Group #13025365 — subgroups

Subgroups — #13025365

Order 1280.

206 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 66 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	#10604068	C₂	1	2	yes	maximal
2	#10600111	C₂	1	1	yes	maximal, characteristic
3	#10598646	C₂	1	1	yes	maximal, characteristic
4	#10598636	C₂	1	1	yes	maximal, characteristic
5	#10598610	C₂	1	2	yes	maximal
6	#65709	—	2	2	no	—
7	C₂×Dic₄₀	C₄	1	1	yes	characteristic
8	#65162	C₂²	1	2	yes	—
9	#64885	C₂²	1	1	yes	characteristic
10	#64882	C₂²	1	2	yes	—
11	C₈×#170	C₂²	1	1	yes	characteristic
12	#64872	—	2	2	no	—
13	C₅×#366	C₄	1	1	yes	characteristic
14	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
15	#64712	C₄	1	2	yes	—
16	#12085	—	5	1	no	maximal, Sylow
17	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
18	#4033	—	2	2	no	—
19	#4002	—	4	2	no	—
20	C₄⋊#170	—	2	2	no	—
21	C₄⋊#170	—	4	2	no	—
22	C₄×#170	D₄	1	1	yes	characteristic
23	C₄×#170	D₄	1	1	yes	characteristic
24	C₂×#674	—	2	1	no	—
25	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₅×#107	C₂×C₄	1	2	yes	—
27	C₈×Dic₅	C₂³	1	1	yes	characteristic
28	C₂×#672	C₂×C₄	1	1	yes	characteristic
29	#3942	—	2	2	no	—
30	Dic₄₀	—	2	2	no	—
31	C₄×Q₃₂	—	5	1	no	—
32	#1591	—	5	2	no	—
33	#1372	—	5	1	no	—
34	#1332	—	5	2	no	—
35	#1311	—	5	1	no	—
36	C₂×Dic₁₀	—	2	2	no	—
37	C₂×#170	—	2	1	no	—
38	C₂×#170	—	4	2	no	—
39	C₂×#170	D₈	1	2	yes	—
40	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₅×C₄⋊C₄	—	2	2	no	—
42	#683	—	4	2	no	—
43	C₄×Dic₅	—	4	2	no	—
44	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
45	C₂×#168	#41	1	1	yes	characteristic
46	Dic₂₀	—	2	2	no	—
47	Dic₂₀	—	2	2	no	—
48	#674	—	2	2	no	—
49	C₅×C₁₆	M₁₆	1	2	yes	—
50	#672	—	2	1	no	—
51	C₂×Q₃₂	—	5	1	no	—
52	#446	—	10	2	no	—
53	C₄×Q₁₆	—	5	2	no	—
54	#366	AGL(1,𝔽₅)	1	1	yes	characteristic
55	#358	—	5	2	no	—
56	C₄×C₁₆	—	5	1	no	—
57	C₄×C₁₆	—	5	1	no	—
58	#335	—	5	2	no	—
59	Q₈⋊C₈	—	10	2	no	—
60	C₈²	—	5	1	no	—
61	C₂×C₄×C₅	—	4	2	no	—
62	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
63	C₂×Dic₅	—	4	2	no	—
64	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
65	C₂×Dic₅	#135	1	1	yes	characteristic
66	Dic₁₀	—	4	2	no	—
67	Dic₁₀	—	4	2	no	—
68	#170	—	2	2	no	—
69	#170	—	4	1	no	—
70	#170	—	8	2	no	—
71	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
72	C₅×C₈	#131	1	1	yes	characteristic
73	#168	—	2	1	no	—
74	C₂×Q₁₆	—	5	2	no	—
75	C₄×Q₈	—	10	2	no	—
76	Q₃₂	—	10	2	no	—
77	C₂×C₁₆	—	5	1	no	—
78	C₂×C₁₆	—	10	1	no	—
79	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
80	#107	C₂×AGL(1,𝔽₅)	1	2	yes	—
81	C₄⋊C₈	—	10	2	no	—
82	C₄⋊C₈	—	20	2	no	—
83	#103	—	10	2	no	—
84	#97	—	20	2	no	—
85	C₄×C₈	—	5	1	no	—
86	C₄×C₈	—	5	1	no	—
87	C₄×C₈	—	5	1	no	—
88	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
89	C₄×C₅	—	8	2	no	—
90	C₄×C₅	#436	1	1	yes	characteristic
91	C₄×C₅	#443	1	1	yes	characteristic
92	Dic₅	—	2	1	no	—
93	Dic₅	—	8	2	no	—
94	Dic₅	#510	1	2	yes	—
95	C₂×Q₈	—	10	2	no	—
96	Q₁₆	—	10	2	no	—
97	Q₁₆	—	10	2	no	—
98	C₂×C₈	—	5	2	no	—
99	C₂×C₈	—	5	1	no	—
100	C₂×C₈	—	10	1	no	—
101	C₂×C₈	—	20	2	no	—
102	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	20	2	no	—
105	C₄²	—	5	1	no	—
106	C₄²	—	20	2	no	—
107	C₁₆	—	10	2	no	—
108	C₁₆	—	10	1	no	—
109	C₁₆	#700	1	2	yes	—
110	C₂×C₅	#1589	1	1	yes	characteristic
111	C₂×C₅	#2176	1	1	yes	characteristic
112	C₂×C₅	#2179	1	1	yes	characteristic
113	Q₈	—	20	2	no	—
114	Q₈	—	20	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	5	1	no	—
117	C₂×C₄	—	5	1	no	—
118	C₂×C₄	—	20	2	no	—
119	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₈	—	10	2	no	—
121	C₈	—	10	1	no	—
122	C₈	—	20	1	no	—
123	C₈	—	40	2	no	—
124	C₈	C₂×#700	1	1	yes	characteristic
125	C₈	#4133	1	1	yes	characteristic
126	C₅	#12085	1	1	yes	characteristic, complemented, Sylow
127	C₂²	D₈×AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₄	—	5	2	no	—
129	C₄	—	8	2	no	—
130	C₄	—	10	1	no	—
131	C₄	—	40	2	no	—
132	C₄	#65705	1	1	yes	characteristic
133	C₄	#65740	1	1	yes	characteristic
134	C₂	#10604067	1	1	yes	characteristic
135	C₂	#10604123	1	1	yes	characteristic
136	C₂	#10604135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604068

