Group #44549744 — subgroups

Subgroups — #44549744

Order 32128.

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

normal subgroups only

This group has 150 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	#30631979	C₂	1	1	yes	maximal, characteristic
2	#30631894	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic
4	#30631438	C₂	1	1	yes	maximal, characteristic
5	#30631436	C₂	1	1	yes	maximal, characteristic
6	#30631433	C₂	1	1	yes	maximal, characteristic
7	#498×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#16860800	—	2	1	no	—
9	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
10	#22548648	—	2	1	no	—
11	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₂₅₁⋊C₈	—	2	1	no	—
15	#22548639	—	2	1	no	—
16	#22548638	—	2	1	no	—
17	C₈×Dic₂₅₁	—	2	1	no	—
18	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
20	#106×C₂₅₁	—	2	1	no	—
21	#106×C₂₅₁	—	2	1	no	—
22	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
24	#16860800	—	4	2	no	—
25	C₂×C₂₅₁⋊C₈	—	2	1	no	—
26	C₂×C₂₅₁⋊C₈	—	2	1	no	—
27	C₂×C₂₅₁⋊C₈	—	2	1	no	—
28	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
29	Dic₂₅₁⋊C₄	—	2	1	no	—
30	C₄×Dic₂₅₁	—	2	1	no	—
31	C₂×Q₈×C₂₅₁	—	2	1	no	—
32	C₂×Q₈×C₂₅₁	—	2	1	no	—
33	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₂₅₁	—	2	1	no	—
36	C₄⋊C₄×C₂₅₁	—	2	1	no	—
37	C₄⋊C₄×C₂₅₁	—	2	1	no	—
38	C₄⋊C₄×C₂₅₁	—	2	1	no	—
39	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
40	Dic₅₀₂	—	2	1	no	—
41	Dic₅₀₂	SD₁₆	1	2	yes	—
42	C₂×Dic₂₅₁	—	2	1	no	—
43	C₂×Dic₂₅₁	—	2	1	no	—
44	C₂₅₁⋊C₈	—	4	1	no	—
45	C₂₅₁⋊C₈	—	4	1	no	—
46	C₂₅₁⋊C₈	—	4	1	no	—
47	Q₈×C₂₅₁	—	4	1	no	—
48	Q₈×C₂₅₁	—	4	1	no	—
49	Q₈×C₂₅₁	—	8	1	no	—
50	C₂×C₄×C₂₅₁	—	4	1	no	—
51	C₂×C₄×C₂₅₁	—	4	1	no	—
52	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₁	#41	1	1	yes	characteristic
55	C₈×C₂₅₁	—	2	1	no	—
56	C₈×C₂₅₁	Q₁₆	1	2	yes	complemented
57	C₂²×C₂₅₁	#121	1	1	yes	characteristic
58	C₄×C₂₅₁	—	2	1	no	—
59	C₄×C₂₅₁	—	8	1	no	—
60	C₄×C₂₅₁	—	8	1	no	—
61	C₄×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₄×C₂₅₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₂₅₁	#135	1	1	yes	characteristic
64	C₄×C₂₅₁	#137	1	1	yes	characteristic
65	Dic₂₅₁	—	2	2	no	—
66	Dic₂₅₁	—	4	1	no	—
67	C₂×C₂₅₁	#462	1	1	yes	characteristic
68	C₂×C₂₅₁	#465	1	1	yes	characteristic
69	C₂×C₂₅₁	#471	1	1	yes	characteristic
70	C₂₅₁	#1672	1	1	yes	characteristic, complemented, Sylow
71	#1672	—	251	1	no	maximal, Sylow
72	#498	D₂₅₁	1	1	yes	characteristic
73	#477	—	251	1	no	—
74	#462	—	251	1	no	—
75	C₈×Q₈	—	251	1	no	—
76	#334	—	251	1	no	—
77	#333	—	251	1	no	—
78	#332	—	251	1	no	—
79	C₂×Q₁₆	—	502	1	no	—
80	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
82	C₄×Q₈	—	251	1	no	—
83	#107	—	502	1	no	—
84	#106	—	2	1	no	—
85	#106	—	2	1	no	—
86	C₄⋊C₈	—	502	1	no	—
87	C₄⋊C₈	—	251	1	no	—
88	C₄⋊C₈	—	251	1	no	—
89	C₄⋊C₈	—	251	1	no	—
90	#103	—	502	1	no	—
91	#103	—	502	1	no	—
92	C₄×C₈	—	502	1	no	—
93	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	251	1	no	—
97	Q₁₆	—	1004	2	no	—
98	C₂×C₈	—	502	1	no	—
99	C₂×C₈	—	502	1	no	—
100	C₂×C₈	—	502	1	no	—
101	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	502	1	no	—
108	C₄⋊C₄	—	251	1	no	—
109	C₄²	—	502	1	no	—
110	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	8	1	no	—
114	Q₈	—	502	1	no	—
115	Q₈	—	251	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	502	1	no	—
119	C₂×C₄	—	502	1	no	—
120	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
122	C₂×C₄	#16860809	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	1004	1	no	—
125	C₈	—	1004	1	no	—
126	C₈	—	1004	1	no	—
127	C₈	#16860800	1	2	yes	complemented
128	C₂²	#22548686	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	1004	1	no	—
133	C₄	—	502	2	no	—
134	C₄	SD₁₆×D₂₅₁	1	1	yes	characteristic
135	C₄	C₂×#16860800	1	1	yes	characteristic
136	C₄	#22548736	1	1	yes	characteristic
137	C₄	#22548746	1	1	yes	characteristic
138	C₂	#30631894	1	1	yes	characteristic
139	C₂	#30631913	1	1	yes	characteristic
140	C₂	#30631947	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30631979

