Group #22553413 — subgroups

Subgroups — #22553413

Order 8384.

120 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 57 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	Q₈×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#16862180	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862170	C₂	1	1	yes	maximal, characteristic, complemented
5	#16862167	C₂	1	1	yes	maximal, characteristic
6	#16862156	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₁	—	2	1	no	—
9	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₃₁	—	2	1	no	—
11	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₁₃₁	—	2	1	no	—
14	C₄×Dic₁₃₁	—	2	1	no	—
15	C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
18	SD₁₆×C₁₃₁	C₄	1	4	yes	complemented
19	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₃₁	—	2	1	no	—
21	C₂×Dic₁₃₁	—	2	1	no	—
22	C₂×Dic₁₃₁	—	4	2	no	—
23	C₂×Dic₁₃₁	D₄	1	1	yes	characteristic
24	C₂×Dic₁₃₁	D₄	1	1	yes	characteristic
25	C₁₃₁⋊C₈	—	2	1	no	—
26	C₂³×C₁₃₁	—	2	1	no	—
27	Q₈×C₁₃₁	—	2	1	no	—
28	Q₈×C₁₃₁	C₂×C₄	1	2	yes	complemented
29	D₄×C₁₃₁	—	2	1	no	—
30	D₄×C₁₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₁₃₁	—	2	1	no	—
32	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₈×C₁₃₁	C₂×C₄	1	2	yes	complemented
34	C₂²×C₁₃₁	—	2	2	no	—
35	C₂²×C₁₃₁	—	4	1	no	—
36	C₂²×C₁₃₁	—	4	1	no	—
37	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
38	C₄×C₁₃₁	—	2	2	no	—
39	C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₁₃₁	#41	1	1	yes	characteristic
41	Dic₁₃₁	—	2	1	no	—
42	Dic₁₃₁	—	2	1	no	—
43	Dic₁₃₁	—	4	1	no	—
44	Dic₁₃₁	—	4	1	no	—
45	C₂×C₁₃₁	—	4	2	no	—
46	C₂×C₁₃₁	C₄×D₄	1	1	yes	characteristic
47	C₂×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
48	C₂×C₁₃₁	#137	1	1	yes	characteristic
49	C₁₃₁	#440	1	1	yes	characteristic, complemented, Sylow
50	#440	—	131	1	no	maximal, Sylow
51	C₂×SD₁₆	D₁₃₁	1	1	yes	characteristic
52	C₄×Q₈	—	131	1	no	—
53	C₄×D₄	—	131	1	no	—
54	#107	—	131	1	no	—
55	#103	—	131	1	no	—
56	#102	—	131	1	no	—
57	#97	—	131	1	no	—
58	C₂×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
59	C₂×D₄	C₂×D₁₃₁	1	1	yes	characteristic
60	C₂²×C₄	—	262	1	no	—
61	SD₁₆	Dic₁₃₁	1	4	yes	complemented
62	C₂×C₈	—	131	1	no	—
63	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
64	C₄⋊C₄	—	262	1	no	—
65	C₄⋊C₄	—	131	1	no	—
66	C₄⋊C₄	—	131	1	no	—
67	C₂²⋊C₄	—	262	1	no	—
68	C₄²	—	262	1	no	—
69	C₄²	—	131	1	no	—
70	C₂³	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	C₂×Dic₁₃₁	1	2	yes	complemented
73	D₄	—	2	1	no	—
74	D₄	C₂×Dic₁₃₁	1	2	yes	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	262	1	no	—
77	C₂×C₄	—	262	1	no	—
78	C₂×C₄	—	524	2	no	—
79	C₂×C₄	—	131	1	no	—
80	C₂×C₄	—	131	1	no	—
81	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
82	C₈	—	262	1	no	—
83	C₈	C₂×Dic₁₃₁	1	2	yes	complemented
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	D₄×D₁₃₁	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	262	1	no	—
90	C₄	—	262	1	no	—
91	C₄	—	524	1	no	—
92	C₄	—	524	1	no	—
93	C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
94	C₄	#14523512	1	1	yes	characteristic
95	C₂	—	4	2	no	—
96	C₂	D₄×Dic₁₃₁	1	1	yes	characteristic
97	C₂	#16862273	1	1	yes	characteristic
98	C₂	#16862275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₃₁

