Group #13103032 — subgroups

Subgroups — #13103032

Order 1600.

496 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 127 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×C₅²×SD₁₆	C₂	1	1	yes	maximal, characteristic
2	#11717264	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87812	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717256	C₂	1	1	yes	maximal, characteristic
5	C₂×#87810	C₂	1	1	yes	maximal, characteristic
6	C₂×#87802	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716682	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
10	C₂×#6045	C₂²	1	1	yes	characteristic
11	C₂×#6045	C₂²	1	1	yes	characteristic
12	C₅²×SD₁₆	—	2	2	no	—
13	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
14	C₂²⋊#1039	—	2	2	no	—
15	#87812	—	2	2	no	—
16	#87810	—	2	2	no	—
17	#87806	—	2	2	no	—
18	C₄×#1039	C₂²	1	1	yes	characteristic
19	C₂×#6029	C₂²	1	1	yes	characteristic
20	#87802	—	2	2	no	—
21	#65456	—	5	6	no	maximal
22	C₂³×C₅²	—	2	1	no	—
23	C₅²×Q₈	—	2	1	no	—
24	C₅²×Q₈	—	2	1	no	—
25	C₅²×D₄	—	2	1	no	—
26	C₅²×D₄	—	2	1	no	—
27	C₂×C₄×C₅²	—	2	1	no	—
28	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
29	C₂×#1039	—	2	1	no	—
30	C₂×#1039	—	2	1	no	—
31	C₂×#1039	D₄	1	2	yes	—
32	#6045	—	2	1	no	—
33	#6045	—	2	1	no	—
34	#6045	—	2	1	no	—
35	#6045	—	2	1	no	—
36	C₅²×C₈	D₄	1	2	yes	—
37	#6029	D₄	1	2	yes	—
38	C₂×C₅×SD₁₆	D₅	1	6	yes	—
39	#4090	—	5	6	no	—
40	C₂×#689	—	5	6	no	—
41	#4082	—	5	6	no	—
42	C₂×#687	—	5	6	no	—
43	C₂×Dic₂₀	—	5	6	no	—
44	#3947	—	5	6	no	—
45	C₂²×C₅²	—	4	1	no	—
46	C₂²×C₅²	—	4	2	no	—
47	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
48	C₄×C₅²	—	4	1	no	—
49	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
50	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
51	#1039	—	2	2	no	—
52	#1039	—	4	1	no	—
53	#1039	—	4	1	no	—
54	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
55	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
56	C₂×Dic₁₀	—	5	6	no	—
57	C₂×Dic₁₀	—	5	6	no	—
58	C₅×SD₁₆	—	2	12	no	—
59	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
60	C₂²⋊Dic₅	—	10	12	no	—
61	#689	—	10	12	no	—
62	#687	—	10	12	no	—
63	#683	—	10	12	no	—
64	C₄×Dic₅	—	5	6	no	—
65	C₂×#168	—	5	6	no	—
66	Dic₂₀	—	10	12	no	—
67	#497	—	25	1	no	Sylow
68	C₂×C₅²	—	8	1	no	—
69	C₂×C₅²	#127	1	1	yes	characteristic
70	C₂×C₅²	#137	1	2	yes	—
71	C₂³×C₅	—	2	6	no	—
72	C₅×Q₈	—	2	6	no	—
73	C₅×Q₈	—	2	6	no	—
74	C₅×D₄	—	2	6	no	—
75	C₅×D₄	—	2	6	no	—
76	C₂×C₄×C₅	—	2	6	no	—
77	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
78	C₂×Dic₅	—	5	12	no	—
79	C₂×Dic₅	—	10	6	no	—
80	C₂×Dic₅	—	10	6	no	—
81	Dic₁₀	—	10	6	no	—
82	Dic₁₀	—	10	6	no	—
83	Dic₁₀	—	10	6	no	—
84	Dic₁₀	—	10	6	no	—
85	C₅×C₈	C₂²⋊D₅	1	12	yes	—
86	#168	—	5	12	no	—
87	C₂×Q₁₆	—	25	1	no	—
88	C₂×Q₁₆	—	25	1	no	—
89	C₂×SD₁₆	—	25	1	no	—
90	C₂×SD₁₆	C₅⋊D₅	1	1	yes	characteristic
91	C₄⋊Q₈	—	25	1	no	—
92	#124	—	25	1	no	—
93	#97	—	25	1	no	—
94	C₅²	#497	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₅	—	4	12	no	—
96	C₂²×C₅	—	4	6	no	—
97	C₂²×C₅	D₄×D₅	1	6	yes	—
98	C₄×C₅	—	4	6	no	—
99	C₄×C₅	D₄×D₅	1	6	yes	—
100	C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
101	Dic₅	—	10	12	no	—
102	Dic₅	—	20	6	no	—
103	Dic₅	—	20	6	no	—
104	C₂×Q₈	—	25	1	no	—
105	C₂×Q₈	—	25	1	no	—
106	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
107	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
108	Q₁₆	—	50	2	no	—
109	Q₁₆	—	50	2	no	—
110	SD₁₆	—	2	2	no	—
111	SD₁₆	—	50	2	no	—
112	C₂×C₈	—	25	1	no	—
113	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
114	C₄⋊C₄	—	50	2	no	—
115	C₂²⋊C₄	—	50	2	no	—
116	C₄²	—	25	1	no	—
117	C₂×C₅	—	8	6	no	—
118	C₂×C₅	#4061	1	12	yes	—
119	C₂×C₅	#4086	1	6	yes	—
120	C₂³	—	2	1	no	—
121	Q₈	—	2	1	no	—
122	Q₈	—	2	1	no	—
123	Q₈	—	50	1	no	—
124	Q₈	—	50	1	no	—
125	Q₈	—	50	1	no	—
126	Q₈	—	50	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	25	2	no	—
131	C₂×C₄	—	50	1	no	—
132	C₂×C₄	—	50	1	no	—
133	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
134	C₈	—	25	2	no	—
135	C₈	C₂²⋊(C₅⋊D₅)	1	2	yes	—
136	C₅	#65456	1	6	yes	complemented
137	C₂²	—	4	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	D₄×C₅⋊D₅	1	1	yes	characteristic
140	C₄	—	4	1	no	—
141	C₄	—	50	2	no	—
142	C₄	—	100	1	no	—
143	C₄	—	100	1	no	—
144	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
145	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
146	C₂	—	8	1	no	—
147	C₂	#11717235	1	2	yes	—
148	C₂	#11717260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅²×SD₁₆

