Group #44550019 — subgroups

Subgroups — #44550019

Order 32128.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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	#30632090	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632061	C₂	1	1	yes	maximal, characteristic
3	#30632039	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632035	C₂	1	1	yes	maximal, characteristic
5	#30632034	C₂	1	1	yes	maximal, characteristic
6	#30631408	C₂	1	1	yes	maximal, characteristic, complemented
7	#487×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16860797	—	2	2	no	—
9	#22548715	C₂²	1	1	yes	characteristic
10	#22548712	C₂²	1	1	yes	characteristic
11	#22548648	—	2	2	no	—
12	#22548638	—	2	2	no	—
13	#22548637	—	2	2	no	—
14	#22548635	—	2	2	no	—
15	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
19	#103×C₂₅₁	—	2	2	no	—
20	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₀₂	—	2	1	no	—
22	C₂×D₁₀₀₄	—	2	1	no	—
23	C₂×D₁₀₀₄	—	2	1	no	—
24	C₂×D₁₀₀₄	—	2	1	no	—
25	#16860797	—	4	4	no	—
26	C₂×C₂₅₁⋊C₈	D₄	1	2	yes	—
27	C₂×C₂₅₁⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₂₅₁	—	2	1	no	—
29	C₄⋊Dic₂₅₁	—	2	1	no	—
30	C₂×Q₈×C₂₅₁	—	2	1	no	—
31	C₂×Q₈×C₂₅₁	—	2	1	no	—
32	C₂×C₈×C₂₅₁	D₄	1	2	yes	—
33	C₄⋊C₄×C₂₅₁	—	2	1	no	—
34	C₄⋊C₄×C₂₅₁	—	2	1	no	—
35	C₄⋊C₄×C₂₅₁	—	2	1	no	—
36	C₄⋊C₄×C₂₅₁	—	2	1	no	—
37	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
38	C₂²×D₂₅₁	—	4	1	no	—
39	Dic₅₀₂	—	4	2	no	—
40	D₁₀₀₄	—	4	2	no	—
41	D₁₀₀₄	—	4	2	no	—
42	D₁₀₀₄	—	4	2	no	—
43	C₂×Dic₂₅₁	—	4	1	no	—
44	C₂₅₁⋊C₈	—	2	2	no	—
45	C₂₅₁⋊C₈	—	2	2	no	—
46	Q₈×C₂₅₁	—	4	2	no	—
47	Q₈×C₂₅₁	—	4	2	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	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
53	C₈×C₂₅₁	—	2	2	no	—
54	C₂²×C₂₅₁	#127	1	1	yes	characteristic
55	C₂×D₂₅₁	—	8	1	no	—
56	C₂×D₂₅₁	—	8	1	no	—
57	C₂×D₂₅₁	—	8	1	no	—
58	C₄×C₂₅₁	—	8	1	no	—
59	C₄×C₂₅₁	—	8	1	no	—
60	C₄×C₂₅₁	#135	1	2	yes	—
61	C₄×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₂₅₁	#137	1	2	yes	—
63	Dic₂₅₁	—	8	1	no	—
64	C₂×C₂₅₁	#495	1	1	yes	characteristic
65	C₂×C₂₅₁	#496	1	1	yes	characteristic
66	C₂×C₂₅₁	#497	1	1	yes	characteristic
67	D₂₅₁	—	16	1	no	—
68	C₂₅₁	#1719	1	1	yes	characteristic, complemented, Sylow
69	#1719	—	251	1	no	maximal, Sylow
70	#500	—	251	1	no	—
71	#492	—	251	1	no	—
72	#487	—	251	1	no	—
73	#487	D₂₅₁	1	1	yes	characteristic, complemented
74	#486	—	251	1	no	—
75	#486	—	251	1	no	—
76	#322	—	251	1	no	—
77	C₂×SD₁₆	—	502	2	no	—
78	C₄⋊Q₈	—	251	1	no	—
79	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
80	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
81	#127	—	251	1	no	—
82	#107	—	502	2	no	—
83	#103	—	2	2	no	—
84	#103	—	502	2	no	—
85	#102	—	502	2	no	—
86	#102	—	502	2	no	—
87	C₄×C₈	—	251	1	no	—
88	C₄×C₈	—	251	1	no	—
89	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	502	1	no	—
93	C₂×D₄	—	502	1	no	—
94	C₂×D₄	—	502	1	no	—
95	C₂×D₄	—	502	1	no	—
96	SD₁₆	—	1004	4	no	—
97	C₂×C₈	—	251	2	no	—
98	C₂×C₈	—	251	2	no	—
99	C₂×C₈	C₂²⋊D₂₅₁	1	2	yes	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	502	1	no	—
105	C₄⋊C₄	—	502	1	no	—
106	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
107	C₂³	—	1004	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	4	2	no	—
110	Q₈	—	1004	2	no	—
111	D₄	—	1004	2	no	—
112	D₄	—	1004	2	no	—
113	D₄	—	1004	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	1004	1	no	—
117	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
118	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	502	2	no	—
122	C₈	—	502	2	no	—
123	C₂²	—	2008	1	no	—
124	C₂²	—	2008	1	no	—
125	C₂²	—	2008	1	no	—
126	C₂²	#22548711	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	2008	1	no	—
130	C₄	#22548735	1	2	yes	—
131	C₄	#22548742	1	2	yes	—
132	C₄	#22548747	1	2	yes	—
133	C₂	—	4016	1	no	—
134	C₂	#30632069	1	1	yes	characteristic
135	C₂	#30632073	1	1	yes	characteristic
136	C₂	#30632080	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632090

