—
60	C₄⋊C₈	—	167	1	no	—
61	C₂²⋊C₈	—	167	1	no	—
62	C₂²⋊C₈	D₁₆₇	1	1	yes	characteristic, complemented
63	C₄×C₈	—	167	1	no	—
64	C₂×D₄	—	167	1	no	—
65	C₂²×C₄	—	167	1	no	—
66	C₂²×C₄	C₂×D₁₆₇	1	1	yes	characteristic
67	M₁₆	—	334	2	no	—
68	M₁₆	—	334	2	no	—
69	C₂×C₈	—	167	1	no	—
70	C₂×C₈	—	167	1	no	—
71	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
72	C₂×C₈	C₂×D₁₆₇	1	1	yes	characteristic
73	C₄⋊C₄	—	167	1	no	—
74	C₂²⋊C₄	—	167	1	no	—
75	C₂²⋊C₄	—	167	1	no	—
76	C₄²	—	167	1	no	—
77	C₂³	—	167	1	no	—
78	C₂³	C₄×D₁₆₇	1	1	yes	characteristic
79	D₄	—	334	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	334	1	no	—
83	C₂×C₄	—	334	1	no	—
84	C₂×C₄	—	167	1	no	—
85	C₂×C₄	—	167	1	no	—
86	C₂×C₄	—	167	1	no	—
87	C₂×C₄	C₄×D₁₆₇	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
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	—	334	1	no	—
97	C₂²	—	334	1	no	—
98	C₂²	—	334	1	no	—
99	C₂²	C₂×C₄×D₁₆₇	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	334	1	no	—
102	C₄	—	334	1	no	—
103	C₄	—	167	2	no	—
104	C₄	D₄×D₁₆₇	1	1	yes	characteristic
105	C₄	#14581949	1	1	yes	characteristic
106	C₂	—	4	1	no	—
107	C₂	—	668	1	no	—
108	C₂	(C₂²⋊D₁₆₇)⋊C₄	1	1	yes	characteristic
109	C₂	M₁₆×D₁₆₇	1	1	yes	characteristic
110	C₂	#19128838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581934

C₂

yes

maximal, characteristic, complemented

C₂×#14581933

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₆₇

C₂

yes

maximal, characteristic

Dic₁₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₆₇)⋊C₈

C₂

yes

maximal, characteristic, complemented

C₈×Dic₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₈×C₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₆₇

