Group #13097931 — subgroups

Subgroups — #13097931

Order 1600.

273 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 95 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	C₂²×Dic₅₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₅	C₂	1	2	yes	maximal
3	C₂×#87718	C₂	1	1	yes	maximal, characteristic
4	C₂₅×#95	C₂	1	1	yes	maximal, characteristic
5	#11716379	C₂	1	2	yes	maximal
6	C₂²×C₄×C₂₅	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₂₅	C₂²	1	2	yes	—
8	C₂²×Dic₂₅	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₅	C₂²	1	2	yes	—
10	C₂²×Dic₂₅	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₀	—	2	2	no	—
12	C₂×Dic₅₀	—	2	4	no	—
13	C₄⋊Dic₂₅	—	2	4	no	—
14	#87718	—	2	2	no	—
15	#64945	—	5	1	no	maximal
16	C₂³×C₂₅	C₂³	1	1	yes	characteristic
17	C₂×C₄×C₂₅	—	2	2	no	—
18	C₂×C₄×C₂₅	—	2	2	no	—
19	C₂×C₄×C₂₅	—	2	2	no	—
20	C₂×C₄×C₂₅	D₄	1	2	yes	—
21	C₂×C₄×C₂₅	Q₈	1	4	yes	complemented
22	C₂×Dic₂₅	—	2	1	no	—
23	C₂×Dic₂₅	—	2	2	no	—
24	C₂×Dic₂₅	—	2	2	no	—
25	C₂×Dic₂₅	—	2	1	no	—
26	C₂×Dic₂₅	—	2	1	no	—
27	C₂×Dic₂₅	—	2	2	no	—
28	C₂×Dic₂₅	D₄	1	4	yes	—
29	C₂×Dic₂₅	Q₈	1	2	yes	complemented
30	Dic₅₀	—	2	8	no	—
31	C₂²×Dic₁₀	—	5	1	no	—
32	C₂×C₄⋊Dic₅	—	5	2	no	—
33	C₂×#683	—	5	1	no	—
34	C₅×#95	D₅	1	1	yes	characteristic
35	#3963	—	5	2	no	—
36	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₂₅	C₂×D₄	1	2	yes	—
38	C₂²×C₂₅	C₂×Q₈	1	2	yes	—
39	C₂²×C₂₅	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₂₅	#41	1	1	yes	characteristic
41	C₄×C₂₅	—	2	2	no	—
42	C₄×C₂₅	—	2	4	no	—
43	Dic₂₅	—	2	4	no	—
44	Dic₂₅	—	2	2	no	—
45	Dic₂₅	—	4	1	no	—
46	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
47	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
48	C₂²×Dic₅	—	5	2	no	—
49	C₂²×Dic₅	—	5	1	no	—
50	C₂²×Dic₅	—	5	1	no	—
51	C₂×Dic₁₀	—	10	4	no	—
52	C₂×Dic₁₀	—	10	2	no	—
53	C₄⋊Dic₅	—	10	4	no	—
54	#683	—	10	2	no	—
55	#395	—	25	1	no	Sylow
56	C₂×C₂₅	#120	1	1	yes	characteristic
57	C₂×C₂₅	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₂₅	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₂₅	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₂₅	C₄⋊Q₈	1	2	yes	—
