Group #44552253 — subgroups

Subgroups — #44552253

Order 32128.

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 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	#30632127	C₂	1	1	yes	maximal, characteristic
2	#30631809	C₂	1	1	yes	maximal, characteristic
3	#30631793	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631509	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631504	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631464	C₂	1	1	yes	maximal, characteristic, complemented
7	#360×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
9	#22548722	—	2	1	no	—
10	#22548680	—	2	1	no	—
11	#22548661	C₄	1	2	yes	complemented
12	#22548659	C₄	1	2	yes	complemented
13	#22548655	C₂²	1	1	yes	characteristic
14	#22548651	C₂²	1	1	yes	characteristic
15	#22548642	—	2	1	no	—
16	#22548638	—	2	1	no	—
17	#22548625	C₂²	1	1	yes	characteristic
18	#135×C₂₅₁	C₂²	1	1	yes	characteristic
19	M₃₂×C₂₅₁	C₂²	1	1	yes	characteristic
20	#106×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₀₂	—	2	1	no	—
22	Dic₁₀₀₄	—	2	1	no	—
23	Dic₁₀₀₄	C₂×C₄	1	2	yes	—
24	#16860794	—	2	1	no	—
25	#16860794	—	4	1	no	—
26	C₂×C₂₅₁⋊C₈	—	4	1	no	—
27	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
28	Dic₂₅₁⋊C₄	—	4	1	no	—
29	C₄×Dic₂₅₁	—	4	1	no	—
30	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
31	C₂₅₁⋊C₁₆	C₂×C₄	1	2	yes	—
32	#41×C₂₅₁	—	2	1	no	—
33	Q₁₆×C₂₅₁	C₂×C₄	1	1	yes	characteristic
34	SD₁₆×C₂₅₁	—	2	1	no	—
35	D₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₂₅₁	—	2	1	no	—
38	C₁₆×C₂₅₁	—	2	1	no	—
39	Dic₅₀₂	—	2	2	no	—
40	Dic₅₀₂	—	4	1	no	—
41	C₂×Dic₂₅₁	—	2	1	no	—
42	C₂×Dic₂₅₁	—	4	1	no	—
43	C₂₅₁⋊C₈	—	2	1	no	—
44	C₂₅₁⋊C₈	—	4	1	no	—
45	Q₈×C₂₅₁	—	2	1	no	—
46	D₄×C₂₅₁	—	2	1	no	—
47	D₄×C₂₅₁	—	4	1	no	—
48	C₂×C₄×C₂₅₁	—	4	1	no	—
49	C₂×C₄×C₂₅₁	—	4	1	no	—
50	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₂₅₁	#41	1	1	yes	characteristic
53	C₂²×C₂₅₁	—	4	1	no	—
54	C₂²×C₂₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₄×C₂₅₁	—	4	1	no	—
56	C₄×C₂₅₁	—	8	1	no	—
57	C₄×C₂₅₁	C₄×D₄	1	1	yes	characteristic
58	C₄×C₂₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₂₅₁	—	4	1	no	—
60	Dic₂₅₁	—	4	2	no	—
61	C₂×C₂₅₁	—	2	1	no	—
62	C₂×C₂₅₁	—	8	1	no	—
63	C₂×C₂₅₁	#442	1	1	yes	characteristic
64	C₂₅₁	#2180	1	1	yes	characteristic, complemented, Sylow
65	#2180	—	251	1	no	maximal, Sylow
66	#510	—	251	1	no	—
67	#444	—	251	1	no	—
68	#441	—	251	1	no	—
69	#365	—	251	1	no	—
70	#362	—	251	1	no	—
71	#360	D₂₅₁	1	1	yes	characteristic
72	#347	—	251	1	no	—
73	#135	C₂×D₂₅₁	1	1	yes	characteristic
74	C₂×Q₁₆	—	251	1	no	—
75	#131	—	502	1	no	—
76	C₄×Q₈	—	502	1	no	—
77	Q₃₂	—	251	2	no	—
78	SD₃₂	—	251	2	no	—
79	M₃₂	—	251	1	no	—
80	M₃₂	C₂×D₂₅₁	1	1	yes	characteristic
81	#108	—	251	1	no	—
82	#106	C₂×D₂₅₁	1	1	yes	characteristic
83	#104	—	502	1	no	—
84	#103	—	502	1	no	—
85	#97	—	251	1	no	—
86	#41	—	2	1	no	—
87	C₂×Q₈	—	502	1	no	—
88	Q₁₆	—	502	1	no	—
89	Q₁₆	—	251	2	no	—
90	Q₁₆	C₄×D₂₅₁	1	1	yes	characteristic
91	SD₁₆	—	2	1	no	—
92	D₈	C₄×D₂₅₁	1	1	yes	characteristic
93	M₁₆	—	1004	1	no	—
94	M₁₆	—	502	1	no	—
95	C₂×C₈	—	1004	1	no	—
96	C₂×C₈	—	251	1	no	—
97	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	1004	1	no	—
100	C₄²	—	1004	1	no	—
101	C₄²	—	251	1	no	—
102	C₁₆	—	2	1	no	—
103	C₁₆	—	251	2	no	—
104	Q₈	—	2	1	no	—
105	Q₈	—	1004	1	no	—
106	Q₈	—	502	2	no	—
107	D₄	—	2	1	no	—
108	D₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	1004	1	no	—
112	C₂×C₄	—	502	1	no	—
113	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
114	C₈	—	1004	1	no	—
115	C₈	—	502	1	no	—
116	C₈	C₂×C₄×D₂₅₁	1	1	yes	characteristic
117	C₈	#16860809	1	1	yes	characteristic
118	C₂²	—	4	1	no	—
119	C₂²	#22548739	1	1	yes	characteristic
120	C₄	—	4	1	no	—
121	C₄	—	8	1	no	—
122	C₄	—	1004	1	no	—
123	C₄	—	1004	2	no	—
124	C₄	(C₂²⋊D₂₅₁)⋊C₄	1	1	yes	characteristic
125	C₄	#22548742	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	8	1	no	—
128	C₂	#30631796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632127

