Group #13107247 — subgroups

Subgroups — #13107247

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×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717259	C₂	1	4	yes	maximal, complemented
3	#11717258	C₂	1	2	yes	maximal, complemented
4	#11717255	C₂	1	4	yes	maximal, complemented
5	D₄×#1039	C₂	1	2	yes	maximal, complemented
6	#11717192	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717190	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅²×D₄	C₂²	1	2	yes	—
9	C₂×C₅²×D₄	C₂²	1	4	yes	—
10	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	4	yes	—
11	C₂²×#1039	C₂²	1	4	yes	—
12	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
13	C₄²×C₅²	C₂²	1	1	yes	characteristic
14	C₂²⋊#1039	C₂²	1	4	yes	—
15	C₂²⋊#1039	C₂²	1	4	yes	—
16	#87808	C₂²	1	2	yes	—
17	C₄⋊#1039	C₂²	1	2	yes	—
18	#87806	C₂²	1	4	yes	—
19	#87806	C₂²	1	2	yes	—
20	C₄×#1039	C₂²	1	1	yes	characteristic
21	#66053	—	5	6	no	maximal
22	C₂³×C₅²	C₂³	1	4	yes	—
23	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
24	C₅²×D₄	—	2	4	no	—
25	C₅²×D₄	—	4	4	no	—
26	C₂×C₄×C₅²	C₂³	1	2	yes	—
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	C₂²⋊(C₅⋊D₅)	—	4	4	no	—
29	C₂×#1039	—	4	4	no	—
30	C₂×#1039	C₂³	1	2	yes	—
31	C₂×#1039	C₂³	1	1	yes	characteristic
32	C₂×#1039	C₂³	1	4	yes	—
33	C₄×C₅⋊D₅	—	2	2	no	—
34	C₅×#127	D₅	1	6	yes	complemented
35	#4085	—	5	24	no	—
36	#4084	—	5	12	no	—
37	#4081	—	5	24	no	—
38	D₄×Dic₅	—	5	12	no	—
39	#4018	—	5	6	no	—
40	#4016	—	5	6	no	—
41	C₂²×C₅²	—	2	4	no	—
42	C₂²×C₅²	—	2	4	no	—
43	C₂²×C₅²	—	2	4	no	—
44	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
45	C₂×C₅⋊D₅	—	2	1	no	—
46	C₂×C₅⋊D₅	—	2	2	no	—
47	C₄×C₅²	—	2	2	no	—
48	C₄×C₅²	#41	1	2	yes	—
49	#1039	—	2	1	no	—
50	#1039	—	2	2	no	—
51	#1039	—	2	4	no	—
52	C₂×C₅×D₄	C₂×D₅	1	12	yes	—
53	C₂×C₅×D₄	C₂×D₅	1	24	yes	—
54	C₂×C₂²⋊D₅	—	5	24	no	—
55	C₂²×Dic₅	—	5	24	no	—
56	C₂×C₄×D₅	—	5	6	no	—
57	C₄²×C₅	C₂×D₅	1	6	yes	—
58	C₂²⋊Dic₅	—	5	24	no	—
59	C₂²⋊Dic₅	—	5	24	no	—
60	#685	—	5	12	no	—
61	C₄⋊Dic₅	—	5	12	no	—
62	#683	—	5	24	no	—
63	#683	—	5	12	no	—
64	C₄×Dic₅	—	5	6	no	—
65	#540	—	25	1	no	Sylow
66	C₂×C₅²	—	4	4	no	—
67	C₂×C₅²	C₂×#41	1	1	yes	characteristic
68	C₂×C₅²	H(5,𝔽₂)	1	2	yes	—
69	C₅⋊D₅	—	4	1	no	—
70	C₂³×C₅	C₂²×D₅	1	24	yes	—
71	C₂²×D₅	—	5	6	no	—
72	C₅×D₄	—	2	24	no	—
73	C₅×D₄	—	4	24	no	—
74	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
75	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
76	C₂²⋊D₅	—	20	24	no	—
77	C₂×Dic₅	—	5	12	no	—
78	C₂×Dic₅	—	5	24	no	—
79	C₂×Dic₅	—	5	6	no	—
80	C₂×Dic₅	—	20	24	no	—
81	C₄×D₅	—	10	12	no	—
82	#127	C₅⋊D₅	1	1	yes	characteristic, complemented
83	#125	—	25	1	no	—
84	#123	—	25	4	no	—
85	#121	—	25	2	no	—
86	#121	—	25	4	no	—
87	C₄×D₄	—	25	2	no	—
88	#117	—	25	1	no	—
89	C₅²	#540	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₅	—	2	24	no	—
91	C₂²×C₅	—	2	24	no	—
92	C₂²×C₅	—	2	24	no	—
93	C₂²×C₅	C₂³×D₅	1	6	yes	—
94	C₂×D₅	—	10	6	no	—
95	C₂×D₅	—	10	12	no	—
96	C₄×C₅	—	2	12	no	—
97	C₄×C₅	#711	1	12	yes	—
98	Dic₅	—	10	6	no	—
99	Dic₅	—	10	12	no	—
100	Dic₅	—	10	24	no	—
101	C₂×D₄	—	25	4	no	—
102	C₂×D₄	C₂×C₅⋊D₅	1	2	yes	—
103	C₂×D₄	C₂×C₅⋊D₅	1	4	yes	—
104	C₂²×C₄	—	25	4	no	—
105	C₂²×C₄	—	25	1	no	—
106	C₄⋊C₄	—	25	2	no	—
107	C₄⋊C₄	—	25	2	no	—
108	C₄⋊C₄	—	25	4	no	—
109	C₂²⋊C₄	—	25	4	no	—
110	C₂²⋊C₄	—	25	4	no	—
111	C₂²⋊C₄	—	25	2	no	—
112	C₄²	—	25	1	no	—
113	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
114	C₂×C₅	—	4	24	no	—
115	C₂×C₅	C₂×#711	1	6	yes	—
116	C₂×C₅	#4144	1	12	yes	—
117	D₅	—	20	6	no	—
118	C₂³	—	25	1	no	—
119	C₂³	C₂²×C₅⋊D₅	1	4	yes	—
120	D₄	—	2	4	no	—
121	D₄	—	4	4	no	—
122	D₄	—	100	4	no	—
123	C₂×C₄	—	25	2	no	—
124	C₂×C₄	—	25	1	no	—
125	C₂×C₄	—	25	4	no	—
126	C₂×C₄	—	50	2	no	—
127	C₂×C₄	—	100	4	no	—
128	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
129	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
130	C₅	#66053	1	6	yes	complemented
131	C₂²	—	2	4	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	50	1	no	—
135	C₂²	—	50	2	no	—
136	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	50	1	no	—
139	C₄	—	50	2	no	—
140	C₄	—	50	4	no	—
141	C₄	#87902	1	2	yes	—
142	C₂	—	4	4	no	—
143	C₂	—	100	1	no	—
144	C₂	C₂×#87902	1	1	yes	characteristic
145	C₂	#11717517	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#127