61	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	Dic₁₀	1	4	yes	—
66	C₂×C₄×C₅	D₂₀	1	2	yes	—
67	C₂×Dic₅	—	5	4	no	—
68	C₂×Dic₅	—	5	2	no	—
69	C₂×Dic₅	—	10	1	no	—
70	C₂×Dic₅	—	10	2	no	—
71	C₂×Dic₅	—	10	2	no	—
72	C₂×Dic₅	—	10	1	no	—
73	C₂×Dic₅	—	10	1	no	—
74	C₂×Dic₅	—	10	2	no	—
75	Dic₁₀	—	10	8	no	—
76	C₂²×Q₈	—	25	1	no	—
77	C₂×C₄⋊C₄	—	25	1	no	—
78	C₂×C₄⋊C₄	—	25	2	no	—
79	#95	—	25	2	no	—
80	#95	D₂₅	1	1	yes	characteristic
81	C₂₅	#395	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₅	C₂×Dic₁₀	1	2	yes	—
83	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
84	C₂²×C₅	D₄×D₅	1	2	yes	—
85	C₂²×C₅	#711	1	1	yes	characteristic
86	C₂²×C₅	Q₈×D₅	1	1	yes	characteristic
87	C₄×C₅	—	2	4	no	—
88	C₄×C₅	—	2	2	no	—
89	Dic₅	—	10	4	no	—
90	Dic₅	—	10	2	no	—
91	Dic₅	—	20	1	no	—
92	C₂×Q₈	—	50	4	no	—
93	C₂×Q₈	—	50	2	no	—
94	C₂²×C₄	—	25	1	no	—
95	C₂²×C₄	—	25	2	no	—
96	C₂²×C₄	—	25	1	no	—
97	C₂²×C₄	C₂×D₂₅	1	2	yes	—
98	C₂²×C₄	C₂×D₂₅	1	1	yes	characteristic
99	C₄⋊C₄	—	50	4	no	—
100	C₄⋊C₄	—	50	2	no	—
101	C₂×C₅	#4015	1	1	yes	characteristic
102	C₂×C₅	#4024	1	2	yes	—
103	C₂×C₅	#4028	1	1	yes	characteristic
104	C₂×C₅	#4034	1	2	yes	—
105	C₂×C₅	#4043	1	1	yes	characteristic
106	C₂³	C₂²×D₂₅	1	1	yes	characteristic
107	Q₈	—	50	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	25	4	no	—
112	C₂×C₄	—	25	2	no	—
113	C₂×C₄	—	50	1	no	—
114	C₂×C₄	—	50	2	no	—
115	C₂×C₄	—	50	2	no	—
116	C₂×C₄	—	50	1	no	—
117	C₂×C₄	—	50	1	no	—
118	C₂×C₄	—	50	2	no	—
119	C₂×C₄	Dic₅₀	1	4	yes	complemented
120	C₂×C₄	D₁₀₀	1	2	yes	—
121	C₅	#64945	1	1	yes	characteristic
122	C₂²	C₂×Dic₅₀	1	2	yes	—
123	C₂²	C₂×D₁₀₀	1	1	yes	characteristic
124	C₂²	D₄×D₂₅	1	2	yes	—
125	C₂²	#87746	1	1	yes	characteristic
126	C₂²	Q₈×D₂₅	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	2	no	—
129	C₄	—	50	4	no	—
130	C₄	—	50	2	no	—
131	C₄	—	100	1	no	—
132	C₂	#11716431	1	1	yes	characteristic
133	C₂	#11716440	1	2	yes	—
134	C₂	#11716444	1	1	yes	characteristic
135	C₂	#11716450	1	2	yes	—
136	C₂	#11716459	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₀

