Group #13099676 — subgroups

Subgroups — #13099676

Order 1600.

387 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 153 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	#41×D₂₅	C₂	1	1	yes	maximal, characteristic
2	#11716552	C₂	1	2	yes	maximal, complemented
3	#11716550	C₂	1	6	yes	maximal, complemented
4	D₂₅⋊M₁₆	C₂	1	3	yes	maximal, complemented
5	C₂×#87735	C₂	1	3	yes	maximal, complemented
6	#41×C₂₅	C₄	1	1	yes	characteristic
7	#87748	C₂²	1	1	yes	characteristic
8	Q₈×D₂₅	C₄	1	1	yes	characteristic
9	#87746	C₂²	1	3	yes	—
10	D₄×D₂₅	C₄	1	3	yes	—
11	#87744	C₄	1	3	yes	—
12	C₂×C₄×D₂₅	C₂²	1	3	yes	—
13	#87739	C₂²	1	6	yes	complemented
14	C₂×#6016	C₂²	1	6	yes	complemented
15	#87735	C₂²	1	3	yes	—
16	#87735	C₂²	1	1	yes	characteristic
17	#87735	C₂²	1	6	yes	complemented
18	D₂₅⋊C₈	C₂²	1	6	yes	complemented
19	#66264	—	5	1	no	maximal
20	C₂²×D₂₅	C₂×C₄	1	3	yes	—
21	Q₈×C₂₅	C₂×C₄	1	1	yes	characteristic
22	D₄×C₂₅	C₂×C₄	1	3	yes	—
23	C₂×C₄×C₂₅	C₂×C₄	1	3	yes	—
24	C₂²⋊D₂₅	C₂×C₄	1	6	yes	—
25	C₂×Dic₂₅	C₂³	1	3	yes	—
26	D₁₀₀	C₂×C₄	1	3	yes	—
27	C₄×D₂₅	C₂×C₄	1	3	yes	—
28	C₄×D₂₅	C₂×C₄	1	3	yes	—
29	C₄×D₂₅	C₂³	1	3	yes	—
30	C₄×D₂₅	C₂³	1	1	yes	characteristic
31	Dic₅₀	C₂×C₄	1	3	yes	—
32	#6016	C₂³	1	2	yes	—
33	#6016	C₂³	1	6	yes	—
34	D₅×#41	—	5	1	no	—
35	#4133	—	5	2	no	—
36	#4131	—	5	6	no	—
37	D₅⋊M₁₆	—	5	3	no	—
38	C₂×#700	—	5	3	no	—
39	C₂²×C₂₅	C₂²×C₄	1	3	yes	—
40	C₂×D₂₅	—	4	3	no	—
41	C₂×D₂₅	C₂²×C₄	1	1	yes	characteristic
42	C₂×D₂₅	C₂²×C₄	1	3	yes	—
43	C₄×C₂₅	C₂²×C₄	1	3	yes	—
44	C₄×C₂₅	C₂²×C₄	1	1	yes	characteristic
45	Dic₂₅	C₂²×C₄	1	3	yes	—
46	Dic₂₅	C₂⁴	1	1	yes	characteristic
47	C₅×#41	AGL(1,𝔽₅)	1	1	yes	characteristic
48	#713	—	5	1	no	—
49	Q₈×D₅	—	5	1	no	—
50	#711	—	5	3	no	—
51	D₄×D₅	—	5	3	no	—
52	#709	—	5	3	no	—
53	C₂×C₄×D₅	—	5	3	no	—
54	#704	—	5	6	no	—
55	C₂×#170	—	5	6	no	—
56	#700	—	5	3	no	—
57	#700	—	5	6	no	—
58	#700	—	5	1	no	—
59	D₅⋊C₈	—	5	6	no	—
60	#568	—	25	1	no	Sylow
61	C₂×C₂₅	—	2	3	no	—
62	C₂×C₂₅	C₂³×C₄	1	1	yes	characteristic
