Group #13107201 — subgroups

Subgroups — #13107201

Order 1600.

784 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₅²×#124	C₂	1	1	yes	maximal, characteristic
2	#11717264	C₂	1	2	yes	maximal, complemented
3	#11717255	C₂	1	2	yes	maximal
4	#11717199	C₂	1	4	yes	maximal, complemented
5	#11717198	C₂	1	4	yes	maximal, complemented
6	#11717190	C₂	1	2	yes	maximal, complemented
7	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
9	C₂²×#1039	C₂²	1	2	yes	—
10	C₂×#6045	C₂²	1	2	yes	—
11	C₅²×C₂²⋊C₄	C₂²	1	4	yes	—
12	C₄²×C₅²	C₂²	1	1	yes	characteristic
13	C₂²⋊#1039	C₂²	1	4	yes	—
14	C₂²⋊#1039	C₂²	1	2	yes	—
15	C₄⋊#1039	C₂²	1	2	yes	—
16	C₄⋊#1039	C₂²	1	2	yes	—
17	#87806	C₂²	1	4	yes	—
18	#87806	C₂²	1	4	yes	—
19	#87806	C₂²	1	4	yes	—
20	C₄×#1039	C₂²	1	2	yes	—
21	#66006	—	5	6	no	maximal
22	C₂³×C₅²	C₂³	1	2	yes	—
23	C₅²×Q₈	—	4	1	no	—
24	C₅²×D₄	—	4	1	no	—
25	C₂×C₄×C₅²	C₂³	1	2	yes	—
26	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₅²	C₂³	1	2	yes	—
28	C₂×#1039	—	4	2	no	—
29	C₂×#1039	C₂³	1	4	yes	—
30	C₂×#1039	C₂³	1	4	yes	—
31	#6045	—	4	2	no	—
32	C₅×#124	D₅	1	6	yes	—
33	#4090	—	5	12	no	—
34	#4081	—	5	12	no	—
35	#4025	—	5	24	no	—
36	#4024	—	5	24	no	—
37	#4016	—	5	12	no	—
38	C₂²×C₅²	—	2	2	no	—
39	C₂²×C₅²	—	2	4	no	—
40	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
41	C₄×C₅²	—	2	2	no	—
42	C₄×C₅²	—	2	1	no	—
43	C₄×C₅²	—	2	2	no	—
44	#1039	—	2	4	no	—
45	#1039	—	2	4	no	—
46	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
47	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
48	C₂²×Dic₅	—	5	12	no	—
49	C₂×Dic₁₀	—	5	12	no	—
50	C₅×C₂²⋊C₄	C₂×D₅	1	24	yes	—
51	C₄²×C₅	C₂×D₅	1	6	yes	—
52	C₂²⋊Dic₅	—	5	24	no	—
53	C₂²⋊Dic₅	—	5	12	no	—
54	C₄⋊Dic₅	—	5	12	no	—
55	C₄⋊Dic₅	—	5	12	no	—
56	#683	—	5	24	no	—
57	#683	—	5	24	no	—
58	#683	—	5	24	no	—
59	C₄×Dic₅	—	5	12	no	—
60	#563	—	25	1	no	Sylow
61	C₂×C₅²	—	4	2	no	—
62	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
63	C₂×C₅²	2₋¹⁺⁴	1	2	yes	—
64	C₂³×C₅	C₂²×D₅	1	12	yes	—
65	C₅×Q₈	—	4	6	no	—
66	C₅×D₄	—	4	6	no	—
67	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
68	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
69	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
70	C₂×Dic₅	—	5	24	no	—
71	C₂×Dic₅	—	5	24	no	—
72	C₂×Dic₅	—	20	12	no	—
73	Dic₁₀	—	20	12	no	—
74	C₄⋊Q₈	—	25	2	no	—
75	#126	—	25	4	no	—
76	#125	—	25	2	no	—
77	#124	C₅⋊D₅	1	1	yes	characteristic
78	#123	—	25	2	no	—
79	C₂²⋊Q₈	—	25	4	no	—
80	C₅²	#563	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	—	2	12	no	—
82	C₂²×C₅	—	2	24	no	—
83	C₂²×C₅	C₂³×D₅	1	6	yes	—
84	C₄×C₅	—	2	12	no	—
85	C₄×C₅	—	2	6	no	—
86	C₄×C₅	—	2	12	no	—
87	Dic₅	—	10	24	no	—
88	Dic₅	—	10	24	no	—
89	C₂×Q₈	—	25	2	no	—
90	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
91	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
92	C₂²×C₄	—	25	2	no	—
93	C₄⋊C₄	—	25	2	no	—
94	C₄⋊C₄	—	25	4	no	—
95	C₄⋊C₄	—	25	4	no	—
96	C₄⋊C₄	—	25	4	no	—
97	C₄⋊C₄	—	25	2	no	—
98	C₂²⋊C₄	—	25	4	no	—
99	C₂²⋊C₄	—	25	2	no	—
100	C₂²⋊C₄	C₂×C₅⋊D₅	1	4	yes	—
101	C₄²	—	25	2	no	—
102	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
103	C₂×C₅	—	4	12	no	—
104	C₂×C₅	#4144	1	6	yes	—
105	C₂×C₅	#4150	1	12	yes	—
106	C₂³	C₂²×C₅⋊D₅	1	2	yes	—
107	Q₈	—	4	1	no	—
108	Q₈	—	100	2	no	—
109	D₄	—	4	1	no	—
110	C₂×C₄	—	25	4	no	—
111	C₂×C₄	—	25	4	no	—
112	C₂×C₄	—	100	2	no	—
113	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
114	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
115	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
116	C₅	#66006	1	6	yes	complemented
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	50	4	no	—
124	C₄	—	50	4	no	—
125	C₂	—	4	2	no	—
126	C₂	#11717517	1	1	yes	characteristic
127	C₂	#11717523	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#124

