Group #13101799 — subgroups

Subgroups — #13101799

Order 1600.

286 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 68 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	Q₈×#1039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#689	C₂	1	2	yes	maximal
3	#11716985	C₂	1	1	yes	maximal, characteristic
4	#11716613	C₂	1	1	yes	maximal, characteristic
5	#11716607	C₂	1	2	yes	maximal
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
8	C₄⋊#1039	—	2	1	no	—
9	C₄⋊#1039	C₂²	1	1	yes	characteristic
10	C₄×#1039	—	2	1	no	—
11	C₄×#1039	C₂²	1	1	yes	characteristic
12	C₅×#689	—	2	4	no	—
13	C₂×C₅×#168	C₂²	1	2	yes	—
14	#87781	—	2	1	no	—
15	#87781	—	2	1	no	—
16	#65471	—	5	2	no	maximal
17	C₅²×Q₈	—	2	1	no	—
18	C₅²×Q₈	D₄	1	2	yes	—
19	C₂×C₄×C₅²	—	2	1	no	—
20	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
21	C₂×#1039	—	2	1	no	—
22	C₂×#1039	D₄	1	1	yes	characteristic
23	C₂×#1039	D₄	1	1	yes	characteristic
24	C₂×C₅×Dic₅	—	2	2	no	—
25	C₅×Dic₁₀	—	2	2	no	—
26	C₅×Dic₁₀	—	2	2	no	—
27	C₅×#168	—	2	2	no	—
28	C₂×C₅×Q₁₆	—	5	2	no	—
29	Q₈×Dic₅	—	5	2	no	—
30	Q₈×Dic₅	—	10	2	no	—
31	#4090	—	5	2	no	—
32	C₂×#689	D₅	1	2	yes	—
33	#3967	—	5	2	no	—
34	#3948	—	5	2	no	—
35	#3946	—	5	2	no	—
36	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
37	C₄×C₅²	—	2	2	no	—
38	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅²	#41	1	1	yes	characteristic
40	#1039	—	2	1	no	—
41	#1039	—	4	1	no	—
42	#1039	Q₁₆	1	2	yes	complemented
43	C₅×Dic₅	—	4	2	no	—
44	C₂×C₅×Q₈	—	2	2	no	—
45	C₂×C₅×Q₈	—	5	2	no	—
46	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
47	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
48	C₅×Q₁₆	—	10	4	no	—
49	C₂×C₅×C₈	—	5	2	no	—
50	#689	—	2	4	no	—
51	C₄⋊Dic₅	—	5	2	no	—
52	C₄⋊Dic₅	—	10	2	no	—
53	C₄⋊Dic₅	—	10	2	no	—
54	C₄⋊Dic₅	—	10	4	no	—
55	#683	—	10	2	no	—
56	#683	—	10	2	no	—
57	C₄×Dic₅	—	5	2	no	—
58	C₄×Dic₅	—	10	2	no	—
59	C₄×Dic₅	—	10	2	no	—
60	C₄×Dic₅	—	10	4	no	—
61	C₂×#168	C₂×D₅	1	2	yes	—
62	#462	—	25	1	no	Sylow
63	C₂×C₅²	#121	1	1	yes	characteristic
64	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₅²	#137	1	1	yes	characteristic
66	C₅×Q₈	—	2	4	no	—
67	C₅×Q₈	—	2	2	no	—
68	C₅×Q₈	—	2	4	no	—
69	C₅×Q₈	—	10	2	no	—
70	C₅×Q₈	—	10	2	no	—
71	C₅×Q₈	C₂²⋊D₅	1	4	yes	—
72	C₂×C₄×C₅	—	2	4	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	10	2	no	—
76	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
77	C₂×Dic₅	—	2	2	no	—
78	C₂×Dic₅	—	5	2	no	—
79	C₂×Dic₅	—	5	2	no	—
80	C₂×Dic₅	—	10	2	no	—
81	C₂×Dic₅	—	10	2	no	—
82	C₂×Dic₅	—	10	4	no	—
83	C₂×Dic₅	—	10	2	no	—
84	Dic₁₀	—	2	2	no	—
85	Dic₁₀	—	2	2	no	—
86	C₅×C₈	—	10	2	no	—
87	#168	—	2	2	no	—
88	C₂×Q₁₆	—	5	2	no	—
89	C₄⋊Q₈	—	25	1	no	—
90	C₄×Q₈	—	25	1	no	—
91	C₄⋊C₈	—	25	1	no	—
92	#103	—	25	2	no	—
93	C₅²	#462	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₅	—	2	2	no	—
95	C₂²×C₅	D₄×D₅	1	2	yes	—
96	C₄×C₅	—	2	8	no	—
97	C₄×C₅	—	2	4	no	—
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	—	2	2	no	—
100	C₄×C₅	—	20	2	no	—
101	C₄×C₅	#711	1	2	yes	—
102	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
103	Dic₅	—	4	2	no	—
104	Dic₅	—	5	4	no	—
105	Dic₅	—	10	4	no	—
106	Dic₅	—	10	2	no	—
107	Dic₅	—	20	2	no	—
108	Dic₅	—	20	4	no	—
109	Dic₅	—	20	2	no	—
110	C₂×Q₈	—	5	2	no	—
111	C₂×Q₈	D₅²	1	1	yes	characteristic
112	Q₁₆	—	10	4	no	—
113	C₂×C₈	—	5	2	no	—
114	C₄⋊C₄	—	25	1	no	—
115	C₄⋊C₄	—	50	1	no	—
116	C₄⋊C₄	—	50	1	no	—
117	C₄⋊C₄	—	50	1	no	—
118	C₄²	—	25	1	no	—
119	C₄²	—	50	1	no	—
120	C₂×C₅	—	2	2	no	—
121	C₂×C₅	—	2	2	no	—
122	C₂×C₅	—	2	2	no	—
123	C₂×C₅	D₅×Q₁₆	1	2	yes	—
124	C₂×C₅	#4064	1	2	yes	—
125	C₂×C₅	#4085	1	2	yes	—
126	Q₈	—	2	1	no	—
127	Q₈	—	10	2	no	—
128	Q₈	—	10	2	no	—
129	Q₈	#6037	1	2	yes	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	10	2	no	—
132	C₂×C₄	—	25	1	no	—
133	C₂×C₄	—	25	1	no	—
134	C₂×C₄	—	50	1	no	—
135	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
136	C₈	—	10	2	no	—
137	C₅	—	2	2	no	—
138	C₅	#65471	1	2	yes	complemented
139	C₂²	#87886	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	20	2	no	—
142	C₄	—	25	2	no	—
143	C₄	—	50	1	no	—
144	C₄	—	100	1	no	—
145	C₄	#87880	1	1	yes	characteristic
146	C₄	C₂×#6037	1	1	yes	characteristic
147	C₂	#11717062	1	1	yes	characteristic
148	C₂	#11717063	1	1	yes	characteristic
149	C₂	#11717098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#1039

