Group #13101798 — subgroups

Subgroups — #13101798

Order 1600.

276 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 58 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	#11717267	C₂	1	1	yes	maximal, characteristic
2	#11717037	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3967	C₂	1	2	yes	maximal, complemented
4	#11716605	C₂	1	2	yes	maximal
5	#11716596	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
8	C₂×#6039	—	2	1	no	—
9	#87808	—	2	1	no	—
10	#87808	—	2	1	no	—
11	C₄×#1039	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₂×C₅×#168	C₂²	1	2	yes	—
14	#87781	—	2	1	no	—
15	#65084	—	5	2	no	maximal
16	C₂²×C₅⋊D₅	—	2	1	no	—
17	C₅²×Q₈	—	2	2	no	—
18	C₂×C₄×C₅²	—	2	1	no	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×#1039	D₄	1	1	yes	characteristic
21	C₂×#1039	D₄	1	1	yes	characteristic
22	C₄⋊(C₅⋊D₅)	—	2	2	no	—
23	C₂×C₅×Dic₅	—	2	2	no	—
24	#6039	—	4	2	no	—
25	C₅×#168	—	2	2	no	—
26	#4093	—	5	2	no	—
27	#4093	—	10	2	no	—
28	#4034	—	5	2	no	—
29	C₅×#103	—	5	2	no	—
30	#3967	D₅	1	2	yes	complemented
31	#3953	—	5	2	no	—
32	#3947	—	5	2	no	—
33	#3941	—	5	2	no	—
34	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
35	C₂×C₅⋊D₅	—	4	1	no	—
36	C₂×C₅⋊D₅	—	4	1	no	—
37	C₂×C₅⋊D₅	—	4	1	no	—
38	C₄×C₅²	—	4	1	no	—
39	C₄×C₅²	#41	1	2	yes	—
40	#1039	—	2	1	no	—
41	#1039	—	2	1	no	—
42	C₅×Dic₅	—	4	2	no	—
43	C₂×C₅×Q₈	—	2	2	no	—
44	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
45	C₂×D₂₀	—	5	2	no	—
46	C₂×D₂₀	—	10	2	no	—
47	C₂×Dic₁₀	—	10	2	no	—
48	C₂×C₅×C₈	—	5	2	no	—
49	C₅×C₄⋊C₄	—	5	2	no	—
50	#685	—	10	2	no	—
51	#685	—	10	2	no	—
52	#685	—	10	4	no	—
53	#685	—	10	4	no	—
54	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
55	#683	—	10	2	no	—
56	C₄×Dic₅	—	5	2	no	—
57	C₄×Dic₅	—	10	2	no	—
58	C₂×#168	C₂×D₅	1	2	yes	—
59	#489	—	25	1	no	Sylow
60	C₂×C₅²	#124	1	1	yes	characteristic
61	C₂×C₅²	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×C₅²	#137	1	1	yes	characteristic
63	C₅⋊D₅	—	8	1	no	—
64	C₂²×D₅	—	10	4	no	—
65	C₂²×D₅	—	10	2	no	—
66	C₅×Q₈	—	2	2	no	—
67	C₅×Q₈	—	2	2	no	—
68	C₅×Q₈	—	4	4	no	—
69	C₂×C₄×C₅	—	2	4	no	—
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	10	2	no	—
73	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
74	C₂×Dic₅	—	2	2	no	—
75	C₂×Dic₅	—	5	2	no	—
76	C₂×Dic₅	—	5	2	no	—
77	C₂×Dic₅	—	10	2	no	—
78	C₂×Dic₅	—	10	2	no	—
79	D₂₀	—	10	2	no	—
80	D₂₀	—	10	2	no	—
81	D₂₀	—	20	4	no	—
82	Dic₁₀	—	20	4	no	—
83	C₅×C₈	—	10	2	no	—
84	#168	—	2	2	no	—
85	C₄⋊Q₈	—	25	1	no	—
86	#124	—	25	1	no	—
87	#103	—	5	2	no	—
88	#102	—	25	2	no	—
89	#97	—	25	1	no	—
90	C₅²	#489	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₅	—	2	2	no	—
92	C₂²×C₅	D₄×D₅	1	2	yes	—
93	C₂×D₅	—	20	4	no	—
94	C₂×D₅	—	20	4	no	—
95	C₂×D₅	—	20	4	no	—
96	C₂×D₅	—	20	2	no	—
97	C₂×D₅	—	20	2	no	—
98	C₂×D₅	—	20	2	no	—
99	C₄×C₅	—	2	4	no	—
100	C₄×C₅	—	4	4	no	—
101	C₄×C₅	—	4	2	no	—
102	C₄×C₅	—	20	2	no	—
103	C₄×C₅	#709	1	2	yes	—
104	C₄×C₅	#711	1	2	yes	—
105	Dic₅	—	4	2	no	—
106	Dic₅	—	10	2	no	—
107	Dic₅	—	10	2	no	—
108	Dic₅	—	20	2	no	—
109	Dic₅	—	20	2	no	—
110	C₂×Q₈	—	50	1	no	—
111	C₂×Q₈	D₅²	1	1	yes	characteristic
112	C₂×D₄	—	25	1	no	—
113	C₂×C₈	—	5	2	no	—
114	C₄⋊C₄	—	5	2	no	—
115	C₄⋊C₄	—	50	1	no	—
116	C₂²⋊C₄	—	50	1	no	—
117	C₂²⋊C₄	—	50	1	no	—
118	C₄²	—	25	1	no	—
119	C₂×C₅	—	2	2	no	—
120	C₂×C₅	—	2	2	no	—
121	C₂×C₅	—	2	2	no	—
122	C₂×C₅	#4031	1	2	yes	—
123	C₂×C₅	#4060	1	2	yes	—
124	C₂×C₅	#4064	1	2	yes	—
125	D₅	—	40	4	no	—
126	D₅	—	40	2	no	—
127	C₂³	—	50	1	no	—
128	Q₈	—	2	2	no	—
129	Q₈	—	100	2	no	—
130	D₄	—	50	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	10	2	no	—
133	C₂×C₄	—	25	1	no	—
134	C₂×C₄	—	25	1	no	—
135	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
136	C₈	—	10	2	no	—
137	C₅	—	2	2	no	—
138	C₅	#65084	1	2	yes	complemented
139	C₂²	—	100	1	no	—
140	C₂²	—	100	1	no	—
141	C₂²	—	100	1	no	—
142	C₂²	#87886	1	1	yes	characteristic
143	C₄	—	4	1	no	—
144	C₄	—	20	2	no	—
145	C₄	—	50	1	no	—
146	C₄	—	50	1	no	—
147	C₄	#87879	1	2	yes	—
148	C₂	—	200	1	no	—
149	C₂	#11717059	1	1	yes	characteristic
150	C₂	#11717063	1	1	yes	characteristic
151	C₂	#11717071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717267

