Group #13107225 — subgroups

Subgroups — #13107225

Order 1600.

960 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 227 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×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717215	C₂	1	4	yes	maximal, complemented
3	#11717215	C₂	1	2	yes	maximal, complemented
4	#11717214	C₂	1	4	yes	maximal, complemented
5	#11717213	C₂	1	2	yes	maximal, complemented
6	#11717194	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717192	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	4	yes	—
11	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅⋊D₅	C₂²	1	4	yes	—
13	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
14	C₅²×C₄⋊C₄	C₂²	1	4	yes	—
15	C₄²×C₅²	C₂²	1	1	yes	characteristic
16	#87808	C₂²	1	2	yes	—
17	#87808	C₂²	1	4	yes	—
18	#87808	C₂²	1	4	yes	—
19	#87806	C₂²	1	2	yes	—
20	C₄×#1039	C₂²	1	1	yes	characteristic
21	#66032	—	5	6	no	maximal
22	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅⋊D₅	C₂³	1	4	yes	—
24	C₂×C₄×C₅²	C₂³	1	4	yes	—
25	C₂×C₄×C₅²	C₂³	1	2	yes	—
26	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
27	C₂×#1039	C₂³	1	1	yes	characteristic
28	C₂×#1039	C₂³	1	2	yes	—
29	C₄⋊(C₅⋊D₅)	—	2	4	no	—
30	C₄⋊(C₅⋊D₅)	—	4	4	no	—
31	C₄⋊(C₅⋊D₅)	—	4	4	no	—
32	C₄×C₅⋊D₅	—	2	2	no	—
33	C₄×C₅⋊D₅	—	4	4	no	—
34	C₅×#125	D₅	1	6	yes	complemented
35	#4041	—	5	24	no	—
36	#4041	—	5	12	no	—
37	#4040	—	5	24	no	—
38	#4039	—	5	12	no	—
39	#4020	—	5	6	no	—
40	#4018	—	5	6	no	—
41	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
42	C₂×C₅⋊D₅	—	2	1	no	—
43	C₂×C₅⋊D₅	—	2	4	no	—
44	C₂×C₅⋊D₅	—	2	4	no	—
45	C₂×C₅⋊D₅	—	2	2	no	—
46	C₂×C₅⋊D₅	—	2	4	no	—
47	C₄×C₅²	—	2	4	no	—
48	C₄×C₅²	—	2	2	no	—
49	C₄×C₅²	#41	1	2	yes	—
50	#1039	—	2	1	no	—
51	#1039	—	2	2	no	—
52	C₂×D₂₀	—	5	12	no	—
53	C₂×D₂₀	—	5	24	no	—
54	C₂×D₂₀	—	5	24	no	—
55	C₂×C₄×D₅	—	5	6	no	—
56	C₂×C₄×D₅	—	5	24	no	—
57	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
58	C₅×C₄⋊C₄	C₂×D₅	1	24	yes	—
59	C₄²×C₅	C₂×D₅	1	6	yes	—
60	#685	—	5	12	no	—
61	#685	—	5	24	no	—
62	#685	—	5	24	no	—
63	#683	—	5	12	no	—
64	C₄×Dic₅	—	5	6	no	—
65	#540	—	25	1	no	Sylow
66	C₂×C₅²	C₂×#41	1	1	yes	characteristic
67	C₂×C₅²	H(5,𝔽₂)	1	2	yes	—
68	C₅⋊D₅	—	4	1	no	—
69	C₅⋊D₅	—	4	4	no	—
70	C₂²×D₅	—	5	6	no	—
71	C₂²×D₅	—	5	24	no	—
72	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
73	C₂×C₄×C₅	C₂²×D₅	1	24	yes	—
74	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
75	C₂×Dic₅	—	5	12	no	—
76	C₂×Dic₅	—	5	6	no	—
77	D₂₀	—	10	24	no	—
78	D₂₀	—	20	24	no	—
79	D₂₀	—	20	24	no	—
80	C₄×D₅	—	10	12	no	—
81	C₄×D₅	—	20	24	no	—
82	#127	—	25	1	no	—
83	#125	C₅⋊D₅	1	1	yes	characteristic, complemented
84	#123	—	25	4	no	—
85	#121	—	25	4	no	—
86	#121	—	25	2	no	—
87	C₄×D₄	—	25	2	no	—
88	#117	—	25	1	no	—
89	C₅²	#540	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₅	C₂³×D₅	1	6	yes	—
91	C₂×D₅	—	10	6	no	—
92	C₂×D₅	—	10	24	no	—
93	C₂×D₅	—	10	24	no	—
94	C₂×D₅	—	10	12	no	—
95	C₂×D₅	—	10	24	no	—
96	C₄×C₅	—	2	24	no	—
97	C₄×C₅	—	2	12	no	—
98	C₄×C₅	#713	1	12	yes	—
99	Dic₅	—	10	6	no	—
100	Dic₅	—	10	12	no	—
101	C₂×D₄	—	25	2	no	—
102	C₂×D₄	—	25	4	no	—
103	C₂×D₄	—	25	4	no	—
104	C₂²×C₄	—	25	1	no	—
105	C₂²×C₄	—	25	4	no	—
106	C₄⋊C₄	—	25	2	no	—
107	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
108	C₄⋊C₄	C₂×C₅⋊D₅	1	4	yes	—
109	C₂²⋊C₄	—	25	2	no	—
110	C₂²⋊C₄	—	25	4	no	—
111	C₂²⋊C₄	—	25	4	no	—
112	C₄²	—	25	1	no	—
113	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
114	C₂×C₅	C₂×#713	1	6	yes	—
115	C₂×C₅	#4149	1	12	yes	—
116	D₅	—	20	6	no	—
117	D₅	—	20	24	no	—
118	C₂³	—	25	1	no	—
119	C₂³	—	25	4	no	—
120	D₄	—	50	4	no	—
121	D₄	—	100	4	no	—
122	D₄	—	100	4	no	—
123	C₂×C₄	—	25	2	no	—
124	C₂×C₄	—	25	1	no	—
125	C₂×C₄	—	50	2	no	—
126	C₂×C₄	—	100	4	no	—
127	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
128	C₂×C₄	C₂²×C₅⋊D₅	1	4	yes	—
129	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
130	C₅	#66032	1	6	yes	complemented
131	C₂²	—	50	1	no	—
132	C₂²	—	50	4	no	—
133	C₂²	—	50	4	no	—
134	C₂²	—	50	2	no	—
135	C₂²	—	50	4	no	—
136	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	2	2	no	—
139	C₄	—	50	1	no	—
140	C₄	—	50	2	no	—
141	C₄	#87904	1	2	yes	—
142	C₂	—	100	1	no	—
143	C₂	—	100	4	no	—
144	C₂	C₂×#87904	1	1	yes	characteristic
145	C₂	#11717522	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#125