C₂

yes

maximal, characteristic

C₂×C₅×#689

C₂

yes

maximal

#11716985

C₂

yes

maximal, characteristic

#11716613

C₂

yes

maximal, characteristic

#11716607

C₂

yes

maximal

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₅×Dic₁₀

C₂²

yes

—

C₄⋊#1039

—

C₄⋊#1039

C₂²

yes

characteristic

C₄×#1039

—

C₄×#1039

C₂²

yes

characteristic

C₅×#689

—

C₂×C₅×#168

C₂²

yes

—

—

—

—

C₅²×Q₈

—

C₅²×Q₈

D₄

yes

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₂×#1039

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₂×C₅×Dic₅

—

C₅×Dic₁₀

—

C₅×Dic₁₀

—

C₅×#168

—

C₂×C₅×Q₁₆

—

Q₈×Dic₅

—

Q₈×Dic₅

—

#4090

—

C₂×#689

D₅

yes

—

#3967

—

#3948

—

#3946

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

C₄×C₅²

yes

—

—

Q₁₆

yes

complemented

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×Dic₁₀

C₂×D₅

yes

—

C₅×Q₁₆

—

C₂×C₅×C₈

—

#689

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×Q₁₆

yes

characteristic

C₂×C₅²

#137

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

C₂²⋊D₅

yes

—

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

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄⋊C₈

—

#103

—

C₅²

#462

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

100

C₄×C₅

—

101

C₄×C₅

#711

yes

—

102

C₄×C₅

C₂×C₂²⋊D₅

yes

—

103

Dic₅

—

104

Dic₅

—

105

Dic₅

—

106

Dic₅

—

107

Dic₅

—

108

Dic₅

—

109

Dic₅

—

110

C₂×Q₈

—

111

C₂×Q₈

D₅²

yes

characteristic

112

Q₁₆

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₂×C₅

—

121

C₂×C₅

—

122

C₂×C₅

—

123

C₂×C₅

D₅×Q₁₆

yes

—

124

C₂×C₅

#4064

yes

—

125

C₂×C₅

#4085

yes

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

#6037

yes

—

130

C₂×C₄

—

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

#65471

yes

complemented

139

C₂²

#87886

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

100

—

145

C₄

#87880

yes

characteristic

146

C₄

C₂×#6037

yes

characteristic

147

C₂

#11717062

yes

characteristic

148

C₂

#11717063

yes

characteristic

149

C₂

#11717098

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, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	#462
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, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₂×Q₈

D₅²

hypocenter

C₅²

#462

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

N	H ≅ G/N	type
42. #1039	Q₁₆	semidirect
93. C₅²	#462	semidirect
138. C₅	#65471	semidirect

H ≅ G/N

type

42. #1039

Q₁₆

semidirect

93. C₅²

#462

semidirect

138. C₅

#65471

semidirect

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

subgroup	index
Q₈×#1039	2
C₂×C₅×#689	2
#11716985	2
#11716613	2
#11716607	2
#65471	5

subgroup

index

C₂×C₅×#689

Subgroups — #13101799

Conjugacy classes of subgroups

Complements

Maximal subgroups