Group #22553496 — subgroups

Subgroups — #22553496

Order 8384.

142 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 39 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	#16862293	C₂	1	1	yes	maximal, characteristic
2	#16862279	C₂	1	1	yes	maximal, characteristic, complemented
3	#16862276	C₂	1	1	yes	maximal, characteristic, complemented
4	#16862246	C₂	1	1	yes	maximal, characteristic, complemented
5	#16862174	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862164	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523513	—	2	1	no	—
9	#14523513	C₂²	1	1	yes	characteristic
10	#14523512	—	2	2	no	—
11	#14523512	—	2	1	no	—
12	C₂×Dic₂₆₂	—	2	1	no	—
13	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
15	Q₈×D₁₃₁	—	2	1	no	—
16	Dic₅₂₄	—	2	1	no	—
17	#14523503	—	2	1	no	—
18	#14523502	—	2	1	no	—
19	#14523501	—	2	1	no	—
20	#14523497	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₁₃₁	—	2	2	no	—
22	#41×C₁₃₁	C₂²	1	1	yes	characteristic
23	M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
25	Dic₂₆₂	—	2	2	no	—
26	Dic₂₆₂	—	2	1	no	—
27	Dic₂₆₂	—	2	1	no	—
28	Dic₂₆₂	—	2	1	no	—
29	Dic₂₆₂	—	2	1	no	—
30	Dic₂₆₂	D₄	1	1	yes	characteristic
31	D₅₂₄	D₄	1	1	yes	characteristic
32	C₂²⋊D₁₃₁	—	2	2	no	—
33	C₂²⋊D₁₃₁	—	2	1	no	—
34	C₂×Dic₁₃₁	—	2	2	no	—
35	C₂×Dic₁₃₁	—	2	1	no	—
36	C₂×Dic₁₃₁	D₄	1	2	yes	—
37	C₄×D₁₃₁	—	2	2	no	—
38	C₄×D₁₃₁	—	2	1	no	—
39	C₁₃₁⋊C₈	—	2	1	no	—
40	Q₈×C₁₃₁	D₄	1	1	yes	characteristic
41	D₄×C₁₃₁	—	2	1	no	—
42	D₄×C₁₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₁	—	2	1	no	—
44	C₂×C₄×C₁₃₁	—	2	1	no	—
45	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₈×C₁₃₁	—	2	1	no	—
47	C₂²×C₁₃₁	—	2	1	no	—
48	C₂²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₃₁	—	2	1	no	—
50	C₄×C₁₃₁	—	2	2	no	—
51	C₄×C₁₃₁	—	2	1	no	—
52	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
53	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
54	Dic₁₃₁	—	2	1	no	—
55	Dic₁₃₁	—	2	2	no	—
56	Dic₁₃₁	—	4	1	no	—
57	C₂×C₁₃₁	—	2	1	no	—
58	C₂×C₁₃₁	—	4	1	no	—
59	C₂×C₁₃₁	#120	1	1	yes	characteristic
60	D₁₃₁	—	4	1	no	—
61	C₁₃₁	#456	1	1	yes	characteristic, complemented, Sylow
62	#456	—	131	1	no	maximal, Sylow
63	2₋¹⁺⁴	—	131	1	no	—
64	#137	—	131	1	no	—
65	#137	—	131	1	no	—
66	C₄⋊Q₈	—	131	1	no	—
67	#104	—	131	1	no	—
68	#104	D₁₃₁	1	1	yes	characteristic, complemented
69	#101	—	131	1	no	—
70	#41	—	262	1	no	—
71	#41	—	262	1	no	—
72	#41	—	262	2	no	—
73	#41	—	131	1	no	—
74	#41	C₂×D₁₃₁	1	1	yes	characteristic
75	C₂×Q₈	—	262	1	no	—
76	C₂×Q₈	—	262	1	no	—
77	C₂×Q₈	—	131	1	no	—
78	C₂×Q₈	—	131	1	no	—
79	Q₁₆	—	262	1	no	—
80	Q₁₆	—	262	1	no	—
81	SD₁₆	—	262	1	no	—
82	SD₁₆	—	262	1	no	—
83	M₁₆	—	131	1	no	—
84	M₁₆	C₂×D₁₃₁	1	1	yes	characteristic
85	C₄⋊C₄	—	262	2	no	—
86	C₄²	C₂×D₁₃₁	1	1	yes	characteristic
87	Q₈	—	262	2	no	—
88	Q₈	—	262	1	no	—
89	Q₈	—	262	1	no	—
90	Q₈	—	262	1	no	—
91	Q₈	—	262	1	no	—
92	Q₈	—	131	1	no	—
93	Q₈	D₅₂₄	1	1	yes	characteristic
94	D₄	—	2	1	no	—
95	D₄	—	262	2	no	—
96	D₄	—	262	1	no	—
97	D₄	—	131	1	no	—
98	D₄	D₅₂₄	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	262	2	no	—
102	C₂×C₄	—	262	2	no	—
103	C₂×C₄	—	262	1	no	—
104	C₂×C₄	—	262	1	no	—
105	C₂×C₄	—	131	2	no	—
106	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	262	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	—	262	1	no	—
111	C₂²	D₄×D₁₃₁	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	262	1	no	—
115	C₄	—	262	2	no	—
116	C₄	—	524	1	no	—
117	C₄	D₄×D₁₃₁	1	1	yes	characteristic
118	C₄	C₂×D₅₂₄	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	4	1	no	—
121	C₂	—	524	1	no	—
122	C₂	#16862216	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16862293