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₂₅

C₂

yes

maximal

C₂×#87718

C₂

yes

maximal, characteristic

C₂₅×#95

C₂

yes

maximal, characteristic

#11716379

C₂

yes

maximal

C₂²×C₄×C₂₅

C₂²

yes

characteristic

C₂²×C₄×C₂₅

C₂²

yes

—

C₂²×Dic₂₅

C₂²

yes

characteristic

C₂²×Dic₂₅

C₂²

yes

—

C₂²×Dic₂₅

C₂²

yes

characteristic

C₂×Dic₅₀

—

C₂×Dic₅₀

—

C₄⋊Dic₂₅

—

#87718

—

#64945

—

maximal

C₂³×C₂₅

C₂³

yes

characteristic

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

D₄

yes

—

C₂×C₄×C₂₅

Q₈

yes

complemented

C₂×Dic₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

—

C₂×Dic₂₅

D₄

yes

—

C₂×Dic₂₅

Q₈

yes

complemented

Dic₅₀

—

C₂²×Dic₁₀

—

C₂×C₄⋊Dic₅

—

C₂×#683

—

C₅×#95

D₅

yes

characteristic

#3963

—

C₂²×C₂₅

C₂×D₄

yes

characteristic

C₂²×C₂₅

C₂×D₄

yes

—

C₂²×C₂₅

C₂×Q₈

yes

—

C₂²×C₂₅

C₂×Q₈

yes

characteristic

C₂²×C₂₅

#41

yes

characteristic

C₄×C₂₅

—

C₄×C₂₅

—

Dic₂₅

—

Dic₂₅

—

Dic₂₅

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×C₄×C₅

C₂×D₅

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₄⋊Dic₅

—

—

—

C₂×C₂₅

yes

C₂×C₂₅

C₂²⋊Q₈

yes

—

C₂×C₂₅

C₂²⋊Q₈

yes

characteristic

C₂×C₂₅

C₄⋊Q₈

yes

characteristic

C₂×C₂₅

C₄⋊Q₈

yes

—

C₂³×C₅

C₂²×D₅

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₁₀

yes

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

#95

—

#95

D₂₅

yes

characteristic

C₂₅

#395

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×Dic₁₀

yes

—

C₂²×C₅

C₂×D₂₀

yes

characteristic

C₂²×C₅

D₄×D₅

yes

—

C₂²×C₅

#711

yes

characteristic

C₂²×C₅

Q₈×D₅

yes

characteristic

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₅

yes

—

C₂²×C₄

C₂×D₂₅

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂×C₅

#4015

yes

characteristic

102

C₂×C₅

#4024

yes

—

103

C₂×C₅

#4028

yes

characteristic

104

C₂×C₅

#4034

yes

—

105

C₂×C₅

#4043

yes

characteristic

106

C₂³

C₂²×D₂₅

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

Dic₅₀

yes

complemented

120

C₂×C₄

D₁₀₀

yes

—

121

C₅

#64945

yes

characteristic

122

C₂²

C₂×Dic₅₀

yes

—

123

C₂²

C₂×D₁₀₀

yes

characteristic

124

C₂²

D₄×D₂₅

yes

—

125

C₂²

#87746

yes

characteristic

126

C₂²

Q₈×D₂₅

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

100

—

132

C₂

#11716431

yes

characteristic

133

C₂

#11716440

yes

—

134

C₂

#11716444

yes

characteristic

135

C₂

#11716450

yes

—

136

C₂

#11716459

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₅
commutator, cosocle	C₂³×C₂₅	C₂³
Frattini, socle, prime radical	C₂³×C₅	C₂²×D₅
Fitting	C₂₅×#95	C₂
hypercenter	#95	D₂₅
hypocenter	C₂₅	#395
LCS	G ▷ C₂³×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center

C₂³

C₂²×D₂₅

commutator, cosocle

C₂³×C₂₅

C₂³

Frattini, socle, prime radical

C₂³×C₅

C₂²×D₅

Fitting

C₂₅×#95

C₂

D₂₅

C₂₅

G ▷ C₂³×C₂₅ ▷ C₂₅

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
21. C₂×C₄×C₂₅	Q₈	semidirect
29. C₂×Dic₂₅	Q₈	semidirect
81. C₂₅	#395	semidirect
119. C₂×C₄	Dic₅₀	semidirect

H ≅ G/N

type

21. C₂×C₄×C₂₅

Q₈

semidirect

29. C₂×Dic₂₅

Q₈

semidirect

81. C₂₅

#395

semidirect

119. C₂×C₄

Dic₅₀

semidirect

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

subgroup	index
C₂²×Dic₅₀	2
C₂×C₄⋊Dic₂₅	2
C₂×#87718	2
C₂₅×#95	2
#11716379	2
#64945	5

subgroup

index

C₂²×Dic₅₀

C₂×C₄⋊Dic₂₅

C₂×#87718

C₂₅×#95

#11716379

#64945

Subgroups — #13097931

Conjugacy classes of subgroups

Complements

Maximal subgroups