C₂

yes

maximal, characteristic, complemented

#11717215

C₂

yes

maximal, complemented

#11717215

C₂

yes

maximal, complemented

#11717214

C₂

yes

maximal, complemented

#11717213

C₂

yes

maximal, complemented

#11717194

C₂

yes

maximal, characteristic, complemented

#11717192

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×C₄×C₅⋊D₅

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

#87808

C₂²

yes

—

#87808

C₂²

yes

—

#87808

C₂²

yes

—

#87806

C₂²

yes

—

C₄×#1039

C₂²

yes

characteristic

#66032

—

maximal

C₂²×C₅⋊D₅

C₂³

yes

characteristic

C₂²×C₅⋊D₅

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

—

C₅×#125

D₅

yes

—

—

—

—

—

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

#41

yes

—

#1039

—

#1039

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

—

—

—

—

C₄×Dic₅

—

#540

—

Sylow

C₂×C₅²

C₂×#41

yes

characteristic

C₂×C₅²

H(5,𝔽₂)

yes

—

C₅⋊D₅

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

#127

—

#125

C₅⋊D₅

yes

characteristic, complemented

#123

—

#121

—

#121

—

C₄×D₄

—

#117

—

C₅²

#540

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂³×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#713

yes

—

Dic₅

—

100

Dic₅

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

108

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

C₂×C₅⋊D₅

yes

characteristic

114

C₂×C₅

C₂×#713

yes

—

115

C₂×C₅

#4149

yes

—

116

D₅

—

117

D₅

—

118

C₂³

—

119

C₂³

—

120

D₄

—

121

D₄

—

100

—

122

D₄

—

100

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

100

—

127

C₂×C₄

C₂²×C₅⋊D₅

yes

—

128

C₂×C₄

C₂²×C₅⋊D₅

yes

—

129

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

130

C₅

#66032

yes

complemented

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂³×C₅⋊D₅

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#87904

yes

—

142

C₂

—

100

—

143

C₂

—

100

—

144

C₂

C₂×#87904

yes

characteristic

145

C₂

#11717522

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#125	C₂
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#540
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×C₅⋊D₅

commutator, socle, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₅²×#125

C₂

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. C₅²×#125	C₂	semidirect
2. #11717215	C₂	semidirect
3. #11717215	C₂	semidirect
4. #11717214	C₂	semidirect
5. #11717213	C₂	semidirect
6. #11717194	C₂	semidirect
7. #11717192	C₂	semidirect
34. C₅×#125	D₅	semidirect
83. #125	C₅⋊D₅	semidirect
89. C₅²	#540	semidirect
130. C₅	#66032	semidirect

H ≅ G/N

type

1. C₅²×#125

C₂

semidirect

2. #11717215

C₂

semidirect

3. #11717215

C₂

semidirect

4. #11717214

C₂

semidirect

5. #11717213

C₂

semidirect

6. #11717194

C₂

semidirect

7. #11717192

C₂

semidirect

34. C₅×#125

D₅

semidirect

83. #125

C₅⋊D₅

semidirect

89. C₅²

#540

semidirect

130. C₅

#66032

semidirect

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

subgroup	index
C₅²×#125	2
#11717215	2
#11717214	2
#11717213	2
#11717194	2
#11717192	2
#66032	5

subgroup

index

C₅²×#125

Subgroups — #13107225

Conjugacy classes of subgroups

Complements

Maximal subgroups