C₂

yes

maximal, characteristic, complemented

#30632061

C₂

yes

maximal, characteristic

#30632039

C₂

yes

maximal, characteristic, complemented

#30632035

C₂

yes

maximal, characteristic

#30632034

C₂

yes

maximal, characteristic

#30631408

C₂

yes

maximal, characteristic, complemented

#487×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860797

—

#22548715

C₂²

yes

characteristic

#22548712

C₂²

yes

—

—

—

—

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

#103×C₂₅₁

—

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

#16860797

—

C₂×C₂₅₁⋊C₈

D₄

yes

—

C₂×C₂₅₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×C₈×C₂₅₁

D₄

yes

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂²×D₂₅₁

—

Dic₅₀₂

—

D₁₀₀₄

—

D₁₀₀₄

—

D₁₀₀₄

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

#127

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#135

yes

—

C₄×C₂₅₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₅₁

#137

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

D₂₅₁

—

C₂₅₁

#1719

yes

characteristic, complemented, Sylow

#1719

—

251

maximal, Sylow

#500

—

251

—

#492

—

251

—

#487

—

251

—

#487

D₂₅₁

yes

characteristic, complemented

#486

—

251

—

#486

—

251

—

#322

—

251

—

C₂×SD₁₆

—

502

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

#127

—

251

—

#107

—

502

—

#103

—

#103

—

502

—

#102

—

502

—

#102

—

502

—

C₄×C₈

—

251

—

C₄×C₈

—

251

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

502

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

SD₁₆

—

1004

—

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₂×C₈

C₂²⋊D₂₅₁

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

502

—

105

C₄⋊C₄

—

502

—

106

C₄²

C₂²×D₂₅₁

yes

characteristic

107

C₂³

—

1004

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

1004

—

111

D₄

—

1004

—

112

D₄

—

1004

—

113

D₄

—

1004

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

1004

—

117

C₂×C₄

D₄×D₂₅₁

yes

characteristic

118

C₂×C₄

D₄×D₂₅₁

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

120

C₈

—

121

C₈

—

502

—

122

C₈

—

502

—

123

C₂²

—

2008

—

124

C₂²

—

2008

—

125

C₂²

—

2008

—

126

C₂²

#22548711

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

2008

—

130

C₄

#22548735

yes

—

131

C₄

#22548742

yes

—

132

C₄

#22548747

yes

—

133

C₂

—

4016

—

134

C₂

#30632069

yes

characteristic

135

C₂

#30632073

yes

characteristic

136

C₂

#30632080

yes

characteristic

	subgroup	G/H
center	C₂²	#22548711
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#487×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#127
prime radical	#22548712	C₂²
hypercenter	#487	D₂₅₁
hypocenter	C₂₅₁	#1719
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#22548711

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#487×C₂₅₁

C₂

C₂²×C₂₅₁

C₂²

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
1. #30632090	C₂	semidirect
3. #30632039	C₂	semidirect
6. #30631408	C₂	semidirect
7. #487×C₂₅₁	C₂	semidirect
68. C₂₅₁	#1719	semidirect
73. #487	D₂₅₁	semidirect

H ≅ G/N

type

1. #30632090

C₂

semidirect

3. #30632039

C₂

semidirect

6. #30631408

C₂

semidirect

7. #487×C₂₅₁

C₂

semidirect

68. C₂₅₁

#1719

semidirect

73. #487

D₂₅₁

semidirect

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

subgroup	index
#30632090	2
#30632061	2
#30632039	2
#30632035	2
#30632034	2
#30631408	2
#487×C₂₅₁	2
#1719	251

subgroup

index

#487×C₂₅₁

251

Subgroups — #44550019

Conjugacy classes of subgroups

Complements

Maximal subgroups