Group #22553476 — subgroups

Subgroups — #22553476

Order 8384.

152 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 43 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×#14523513	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14523502	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862225	C₂	1	1	yes	maximal, characteristic, complemented
5	#16862171	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862168	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523513	—	2	2	no	—
9	#14523513	—	2	2	no	—
10	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
12	C₂×D₅₂₄	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
14	C₂×C₄×D₁₃₁	—	2	1	no	—
15	Dic₅₂₄	—	2	2	no	—
16	#14523502	—	2	2	no	—
17	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
18	Dic₁₃₁⋊C₄	—	2	1	no	—
19	C₂²⋊Dic₁₃₁	—	2	1	no	—
20	C₂²×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂²×D₁₃₁	—	2	1	no	—
24	Dic₂₆₂	—	2	1	no	—
25	Dic₂₆₂	—	2	1	no	—
26	Dic₂₆₂	—	2	1	no	—
27	Dic₂₆₂	D₄	1	2	yes	—
28	D₅₂₄	—	2	1	no	—
29	D₅₂₄	D₄	1	2	yes	—
30	C₂²⋊D₁₃₁	—	2	4	no	—
31	C₂×Dic₁₃₁	—	2	1	no	—
32	C₂×Dic₁₃₁	—	2	1	no	—
33	C₂×Dic₁₃₁	—	2	1	no	—
34	C₄×D₁₃₁	—	2	4	no	—
35	C₂³×C₁₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₁	—	2	1	no	—
37	C₂×C₄×C₁₃₁	—	2	1	no	—
38	C₂×C₄×C₁₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₈×C₁₃₁	—	2	1	no	—
41	C₈×C₁₃₁	—	2	1	no	—
42	C₂²×C₁₃₁	—	2	1	no	—
43	C₂²×C₁₃₁	—	2	1	no	—
44	C₂²×C₁₃₁	—	2	1	no	—
45	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×D₁₃₁	—	2	2	no	—
47	C₂×D₁₃₁	—	4	1	no	—
48	C₂×D₁₃₁	—	4	1	no	—
49	C₄×C₁₃₁	—	2	1	no	—
50	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
51	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
52	Dic₁₃₁	—	2	2	no	—
53	Dic₁₃₁	—	4	1	no	—
54	Dic₁₃₁	—	4	1	no	—
55	C₂×C₁₃₁	—	4	1	no	—
56	C₂×C₁₃₁	#120	1	1	yes	characteristic
57	C₂×C₁₃₁	#135	1	1	yes	characteristic
58	C₂×C₁₃₁	#137	1	1	yes	characteristic
59	D₁₃₁	—	4	2	no	—
60	C₁₃₁	#452	1	1	yes	characteristic, complemented, Sylow
61	#452	—	131	1	no	maximal, Sylow
62	C₂×#41	—	131	1	no	—
63	C₂×Q₁₆	—	131	1	no	—
64	C₂×SD₁₆	—	131	1	no	—
65	C₂²⋊Q₈	—	131	1	no	—
66	#103	—	131	1	no	—
67	#102	—	131	1	no	—
68	C₂²⋊C₈	D₁₃₁	1	1	yes	characteristic, complemented
69	#41	—	262	2	no	—
70	#41	—	262	2	no	—
71	C₂×Q₈	—	131	1	no	—
72	C₂×Q₈	—	131	1	no	—
73	C₂×D₄	—	262	1	no	—
74	C₂×D₄	—	131	1	no	—
75	C₂²×C₄	—	262	1	no	—
76	C₂²×C₄	C₂×D₁₃₁	1	1	yes	characteristic
77	Q₁₆	—	262	2	no	—
78	SD₁₆	—	262	2	no	—
79	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
80	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
81	C₄⋊C₄	—	262	1	no	—
82	C₄⋊C₄	—	131	1	no	—
83	C₂²⋊C₄	—	262	1	no	—
84	C₂³	—	262	1	no	—
85	C₂³	D₅₂₄	1	1	yes	characteristic
86	Q₈	—	262	1	no	—
87	Q₈	—	262	1	no	—
88	Q₈	—	262	1	no	—
89	Q₈	—	131	2	no	—
90	D₄	—	262	4	no	—
91	D₄	—	262	1	no	—
92	D₄	—	131	2	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	262	4	no	—
96	C₂×C₄	—	262	1	no	—
97	C₂×C₄	—	262	1	no	—
98	C₂×C₄	—	262	1	no	—
99	C₂×C₄	D₅₂₄	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	262	2	no	—
107	C₂²	—	524	1	no	—
108	C₂²	—	524	1	no	—
109	C₂²	C₂×D₅₂₄	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	262	2	no	—
112	C₄	—	524	1	no	—
113	C₄	—	524	1	no	—
114	C₄	D₄×D₁₃₁	1	1	yes	characteristic
115	C₄	D₄×D₁₃₁	1	1	yes	characteristic
116	C₂	—	4	1	no	—
117	C₂	—	524	2	no	—
118	C₂	#16862216	1	1	yes	characteristic
119	C₂	#16862268	1	1	yes	characteristic
120	C₂	#16862279	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523513

