Group #22553397 — subgroups

Subgroups — #22553397

Order 8384.

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×#14523500	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#16862208	C₂	1	1	yes	maximal, characteristic
4	#16862177	C₂	1	1	yes	maximal, characteristic, complemented
5	#16862168	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₂₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₃₁	—	2	1	no	—
10	#14523500	C₄	1	4	yes	complemented
11	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₁	—	2	1	no	—
13	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
14	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
15	C₄×Dic₁₃₁	—	2	1	no	—
16	C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂²×D₁₃₁	—	2	1	no	—
21	D₅₂₄	—	2	1	no	—
22	D₅₂₄	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₃₁	—	2	1	no	—
24	C₂×Dic₁₃₁	D₄	1	1	yes	characteristic
25	C₂×Dic₁₃₁	D₄	1	1	yes	characteristic
26	C₄×D₁₃₁	—	4	2	no	—
27	C₁₃₁⋊C₈	C₂×C₄	1	2	yes	complemented
28	Q₈×C₁₃₁	—	2	1	no	—
29	Q₈×C₁₃₁	C₂×C₄	1	2	yes	complemented
30	C₂×C₄×C₁₃₁	—	2	1	no	—
31	C₂×C₄×C₁₃₁	—	2	1	no	—
32	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₈×C₁₃₁	—	2	1	no	—
34	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
35	C₂×D₁₃₁	—	2	2	no	—
36	C₂×D₁₃₁	—	4	1	no	—
37	C₂×D₁₃₁	—	4	1	no	—
38	C₄×C₁₃₁	—	2	2	no	—
39	C₄×C₁₃₁	—	4	1	no	—
40	C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₁₃₁	#41	1	1	yes	characteristic
42	Dic₁₃₁	—	2	1	no	—
43	Dic₁₃₁	—	4	1	no	—
44	Dic₁₃₁	SD₁₆	1	2	yes	complemented
45	C₂×C₁₃₁	C₄×D₄	1	1	yes	characteristic
46	C₂×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
47	C₂×C₁₃₁	#135	1	1	yes	characteristic
48	D₁₃₁	—	4	2	no	—
49	C₁₃₁	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
50	C₄×SD₁₆	—	131	1	no	maximal, Sylow
51	C₂×SD₁₆	—	131	1	no	—
52	C₄×Q₈	—	131	1	no	—
53	C₄×D₄	—	131	1	no	—
54	#106	—	131	1	no	—
55	#103	D₁₃₁	1	1	yes	characteristic, complemented
56	#102	—	131	1	no	—
57	C₄×C₈	—	131	1	no	—
58	C₂×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
59	C₂×D₄	—	131	1	no	—
60	C₂²×C₄	—	262	1	no	—
61	SD₁₆	—	131	4	no	—
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	C₂×D₁₃₁	1	1	yes	characteristic
67	C₂²⋊C₄	—	262	1	no	—
68	C₄²	—	262	1	no	—
69	C₄²	—	131	1	no	—
70	C₂³	—	262	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	C₄×D₁₃₁	1	2	yes	complemented
73	D₄	—	262	1	no	—
74	D₄	—	131	2	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	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	—	2	1	no	—
83	C₈	—	131	2	no	—
84	C₂²	—	262	2	no	—
85	C₂²	—	524	1	no	—
86	C₂²	—	524	1	no	—
87	C₂²	D₄×D₁₃₁	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	4	1	no	—
90	C₄	—	262	1	no	—
91	C₄	—	524	1	no	—
92	C₄	—	131	2	no	—
93	C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
94	C₄	#14523512	1	1	yes	characteristic
95	C₂	—	524	2	no	—
96	C₂	(C₂²⋊D₁₃₁)⋊C₄	1	1	yes	characteristic
97	C₂	SD₁₆×D₁₃₁	1	1	yes	characteristic
98	C₂	#16862269	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523500