C₂

yes

maximal

#10600111

C₂

yes

maximal, characteristic

#10598646

C₂

yes

maximal, characteristic

#10598636

C₂

yes

maximal, characteristic

#10598610

C₂

yes

maximal

#65709

—

C₂×Dic₄₀

C₄

yes

characteristic

#65162

C₂²

yes

—

#64885

C₂²

yes

characteristic

#64882

C₂²

yes

—

C₈×#170

C₂²

yes

characteristic

#64872

—

C₅×#366

C₄

yes

characteristic

C₁₆×Dic₅

C₂²

yes

characteristic

#64712

C₄

yes

—

#12085

—

maximal, Sylow

C₂×Dic₂₀

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#170

D₄

yes

characteristic

C₂×#674

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₅×#107

C₂×C₄

yes

—

C₈×Dic₅

C₂³

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

#3942

—

Dic₄₀

—

C₄×Q₃₂

—

—

—

—

—

C₂×Dic₁₀

—

C₂×#170

—

C₂×#170

—

C₂×#170

D₈

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₅×C₄⋊C₄

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

C₂×D₄

yes

yes

Dic₂₀

—

Dic₂₀

—

#674

—

C₅×C₁₆

M₁₆

yes

—

#672

—

C₂×Q₃₂

—

#446

—

C₄×Q₁₆

—

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂×D₈

yes

characteristic

C₂×Dic₅

#135

yes

characteristic

Dic₁₀

—

Dic₁₀

—

#170

—

#170

—

#170

—

C₅×C₈

C₂×M₁₆

yes

C₅×C₈

yes

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₄×D₈

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

—

Dic₅

#510

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

#700

yes

—

110

C₂×C₅

#1589

yes

characteristic

111

C₂×C₅

#2176

yes

characteristic

112

C₂×C₅

#2179

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×#700

yes

characteristic

125

C₈

#4133

yes

characteristic

126

C₅

#12085

yes

characteristic, complemented, Sylow

127

C₂²

D₈×AGL(1,𝔽₅)

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#65705

yes

characteristic

133

C₄

#65740

yes

characteristic

134

C₂

#10604067

yes

characteristic

135

C₂

#10604123

yes

characteristic

136

C₂

#10604135

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×AGL(1,𝔽₅)
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#366	C₄
socle, prime radical	C₂²×C₅	C₄×D₈
cosocle	C₈×Dic₅	C₂³
hypercenter	#366	AGL(1,𝔽₅)
hypocenter	C₅	#12085
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×AGL(1,𝔽₅)

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#366

C₄

socle, prime radical

C₂²×C₅

C₄×D₈

cosocle

C₈×Dic₅

C₂³

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
126. C₅	#12085	semidirect

H ≅ G/N

type

126. C₅

#12085

semidirect

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

subgroup	index
#10604068	2
#10600111	2
#10598646	2
#10598636	2
#10598610	2
#12085	5

subgroup

index

Subgroups — #13025365

Conjugacy classes of subgroups

Complements

Maximal subgroups