C₂

yes

maximal, characteristic

#30631809

C₂

yes

maximal, characteristic

#30631793

C₂

yes

maximal, characteristic, complemented

#30631509

C₂

yes

maximal, characteristic, complemented

#30631504

C₂

yes

maximal, characteristic, complemented

#30631464

C₂

yes

maximal, characteristic, complemented

#360×C₂₅₁

C₂

yes

maximal, characteristic

C₂×Dic₁₀₀₄

C₂²

yes

—

—

C₄

yes

complemented

#22548659

C₄

yes

complemented

#22548655

C₂²

yes

characteristic

#22548651

C₂²

yes

—

—

C₂²

yes

characteristic

#135×C₂₅₁

C₂²

yes

characteristic

M₃₂×C₂₅₁

C₂²

yes

characteristic

#106×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

Dic₁₀₀₄

—

Dic₁₀₀₄

C₂×C₄

yes

—

#16860794

—

#16860794

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

D₄

yes

characteristic

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

D₄

yes

characteristic

C₂₅₁⋊C₁₆

C₂×C₄

yes

—

#41×C₂₅₁

—

Q₁₆×C₂₅₁

C₂×C₄

yes

characteristic

SD₁₆×C₂₅₁

—

D₈×C₂₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₁₆×C₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

Q₈×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

C₂²×C₄

yes

characteristic

C₈×C₂₅₁

#41

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

AGL(1,ℤ/8)

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×D₄

yes

characteristic

C₄×C₂₅₁

AGL(1,ℤ/8)

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#442

yes

characteristic

C₂₅₁

#2180

yes

characteristic, complemented, Sylow

#2180

—

251

maximal, Sylow

#510

—

251

—

#444

—

251

—

#441

—

251

—

#365

—

251

—

#362

—

251

—

#360

D₂₅₁

yes

characteristic

#347

—

251

—

#135

C₂×D₂₅₁

yes

characteristic

C₂×Q₁₆

—

251

—

#131

—

502

—

C₄×Q₈

—

502

—

Q₃₂

—

251

—

SD₃₂

—

251

—

M₃₂

—

251

—

M₃₂

C₂×D₂₅₁

yes

characteristic

#108

—

251

—

#106

C₂×D₂₅₁

yes

characteristic

#104

—

502

—

#103

—

502

—

#97

—

251

—

#41

—

C₂×Q₈

—

502

—

Q₁₆

—

502

—

Q₁₆

—

251

—

Q₁₆

C₄×D₂₅₁

yes

characteristic

SD₁₆

—

D₈

C₄×D₂₅₁

yes

characteristic

M₁₆

—

1004

—

M₁₆

—

502

—

C₂×C₈

—

1004

—

C₂×C₈

—

251

—

C₂×C₈

C₂²×D₂₅₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

1004

—

100

C₄²

—

1004

—

101

C₄²

—

251

—

102

C₁₆

—

103

C₁₆

—

251

—

104

Q₈

—

105

Q₈

—

1004

—

106

Q₈

—

502

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

1004

—

112

C₂×C₄

—

502

—

113

C₂×C₄

D₄×D₂₅₁

yes

characteristic

114

C₈

—

1004

—

115

C₈

—

502

—

116

C₈

C₂×C₄×D₂₅₁

yes

characteristic

117

C₈

#16860809

yes

characteristic

118

C₂²

—

119

C₂²

#22548739

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

1004

—

123

C₄

—

1004

—

124

C₄

(C₂²⋊D₂₅₁)⋊C₄

yes

characteristic

125

C₄

#22548742

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#30631796

yes

characteristic

	subgroup	G/H
center	C₂	#30631796
commutator	C₈×C₂₅₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#360×C₂₅₁	C₂
socle	C₂×C₂₅₁	#442
prime radical	#135×C₂₅₁	C₂²
cosocle	C₂×C₈×C₂₅₁	C₂³
hypercenter	#360	D₂₅₁
hypocenter	C₂₅₁	#2180
LCS	G ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#30631796

commutator

C₈×C₂₅₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₂₅₁

Fitting

#360×C₂₅₁

C₂

socle

C₂×C₂₅₁

#442

prime radical

#135×C₂₅₁

C₂²

cosocle

C₂×C₈×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
3. #30631793	C₂	semidirect
4. #30631509	C₂	semidirect
5. #30631504	C₂	semidirect
6. #30631464	C₂	semidirect
11. #22548661	C₄	semidirect
12. #22548659	C₄	semidirect
64. C₂₅₁	#2180	semidirect

H ≅ G/N

type

3. #30631793

C₂

semidirect

4. #30631509

C₂

semidirect

5. #30631504

C₂

semidirect

6. #30631464

C₂

semidirect

11. #22548661

C₄

semidirect

12. #22548659

C₄

semidirect

64. C₂₅₁

#2180

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#30632127	2
#30631809	2
#30631793	2
#30631509	2
#30631504	2
#30631464	2
#360×C₂₅₁	2
#2180	251

subgroup

index

#360×C₂₅₁

251

Subgroups — #44552253

Conjugacy classes of subgroups

Complements

Maximal subgroups