Group #13100717 — subgroups

Subgroups — #13100717

Order 1600.

264 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 70 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	#11717208	C₂	1	1	yes	maximal, characteristic
2	#11717037	C₂	1	1	yes	maximal, characteristic
3	C₅×#3940	C₂	1	2	yes	maximal
4	#11716614	C₂	1	2	yes	maximal
5	#11716594	C₂	1	1	yes	maximal, characteristic
6	C₂×#6045	—	2	1	no	—
7	C₂×#6039	—	2	1	no	—
8	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₄⋊#1039	C₂²	1	1	yes	characteristic
10	#87806	—	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	#65144	—	5	2	no	maximal
16	C₂×C₄×C₅²	—	2	1	no	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	—	2	1	no	—
19	C₂×#1039	D₄	1	1	yes	characteristic
20	C₂×#1039	D₄	1	1	yes	characteristic
21	#6045	—	4	2	no	—
22	C₂×C₅×Dic₅	—	2	2	no	—
23	#6039	—	4	2	no	—
24	C₅×#168	Q₈	1	4	yes	complemented
25	#4034	—	5	2	no	—
26	#4034	—	5	2	no	—
27	#4034	—	10	2	no	—
28	C₅×#106	—	5	2	no	—
29	#3949	—	5	2	no	—
30	C₈×Dic₅	—	5	2	no	—
31	#3940	—	5	2	no	—
32	#3940	D₅	1	2	yes	—
33	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅²	—	4	1	no	—
35	C₄×C₅²	C₂×Q₈	1	2	yes	—
36	#1039	—	4	1	no	—
37	#1039	SD₁₆	1	2	yes	—
38	#1039	SD₁₆	1	2	yes	—
39	C₅×Dic₅	—	4	2	no	—
40	C₂×Dic₁₀	—	10	2	no	—
41	C₂×Dic₁₀	—	10	2	no	—
42	C₂×Dic₁₀	—	10	4	no	—
43	C₂×C₅×C₈	—	5	2	no	—
44	C₅×C₄⋊C₄	—	2	2	no	—
45	C₅×C₄⋊C₄	—	5	2	no	—
46	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
47	C₄⋊Dic₅	—	5	2	no	—
48	C₄⋊Dic₅	—	10	2	no	—
49	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
50	#683	—	10	2	no	—
51	#683	—	10	2	no	—
52	#683	—	10	4	no	—
53	C₄×Dic₅	—	5	2	no	—
54	C₄×Dic₅	—	10	2	no	—
55	C₂×#168	C₂×D₅	1	2	yes	—
56	#498	—	25	1	no	Sylow
57	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅	—	2	4	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	—	10	2	no	—
64	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
65	C₂×Dic₅	—	2	2	no	—
66	C₂×Dic₅	—	5	2	no	—
67	C₂×Dic₅	—	5	2	no	—
68	C₂×Dic₅	—	10	2	no	—
69	C₂×Dic₅	—	10	2	no	—
70	C₂×Dic₅	—	10	4	no	—
71	C₂×Dic₅	—	10	2	no	—
72	Dic₁₀	—	20	4	no	—
73	Dic₁₀	—	20	4	no	—
74	Dic₁₀	—	20	8	no	—
75	C₅×C₈	—	5	4	no	—
76	#168	Dic₁₀	1	4	yes	complemented
77	C₄⋊Q₈	—	25	1	no	—
78	C₄⋊Q₈	—	25	1	no	—
79	#106	—	5	2	no	—
80	#106	—	25	2	no	—
81	C₄×C₈	—	25	1	no	—
82	C₅²	#498	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₅	—	2	2	no	—
84	C₂²×C₅	D₄×D₅	1	2	yes	—
85	C₄×C₅	—	2	4	no	—
86	C₄×C₅	—	4	4	no	—
87	C₄×C₅	—	4	2	no	—
88	C₄×C₅	—	20	2	no	—
89	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
90	C₄×C₅	Q₈×D₅	1	2	yes	—
91	Dic₅	—	4	2	no	—
92	Dic₅	—	5	4	no	—
93	Dic₅	—	5	4	no	—
94	Dic₅	—	10	4	no	—
95	Dic₅	—	10	4	no	—
96	Dic₅	—	20	4	no	—
97	Dic₅	—	20	2	no	—
98	C₂×Q₈	—	50	1	no	—
99	C₂×Q₈	—	50	1	no	—
100	C₂×C₈	—	5	2	no	—
101	C₄⋊C₄	—	5	2	no	—
102	C₄⋊C₄	—	25	1	no	—
103	C₄⋊C₄	—	50	1	no	—
104	C₄⋊C₄	—	50	1	no	—
105	C₄⋊C₄	D₅²	1	1	yes	characteristic
106	C₄²	—	25	1	no	—
107	C₂×C₅	—	2	2	no	—
108	C₂×C₅	—	2	2	no	—
109	C₂×C₅	—	2	2	no	—
110	C₂×C₅	#4034	1	2	yes	—
111	C₂×C₅	D₅×SD₁₆	1	2	yes	—
112	C₂×C₅	D₅×SD₁₆	1	2	yes	—
113	Q₈	—	100	2	no	—
114	Q₈	—	100	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	10	2	no	—
117	C₂×C₄	—	25	1	no	—
118	C₂×C₄	—	25	1	no	—
119	C₂×C₄	—	50	1	no	—
120	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
121	C₈	—	5	4	no	—
122	C₅	—	2	2	no	—
123	C₅	#65144	1	2	yes	complemented
124	C₂²	#87886	1	1	yes	characteristic
125	C₄	—	4	1	no	—
126	C₄	—	20	2	no	—
127	C₄	—	25	2	no	—
128	C₄	—	25	2	no	—
129	C₄	—	100	1	no	—
130	C₄	D₅×Dic₁₀	1	2	yes	—
131	C₂	#11716958	1	1	yes	characteristic
132	C₂	#11717048	1	1	yes	characteristic
133	C₂	#11717058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717208

