Group #72269200 — subgroups

Subgroups — #72269200

Order 6400.

256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 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	#72269154	C₂	1	1	yes	maximal, characteristic
2	#72269154	C₂	1	1	yes	maximal, characteristic
3	#72242123	C₂	1	1	yes	maximal, characteristic
4	#13100347	C₄	1	2	yes	—
5	#13100347	C₄	1	2	yes	—
6	C₅²×#346	C₄	1	1	yes	characteristic
7	#13099493	C₂²	1	1	yes	characteristic
8	#13099489	C₄	1	1	yes	characteristic
9	#13014080	—	5	2	no	maximal
10	C₂×C₅²×C₁₆	—	2	1	no	—
11	C₄×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₂×#87756	C₂×C₄	1	1	yes	characteristic
13	C₂×#87756	C₂×C₄	1	1	yes	characteristic
14	C₄×#6029	—	2	1	no	—
15	#11716581	—	2	2	no	—
16	#11716581	C₈	1	4	yes	complemented
17	#10598537	—	5	2	no	—
18	#10598537	—	5	2	no	—
19	#10598003	—	5	2	no	—
20	#10598003	—	10	2	no	—
21	C₂×C₅²×C₈	C₄²	1	1	yes	characteristic
22	C₂×C₅²×C₈	M₁₆	1	1	yes	characteristic
23	C₄²×C₅²	M₁₆	1	1	yes	characteristic
24	C₂×#6029	—	4	1	no	—
25	C₅²×C₁₆	—	4	1	no	—
26	#87756	C₂×C₈	1	2	yes	—
27	#87756	M₁₆	1	2	yes	—
28	#64877	—	5	4	no	—
29	#64877	—	5	4	no	—
30	C₅×#346	—	2	2	no	—
31	C₅×#346	AGL(1,𝔽₅)	1	2	yes	—
32	#64681	—	5	2	no	—
33	#64681	—	10	2	no	—
34	#64675	—	5	2	no	—
35	#64675	—	10	2	no	—
36	#7333	—	25	1	no	Sylow
37	C₂×C₄×C₅²	—	2	1	no	—
38	C₂×C₄×C₅²	#97	1	1	yes	characteristic
39	C₅²×C₈	—	2	1	no	—
40	C₅²×C₈	C₄×C₈	1	1	yes	characteristic
41	C₅²×C₈	#97	1	1	yes	characteristic
42	#6029	—	8	1	no	—
43	C₂×C₅×C₁₆	—	2	4	no	—
44	C₂×C₅×C₁₆	—	2	2	no	—
45	C₄×C₅×C₈	—	2	2	no	—
46	C₄×C₅×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
47	C₂×#672	—	5	2	no	—
48	C₂×#672	—	5	2	no	—
49	C₂×#672	—	10	2	no	—
50	C₂×#672	—	10	2	no	—
51	C₄×#168	—	10	4	no	—
52	C₄×#168	—	10	2	no	—
53	#3928	—	5	8	no	—
54	#3928	—	10	4	no	—
55	#1398	—	25	1	no	—
56	#1398	—	25	1	no	—
57	#1313	—	25	1	no	—
58	C₂²×C₅²	#347	1	1	yes	characteristic
59	C₄×C₅²	—	4	1	no	—
60	C₄×C₅²	#322	1	1	yes	characteristic
61	C₄×C₅²	#347	1	1	yes	characteristic
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	—	2	2	no	—
64	C₂×C₅×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
65	C₂×C₅×C₈	#704	1	2	yes	—
66	C₄²×C₅	—	2	2	no	—
67	C₄²×C₅	#700	1	2	yes	—
68	C₂×#168	—	20	4	no	—
69	C₂×#168	—	20	2	no	—
70	C₅×C₁₆	—	4	4	no	—
71	C₅×C₁₆	—	4	2	no	—
72	#672	—	5	4	no	—
73	#672	—	5	4	no	—
74	#672	—	10	4	no	—
75	#672	—	10	4	no	—
76	M₆₄	—	25	2	no	—
77	M₆₄	—	25	2	no	—
78	#346	—	25	1	no	—
79	#346	#1044	1	1	yes	characteristic
80	#322	—	25	1	no	—
81	C₂×C₅²	—	2	1	no	—
82	C₂×C₅²	#1313	1	1	yes	characteristic
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	#4002	1	2	yes	—
87	C₅×C₈	—	2	4	no	—
88	C₅×C₈	—	2	2	no	—
89	C₅×C₈	—	2	2	no	—
90	C₅×C₈	—	2	2	no	—
91	C₅×C₈	C₈×AGL(1,𝔽₅)	1	2	yes	—
92	C₅×C₈	#3992	1	2	yes	—
93	#168	—	40	4	no	—
94	#168	—	40	2	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	25	1	no	—
97	C₂×C₁₆	—	25	1	no	—
98	C₄×C₈	—	50	1	no	—
99	C₄×C₈	C₂×#1044	1	1	yes	characteristic
100	C₃₂	—	25	4	no	—
101	C₃₂	—	50	2	no	—
102	C₅²	#7333	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₅	—	2	2	no	—
104	C₂²×C₅	#64890	1	2	yes	—
105	C₄×C₅	—	2	2	no	—
106	C₄×C₅	—	2	2	no	—
107	C₄×C₅	—	4	4	no	—
108	C₄×C₅	—	4	2	no	—
109	C₄×C₅	#64860	1	2	yes	—
110	C₄×C₅	#170⋊C₈	1	2	yes	—
111	C₂×C₈	—	100	1	no	—
112	C₂×C₈	C₄×#1044	1	1	yes	characteristic
113	C₂×C₈	#87867	1	1	yes	characteristic
114	C₄²	#87863	1	1	yes	characteristic
115	C₁₆	—	4	1	no	—
116	C₁₆	—	25	2	no	—
117	C₁₆	—	25	2	no	—
118	C₂×C₅	—	2	4	no	—
119	C₂×C₅	—	2	2	no	—
120	C₂×C₅	—	2	2	no	—
121	C₂×C₅	#10598515	1	2	yes	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	#11716895	1	1	yes	characteristic
124	C₈	—	2	1	no	—
125	C₈	—	200	1	no	—
126	C₈	C₈×#1044	1	1	yes	characteristic
127	C₈	#11716885	1	1	yes	characteristic
128	C₅	—	2	2	no	—
129	C₅	#13014080	1	2	yes	complemented
130	C₂²	#13100359	1	1	yes	characteristic
131	C₄	—	4	1	no	—
132	C₄	#13100330	1	1	yes	characteristic
133	C₄	#13100349	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	#72242185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#72269154

