Group #22553559 — subgroups

Subgroups — #22553559

Order 8384.

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×#14523502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523496	C₂	1	1	yes	maximal, characteristic, complemented
3	#16862226	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862221	C₂	1	1	yes	maximal, characteristic
5	#16862169	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862166	C₂	1	1	yes	maximal, characteristic, complemented
7	#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
9	C₂×D₅₂₄	—	2	1	no	—
10	C₂×D₅₂₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₃₁	C₂²	1	1	yes	characteristic
12	#14523502	—	2	2	no	—
13	#14523496	—	2	2	no	—
14	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₁	—	2	1	no	—
16	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
17	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
18	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₁	—	2	1	no	—
22	C₂²×D₁₃₁	D₄	1	1	yes	characteristic
23	Dic₂₆₂	—	2	1	no	—
24	Dic₂₆₂	—	2	1	no	—
25	D₅₂₄	—	2	1	no	—
26	D₅₂₄	—	2	1	no	—
27	D₅₂₄	—	4	2	no	—
28	C₂×Dic₁₃₁	—	2	1	no	—
29	C₂×Dic₁₃₁	D₄	1	1	yes	characteristic
30	C₄×D₁₃₁	—	2	1	no	—
31	C₄×D₁₃₁	—	2	1	no	—
32	C₁₃₁⋊C₈	—	2	1	no	—
33	C₂×C₄×C₁₃₁	—	2	1	no	—
34	C₂×C₄×C₁₃₁	—	2	1	no	—
35	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₈×C₁₃₁	D₄	1	2	yes	complemented
37	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×D₁₃₁	—	2	1	no	—
39	C₂×D₁₃₁	—	2	1	no	—
40	C₂×D₁₃₁	—	2	1	no	—
41	C₂×D₁₃₁	—	4	1	no	—
42	C₂×D₁₃₁	—	4	1	no	—
43	C₂×D₁₃₁	—	4	1	no	—
44	C₄×C₁₃₁	—	4	1	no	—
45	C₄×C₁₃₁	—	4	1	no	—
46	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₃₁	#41	1	1	yes	characteristic
48	Dic₁₃₁	—	2	1	no	—
49	Dic₁₃₁	—	4	1	no	—
50	C₂×C₁₃₁	#121	1	1	yes	characteristic
51	C₂×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
52	C₂×C₁₃₁	#137	1	1	yes	characteristic
53	D₁₃₁	—	4	1	no	—
54	D₁₃₁	—	8	1	no	—
55	C₁₃₁	#468	1	1	yes	characteristic, complemented, Sylow
56	#468	—	131	1	no	maximal, Sylow
57	C₂×SD₁₆	—	131	1	no	—
58	C₂×M₁₆	—	131	1	no	—
59	C₂²⋊Q₈	—	131	1	no	—
60	#121	—	131	1	no	—
61	#107	D₁₃₁	1	1	yes	characteristic, complemented
62	#103	—	131	1	no	—
63	#102	—	131	1	no	—
64	C₂×Q₈	—	131	1	no	—
65	C₂×D₄	—	262	1	no	—
66	C₂×D₄	—	131	1	no	—
67	C₂²×C₄	—	131	1	no	—
68	SD₁₆	—	262	2	no	—
69	M₁₆	—	262	2	no	—
70	C₂×C₈	—	131	1	no	—
71	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
72	C₄⋊C₄	—	262	1	no	—
73	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
74	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
75	C₂²⋊C₄	—	262	1	no	—
76	C₂²⋊C₄	—	262	1	no	—
77	C₂³	—	262	1	no	—
78	C₂³	—	131	1	no	—
79	Q₈	—	262	1	no	—
80	Q₈	—	262	1	no	—
81	D₄	—	262	1	no	—
82	D₄	—	262	1	no	—
83	D₄	—	524	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	262	1	no	—
87	C₂×C₄	—	262	1	no	—
88	C₂×C₄	—	262	1	no	—
89	C₂×C₄	—	131	1	no	—
90	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
91	C₈	—	262	1	no	—
92	C₈	D₅₂₄	1	2	yes	complemented
93	C₂²	—	262	1	no	—
94	C₂²	—	262	1	no	—
95	C₂²	—	262	1	no	—
96	C₂²	—	524	1	no	—
97	C₂²	—	524	1	no	—
98	C₂²	—	524	1	no	—
99	C₂²	D₄×D₁₃₁	1	1	yes	characteristic
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	262	1	no	—
103	C₄	—	524	1	no	—
104	C₄	C₂×D₅₂₄	1	1	yes	characteristic
105	C₄	#14523514	1	1	yes	characteristic
106	C₂	—	524	1	no	—
107	C₂	—	1048	1	no	—
108	C₂	#16862221	1	1	yes	characteristic
109	C₂	#16862270	1	1	yes	characteristic
110	C₂	#16862278	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523502

