Group #24923270 — subgroups

Subgroups — #24923270

Order 10688.

162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 71 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×C₄⋊Dic₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128756	C₂	1	2	yes	maximal, complemented
4	#19128753	C₂	1	2	yes	maximal, complemented
5	C₂×#41×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₇	—	2	1	no	—
7	#14581934	—	2	2	no	—
8	C₂×C₁₆₇⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₆₇	—	2	1	no	—
10	C₄⋊Dic₁₆₇	C₂²	1	2	yes	—
11	#41×C₁₆₇	—	2	2	no	—
12	#41×C₁₆₇	C₄	1	4	yes	complemented
13	C₂×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₁₆₇	—	2	1	no	—
15	C₂×D₄×C₁₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₆₇	—	2	1	no	—
17	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₆₇	—	2	2	no	—
19	C₂×Dic₁₆₇	—	4	2	no	—
20	C₁₆₇⋊C₈	—	2	2	no	—
21	C₂³×C₁₆₇	—	2	1	no	—
22	C₂³×C₁₆₇	D₄	1	1	yes	characteristic
23	Q₈×C₁₆₇	—	2	1	no	—
24	Q₈×C₁₆₇	C₂×C₄	1	2	yes	complemented
25	D₄×C₁₆₇	—	2	4	no	—
26	D₄×C₁₆₇	—	2	1	no	—
27	D₄×C₁₆₇	C₂×C₄	1	2	yes	complemented
28	C₂×C₄×C₁₆₇	—	2	1	no	—
29	C₂×C₄×C₁₆₇	—	2	4	no	—
30	C₂×C₄×C₁₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₆₇	D₄	1	2	yes	—
33	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₁₆₇	—	2	1	no	—
35	C₂²×C₁₆₇	—	2	1	no	—
36	C₂²×C₁₆₇	—	2	2	no	—
37	C₂²×C₁₆₇	—	4	1	no	—
38	C₂²×C₁₆₇	—	4	1	no	—
39	C₂²×C₁₆₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
41	C₄×C₁₆₇	—	2	2	no	—
42	C₄×C₁₆₇	C₂²⋊C₄	1	2	yes	—
43	C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
45	Dic₁₆₇	—	4	2	no	—
46	C₂×C₁₆₇	—	2	2	no	—
47	C₂×C₁₆₇	—	4	2	no	—
48	C₂×C₁₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₁₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×C₁₆₇	#137	1	1	yes	characteristic
51	C₁₆₇	#417	1	1	yes	characteristic, complemented, Sylow
52	#417	—	167	1	no	maximal, Sylow
53	C₂×#41	D₁₆₇	1	1	yes	characteristic
54	C₂×M₁₆	—	167	1	no	—
55	C₂×C₄⋊C₄	—	167	1	no	—
56	#103	—	167	2	no	—
57	#102	—	167	2	no	—
58	#41	—	2	2	no	—
59	#41	Dic₁₆₇	1	4	yes	complemented
60	C₂×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	C₂×D₁₆₇	1	1	yes	characteristic
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	334	1	no	—
65	C₂²×C₄	C₂×D₁₆₇	1	1	yes	characteristic
66	M₁₆	—	334	2	no	—
67	C₂×C₈	—	167	2	no	—
68	C₄⋊C₄	—	334	1	no	—
69	C₄⋊C₄	—	167	2	no	—
70	C₂³	—	2	1	no	—
71	C₂³	C₂²⋊D₁₆₇	1	1	yes	characteristic
72	Q₈	—	2	1	no	—
73	Q₈	C₂×Dic₁₆₇	1	2	yes	complemented
74	D₄	—	2	4	no	—
75	D₄	—	2	1	no	—
76	D₄	C₂×Dic₁₆₇	1	2	yes	complemented
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	334	2	no	—
80	C₂×C₄	—	668	2	no	—
81	C₂×C₄	C₂×Dic₁₆₇	1	2	yes	—
82	C₂×C₄	C₂²⋊D₁₆₇	1	1	yes	characteristic
83	C₂×C₄	C₂²⋊D₁₆₇	1	2	yes	—
84	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
85	C₈	—	334	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	2	no	—
89	C₂²	—	4	1	no	—
90	C₂²	—	4	1	no	—
91	C₂²	C₂²⋊Dic₁₆₇	1	2	yes	—
92	C₂²	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	668	2	no	—
95	C₄	C₂²⋊Dic₁₆₇	1	2	yes	—
96	C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
97	C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	—	4	2	no	—
100	C₂	C₂×C₂²⋊Dic₁₆₇	1	1	yes	characteristic
101	C₂	#19128857	1	1	yes	characteristic
102	C₂	#19128862	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581934

