Group #22553552 — subgroups

Subgroups — #22553552

Order 8384.

114 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 114 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×#14523503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#16862226	C₂	1	1	yes	maximal, characteristic
4	#16862222	C₂	1	1	yes	maximal, characteristic
5	#16862179	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862171	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₃₁	C₂²	1	1	yes	characteristic
10	#14523503	—	2	2	no	—
11	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₁₃₁	—	2	2	no	—
13	C₄⋊Dic₁₃₁	—	2	1	no	—
14	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
15	Dic₁₃₁⋊C₄	—	2	1	no	—
16	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
17	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
18	C₂×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₁	D₄	1	1	yes	characteristic
22	Dic₂₆₂	—	2	1	no	—
23	Dic₂₆₂	—	2	1	no	—
24	C₂×Dic₁₃₁	—	2	1	no	—
25	C₂×Dic₁₃₁	—	2	1	no	—
26	C₂×Dic₁₃₁	D₄	1	1	yes	characteristic
27	C₄×D₁₃₁	—	2	1	no	—
28	C₄×D₁₃₁	—	2	1	no	—
29	C₁₃₁⋊C₈	D₄	1	2	yes	—
30	Q₈×C₁₃₁	—	2	1	no	—
31	Q₈×C₁₃₁	—	2	1	no	—
32	C₂×C₄×C₁₃₁	—	2	1	no	—
33	C₂×C₄×C₁₃₁	—	2	1	no	—
34	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₈×C₁₃₁	—	2	1	no	—
36	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
37	C₂×D₁₃₁	—	2	1	no	—
38	C₂×D₁₃₁	—	2	1	no	—
39	C₂×D₁₃₁	Q₁₆	1	2	yes	complemented
40	C₄×C₁₃₁	—	4	1	no	—
41	C₄×C₁₃₁	—	4	1	no	—
42	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
43	C₄×C₁₃₁	#41	1	1	yes	characteristic
44	Dic₁₃₁	—	2	1	no	—
45	Dic₁₃₁	—	4	1	no	—
46	Dic₁₃₁	—	4	1	no	—
47	C₂×C₁₃₁	#121	1	1	yes	characteristic
48	C₂×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₁₃₁	#135	1	1	yes	characteristic
50	D₁₃₁	—	2	2	no	—
51	C₁₃₁	#467	1	1	yes	characteristic, complemented, Sylow
52	#467	—	131	1	no	maximal, Sylow
53	C₂×Q₁₆	—	131	1	no	—
54	C₂²×C₈	—	131	1	no	—
55	C₂²⋊Q₈	—	131	1	no	—
56	C₂²⋊Q₈	—	131	1	no	—
57	#107	—	131	1	no	—
58	#103	—	131	1	no	—
59	#103	D₁₃₁	1	1	yes	characteristic, complemented
60	C₂×Q₈	—	131	1	no	—
61	C₂×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
62	C₂²×C₄	—	131	1	no	—
63	Q₁₆	—	262	2	no	—
64	C₂×C₈	—	262	2	no	—
65	C₂×C₈	—	131	1	no	—
66	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
67	C₄⋊C₄	—	262	1	no	—
68	C₄⋊C₄	—	262	1	no	—
69	C₄⋊C₄	—	131	1	no	—
70	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
71	C₂²⋊C₄	—	262	1	no	—
72	C₂²⋊C₄	—	262	1	no	—
73	C₂³	—	131	1	no	—
74	Q₈	—	2	1	no	—
75	Q₈	—	2	1	no	—
76	Q₈	—	262	1	no	—
77	Q₈	—	262	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	262	1	no	—
81	C₂×C₄	—	262	1	no	—
82	C₂×C₄	—	262	1	no	—
83	C₂×C₄	—	262	1	no	—
84	C₂×C₄	—	131	1	no	—
85	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	131	2	no	—
88	C₂²	—	262	1	no	—
89	C₂²	—	262	1	no	—
90	C₂²	—	131	2	no	—
91	C₂²	D₄×D₁₃₁	1	1	yes	characteristic
92	C₄	—	4	1	no	—
93	C₄	—	4	1	no	—
94	C₄	—	262	1	no	—
95	C₄	—	524	1	no	—
96	C₄	—	524	1	no	—
97	C₄	D₄×D₁₃₁	1	1	yes	characteristic
98	C₄	#14523513	1	1	yes	characteristic
99	C₂	—	262	2	no	—
100	C₂	#16862219	1	1	yes	characteristic
101	C₂	Q₁₆×D₁₃₁	1	1	yes	characteristic
102	C₂	#16862267	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523503