63	D₂₅	—	2	1	no	—
64	D₂₅	—	2	3	no	—
65	C₂²×D₅	—	5	3	no	—
66	C₅×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₅×D₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
68	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	—
69	C₂²⋊D₅	—	5	6	no	—
70	C₂×Dic₅	—	5	3	no	—
71	D₂₀	—	5	3	no	—
72	C₄×D₅	—	5	3	no	—
73	C₄×D₅	—	5	3	no	—
74	C₄×D₅	—	5	3	no	—
75	C₄×D₅	—	5	1	no	—
76	Dic₁₀	—	5	3	no	—
77	#170	—	5	2	no	—
78	#170	—	5	6	no	—
79	C₂×#41	—	25	1	no	—
80	#131	—	25	6	no	—
81	#131	—	25	2	no	—
82	C₂×M₁₆	—	25	3	no	—
83	C₂×M₁₆	—	25	3	no	—
84	C₂₅	#568	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	3	yes	—
86	C₂×D₅	—	5	1	no	—
87	C₂×D₅	—	5	3	no	—
88	C₂×D₅	—	20	3	no	—
89	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	3	yes	—
90	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
91	Dic₅	—	5	3	no	—
92	Dic₅	—	5	1	no	—
93	#41	—	25	1	no	—
94	#41	—	25	3	no	—
95	#41	—	25	3	no	—
96	#41	#1035	1	1	yes	characteristic
97	C₂×Q₈	—	25	1	no	—
98	C₂×D₄	—	25	3	no	—
99	C₂²×C₄	—	25	3	no	—
100	M₁₆	—	25	3	no	—
101	M₁₆	—	25	1	no	—
102	M₁₆	—	25	6	no	—
103	M₁₆	—	25	6	no	—
104	C₂×C₈	—	25	6	no	—
105	C₂×C₈	—	25	6	no	—
106	C₂×C₅	—	2	3	no	—
107	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
108	D₅	—	10	1	no	—
109	D₅	—	10	3	no	—
110	C₂³	—	25	3	no	—
111	Q₈	—	25	3	no	—
112	Q₈	C₂×#1035	1	1	yes	characteristic
113	D₄	—	25	6	no	—
114	D₄	—	25	3	no	—
115	D₄	C₂×#1035	1	3	yes	—
116	C₂×C₄	—	25	3	no	—
117	C₂×C₄	—	25	3	no	—
118	C₂×C₄	—	25	1	no	—
119	C₂×C₄	—	25	3	no	—
120	C₂×C₄	—	25	3	no	—
121	C₂×C₄	C₂×#1035	1	3	yes	—
122	C₈	—	25	2	no	—
123	C₈	—	25	6	no	—
124	C₅	#66264	1	1	yes	characteristic
125	C₂²	—	25	1	no	—
126	C₂²	—	25	3	no	—
127	C₂²	—	100	3	no	—
128	C₂²	C₂²×#1035	1	3	yes	—
129	C₄	—	25	1	no	—
130	C₄	—	25	3	no	—
131	C₄	C₂²×#1035	1	1	yes	characteristic
132	C₄	C₂²×#1035	1	3	yes	—
133	C₂	—	2	3	no	—
134	C₂	—	50	1	no	—
135	C₂	—	50	3	no	—
136	C₂	C₂³×#1035	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×D₂₅