C₂

yes

maximal, characteristic, complemented

#11717259

C₂

yes

maximal, complemented

#11717258

C₂

yes

maximal, complemented

#11717255

C₂

yes

maximal, complemented

D₄×#1039

C₂

yes

maximal, complemented

#11717192

C₂

yes

maximal, characteristic, complemented

#11717190

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₄

C₂²

yes

—

C₂×C₅²×D₄

C₂²

yes

—

C₂×C₂²⋊(C₅⋊D₅)

C₂²

yes

—

C₂²×#1039

C₂²

yes

—

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₄²×C₅²

C₂²

yes

characteristic

C₂²⋊#1039

C₂²

yes

—

C₂²⋊#1039

C₂²

yes

—

#87808

C₂²

yes

—

C₄⋊#1039

C₂²

yes

—

#87806

C₂²

yes

—

#87806

C₂²

yes

—

C₄×#1039

C₂²

yes

characteristic

#66053

—

maximal

C₂³×C₅²

C₂³

yes

—

C₂²×C₅⋊D₅

C₂³

yes

characteristic

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂²⋊(C₅⋊D₅)

—

C₂×#1039

—

C₂×#1039

C₂³

yes

—

C₂×#1039

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

—

C₄×C₅⋊D₅

—

C₅×#127

D₅

yes

—

—

—

D₄×Dic₅

—

#4018

—

#4016

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

#41

yes

—

#1039

—

#1039

—

#1039

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×C₂²⋊D₅

—

C₂²×Dic₅

—

C₂×C₄×D₅

—

C₄²×C₅

C₂×D₅

yes

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

#685

—

C₄⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

#540

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×#41

yes

characteristic

C₂×C₅²

H(5,𝔽₂)

yes

—

C₅⋊D₅

—

C₂³×C₅

C₂²×D₅

yes

—

C₂²×D₅

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

#127

C₅⋊D₅

yes

characteristic, complemented

—

—

—

—

C₄×D₄

—

#117

—

C₅²

#540

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂³×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#711

yes

—

Dic₅

—

Dic₅

—

100

Dic₅

—

101

C₂×D₄

—

102

C₂×D₄

C₂×C₅⋊D₅

yes

—

103

C₂×D₄

C₂×C₅⋊D₅

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

C₂×C₅⋊D₅

yes

characteristic

114

C₂×C₅

—

115

C₂×C₅

C₂×#711

yes

—

116

C₂×C₅

#4144

yes

—

117

D₅

—

118

C₂³

—

119

C₂³

C₂²×C₅⋊D₅

yes

—

120

D₄

—

121

D₄

—

122

D₄

—

100

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

100

—

128

C₂×C₄

C₂²×C₅⋊D₅

yes

—

129

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

130

C₅

#66053

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

#87902

yes

—

142

C₂

—

143

C₂

—

100

—

144

C₂

C₂×#87902

yes

characteristic

145

C₂

#11717517

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#127	C₂
hypercenter	#127	C₅⋊D₅
hypocenter	C₅²	#540
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×C₅⋊D₅

commutator, socle, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₅²×#127

C₂

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. C₅²×#127	C₂	semidirect
2. #11717259	C₂	semidirect
3. #11717258	C₂	semidirect
4. #11717255	C₂	semidirect
5. D₄×#1039	C₂	semidirect
6. #11717192	C₂	semidirect
7. #11717190	C₂	semidirect
34. C₅×#127	D₅	semidirect
82. #127	C₅⋊D₅	semidirect
89. C₅²	#540	semidirect
130. C₅	#66053	semidirect

H ≅ G/N

type

1. C₅²×#127

C₂

semidirect

2. #11717259

C₂

semidirect

3. #11717258

C₂

semidirect

4. #11717255

C₂

semidirect

5. D₄×#1039

C₂

semidirect

6. #11717192

C₂

semidirect

7. #11717190

C₂

semidirect

34. C₅×#127

D₅

semidirect

82. #127

C₅⋊D₅

semidirect

89. C₅²

#540

semidirect

130. C₅

#66053

semidirect

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

subgroup	index
C₅²×#127	2
#11717259	2
#11717258	2
#11717255	2
D₄×#1039	2
#11717192	2
#11717190	2
#66053	5

subgroup

index

C₅²×#127

Subgroups — #13107247

Conjugacy classes of subgroups

Complements

Maximal subgroups