C₂

yes

maximal, characteristic

#30631894

C₂

yes

maximal, characteristic

C₈×Dic₅₀₂

C₂

yes

maximal, characteristic

#30631438

C₂

yes

maximal, characteristic

#30631436

C₂

yes

maximal, characteristic

#30631433

C₂

yes

maximal, characteristic

#498×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860800

—

C₄×Dic₅₀₂

C₂²

yes

characteristic

#22548648

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

characteristic

Dic₂₅₁⋊C₈

—

#22548639

—

#22548638

—

C₈×Dic₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

#106×C₂₅₁

—

#106×C₂₅₁

—

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

D₄

yes

characteristic

#16860800

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

D₄

yes

characteristic

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×C₈×C₂₅₁

D₄

yes

characteristic

C₂×C₈×C₂₅₁

D₄

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

Dic₅₀₂

SD₁₆

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

Q₈×C₂₅₁

—

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

#41

yes

characteristic

C₈×C₂₅₁

—

C₈×C₂₅₁

Q₁₆

yes

complemented

C₂²×C₂₅₁

#121

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×SD₁₆

yes

characteristic

C₄×C₂₅₁

C₂×Q₁₆

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1672

—

251

maximal, Sylow

#498

D₂₅₁

yes

characteristic

#477

—

251

—

#462

—

251

—

C₈×Q₈

—

251

—

#334

—

251

—

#333

—

251

—

#332

—

251

—

C₂×Q₁₆

—

502

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

C₄×Q₈

—

251

—

#107

—

502

—

#106

—

#106

—

C₄⋊C₈

—

502

—

C₄⋊C₈

—

251

—

C₄⋊C₈

—

251

—

C₄⋊C₈

—

251

—

#103

—

502

—

#103

—

502

—

C₄×C₈

—

502

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

251

—

Q₁₆

—

1004

—

C₂×C₈

—

502

—

C₂×C₈

—

502

—

100

C₂×C₈

—

502

—

101

C₂×C₈

C₂²⋊D₂₅₁

yes

characteristic

102

C₂×C₈

C₂²⋊D₂₅₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

502

—

108

C₄⋊C₄

—

251

—

109

C₄²

—

502

—

110

C₄²

C₂²×D₂₅₁

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

502

—

115

Q₈

—

251

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

502

—

119

C₂×C₄

—

502

—

120

C₂×C₄

D₄×D₂₅₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

122

C₂×C₄

#16860809

yes

characteristic

123

C₈

—

124

C₈

—

1004

—

125

C₈

—

1004

—

126

C₈

—

1004

—

127

C₈

#16860800

yes

complemented

128

C₂²

#22548686

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

1004

—

133

C₄

—

502

—

134

C₄

SD₁₆×D₂₅₁

yes

characteristic

135

C₄

C₂×#16860800

yes

characteristic

136

C₄

#22548736

yes

characteristic

137

C₄

#22548746

yes

characteristic

138

C₂

#30631894

yes

characteristic

139

C₂

#30631913

yes

characteristic

140

C₂

#30631947

yes

characteristic

	subgroup	G/H
center	C₂²	#22548686
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#498×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#121
hypercenter	#498	D₂₅₁
hypocenter	C₂₅₁	#1672
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#22548686

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#498×C₂₅₁

C₂

C₂²×C₂₅₁

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
56. C₈×C₂₅₁	Q₁₆	semidirect
70. C₂₅₁	#1672	semidirect
127. C₈	#16860800	semidirect

H ≅ G/N

type

56. C₈×C₂₅₁

Q₁₆

semidirect

70. C₂₅₁

#1672

semidirect

127. C₈

#16860800

semidirect

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

subgroup	index
#30631979	2
#30631894	2
C₈×Dic₅₀₂	2
#30631438	2
#30631436	2
#30631433	2
#498×C₂₅₁	2
#1672	251

subgroup

index

C₈×Dic₅₀₂

#498×C₂₅₁

251

Subgroups — #44549744

Conjugacy classes of subgroups

Complements

Maximal subgroups