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₃₁

C₂

yes

maximal, characteristic

#16862226

C₂

yes

maximal, characteristic

#16862222

C₂

yes

maximal, characteristic

#16862179

C₂

yes

maximal, characteristic, complemented

#16862171

C₂

yes

maximal, characteristic, complemented

#103×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₂

C₂²

yes

characteristic

C₂×C₄×D₁₃₁

C₂²

yes

characteristic

#14523503

—

C₂×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₈×D₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

Dic₁₃₁⋊C₄

—

(C₂×D₁₃₁)⋊C₄

—

(C₂×D₁₃₁)⋊C₄

—

C₂×Q₈×C₁₃₁

C₂²

yes

characteristic

C₂×C₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₂²×D₁₃₁

D₄

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

D₄

yes

characteristic

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₁₃₁⋊C₈

D₄

yes

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

C₂×D₄

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

Q₁₆

yes

complemented

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₄×C₁₃₁

#41

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

#121

yes

characteristic

C₂×C₁₃₁

C₂×Q₁₆

yes

characteristic

C₂×C₁₃₁

#135

yes

characteristic

D₁₃₁

—

C₁₃₁

#467

yes

characteristic, complemented, Sylow

#467

—

131

maximal, Sylow

C₂×Q₁₆

—

131

—

C₂²×C₈

—

131

—

C₂²⋊Q₈

—

131

—

C₂²⋊Q₈

—

131

—

#107

—

131

—

#103

—

131

—

#103

D₁₃₁

yes

characteristic, complemented

C₂×Q₈

—

131

—

C₂×Q₈

C₂×D₁₃₁

yes

characteristic

C₂²×C₄

—

131

—

Q₁₆

—

262

—

C₂×C₈

—

262

—

C₂×C₈

—

131

—

C₂×C₈

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

—

262

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

131

—

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₂²⋊C₄

—

262

—

C₂²⋊C₄

—

262

—

C₂³

—

131

—

Q₈

—

Q₈

—

Q₈

—

262

—

Q₈

—

262

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

131

—

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

C₈

—

C₈

—

131

—

C₂²

—

262

—

C₂²

—

262

—

C₂²

—

131

—

C₂²

D₄×D₁₃₁

yes

characteristic

C₄

—

C₄

—

C₄

—

262

—

C₄

—

524

—

C₄

—

524

—

C₄

D₄×D₁₃₁

yes

characteristic

C₄

#14523513

yes

characteristic

C₂

—

262

—

100

C₂

#16862219

yes

characteristic

101

C₂

Q₁₆×D₁₃₁

yes

characteristic

102

C₂

#16862267

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₁
commutator	C₂×C₄×C₁₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	#103×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	#103	D₁₃₁
hypocenter	C₁₃₁	#467
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₃₁

commutator

C₂×C₄×C₁₃₁

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₁

Fitting

#103×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂×D₄

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
1. C₂×#14523503	C₂	semidirect
5. #16862179	C₂	semidirect
6. #16862171	C₂	semidirect
7. #103×C₁₃₁	C₂	semidirect
39. C₂×D₁₃₁	Q₁₆	semidirect
51. C₁₃₁	#467	semidirect
59. #103	D₁₃₁	semidirect

H ≅ G/N

type

1. C₂×#14523503

C₂

semidirect

5. #16862179

C₂

semidirect

6. #16862171

C₂

semidirect

7. #103×C₁₃₁

C₂

semidirect

39. C₂×D₁₃₁

Q₁₆

semidirect

51. C₁₃₁

#467

semidirect

59. #103

D₁₃₁

semidirect

Subgroups — #22553552

Conjugacy classes of subgroups

Complements