C₂

yes

maximal, characteristic

#11717037

C₂

yes

maximal, characteristic, complemented

C₅×#3967

C₂

yes

maximal, complemented

#11716605

C₂

yes

maximal

#11716596

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87781

—

#65084

—

maximal

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₄⋊(C₅⋊D₅)

—

C₂×C₅×Dic₅

—

—

—

—

—

—

—

D₅

yes

—

—

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

#41

yes

—

#1039

—

#1039

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

—

—

—

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

—

C₂×C₅²

yes

C₂×C₅²

yes

C₂×C₅²

yes

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₄⋊Q₈

—

—

—

—

—

C₅²

#489

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

—

103

C₄×C₅

#709

yes

—

104

C₄×C₅

#711

yes

—

105

Dic₅

—

106

Dic₅

—

107

Dic₅

—

108

Dic₅

—

109

Dic₅

—

110

C₂×Q₈

—

111

C₂×Q₈

D₅²

yes

characteristic

112

C₂×D₄

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₂×C₅

—

120

C₂×C₅

—

121

C₂×C₅

—

122

C₂×C₅

#4031

yes

—

123

C₂×C₅

#4060

yes

—

124

C₂×C₅

#4064

yes

—

125

D₅

—

126

D₅

—

127

C₂³

—

128

Q₈

—

129

Q₈

—

100

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×D₅²

yes

characteristic

136

C₈

—

137

C₅

—

138

C₅

#65084

yes

complemented

139

C₂²

—

100

—

140

C₂²

—

100

—

141

C₂²

—

100

—

142

C₂²

#87886

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#87879

yes

—

148

C₂

—

200

—

149

C₂

#11717059

yes

characteristic

150

C₂

#11717063

yes

characteristic

151

C₂

#11717071

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₂×C₅²×Q₈	C₂²
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₂²
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	#489
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₂×C₅²×Q₈

C₂²

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂×C₄⋊(C₅⋊D₅)

C₂²

hypercenter

C₂×Q₈

D₅²

hypocenter

C₅²

#489

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

N	H ≅ G/N	type
2. #11717037	C₂	semidirect
3. C₅×#3967	C₂	semidirect
5. #11716596	C₂	semidirect
30. #3967	D₅	semidirect
90. C₅²	#489	semidirect
138. C₅	#65084	semidirect

H ≅ G/N

type

2. #11717037

C₂

semidirect

3. C₅×#3967

C₂

semidirect

5. #11716596

C₂

semidirect

30. #3967

D₅

semidirect

90. C₅²

#489

semidirect

138. C₅

#65084

semidirect

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

subgroup	index
#11717267	2
#11717037	2
C₅×#3967	2
#11716605	2
#11716596	2
#65084	5

subgroup

index

Subgroups — #13101798

Conjugacy classes of subgroups

Complements

Maximal subgroups