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Dic₁₆₇

C₂

yes

maximal, characteristic, complemented

#19128756

C₂

yes

maximal, complemented

#19128753

C₂

yes

maximal, complemented

C₂×#41×C₁₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₇

—

#14581934

—

C₂×C₁₆₇⋊C₈

C₂²

yes

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

C₂²

yes

—

#41×C₁₆₇

—

#41×C₁₆₇

C₄

yes

complemented

C₂×Q₈×C₁₆₇

C₂²

yes

characteristic

C₂×D₄×C₁₆₇

—

C₂×D₄×C₁₆₇

C₂²

yes

characteristic

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

C₂²

yes

characteristic

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

—

C₂³×C₁₆₇

D₄

yes

characteristic

Q₈×C₁₆₇

—

Q₈×C₁₆₇

C₂×C₄

yes

complemented

D₄×C₁₆₇

—

D₄×C₁₆₇

—

D₄×C₁₆₇

C₂×C₄

yes

complemented

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×C₄

yes

—

C₂×C₄×C₁₆₇

D₄

yes

characteristic

C₂×C₄×C₁₆₇

D₄

yes

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂²⋊C₄

yes

—

C₂²×C₁₆₇

C₂×D₄

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂²⋊C₄

yes

—

C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₄×C₁₆₇

C₂×D₄

yes

characteristic

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

C₂×C₂²⋊C₄

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#417

—

167

maximal, Sylow

C₂×#41

D₁₆₇

yes

characteristic

C₂×M₁₆

—

167

—

C₂×C₄⋊C₄

—

167

—

#103

—

167

—

#102

—

167

—

#41

—

#41

Dic₁₆₇

yes

complemented

C₂×Q₈

C₂×D₁₆₇

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₁₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

334

—

C₂²×C₄

C₂×D₁₆₇

yes

characteristic

M₁₆

—

334

—

C₂×C₈

—

167

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

167

—

C₂³

—

C₂³

C₂²⋊D₁₆₇

yes

characteristic

Q₈

—

Q₈

C₂×Dic₁₆₇

yes

complemented

D₄

—

D₄

—

D₄

C₂×Dic₁₆₇

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

668

—

C₂×C₄

C₂×Dic₁₆₇

yes

—

C₂×C₄

C₂²⋊D₁₆₇

yes

characteristic

C₂×C₄

C₂²⋊D₁₆₇

yes

—

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₈

—

334

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²⋊Dic₁₆₇

yes

—

C₂²

C₂×C₂²⋊D₁₆₇

yes

characteristic

C₄

—

C₄

—

668

—

C₄

C₂²⋊Dic₁₆₇

yes

—

C₄

C₂²×Dic₁₆₇

yes

characteristic

C₄

C₂×C₂²⋊D₁₆₇

yes

characteristic

C₂

—

C₂

—

100

C₂

C₂×C₂²⋊Dic₁₆₇

yes

characteristic

101

C₂

#19128857

yes

characteristic

102

C₂

#19128862

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₆₇
commutator	C₄×C₁₆₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₆₇
Fitting, prime radical	C₂×#41×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₄
cosocle	C₂×C₄×C₁₆₇	C₂³
hypercenter	C₂×#41	D₁₆₇
hypocenter	C₁₆₇	#417
LCS	G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₆₇

commutator

C₄×C₁₆₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₆₇

Fitting, prime radical

C₂×#41×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂×D₄

cosocle

C₂×C₄×C₁₆₇

C₂³

hypercenter

C₂×#41

D₁₆₇

hypocenter

C₁₆₇

#417

LCS

G ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#14581934	C₂	semidirect
2. C₂×C₄⋊Dic₁₆₇	C₂	semidirect
3. #19128756	C₂	semidirect
4. #19128753	C₂	semidirect
12. #41×C₁₆₇	C₄	semidirect
24. Q₈×C₁₆₇	C₂×C₄	semidirect
27. D₄×C₁₆₇	C₂×C₄	semidirect
51. C₁₆₇	#417	semidirect
59. #41	Dic₁₆₇	semidirect
73. Q₈	C₂×Dic₁₆₇	semidirect
76. D₄	C₂×Dic₁₆₇	semidirect

H ≅ G/N

type

1. C₂×#14581934

C₂

semidirect

2. C₂×C₄⋊Dic₁₆₇

C₂

semidirect

3. #19128756

C₂

semidirect

4. #19128753

C₂

semidirect

12. #41×C₁₆₇

C₄

semidirect

24. Q₈×C₁₆₇

C₂×C₄

semidirect

27. D₄×C₁₆₇

C₂×C₄

semidirect

51. C₁₆₇

#417

semidirect

59. #41

Dic₁₆₇

semidirect

73. Q₈

C₂×Dic₁₆₇

semidirect

76. D₄

C₂×Dic₁₆₇

semidirect

Subgroups — #24923270

Conjugacy classes of subgroups

Complements