C₂

yes

maximal, characteristic

#72269154

C₂

yes

maximal, characteristic

#72242123

C₂

yes

maximal, characteristic

#13100347

C₄

yes

—

#13100347

C₄

yes

—

C₅²×#346

C₄

yes

characteristic

#13099493

C₂²

yes

characteristic

#13099489

C₄

yes

characteristic

#13014080

—

maximal

C₂×C₅²×C₁₆

—

C₄×C₅²×C₈

C₂×C₄

yes

characteristic

C₂×#87756

C₂×C₄

yes

characteristic

C₂×#87756

C₂×C₄

yes

—

—

C₈

yes

—

—

—

—

C₂×C₅²×C₈

C₄²

yes

characteristic

C₂×C₅²×C₈

M₁₆

yes

characteristic

C₄²×C₅²

M₁₆

yes

characteristic

C₂×#6029

—

C₅²×C₁₆

—

#87756

C₂×C₈

yes

—

#87756

M₁₆

yes

—

—

—

—

yes

—

—

—

—

—

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

#97

yes

characteristic

C₅²×C₈

—

C₅²×C₈

C₄×C₈

yes

C₅²×C₈

yes

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×AGL(1,𝔽₅)

yes

—

—

—

—

—

—

—

—

—

—

—

—

C₂²×C₅²

#347

yes

characteristic

C₄×C₅²

—

C₄×C₅²

yes

C₄×C₅²

yes

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄×AGL(1,𝔽₅)

yes

—

C₂×C₅×C₈

#704

yes

—

C₄²×C₅

—

C₄²×C₅

#700

yes

—

C₂×#168

—

C₂×#168

—

C₅×C₁₆

—

C₅×C₁₆

—

—

—

—

—

M₆₄

—

M₆₄

—

—

yes

—

C₂×C₅²

—

C₂×C₅²

#1313

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#4002

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₈×AGL(1,𝔽₅)

yes

—

C₅×C₈

#3992

yes

—

#168

—

#168

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

C₂×#1044

yes

characteristic

100

C₃₂

—

101

C₃₂

—

102

C₅²

#7333

yes

characteristic, complemented, Sylow

103

C₂²×C₅

—

104

C₂²×C₅

#64890

yes

—

105

C₄×C₅

—

106

C₄×C₅

—

107

C₄×C₅

—

108

C₄×C₅

—

109

C₄×C₅

#64860

yes

—

110

C₄×C₅

#170⋊C₈

yes

—

111

C₂×C₈

—

100

—

112

C₂×C₈

C₄×#1044

yes

characteristic

113

C₂×C₈

#87867

yes

characteristic

114

C₄²

#87863

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₂×C₅

—

119

C₂×C₅

—

120

C₂×C₅

—

121

C₂×C₅

#10598515

yes

—

122

C₂×C₄

—

123

C₂×C₄

#11716895

yes

characteristic

124

C₈

—

125

C₈

—

200

—

126

C₈

C₈×#1044

yes

characteristic

127

C₈

#11716885

yes

characteristic

128

C₅

—

129

C₅

#13014080

yes

complemented

130

C₂²

#13100359

yes

characteristic

131

C₄

—

132

C₄

#13100330

yes

characteristic

133

C₄

#13100349

yes

characteristic

134

C₂

—

135

C₂

#72242185

yes

characteristic

	subgroup	G/H
center	C₄	#13100349
commutator	C₅²×C₈	C₄×C₈
Frattini	C₄×C₈	C₂×#1044
Fitting	C₅²×#346	C₄
socle	C₂×C₅²	#1313
prime radical	C₂²×C₅²	#347
cosocle	#13099493	C₂²
hypercenter	#346	#1044
hypocenter	C₅²	#7333
LCS	G ▷ C₅²×C₈ ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₄

#13100349

commutator

C₅²×C₈

C₄×C₈

Frattini

C₄×C₈

C₂×#1044

Fitting

C₅²×#346

C₄

C₂×C₅²

C₂²×C₅²

C₂²

C₅²

G ▷ C₅²×C₈ ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₄ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
16. #11716581	C₈	semidirect
102. C₅²	#7333	semidirect
129. C₅	#13014080	semidirect

H ≅ G/N

type

16. #11716581

C₈

semidirect

102. C₅²

#7333

semidirect

129. C₅

#13014080

semidirect

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

subgroup	index
#72269154	2
#72242123	2
#13014080	5

subgroup

index

#72269154

#72242123

#13014080

Subgroups — #72269200

Conjugacy classes of subgroups

Complements

Maximal subgroups