C₄

yes

characteristic

C₂×C₄×D₁₆₇

C₂²

yes

characteristic

#14581934

—

#14581933

—

C₂×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₆₇⋊C₈

C₂²

yes

characteristic

Dic₁₆₇⋊C₄

C₄

yes

characteristic

C₂²⋊Dic₁₆₇

C₄

yes

characteristic

(C₂×D₁₆₇)⋊C₄

C₄

yes

characteristic

C₄×Dic₁₆₇

C₂²

yes

characteristic

C₂²×C₄×C₁₆₇

C₂²

yes

characteristic

C₂×C₈×C₁₆₇

C₂²

yes

characteristic

C₂×C₈×C₁₆₇

C₂²

yes

characteristic

C₂²×D₁₆₇

C₂×C₄

yes

characteristic

C₂²⋊D₁₆₇

—

C₂×Dic₁₆₇

C₂×C₄

yes

characteristic

C₂×Dic₁₆₇

C₂×C₄

yes

characteristic

C₂×Dic₁₆₇

C₂×C₄

yes

characteristic

C₄×D₁₆₇

—

C₄×D₁₆₇

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

D₄

yes

complemented

C₂³×C₁₆₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₈×C₁₆₇

—

C₈×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₄×C₁₆₇

#41

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

Dic₁₆₇

M₁₆

yes

complemented

C₂×C₁₆₇

—

C₂×C₁₆₇

C₄×D₄

yes

characteristic

C₂×C₁₆₇

C₂×M₁₆

yes

characteristic

C₂×C₁₆₇

#131

yes

characteristic

D₁₆₇

—

C₁₆₇

#436

yes

characteristic, complemented, Sylow

#436

—

167

maximal, Sylow

C₂×M₁₆

—

167

—

C₂×M₁₆

—

167

—

C₄×D₄

—

167

—

C₄⋊C₈

—

167

—

C₂²⋊C₈

—

167

—

C₂²⋊C₈

D₁₆₇

yes

characteristic, complemented

C₄×C₈

—

167

—

C₂×D₄

—

167

—

C₂²×C₄

—

167

—

C₂²×C₄

C₂×D₁₆₇

yes

characteristic

M₁₆

—

334

—

M₁₆

—

334

—

C₂×C₈

—

167

—

C₂×C₈

—

167

—

C₂×C₈

C₂×D₁₆₇

yes

characteristic

C₂×C₈

C₂×D₁₆₇

yes

characteristic

C₄⋊C₄

—

167

—

C₂²⋊C₄

—

167

—

C₂²⋊C₄

—

167

—

C₄²

—

167

—

C₂³

—

167

—

C₂³

C₄×D₁₆₇

yes

characteristic

D₄

—

334

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

334

—

C₂×C₄

—

167

—

C₂×C₄

—

167

—

C₂×C₄

—

167

—

C₂×C₄

C₄×D₁₆₇

yes

characteristic

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₈

—

C₈

—

C₈

—

334

—

C₈

—

167

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

334

—

C₂²

—

334

—

C₂²

—

334

—

C₂²

C₂×C₄×D₁₆₇

yes

characteristic

100

C₄

—

101

C₄

—

334

—

102

C₄

—

334

—

103

C₄

—

167

—

104

C₄

D₄×D₁₆₇

yes

characteristic

105

C₄

#14581949

yes

characteristic

106

C₂

—

107

C₂

—

668

—

108

C₂

(C₂²⋊D₁₆₇)⋊C₄

yes

characteristic

109

C₂

M₁₆×D₁₆₇

yes

characteristic

110

C₂

#19128838

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₆₇
commutator, socle	C₂²×C₁₆₇	C₂²×C₄
Fitting	C₂²⋊C₈×C₁₆₇	C₂
prime radical	C₂×C₂²⋊D₁₆₇	C₄
cosocle	C₂×C₄×C₁₆₇	C₂³
hypercenter	C₂²⋊C₈	D₁₆₇
hypocenter	C₁₆₇	#436
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₆₇

commutator, socle

C₂²×C₁₆₇

C₂²×C₄

Fitting

C₂²⋊C₈×C₁₆₇

C₂

prime radical

C₂×C₂²⋊D₁₆₇

C₄

cosocle

C₂×C₄×C₁₆₇

C₂³

hypercenter

C₂²⋊C₈

D₁₆₇

C₁₆₇

G ▷ C₂²×C₁₆₇ ▷ C₁₆₇

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
1. C₂×#14581934	C₂	semidirect
2. C₂×#14581933	C₂	semidirect
4. Dic₁₆₇⋊C₈	C₂	semidirect
5. (C₂×D₁₆₇)⋊C₈	C₂	semidirect
6. C₈×Dic₁₆₇	C₂	semidirect
7. C₂²⋊C₈×C₁₆₇	C₂	semidirect
29. C₁₆₇⋊C₈	D₄	semidirect
49. Dic₁₆₇	M₁₆	semidirect
55. C₁₆₇	#436	semidirect
62. C₂²⋊C₈	D₁₆₇	semidirect

H ≅ G/N

type

1. C₂×#14581934

C₂

semidirect

2. C₂×#14581933

C₂

semidirect

4. Dic₁₆₇⋊C₈

C₂

semidirect

5. (C₂×D₁₆₇)⋊C₈

C₂

semidirect

6. C₈×Dic₁₆₇

C₂

semidirect

7. C₂²⋊C₈×C₁₆₇

C₂

semidirect

29. C₁₆₇⋊C₈

D₄

semidirect

49. Dic₁₆₇

M₁₆

semidirect

55. C₁₆₇

#436

semidirect

62. C₂²⋊C₈

D₁₆₇

semidirect

Subgroups — #24923344

Conjugacy classes of subgroups

Complements