Group #13107211 — subgroups

Subgroups — #13107211

Order 1600.

1040 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 1040 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, complemented
2	#11717267	C₂	1	2	yes	maximal, complemented
3	#11717257	C₂	1	2	yes	maximal, complemented
4	#11717205	C₂	1	4	yes	maximal, complemented
5	#11717202	C₂	1	4	yes	maximal, complemented
6	#11717195	C₂	1	2	yes	maximal, complemented
7	C₂³×C₅⋊D₅	C₂²	1	2	yes	complemented
8	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
13	C₂×#6045	C₂²	1	2	yes	complemented
14	C₅²×C₂²⋊C₄	C₂²	1	4	yes	complemented
15	C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
16	C₂²⋊#1039	C₂²	1	2	yes	complemented
17	#87808	C₂²	1	4	yes	complemented
18	#87808	C₂²	1	4	yes	complemented
19	#87808	C₂²	1	4	yes	complemented
20	C₄×#1039	C₂²	1	2	yes	complemented
21	#66017	—	5	6	no	maximal
22	C₂³×C₅²	C₂³	1	2	yes	—
23	C₂²×C₅⋊D₅	—	4	2	no	—
24	C₂²×C₅⋊D₅	—	4	4	no	—
25	C₂²×C₅⋊D₅	C₂³	1	4	yes	—
26	C₅²×Q₈	—	4	1	no	—
27	C₅²×D₄	—	4	1	no	—
28	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₅²	C₂³	1	2	yes	—
30	C₂×C₄×C₅²	C₂³	1	2	yes	—
31	C₂²⋊(C₅⋊D₅)	—	4	4	no	—
32	C₂×#1039	C₂³	1	4	yes	—
33	C₄⋊(C₅⋊D₅)	—	4	2	no	—
34	C₄⋊(C₅⋊D₅)	—	4	2	no	—
35	#6045	—	4	2	no	—
36	C₅×#124	D₅	1	6	yes	complemented
37	#4093	—	5	12	no	—
38	#4083	—	5	12	no	—
39	#4031	—	5	24	no	—
40	#4028	—	5	24	no	—
41	#4021	—	5	12	no	—
42	C₂²×C₅²	—	2	4	no	—
43	C₂²×C₅²	—	2	2	no	—
44	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
45	C₂×C₅⋊D₅	—	2	4	no	—
46	C₂×C₅⋊D₅	—	2	4	no	—
47	C₂×C₅⋊D₅	—	2	4	no	—
48	C₂×C₅⋊D₅	—	4	8	no	—
49	C₄×C₅²	—	2	2	no	—
50	C₄×C₅²	—	2	2	no	—
51	C₄×C₅²	—	2	1	no	—
52	#1039	—	2	4	no	—
53	C₂³×D₅	—	5	12	no	—
54	C₂×C₅×Q₈	C₂×D₅	1	6	yes	complemented
55	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
56	C₂×C₂²⋊D₅	—	5	24	no	—
57	C₂×D₂₀	—	5	12	no	—
58	C₂×D₂₀	—	5	12	no	—
59	C₂×Dic₁₀	—	5	12	no	—
60	C₅×C₂²⋊C₄	C₂×D₅	1	24	yes	complemented
61	C₄²×C₅	C₂×D₅	1	6	yes	complemented
62	C₂²⋊Dic₅	—	5	12	no	—
63	#685	—	5	24	no	—
64	#685	—	5	24	no	—
65	#685	—	5	24	no	—
66	C₄×Dic₅	—	5	12	no	—
67	H(3,𝔽₄)	—	25	1	no	Sylow
68	C₂×C₅²	—	4	2	no	—
69	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
70	C₂×C₅²	H(5,𝔽₂)	1	2	yes	—
71	C₅⋊D₅	—	4	4	no	—
72	C₂³×C₅	C₂²×D₅	1	12	yes	—
73	C₂²×D₅	—	5	24	no	—
74	C₂²×D₅	—	20	12	no	—
75	C₂²×D₅	—	20	24	no	—
76	C₅×Q₈	—	4	6	no	—
77	C₅×D₄	—	4	6	no	—
78	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
79	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
80	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
81	C₂²⋊D₅	—	20	24	no	—
82	C₂×Dic₅	—	5	24	no	—
83	D₂₀	—	20	12	no	—
84	D₂₀	—	20	12	no	—
85	Dic₁₀	—	20	12	no	—
86	#124	—	25	2	no	—
87	#124	—	25	4	no	—
88	#124	—	25	2	no	—
89	#124	C₅⋊D₅	1	1	yes	characteristic, complemented
90	#120	—	25	2	no	—
91	#120	—	25	4	no	—
92	C₅²	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₅	—	2	12	no	—
94	C₂²×C₅	—	2	24	no	—
95	C₂²×C₅	C₂³×D₅	1	6	yes	—
96	C₂×D₅	—	10	24	no	—
97	C₂×D₅	—	10	24	no	—
98	C₂×D₅	—	10	24	no	—
99	C₂×D₅	—	20	48	no	—
100	C₄×C₅	—	2	12	no	—
101	C₄×C₅	—	2	6	no	—
102	C₄×C₅	—	2	12	no	—
103	Dic₅	—	10	24	no	—
104	C₂⁴	—	25	2	no	—
105	C₂×Q₈	—	25	2	no	—
106	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
107	C₂×D₄	—	25	4	no	—
108	C₂×D₄	—	25	2	no	—
109	C₂×D₄	—	25	2	no	—
110	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
111	C₂²⋊C₄	—	25	2	no	—
112	C₂²⋊C₄	—	25	4	no	—
113	C₂²⋊C₄	—	25	4	no	—
114	C₂²⋊C₄	—	25	4	no	—
115	C₂²⋊C₄	C₂×C₅⋊D₅	1	4	yes	complemented
116	C₄²	—	25	2	no	—
117	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
118	C₂×C₅	—	4	12	no	—
119	C₂×C₅	#4144	1	6	yes	—
120	C₂×C₅	#4149	1	12	yes	—
121	D₅	—	20	24	no	—
122	C₂³	—	25	4	no	—
123	C₂³	—	100	2	no	—
124	C₂³	—	100	4	no	—
125	C₂³	C₂²×C₅⋊D₅	1	2	yes	—
126	Q₈	—	4	1	no	—
127	Q₈	—	100	2	no	—
128	D₄	—	4	1	no	—
129	D₄	—	100	2	no	—
130	D₄	—	100	2	no	—
131	D₄	—	100	4	no	—
132	C₂×C₄	—	25	4	no	—
133	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
134	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
136	C₅	#66017	1	6	yes	complemented
137	C₂²	—	2	2	no	—
138	C₂²	—	2	4	no	—
139	C₂²	—	50	4	no	—
140	C₂²	—	50	4	no	—
141	C₂²	—	50	4	no	—
142	C₂²	—	100	8	no	—
143	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	50	4	no	—
148	C₂	—	4	2	no	—
149	C₂	—	100	4	no	—
150	C₂	#11717517	1	1	yes	characteristic
151	C₂	#11717522	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#124

