Group #22553539 — subgroups

Subgroups — #22553539

Order 8384.

124 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 45 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×#14523501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16862225	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862177	C₂	1	1	yes	maximal, characteristic, complemented
5	#16862169	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862167	C₂	1	1	yes	maximal, characteristic, complemented
7	#121×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
9	#14523501	—	2	2	no	—
10	C₂×C₁₃₁⋊C₈	—	2	2	no	—
11	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
14	Dic₁₃₁⋊C₄	—	2	1	no	—
15	C₂²⋊Dic₁₃₁	—	2	1	no	—
16	C₂×D₄×C₁₃₁	—	2	1	no	—
17	C₂×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
21	Dic₂₆₂	—	2	1	no	—
22	Dic₂₆₂	—	2	1	no	—
23	C₂×Dic₁₃₁	—	2	1	no	—
24	C₂×Dic₁₃₁	—	2	1	no	—
25	C₁₃₁⋊C₈	—	2	1	no	—
26	C₁₃₁⋊C₈	D₄	1	2	yes	complemented
27	C₂³×C₁₃₁	—	2	1	no	—
28	C₂³×C₁₃₁	D₄	1	1	yes	characteristic
29	D₄×C₁₃₁	—	2	1	no	—
30	D₄×C₁₃₁	—	2	1	no	—
31	D₄×C₁₃₁	—	4	2	no	—
32	C₂×C₄×C₁₃₁	—	2	1	no	—
33	C₂×C₄×C₁₃₁	—	2	1	no	—
34	C₂×C₄×C₁₃₁	—	2	1	no	—
35	C₂×C₄×C₁₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
37	C₂²×C₁₃₁	—	2	1	no	—
38	C₂²×C₁₃₁	—	2	1	no	—
39	C₂²×C₁₃₁	—	4	1	no	—
40	C₂²×C₁₃₁	—	4	1	no	—
41	C₂²×C₁₃₁	—	4	1	no	—
42	C₂²×C₁₃₁	SD₁₆	1	2	yes	complemented
43	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
44	C₄×C₁₃₁	—	2	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	—	4	1	no	—
49	Dic₁₃₁	—	4	1	no	—
50	C₂×C₁₃₁	—	2	2	no	—
51	C₂×C₁₃₁	—	8	1	no	—
52	C₂×C₁₃₁	#121	1	1	yes	characteristic
53	C₂×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₁₃₁	#135	1	1	yes	characteristic
55	C₁₃₁	#465	1	1	yes	characteristic, complemented, Sylow
56	#465	—	131	1	no	maximal, Sylow
57	C₂×SD₁₆	—	131	1	no	—
58	C₂²×C₈	—	131	1	no	—
59	C₂²⋊Q₈	—	131	1	no	—
60	#121	D₁₃₁	1	1	yes	characteristic
61	#106	—	131	1	no	—
62	#103	—	131	1	no	—
63	#102	—	131	1	no	—
64	C₂×Q₈	—	131	1	no	—
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	C₂×D₁₃₁	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₁₃₁	1	1	yes	characteristic
68	SD₁₆	—	262	2	no	—
69	C₂×C₈	—	262	2	no	—
70	C₂×C₈	—	131	1	no	—
71	C₂×C₈	—	131	1	no	—
72	C₄⋊C₄	—	262	1	no	—
73	C₄⋊C₄	—	131	1	no	—
74	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
75	C₂²⋊C₄	—	2	1	no	—
76	C₂²⋊C₄	—	262	1	no	—
77	C₂³	—	2	1	no	—
78	C₂³	C₂²⋊D₁₃₁	1	1	yes	characteristic
79	Q₈	—	262	1	no	—
80	Q₈	—	262	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	4	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	262	1	no	—
88	C₂×C₄	—	262	1	no	—
89	C₂×C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
91	C₈	—	262	1	no	—
92	C₈	—	131	2	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	#14523501	1	2	yes	complemented
99	C₂²	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	524	1	no	—
103	C₄	—	524	1	no	—
104	C₄	D₄×D₁₃₁	1	1	yes	characteristic
105	C₄	#14523512	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	8	1	no	—
108	C₂	#16862218	1	1	yes	characteristic
109	C₂	C₂×#14523501	1	1	yes	characteristic
110	C₂	#16862266	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523501