C₂

yes

maximal, characteristic

C₂×Dic₅₂₄

C₂

yes

maximal, characteristic, complemented

C₂×#14523502

C₂

yes

maximal, characteristic, complemented

#16862225

C₂

yes

maximal, characteristic, complemented

#16862171

C₂

yes

maximal, characteristic, complemented

#16862168

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₈×C₁₃₁

C₂

yes

maximal, characteristic, complemented

#14523513

—

#14523513

—

C₂×Dic₂₆₂

C₂²

yes

characteristic

C₂×Dic₂₆₂

C₂²

yes

characteristic

C₂×D₅₂₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₁

—

C₂×C₄×D₁₃₁

—

Dic₅₂₄

—

#14523502

—

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

Dic₁₃₁⋊C₄

—

C₂²⋊Dic₁₃₁

—

C₂²×C₄×C₁₃₁

C₂²

yes

characteristic

C₂×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×C₈×C₁₃₁

C₂²

yes

characteristic

C₂²×D₁₃₁

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

D₄

yes

—

D₅₂₄

—

D₅₂₄

D₄

yes

—

C₂²⋊D₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₄×D₁₃₁

—

C₂³×C₁₃₁

D₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂×D₄

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₄×C₁₃₁

C₂×D₄

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

D₁₃₁

—

C₁₃₁

#452

yes

characteristic, complemented, Sylow

#452

—

131

maximal, Sylow

C₂×#41

—

131

—

C₂×Q₁₆

—

131

—

C₂×SD₁₆

—

131

—

C₂²⋊Q₈

—

131

—

#103

—

131

—

#102

—

131

—

C₂²⋊C₈

D₁₃₁

yes

characteristic, complemented

#41

—

262

—

#41

—

262

—

C₂×Q₈

—

131

—

C₂×Q₈

—

131

—

C₂×D₄

—

262

—

C₂×D₄

—

131

—

C₂²×C₄

—

262

—

C₂²×C₄

C₂×D₁₃₁

yes

characteristic

Q₁₆

—

262

—

SD₁₆

—

262

—

C₂×C₈

C₂×D₁₃₁

yes

characteristic

C₂×C₈

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

—

262

—

C₄⋊C₄

—

131

—

C₂²⋊C₄

—

262

—

C₂³

—

262

—

C₂³

D₅₂₄

yes

characteristic

Q₈

—

262

—

Q₈

—

262

—

Q₈

—

262

—

Q₈

—

131

—

D₄

—

262

—

D₄

—

262

—

D₄

—

131

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

D₅₂₄

yes

characteristic

100

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

101

C₈

—

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

262

—

107

C₂²

—

524

—

108

C₂²

—

524

—

109

C₂²

C₂×D₅₂₄

yes

characteristic

110

C₄

—

111

C₄

—

262

—

112

C₄

—

524

—

113

C₄

—

524

—

114

C₄

D₄×D₁₃₁

yes

characteristic

115

C₄

D₄×D₁₃₁

yes

characteristic

116

C₂

—

117

C₂

—

524

—

118

C₂

#16862216

yes

characteristic

119

C₂

#16862268

yes

characteristic

120

C₂

#16862279

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₂₄
commutator	C₂×C₄×C₁₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	C₂²⋊C₈×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×D₄
hypercenter	C₂²⋊C₈	D₁₃₁
hypocenter	C₁₃₁	#452
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center

C₂²

C₂×D₅₂₄

commutator

C₂×C₄×C₁₃₁

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₁

Fitting

C₂²⋊C₈×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂×D₄

hypercenter

C₂²⋊C₈

D₁₃₁

hypocenter

C₁₃₁

#452

LCS

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×Dic₅₂₄	C₂	semidirect
3. C₂×#14523502	C₂	semidirect
4. #16862225	C₂	semidirect
5. #16862171	C₂	semidirect
6. #16862168	C₂	semidirect
7. C₂²⋊C₈×C₁₃₁	C₂	semidirect
60. C₁₃₁	#452	semidirect
68. C₂²⋊C₈	D₁₃₁	semidirect

H ≅ G/N

type

2. C₂×Dic₅₂₄

C₂

semidirect

3. C₂×#14523502

C₂

semidirect

4. #16862225

C₂

semidirect

5. #16862171

C₂

semidirect

6. #16862168

C₂

semidirect

7. C₂²⋊C₈×C₁₃₁

C₂

semidirect

60. C₁₃₁

#452

semidirect

68. C₂²⋊C₈

D₁₃₁

semidirect

Subgroups — #22553476

Conjugacy classes of subgroups

Complements