C₂

yes

maximal, characteristic, complemented

#11717267

C₂

yes

maximal, complemented

#11717257

C₂

yes

maximal, complemented

#11717205

C₂

yes

maximal, complemented

#11717202

C₂

yes

maximal, complemented

#11717195

C₂

yes

maximal, complemented

C₂³×C₅⋊D₅

C₂²

yes

complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic, complemented

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂×C₂²⋊(C₅⋊D₅)

C₂²

yes

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₂×#6045

C₂²

yes

complemented

C₅²×C₂²⋊C₄

C₂²

yes

complemented

C₄²×C₅²

C₂²

yes

characteristic, complemented

C₂²⋊#1039

C₂²

yes

complemented

#87808

C₂²

yes

complemented

#87808

C₂²

yes

complemented

#87808

C₂²

yes

complemented

C₄×#1039

C₂²

yes

complemented

#66017

—

maximal

C₂³×C₅²

C₂³

yes

—

C₂²×C₅⋊D₅

—

C₂²×C₅⋊D₅

—

C₂²×C₅⋊D₅

C₂³

yes

—

C₅²×Q₈

—

C₅²×D₄

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

—

C₂²⋊(C₅⋊D₅)

—

C₂×#1039

C₂³

yes

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

#6045

—

C₅×#124

D₅

yes

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

—

#1039

—

C₂³×D₅

—

C₂×C₅×Q₈

C₂×D₅

yes

complemented

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×C₂²⋊D₅

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₅×C₂²⋊C₄

C₂×D₅

yes

complemented

C₄²×C₅

C₂×D₅

yes

complemented

C₂²⋊Dic₅

—

#685

—

#685

—

#685

—

C₄×Dic₅

—

H(3,𝔽₄)

—

Sylow

C₂×C₅²

—

C₂×C₅²

H(5,𝔽₂)

yes

characteristic

C₂×C₅²

H(5,𝔽₂)

yes

—

C₅⋊D₅

—

C₂³×C₅

C₂²×D₅

yes

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

Dic₁₀

—

—

—

—

C₅⋊D₅

yes

characteristic, complemented

#120

—

#120

—

C₅²

H(3,𝔽₄)