C₂

yes

maximal, characteristic

#11717264

C₂

yes

maximal, complemented

#11717255

C₂

yes

maximal

#11717199

C₂

yes

maximal, complemented

#11717198

C₂

yes

maximal, complemented

#11717190

C₂

yes

maximal, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂²×#1039

C₂²

yes

—

C₂×#6045

C₂²

yes

—

C₅²×C₂²⋊C₄

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

C₂²⋊#1039

C₂²

yes

—

C₂²⋊#1039

C₂²

yes

—

C₄⋊#1039

C₂²

yes

—

C₄⋊#1039

C₂²

yes

—

#87806

C₂²

yes

—

#87806

C₂²

yes

—

#87806

C₂²

yes

—

C₄×#1039

C₂²

yes

—

#66006

—

maximal

C₂³×C₅²

C₂³

yes

—

C₅²×Q₈

—

C₅²×D₄

—

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

C₂×#1039

—

C₂×#1039

C₂³

yes

—

C₂×#1039

C₂³

yes

—

#6045

—

C₅×#124

D₅

yes

—

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

—

#1039

—

#1039

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₅×C₂²⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

#683

—

#683

—

C₄×Dic₅

—

#563

—

Sylow

C₂×C₅²

—

C₂×C₅²

H(5,𝔽₂)

yes

characteristic

C₂×C₅²

2₋¹⁺⁴

yes

—

C₂³×C₅

C₂²×D₅

yes

—

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₄⋊Q₈

—

#126

—

#125

—

#124

C₅⋊D₅

yes

characteristic

#123

—

C₂²⋊Q₈

—

C₅²

#563

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂³×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

C₂×C₅⋊D₅

yes

—

101

C₄²

—

102

C₄²

C₂×C₅⋊D₅

yes

characteristic

103

C₂×C₅

—

104

C₂×C₅

#4144

yes

—

105

C₂×C₅

#4150

yes

—

106

C₂³

C₂²×C₅⋊D₅

yes

—

107

Q₈

—

108

Q₈

—

100

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

100

—

113

C₂×C₄

C₂²×C₅⋊D₅

yes

—

114

C₂×C₄

C₂²×C₅⋊D₅

yes

—

115

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

116

C₅

#66006

yes

complemented

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×C₅⋊D₅

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₂

—

126

C₂

#11717517

yes

characteristic

127

C₂

#11717523

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#124	C₂
prime radical	C₂×C₅²×D₄	C₂²
hypercenter	#124	C₅⋊D₅
hypocenter	C₅²	#563
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×C₅⋊D₅

commutator, socle, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₅²×#124

C₂

prime radical

C₂×C₅²×D₄

C₂²

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. #11717264	C₂	semidirect
4. #11717199	C₂	semidirect
5. #11717198	C₂	semidirect
6. #11717190	C₂	semidirect
80. C₅²	#563	semidirect
116. C₅	#66006	semidirect

H ≅ G/N

type

2. #11717264

C₂

semidirect

4. #11717199

C₂

semidirect

5. #11717198

C₂

semidirect

6. #11717190

C₂

semidirect

80. C₅²

#563

semidirect

116. C₅

#66006

semidirect

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

subgroup	index
C₅²×#124	2
#11717264	2
#11717255	2
#11717199	2
#11717198	2
#11717190	2
#66006	5

subgroup

index

C₅²×#124

Subgroups — #13107201

Conjugacy classes of subgroups

Complements

Maximal subgroups