C₂

yes

maximal, characteristic

#11716552

C₂

yes

maximal, complemented

#11716550

C₂

yes

maximal, complemented

D₂₅⋊M₁₆

C₂

yes

maximal, complemented

C₂×#87735

C₂

yes

maximal, complemented

#41×C₂₅

C₄

yes

characteristic

#87748

C₂²

yes

characteristic

Q₈×D₂₅

C₄

yes

characteristic

#87746

C₂²

yes

—

D₄×D₂₅

C₄

yes

—

#87744

C₄

yes

—

C₂×C₄×D₂₅

C₂²

yes

—

#87739

C₂²

yes

complemented

C₂×#6016

C₂²

yes

complemented

#87735

C₂²

yes

—

#87735

C₂²

yes

characteristic

#87735

C₂²

yes

complemented

D₂₅⋊C₈

C₂²

yes

complemented

#66264

—

maximal

C₂²×D₂₅

C₂×C₄

yes

—

Q₈×C₂₅

C₂×C₄

yes

characteristic

D₄×C₂₅

C₂×C₄

yes

—

C₂×C₄×C₂₅

C₂×C₄

yes

—

C₂²⋊D₂₅

C₂×C₄

yes

—

C₂×Dic₂₅

C₂³

yes

—

D₁₀₀

C₂×C₄

yes

—

C₄×D₂₅

C₂×C₄

yes

—

C₄×D₂₅

C₂×C₄

yes

—

C₄×D₂₅

C₂³

yes

—

C₄×D₂₅

C₂³

yes

characteristic

Dic₅₀

C₂×C₄

yes

—

#6016

C₂³

yes

—

#6016

C₂³

yes

—

D₅×#41

—

#4133

—

#4131

—

D₅⋊M₁₆

—

C₂×#700

—

C₂²×C₂₅

C₂²×C₄

yes

—

C₂×D₂₅

—

C₂×D₂₅

C₂²×C₄

yes

characteristic

C₂×D₂₅

C₂²×C₄

yes

—

C₄×C₂₅

C₂²×C₄

yes

—

C₄×C₂₅

C₂²×C₄

yes

characteristic

Dic₂₅

C₂²×C₄

yes

—

Dic₂₅

C₂⁴

yes

C₅×#41

yes

—

Q₈×D₅

—

#711

—

D₄×D₅

—

#709

—

C₂×C₄×D₅

—

—

—

—

—

—

D₅⋊C₈

—

#568

—

Sylow

C₂×C₂₅

—

C₂×C₂₅

C₂³×C₄

yes

characteristic

D₂₅

—

D₂₅

—

C₂²×D₅

—

C₅×Q₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₅×D₄

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

#170

—

#170

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂₅

#568

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

—

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

characteristic

Dic₅

—

Dic₅

—

—

—

—

yes

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₅

—

107

C₂×C₅

C₂³×AGL(1,𝔽₅)

yes

characteristic

108

D₅

—

109

D₅

—

110

C₂³

—

111

Q₈

—

112

Q₈

C₂×#1035

yes

characteristic

113

D₄

—

114

D₄

—

115

D₄

C₂×#1035

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂×#1035

yes

—

122

C₈

—

123

C₈

—

124

C₅

#66264

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

100

—

128

C₂²

C₂²×#1035

yes

—

129

C₄

—

130

C₄

—

131

C₄

C₂²×#1035

yes

characteristic

132

C₄

C₂²×#1035

yes

—

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

C₂³×#1035

yes

characteristic

	subgroup	G/H
center	C₂	C₂³×#1035
commutator	C₂×C₂₅	C₂³×C₄
Frattini, socle	C₂×C₅	C₂³×AGL(1,𝔽₅)
Fitting	#41×C₂₅	C₄
prime radical	#41×D₂₅	C₂
cosocle	Dic₂₅	C₂⁴
hypercenter	#41	#1035
hypocenter	C₂₅	#568
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂ ◁ #41

subgroup

G/H

center

C₂

C₂³×#1035

commutator

C₂×C₂₅

C₂³×C₄

Frattini, socle

C₂×C₅

C₂³×AGL(1,𝔽₅)

Fitting

#41×C₂₅

C₄

prime radical

#41×D₂₅

C₂

cosocle

Dic₂₅

C₂⁴

C₂₅

G ▷ C₂×C₂₅ ▷ C₂₅

C₁ ◁ C₂ ◁ #41

N	H ≅ G/N	type
2. #11716552	C₂	semidirect
3. #11716550	C₂	semidirect
4. D₂₅⋊M₁₆	C₂	semidirect
5. C₂×#87735	C₂	semidirect
13. #87739	C₂²	semidirect
14. C₂×#6016	C₂²	semidirect
17. #87735	C₂²	semidirect
18. D₂₅⋊C₈	C₂²	semidirect
84. C₂₅	#568	semidirect

H ≅ G/N

type

2. #11716552

C₂

semidirect

3. #11716550

C₂

semidirect

4. D₂₅⋊M₁₆

C₂

semidirect

5. C₂×#87735

C₂

semidirect

13. #87739

C₂²

semidirect

14. C₂×#6016

C₂²

semidirect

17. #87735

C₂²

semidirect

18. D₂₅⋊C₈

C₂²

semidirect

84. C₂₅

#568

semidirect

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

subgroup	index
#41×D₂₅	2
#11716552	2
#11716550	2
D₂₅⋊M₁₆	2
C₂×#87735	2
#66264	5

subgroup

index

#41×D₂₅

D₂₅⋊M₁₆

Subgroups — #13099676

Conjugacy classes of subgroups

Complements

Maximal subgroups