C₂

yes

maximal, characteristic

#11717037

C₂

yes

maximal, characteristic

C₅×#3940

C₂

yes

maximal

#11716614

C₂

yes

maximal

#11716594

C₂

yes

maximal, characteristic

C₂×#6045

—

C₂×#6039

—

C₅²×C₄⋊C₄

C₂²

yes

characteristic

C₄⋊#1039

C₂²

yes

characteristic

#87806

—

C₄×#1039

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87781

—

#65144

—

maximal

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₂×#1039

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

#6045

—

C₂×C₅×Dic₅

—

#6039

—

C₅×#168

Q₈

yes

—

—

—

—

—

C₈×Dic₅

—

#3940

—

#3940

D₅

yes

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

—

#1039

—

#1039

SD₁₆

yes

—

#1039

SD₁₆

yes

—

C₅×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

#498

—

Sylow

C₂×C₅²

C₄⋊Q₈

yes

characteristic

C₂×C₅²

C₂×SD₁₆

yes

characteristic

C₂×C₅²

C₂×SD₁₆

yes

characteristic

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

—

Dic₁₀

—

C₅×C₈

—

#168

Dic₁₀

yes

complemented

C₄⋊Q₈

—

C₄⋊Q₈

—

#106

—

#106

—

C₄×C₈

—

C₅²

#498

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×Dic₁₀

yes

—

C₄×C₅

Q₈×D₅

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

D₅²

yes

characteristic

106

C₄²

—

107

C₂×C₅

—

108

C₂×C₅

—

109

C₂×C₅

—

110

C₂×C₅

#4034

yes

—

111

C₂×C₅

D₅×SD₁₆

yes

—

112

C₂×C₅

D₅×SD₁₆

yes

—

113

Q₈

—

100

—

114

Q₈

—

100

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×D₅²

yes

characteristic

121

C₈

—

122

C₅

—

123

C₅

#65144

yes

complemented

124

C₂²

#87886

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

100

—

130

C₄

D₅×Dic₁₀

yes

—

131

C₂

#11716958

yes

characteristic

132

C₂

#11717048

yes

characteristic

133

C₂

#11717058

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₅²×C₄⋊C₄	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄⋊C₄	D₅²
hypocenter	C₅²	#498
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₅²×C₄⋊C₄

C₂²

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₄⋊C₄

D₅²

hypocenter

C₅²

#498

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

N	H ≅ G/N	type
24. C₅×#168	Q₈	semidirect
76. #168	Dic₁₀	semidirect
82. C₅²	#498	semidirect
123. C₅	#65144	semidirect

H ≅ G/N

type

24. C₅×#168

Q₈

semidirect

76. #168

Dic₁₀

semidirect

82. C₅²

#498

semidirect

123. C₅

#65144

semidirect

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

subgroup	index
#11717208	2
#11717037	2
C₅×#3940	2
#11716614	2
#11716594	2
#65144	5

subgroup

index

Subgroups — #13100717

Conjugacy classes of subgroups

Complements

Maximal subgroups