C₂

yes

maximal, characteristic

#16862279

C₂

yes

maximal, characteristic, complemented

#16862276

C₂

yes

maximal, characteristic, complemented

#16862246

C₂

yes

maximal, characteristic, complemented

#16862174

C₂

yes

maximal, characteristic, complemented

#16862164

C₂

yes

maximal, characteristic, complemented

#104×C₁₃₁

C₂

yes

maximal, characteristic, complemented

#14523513

—

#14523513

C₂²

yes

characteristic

#14523512

—

#14523512

—

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

C₂²

yes

characteristic

C₂×Dic₂₆₂

C₂²

yes

characteristic

Q₈×D₁₃₁

—

Dic₅₂₄

—

—

—

—

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

—

#41×C₁₃₁

C₂²

yes

characteristic

M₁₆×C₁₃₁

C₂²

yes

characteristic

C₄²×C₁₃₁

C₂²

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

D₄

yes

characteristic

D₅₂₄

D₄

yes

characteristic

C₂²⋊D₁₃₁

—

C₂²⋊D₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

D₄

yes

—

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₁₃₁⋊C₈

—

Q₈×C₁₃₁

D₄

yes

characteristic

D₄×C₁₃₁

—

D₄×C₁₃₁

D₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂×D₄

yes

characteristic

C₂×D₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₄×C₁₃₁

C₂×D₄

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

#120

yes

characteristic

D₁₃₁

—

C₁₃₁

#456

yes

characteristic, complemented, Sylow

#456

—

131

maximal, Sylow

2₋¹⁺⁴

—

131

—

#137

—

131

—

#137

—

131

—

C₄⋊Q₈

—

131

—

#104

—

131

—

#104

D₁₃₁

yes

characteristic, complemented

#101

—

131

—

#41

—

262

—

#41

—

262

—

#41

—

262

—

#41

—

131

—

#41

C₂×D₁₃₁

yes

characteristic

C₂×Q₈

—

262

—

C₂×Q₈

—

262

—

C₂×Q₈

—

131

—

C₂×Q₈

—

131

—

Q₁₆

—

262

—

Q₁₆

—

262

—

SD₁₆

—

262

—

SD₁₆

—

262

—

M₁₆

—

131

—

M₁₆

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

—

262

—

C₄²

C₂×D₁₃₁

yes

characteristic

Q₈

—

262

—

Q₈

—

262

—

Q₈

—

262

—

Q₈

—

262

—

Q₈

—

262

—

Q₈

—

131

—

Q₈

D₅₂₄

yes

characteristic

D₄

—

D₄

—

262

—

D₄

—

262

—

D₄

—

131

—

D₄

D₅₂₄

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

262

—

102

C₂×C₄

—

262

—

103

C₂×C₄

—

262

—

104

C₂×C₄

—

262

—

105

C₂×C₄

—

131

—

106

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

107

C₈

—

108

C₈

—

262

—

109

C₂²

—

110

C₂²

—

262

—

111

C₂²

D₄×D₁₃₁

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

262

—

115

C₄

—

262

—

116

C₄

—

524

—

117

C₄

D₄×D₁₃₁

yes

characteristic

118

C₄

C₂×D₅₂₄

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

524

—

122

C₂

#16862216

yes

characteristic

	subgroup	G/H
center	C₂	#16862216
commutator	C₂×C₄×C₁₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	#104×C₁₃₁	C₂
socle	C₂×C₁₃₁	#120
hypercenter	#104	D₁₃₁
hypocenter	C₁₃₁	#456
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

subgroup

G/H

center

C₂

#16862216

commutator

C₂×C₄×C₁₃₁

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₁

Fitting

#104×C₁₃₁

C₂

C₂×C₁₃₁

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

N	H ≅ G/N	type
2. #16862279	C₂	semidirect
3. #16862276	C₂	semidirect
4. #16862246	C₂	semidirect
5. #16862174	C₂	semidirect
6. #16862164	C₂	semidirect
7. #104×C₁₃₁	C₂	semidirect
61. C₁₃₁	#456	semidirect
68. #104	D₁₃₁	semidirect

H ≅ G/N

type

2. #16862279

C₂

semidirect

3. #16862276

C₂

semidirect

4. #16862246

C₂

semidirect

5. #16862174

C₂

semidirect

6. #16862164

C₂

semidirect

7. #104×C₁₃₁

C₂

semidirect

61. C₁₃₁

#456

semidirect

68. #104

D₁₃₁

semidirect

Subgroups — #22553496

Conjugacy classes of subgroups

Complements