Group #13023074 — subgroups

Subgroups — #13023074

Order 1280.

228 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 100 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	#10604152	C₂	1	2	yes	maximal
2	#10601533	C₂	1	1	yes	maximal, characteristic
3	#10598515	C₂	1	4	yes	maximal, complemented
4	C₅×#431	C₄	1	1	yes	characteristic
5	#65214	C₄	1	1	yes	characteristic
6	C₂×C₄×#168	C₂²	1	1	yes	characteristic
7	C₂²⋊#674	C₄	1	4	yes	—
8	#64859	C₂²	1	4	yes	—
9	C₄⋊#674	C₄	1	4	yes	—
10	#64674	C₂²	1	2	yes	—
11	C₈×#168	C₄	1	2	yes	—
12	#11416	—	5	1	no	maximal, Sylow
13	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂²×#168	C₂×C₄	1	2	yes	—
15	C₂×#674	C₂×C₄	1	8	yes	—
16	C₅×C₄⋊C₈	—	2	1	no	—
17	C₅×C₂²⋊C₈	—	2	1	no	—
18	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
19	C₂²⋊#168	—	2	1	no	—
20	C₄⋊#168	—	2	1	no	—
21	C₄×#168	C₂×C₄	1	2	yes	—
22	C₄×#168	C₂×C₄	1	1	yes	characteristic
23	C₄×#168	C₂×C₄	1	2	yes	—
24	C₄×#168	C₂³	1	1	yes	characteristic
25	#2163	—	5	2	no	—
26	#1450	—	5	1	no	—
27	#1313	—	5	4	no	—
28	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
31	C₂×C₅×C₈	—	2	2	no	—
32	C₄²×C₅	C₄²	1	1	yes	characteristic
33	C₄²×C₅	M₁₆	1	1	yes	characteristic
34	C₄²×C₅	M₁₆	1	1	yes	characteristic
35	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₂×#168	—	2	4	no	—
37	C₂×#168	—	2	2	no	—
38	C₂×#168	C₄²	1	2	yes	—
39	C₂×#168	C₂²×C₄	1	2	yes	—
40	#674	—	2	8	no	—
41	#431	—	5	1	no	—
42	#431	AGL(1,𝔽₅)	1	1	yes	characteristic
43	C₂×C₄×C₈	—	5	1	no	—
44	C₄⋊C₁₆	—	5	4	no	—
45	C₂²⋊C₁₆	—	5	4	no	—
46	#346	—	5	4	no	—
47	#322	—	5	2	no	—
48	C₈²	—	5	2	no	—
49	C₂³×C₅	#97	1	1	yes	characteristic
50	C₂×C₄×C₅	—	2	2	no	—
51	C₂×C₄×C₅	—	4	1	no	—
52	C₂×C₄×C₅	—	4	1	no	—
53	C₂×C₄×C₅	#97	1	1	yes	characteristic
54	C₂×C₄×C₅	#97	1	1	yes	characteristic
55	C₂×C₄×C₅	#97	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
59	C₅×C₈	—	4	2	no	—
60	#168	—	2	2	no	—
61	#168	—	4	2	no	—
62	#168	#131	1	4	yes	—
63	C₂²×C₈	—	5	2	no	—
64	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂×C₁₆	—	5	8	no	—
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	—	10	1	no	—
68	C₂²⋊C₈	—	2	1	no	—
69	C₂²⋊C₈	—	10	1	no	—
70	C₄×C₈	—	5	2	no	—
71	C₄×C₈	—	5	1	no	—
72	C₄×C₈	—	5	1	no	—
73	C₄×C₈	—	5	2	no	—
74	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
75	C₂²×C₅	—	2	1	no	—
76	C₂²×C₅	—	2	1	no	—
77	C₂²×C₅	—	2	1	no	—
78	C₂²×C₅	C₂×#97	1	1	yes	characteristic
79	C₄×C₅	—	2	1	no	—
80	C₄×C₅	—	2	1	no	—
81	C₄×C₅	—	2	1	no	—
82	C₄×C₅	—	2	1	no	—
83	C₄×C₅	—	2	1	no	—
84	C₄×C₅	#405	1	2	yes	—
85	C₂²×C₄	#700	1	1	yes	characteristic
86	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂²×C₄	#704	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	5	2	no	—
90	C₂×C₈	—	5	2	no	—
91	C₂×C₈	—	10	4	no	—
92	C₂×C₈	—	10	2	no	—
93	C₄²	#700	1	1	yes	characteristic
94	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₄²	#704	1	1	yes	characteristic
96	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₁₆	—	10	8	no	—
98	C₂×C₅	—	4	1	no	—
99	C₂×C₅	#1452	1	1	yes	characteristic
100	C₂×C₅	#2110	1	1	yes	characteristic
101	C₂×C₅	#2110	1	1	yes	characteristic
102	C₂³	#4002	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	#4002	1	1	yes	characteristic
107	C₂×C₄	#4002	1	1	yes	characteristic
108	C₂×C₄	#4002	1	1	yes	characteristic
109	C₂×C₄	C₂×#700	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₂×C₄	C₂×#704	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	5	4	no	—
114	C₈	—	10	2	no	—
115	C₈	—	20	2	no	—
116	C₅	#11416	1	1	yes	characteristic, complemented, Sylow
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×#4002	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	#65718	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	#10604369	1	1	yes	characteristic
129	C₂	#10604689	1	1	yes	characteristic
130	C₂	#10604748	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604152