C₂

yes

maximal, characteristic

#11717264

C₂

yes

maximal, characteristic, complemented

C₂×#87812

C₂

yes

maximal, characteristic, complemented

#11717256

C₂

yes

maximal, characteristic

C₂×#87810

C₂

yes

maximal, characteristic

C₂×#87802

C₂

yes

maximal, characteristic, complemented

#11716682

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₅²×SD₁₆

—

C₂×C₅²×C₈

C₂²

yes

characteristic

C₂²⋊#1039

—

—

—

—

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

—

—

C₂³×C₅²

—

C₅²×Q₈

—

C₅²×Q₈

—

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

—

—

D₄

yes

—

—

—

—

—

C₅²×C₈

D₄

yes

—

#6029

D₄

yes

—

C₂×C₅×SD₁₆

D₅

yes

—

—

—

—

—

C₂×Dic₂₀

—

#3947

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

—

—

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×SD₁₆

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₂²⋊Dic₅

—

#689

—

#687

—

#683

—

C₄×Dic₅

—

C₂×#168

—

Dic₂₀

—

#497

—

Sylow

C₂×C₅²

—

C₂×C₅²

#127

yes

characteristic

C₂×C₅²

#137

yes

—

C₂³×C₅

—

C₅×Q₈

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

C₂²⋊D₅

yes

—

#168

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

C₅⋊D₅

yes

characteristic

C₄⋊Q₈

—

#124

—

#97

—

C₅²

#497

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

D₄×D₅

yes

—

100

C₄×C₅

C₂×C₂²⋊D₅

yes

—

101

Dic₅

—

102

Dic₅

—

103

Dic₅

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

107

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

108

Q₁₆

—

109

Q₁₆

—

110

SD₁₆

—

111

SD₁₆

—

112

C₂×C₈

—

113

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₂×C₅

—

118

C₂×C₅

#4061

yes

—

119

C₂×C₅

#4086

yes

—

120

C₂³

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

134

C₈

—

135

C₈

C₂²⋊(C₅⋊D₅)

yes

—

136

C₅

#65456

yes

complemented

137

C₂²

—

138

C₂²

—

139

C₂²

D₄×C₅⋊D₅

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

100

—

143

C₄

—

100

—

144

C₄

D₄×C₅⋊D₅

yes

characteristic

145

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

146

C₂

—

147

C₂

#11717235

yes

—

148

C₂

#11717260

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×C₅⋊D₅
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₂×C₅²×SD₁₆	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₅²×D₄	C₂²
hypercenter	C₂×SD₁₆	C₅⋊D₅
hypocenter	C₅²	#497
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×C₅⋊D₅

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₂×C₅²×SD₁₆

C₂

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂×C₅²×D₄

C₂²

hypercenter

C₂×SD₁₆

C₅⋊D₅

hypocenter

C₅²

#497

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
2. #11717264	C₂	semidirect
3. C₂×#87812	C₂	semidirect
6. C₂×#87802	C₂	semidirect
7. #11716682	C₂	semidirect
94. C₅²	#497	semidirect
136. C₅	#65456	semidirect

H ≅ G/N

type

2. #11717264

C₂

semidirect

3. C₂×#87812

C₂

semidirect

6. C₂×#87802

C₂

semidirect

7. #11716682

C₂

semidirect

94. C₅²

#497

semidirect

136. C₅

#65456

semidirect

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

subgroup	index
C₂×C₅²×SD₁₆	2
#11717264	2
C₂×#87812	2
#11717256	2
C₂×#87810	2
C₂×#87802	2
#11716682	2
#65456	5

subgroup

index

C₂×C₅²×SD₁₆

Subgroups — #13103032

Conjugacy classes of subgroups

Complements

Maximal subgroups