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂³×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

—

103

Dic₅

—

104

C₂⁴

—

105

C₂×Q₈

—

106

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic, complemented

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

C₂×C₅⋊D₅

yes

complemented

116

C₄²

—

117

C₄²

C₂×C₅⋊D₅

yes

characteristic, complemented

118

C₂×C₅

—

119

C₂×C₅

#4144

yes

—

120

C₂×C₅

#4149

yes

—

121

D₅

—

122

C₂³

—

123

C₂³

—

100

—

124

C₂³

—

100

—

125

C₂³

C₂²×C₅⋊D₅

yes

—

126

Q₈

—

127

Q₈

—

100

—

128

D₄

—

129

D₄

—

100

—

130

D₄

—

100

—

131

D₄

—

100

—

132

C₂×C₄

—

133

C₂×C₄

C₂²×C₅⋊D₅

yes

—

134

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

135

C₂×C₄

C₂²×C₅⋊D₅

yes

—

136

C₅

#66017

yes

complemented

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

100

—

143

C₂²

C₂³×C₅⋊D₅

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₂

—

149

C₂

—

100

—

150

C₂

#11717517

yes

characteristic

151

C₂

#11717522

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#124	C₂
hypercenter	#124	C₅⋊D₅
hypocenter	C₅²	H(3,𝔽₄)
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

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₅²×#124	C₂	semidirect
2. #11717267	C₂	semidirect
3. #11717257	C₂	semidirect
4. #11717205	C₂	semidirect
5. #11717202	C₂	semidirect
6. #11717195	C₂	semidirect
7. C₂³×C₅⋊D₅	C₂²	semidirect
8. C₂×C₅²×Q₈	C₂²	semidirect
13. C₂×#6045	C₂²	semidirect
14. C₅²×C₂²⋊C₄	C₂²	semidirect
15. C₄²×C₅²	C₂²	semidirect
16. C₂²⋊#1039	C₂²	semidirect
17. #87808	C₂²	semidirect
18. #87808	C₂²	semidirect
19. #87808	C₂²	semidirect
20. C₄×#1039	C₂²	semidirect
36. C₅×#124	D₅	semidirect
54. C₂×C₅×Q₈	C₂×D₅	semidirect
60. C₅×C₂²⋊C₄	C₂×D₅	semidirect
61. C₄²×C₅	C₂×D₅	semidirect
89. #124	C₅⋊D₅	semidirect
92. C₅²	H(3,𝔽₄)	semidirect
106. C₂×Q₈	C₂×C₅⋊D₅	semidirect
115. C₂²⋊C₄	C₂×C₅⋊D₅	semidirect
117. C₄²	C₂×C₅⋊D₅	semidirect
136. C₅	#66017	semidirect

H ≅ G/N

type

1. C₅²×#124

C₂

semidirect

2. #11717267

C₂

semidirect

3. #11717257

C₂

semidirect

4. #11717205

C₂

semidirect

5. #11717202

C₂

semidirect

6. #11717195

C₂

semidirect

7. C₂³×C₅⋊D₅

C₂²

semidirect

8. C₂×C₅²×Q₈

C₂²

semidirect

13. C₂×#6045

C₂²

semidirect

14. C₅²×C₂²⋊C₄

C₂²

semidirect

15. C₄²×C₅²

C₂²

semidirect

16. C₂²⋊#1039

C₂²

semidirect

17. #87808

C₂²

semidirect

18. #87808

C₂²

semidirect

19. #87808

C₂²

semidirect

20. C₄×#1039

C₂²

semidirect

36. C₅×#124

D₅

semidirect

54. C₂×C₅×Q₈

C₂×D₅

semidirect

60. C₅×C₂²⋊C₄

C₂×D₅

semidirect

61. C₄²×C₅

C₂×D₅

semidirect

89. #124

C₅⋊D₅

semidirect

92. C₅²

H(3,𝔽₄)

semidirect

106. C₂×Q₈

C₂×C₅⋊D₅

semidirect

115. C₂²⋊C₄

C₂×C₅⋊D₅

semidirect

117. C₄²

C₂×C₅⋊D₅

semidirect

136. C₅

#66017

semidirect

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

subgroup	index
C₅²×#124	2
#11717267	2
#11717257	2
#11717205	2
#11717202	2
#11717195	2
#66017	5

subgroup

index

C₅²×#124

Subgroups — #13107211

Conjugacy classes of subgroups

Complements

Maximal subgroups