C₂

yes

maximal

#10601533

C₂

yes

maximal, characteristic

#10598515

C₂

yes

maximal, complemented

C₅×#431

C₄

yes

characteristic

#65214

C₄

yes

characteristic

C₂×C₄×#168

C₂²

yes

characteristic

C₂²⋊#674

C₄

yes

—

#64859

C₂²

yes

—

C₄⋊#674

C₄

yes

—

#64674

C₂²

yes

—

C₈×#168

C₄

yes

—

#11416

—

maximal, Sylow

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂²×#168

C₂×C₄

yes

—

C₂×#674

C₂×C₄

yes

—

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₂²⋊#168

—

C₄⋊#168

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

characteristic

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂³

yes

—

—

—

C₂²×C₄×C₅

C₄²

yes

characteristic

C₂²×C₄×C₅

M₁₆

yes

characteristic

C₂²×C₄×C₅

M₁₆

yes

characteristic

C₂×C₅×C₈

—

C₄²×C₅

C₄²

yes

characteristic

C₄²×C₅

M₁₆

yes

characteristic

C₄²×C₅

M₁₆

yes

characteristic

C₄²×C₅

C₂²×C₄

yes

—

—

C₄²

yes

—

C₂×#168

C₂²×C₄

yes

—

—

—

yes

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₈²

—

C₂³×C₅

#97

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₅×C₈

—

—

—

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×#97

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#405

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₁₆

—

C₂×C₅

—

C₂×C₅

#1452

yes

characteristic

100

C₂×C₅

#2110

yes

characteristic

101

C₂×C₅

#2110

yes

characteristic

102

C₂³

#4002

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#4002

yes

characteristic

107

C₂×C₄

#4002

yes

characteristic

108

C₂×C₄

#4002

yes

characteristic

109

C₂×C₄

C₂×#700

yes

characteristic

110

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

111

C₂×C₄

C₂×#704

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₅

#11416

yes

characteristic, complemented, Sylow

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#4002

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#65718

yes

—

127

C₂

—

128

C₂

#10604369

yes

characteristic

129

C₂

#10604689

yes

characteristic

130

C₂

#10604748

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#431	C₄
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	#97
cosocle	C₄×#168	C₂³
hypercenter	#431	AGL(1,𝔽₅)
hypocenter	C₅	#11416
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#431

C₄

socle

C₂²×C₅

C₂×#97

C₂³×C₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₅

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
3. #10598515	C₂	semidirect
116. C₅	#11416	semidirect

H ≅ G/N

type

3. #10598515

C₂

semidirect

116. C₅

#11416

semidirect

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

subgroup	index
#10604152	2
#10601533	2
#10598515	2
#11416	5

subgroup

index

Subgroups — #13023074

Conjugacy classes of subgroups

Complements

Maximal subgroups