C₂

yes

maximal, characteristic

Q₈×Dic₁₃₁

C₂

yes

maximal, characteristic, complemented

#16862208

C₂

yes

maximal, characteristic

#16862177

C₂

yes

maximal, characteristic, complemented

#16862168

C₂

yes

maximal, characteristic

C₈×Dic₁₃₁

C₂

yes

maximal, characteristic, complemented

#103×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₂₄

C₂²

yes

characteristic

C₂×C₄×D₁₃₁

—

#14523500

C₄

yes

complemented

C₂×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

(C₂×D₁₃₁)⋊C₄

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

C₂²

yes

characteristic

C₂×Q₈×C₁₃₁

C₂²

yes

characteristic

C₂×C₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₂²×D₁₃₁

—

D₅₂₄

—

D₅₂₄

C₂×C₄

yes

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

D₄

yes

characteristic

C₂×Dic₁₃₁

D₄

yes

characteristic

C₄×D₁₃₁

—

C₁₃₁⋊C₈

C₂×C₄

yes

complemented

Q₈×C₁₃₁

—

Q₈×C₁₃₁

C₂×C₄

yes

complemented

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

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₄×C₁₃₁

#41

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

SD₁₆

yes

complemented

C₂×C₁₃₁

C₄×D₄

yes

characteristic

C₂×C₁₃₁

C₂×SD₁₆

yes

characteristic

C₂×C₁₃₁

#135

yes

characteristic

D₁₃₁

—

C₁₃₁

C₄×SD₁₆

yes

characteristic, complemented, Sylow

C₄×SD₁₆

—

131

maximal, Sylow

C₂×SD₁₆

—

131

—

C₄×Q₈

—

131

—

C₄×D₄

—

131

—

#106

—

131

—

#103

D₁₃₁

yes

characteristic, complemented

#102

—

131

—

C₄×C₈

—

131

—

C₂×Q₈

C₂×D₁₃₁

yes

characteristic

C₂×D₄

—

131

—

C₂²×C₄

—

262

—

SD₁₆

—

131

—

C₂×C₈

—

131

—

C₂×C₈

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

—

262

—

C₄⋊C₄

—

131

—

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₂²⋊C₄

—

262

—

C₄²

—

262

—

C₄²

—

131

—

C₂³

—

262

—

Q₈

—

Q₈

C₄×D₁₃₁

yes

complemented

D₄

—

262

—

D₄

—

131

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

524

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

C₈

—

C₈

—

131

—

C₂²

—

262

—

C₂²

—

524

—

C₂²

—

524

—

C₂²

D₄×D₁₃₁

yes

characteristic

C₄

—

C₄

—

C₄

—

262

—

C₄

—

524

—

C₄

—

131

—

C₄

C₂×C₄×D₁₃₁

yes

characteristic

C₄

#14523512

yes

characteristic

C₂

—

524

—

C₂

(C₂²⋊D₁₃₁)⋊C₄

yes

characteristic

C₂

SD₁₆×D₁₃₁

yes

characteristic

C₂

#16862269

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₁
commutator	C₄×C₁₃₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	#103×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	#103	D₁₃₁
hypocenter	C₁₃₁	C₄×SD₁₆
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₃₁

commutator

C₄×C₁₃₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₃₁

Fitting

#103×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂×D₄

hypercenter

#103

D₁₃₁

hypocenter

C₁₃₁

C₄×SD₁₆

LCS

G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
2. Q₈×Dic₁₃₁	C₂	semidirect
4. #16862177	C₂	semidirect
6. C₈×Dic₁₃₁	C₂	semidirect
7. #103×C₁₃₁	C₂	semidirect
10. #14523500	C₄	semidirect
27. C₁₃₁⋊C₈	C₂×C₄	semidirect
29. Q₈×C₁₃₁	C₂×C₄	semidirect
44. Dic₁₃₁	SD₁₆	semidirect
49. C₁₃₁	C₄×SD₁₆	semidirect
55. #103	D₁₃₁	semidirect
72. Q₈	C₄×D₁₃₁	semidirect

H ≅ G/N

type

2. Q₈×Dic₁₃₁

C₂

semidirect

4. #16862177

C₂

semidirect

6. C₈×Dic₁₃₁

C₂

semidirect

7. #103×C₁₃₁

C₂

semidirect

10. #14523500

C₄

semidirect

27. C₁₃₁⋊C₈

C₂×C₄

semidirect

29. Q₈×C₁₃₁

C₂×C₄

semidirect

44. Dic₁₃₁

SD₁₆

semidirect

49. C₁₃₁

C₄×SD₁₆

semidirect

55. #103

D₁₃₁

semidirect

72. Q₈

C₄×D₁₃₁

semidirect

Subgroups — #22553397

Conjugacy classes of subgroups

Complements