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₃₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#16862225

C₂

yes

maximal, characteristic, complemented

#16862177

C₂

yes

maximal, characteristic, complemented

#16862169

C₂

yes

maximal, characteristic, complemented

#16862167

C₂

yes

maximal, characteristic, complemented

#121×C₁₃₁

C₂

yes

maximal, characteristic

C₂×Dic₂₆₂

C₂²

yes

characteristic

#14523501

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

Dic₁₃₁⋊C₄

—

C₂²⋊Dic₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

C₂²

yes

characteristic

C₂²×C₄×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₁

—

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

D₄

yes

complemented

C₂³×C₁₃₁

—

C₂³×C₁₃₁

D₄

yes

characteristic

D₄×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

SD₁₆

yes

complemented

C₂²×C₁₃₁

C₂×D₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₄×C₁₃₁

#41

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

#121

yes

characteristic

C₂×C₁₃₁

C₂×SD₁₆

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#465

—

131

maximal, Sylow

C₂×SD₁₆

—

131

—

C₂²×C₈

—

131

—

C₂²⋊Q₈

—

131

—

#121

D₁₃₁

yes

characteristic

#106

—

131

—

#103

—

131

—

#102

—

131

—

C₂×Q₈

—

131

—

C₂×D₄

—

C₂×D₄

C₂×D₁₃₁

yes

characteristic

C₂²×C₄

C₂×D₁₃₁

yes

characteristic

SD₁₆

—

262

—

C₂×C₈

—

262

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

131

—

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

262

—

C₂³

—

C₂³

C₂²⋊D₁₃₁

yes

characteristic

Q₈

—

262

—

Q₈

—

262

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

C₂²⋊D₁₃₁

yes

characteristic

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

C₈

—

262

—

C₈

—

131

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14523501

yes

complemented

C₂²

C₂×C₂²⋊D₁₃₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

524

—

103

C₄

—

524

—

104

C₄

D₄×D₁₃₁

yes

characteristic

105

C₄

#14523512

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#16862218

yes

characteristic

109

C₂

C₂×#14523501

yes

characteristic

110

C₂

#16862266

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₁
commutator	C₂×C₄×C₁₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	#121×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	#121	D₁₃₁
hypocenter	C₁₃₁	#465
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₃₁

commutator

C₂×C₄×C₁₃₁

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₁

Fitting

#121×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂×D₄

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

N	H ≅ G/N	type
1. C₂×#14523501	C₂	semidirect
2. C₂²×C₁₃₁⋊C₈	C₂	semidirect
3. #16862225	C₂	semidirect
4. #16862177	C₂	semidirect
5. #16862169	C₂	semidirect
6. #16862167	C₂	semidirect
26. C₁₃₁⋊C₈	D₄	semidirect
42. C₂²×C₁₃₁	SD₁₆	semidirect
55. C₁₃₁	#465	semidirect
98. C₂²	#14523501	semidirect

H ≅ G/N

type

1. C₂×#14523501

C₂

semidirect

2. C₂²×C₁₃₁⋊C₈

C₂

semidirect

3. #16862225

C₂

semidirect

4. #16862177

C₂

semidirect

5. #16862169

C₂

semidirect

6. #16862167

C₂

semidirect

26. C₁₃₁⋊C₈

D₄

semidirect

42. C₂²×C₁₃₁

SD₁₆

semidirect

55. C₁₃₁

#465

semidirect

98. C₂²

#14523501

semidirect

Subgroups — #22553539

Conjugacy classes of subgroups

Complements