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₃₁

C₂

yes

maximal, characteristic

#16862180

C₂

yes

maximal, characteristic, complemented

#16862170

C₂

yes

maximal, characteristic, complemented

#16862167

C₂

yes

maximal, characteristic

#16862156

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₁

—

C₂×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

C₂²⋊Dic₁₃₁

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

C₂²

yes

characteristic

C₂×Q₈×C₁₃₁

C₂²

yes

characteristic

C₂×D₄×C₁₃₁

C₂²

yes

characteristic

SD₁₆×C₁₃₁

C₄

yes

complemented

C₂×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

D₄

yes

characteristic

C₂×Dic₁₃₁

D₄

yes

characteristic

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

C₂×C₄

yes

complemented

D₄×C₁₃₁

—

D₄×C₁₃₁

C₂×C₄

yes

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₈×C₁₃₁

C₂×C₄

yes

complemented

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂×D₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₄×C₁₃₁

#41

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

C₄×D₄

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#440

—

131

maximal, Sylow

C₂×SD₁₆

D₁₃₁

yes

characteristic

C₄×Q₈

—

131

—

C₄×D₄

—

131

—

#107

—

131

—

#103

—

131

—

#102

—

131

—

#97

—

131

—

C₂×Q₈

C₂×D₁₃₁

yes

characteristic

C₂×D₄

C₂×D₁₃₁

yes

characteristic

C₂²×C₄

—

262

—

SD₁₆

Dic₁₃₁

yes

complemented

C₂×C₈

—

131

—

C₂×C₈

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

—

262

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₂²⋊C₄

—

262

—

C₄²

—

262

—

C₄²

—

131

—

C₂³

—

Q₈

—

Q₈

C₂×Dic₁₃₁

yes

complemented

D₄

—

D₄

C₂×Dic₁₃₁

yes

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

524

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

C₈

—

262

—

C₈

C₂×Dic₁₃₁

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₃₁

yes

characteristic

C₄

—

C₄

—

262

—

C₄

—

262

—

C₄

—

524

—

C₄

—

524

—

C₄

C₂²×Dic₁₃₁

yes

characteristic

C₄

#14523512

yes

characteristic

C₂

—

C₂

D₄×Dic₁₃₁

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×D₁₃₁
commutator	C₄×C₁₃₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	C₂×SD₁₆×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₃₁
hypocenter	C₁₃₁	#440
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₃₁

commutator

C₄×C₁₃₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₃₁

Fitting

C₂×SD₁₆×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₃₁

hypocenter

C₁₃₁

#440

LCS

G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. Q₈×Dic₁₃₁	C₂	semidirect
3. #16862180	C₂	semidirect
4. #16862170	C₂	semidirect
6. #16862156	C₂	semidirect
18. SD₁₆×C₁₃₁	C₄	semidirect
28. Q₈×C₁₃₁	C₂×C₄	semidirect
33. C₈×C₁₃₁	C₂×C₄	semidirect
49. C₁₃₁	#440	semidirect
61. SD₁₆	Dic₁₃₁	semidirect
72. Q₈	C₂×Dic₁₃₁	semidirect
83. C₈	C₂×Dic₁₃₁	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₃₁

C₂

semidirect

3. #16862180

C₂

semidirect

4. #16862170

C₂

semidirect

6. #16862156

C₂

semidirect

18. SD₁₆×C₁₃₁

C₄

semidirect

28. Q₈×C₁₃₁

C₂×C₄

semidirect

33. C₈×C₁₃₁

C₂×C₄

semidirect

49. C₁₃₁

#440

semidirect

61. SD₁₆

Dic₁₃₁

semidirect

72. Q₈

C₂×Dic₁₃₁

semidirect

83. C₈

C₂×Dic₁₃₁

semidirect

Subgroups — #22553413

Conjugacy classes of subgroups

Complements