C₂

yes

maximal, characteristic, complemented

C₂×#14523496

C₂

yes

maximal, characteristic, complemented

#16862226

C₂

yes

maximal, characteristic, complemented

#16862221

C₂

yes

maximal, characteristic

#16862169

C₂

yes

maximal, characteristic, complemented

#16862166

C₂

yes

maximal, characteristic, complemented

#107×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₂

C₂²

yes

characteristic

C₂×D₅₂₄

—

C₂×D₅₂₄

C₂²

yes

characteristic

C₂×C₄×D₁₃₁

C₂²

yes

characteristic

#14523502

—

#14523496

—

C₂×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

—

(C₂×D₁₃₁)⋊C₄

—

(C₂×D₁₃₁)⋊C₄

—

C₂×C₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₂²×D₁₃₁

—

C₂²×D₁₃₁

D₄

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

D₅₂₄

—

D₅₂₄

—

D₅₂₄

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

D₄

yes

characteristic

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₁₃₁⋊C₈

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₈×C₁₃₁

D₄

yes

complemented

C₂²×C₁₃₁

C₂×D₄

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₄×C₁₃₁

#41

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

D₁₃₁

—

D₁₃₁

—

C₁₃₁

#468

yes

characteristic, complemented, Sylow

#468

—

131

maximal, Sylow

C₂×SD₁₆

—

131

—

C₂×M₁₆

—

131

—

C₂²⋊Q₈

—

131

—

#121

—

131

—

#107

D₁₃₁

yes

characteristic, complemented

#103

—

131

—

#102

—

131

—

C₂×Q₈

—

131

—

C₂×D₄

—

262

—

C₂×D₄

—

131

—

C₂²×C₄

—

131

—

SD₁₆

—

262

—

M₁₆

—

262

—

C₂×C₈

—

131

—

C₂×C₈

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

—

262

—

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₂²⋊C₄

—

262

—

C₂²⋊C₄

—

262

—

C₂³

—

262

—

C₂³

—

131

—

Q₈

—

262

—

Q₈

—

262

—

D₄

—

262

—

D₄

—

262

—

D₄

—

524

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

131

—

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

C₈

—

262

—

C₈

D₅₂₄

yes

complemented

C₂²

—

262

—

C₂²

—

262

—

C₂²

—

262

—

C₂²

—

524

—

C₂²

—

524

—

C₂²

—

524

—

C₂²

D₄×D₁₃₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

262

—

103

C₄

—

524

—

104

C₄

C₂×D₅₂₄

yes

characteristic

105

C₄

#14523514

yes

characteristic

106

C₂

—

524

—

107

C₂

—

1048

—

108

C₂

#16862221

yes

characteristic

109

C₂

#16862270

yes

characteristic

110

C₂

#16862278

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₁
commutator	C₂×C₄×C₁₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	#107×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	#107	D₁₃₁
hypocenter	C₁₃₁	#468
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×D₁₃₁

commutator

C₂×C₄×C₁₃₁

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₁

Fitting

#107×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂×D₄

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
1. C₂×#14523502	C₂	semidirect
2. C₂×#14523496	C₂	semidirect
3. #16862226	C₂	semidirect
5. #16862169	C₂	semidirect
6. #16862166	C₂	semidirect
7. #107×C₁₃₁	C₂	semidirect
36. C₈×C₁₃₁	D₄	semidirect
55. C₁₃₁	#468	semidirect
61. #107	D₁₃₁	semidirect
92. C₈	D₅₂₄	semidirect

H ≅ G/N

type

1. C₂×#14523502

C₂

semidirect

2. C₂×#14523496

C₂

semidirect

3. #16862226

C₂

semidirect

5. #16862169

C₂

semidirect

6. #16862166

C₂

semidirect

7. #107×C₁₃₁

C₂

semidirect

36. C₈×C₁₃₁

D₄

semidirect

55. C₁₃₁

#468

semidirect

61. #107

D₁₃₁

semidirect

92. C₈

D₅₂₄

semidirect

Subgroups — #22553559

Conjugacy classes of subgroups

Complements