Group #13023067 — subgroups

Subgroups — #13023067

Order 1280.

246 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₂×#64674	C₂	1	1	yes	maximal, characteristic
3	#10598515	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×#168	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	C₄	1	2	yes	—
11	#64674	C₂²	1	2	yes	—
12	#11415	—	5	1	no	maximal, Sylow
13	C₂²×C₅×C₈	—	2	1	no	—
14	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×#168	—	2	1	no	—
16	C₂²×#168	C₂×C₄	1	2	yes	—
17	C₂×#674	C₂×C₄	1	8	yes	—
18	C₄×C₅×C₈	—	2	1	no	—
19	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
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	C₂×#322	—	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	C₂×C₅×C₈	—	4	2	no	—
33	C₄²×C₅	C₄²	1	1	yes	characteristic
34	C₄²×C₅	M₁₆	1	1	yes	characteristic
35	C₄²×C₅	M₁₆	1	1	yes	characteristic
36	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
37	C₂×#168	—	2	4	no	—
38	C₂×#168	—	2	2	no	—
39	C₂×#168	—	4	2	no	—
40	C₂×#168	C₄²	1	2	yes	—
41	C₂×#168	C₂²×C₄	1	2	yes	—
42	#674	—	2	8	no	—
43	C₂×C₄×C₈	—	5	1	no	—
44	C₂×C₄×C₈	—	5	1	no	—
45	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₄⋊C₁₆	—	5	4	no	—
47	C₂²⋊C₁₆	—	5	4	no	—
48	#346	—	5	4	no	—
49	#322	—	5	2	no	—
50	#322	—	5	2	no	—
51	C₂³×C₅	#97	1	1	yes	characteristic
52	C₂×C₄×C₅	—	2	2	no	—
53	C₂×C₄×C₅	—	2	2	no	—
54	C₂×C₄×C₅	—	2	2	no	—
55	C₂×C₄×C₅	#97	1	1	yes	characteristic
56	C₂×C₄×C₅	#97	1	1	yes	characteristic
57	C₂×C₄×C₅	#97	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
61	C₅×C₈	—	4	2	no	—
62	#168	—	2	2	no	—
63	#168	—	4	2	no	—
64	#168	#131	1	4	yes	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	5	2	no	—
67	C₂²×C₈	—	10	1	no	—
68	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂×C₁₆	—	5	8	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	5	2	no	—
72	C₄×C₈	—	5	1	no	—
73	C₄×C₈	—	5	1	no	—
74	C₄×C₈	—	5	2	no	—
75	C₄×C₈	—	10	1	no	—
76	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
77	C₂²×C₅	—	2	1	no	—
78	C₂²×C₅	—	2	1	no	—
79	C₂²×C₅	#347	1	2	yes	—
80	C₂²×C₅	C₂×#97	1	1	yes	characteristic
81	C₄×C₅	—	2	1	no	—
82	C₄×C₅	—	2	1	no	—
83	C₄×C₅	—	2	1	no	—
84	C₄×C₅	—	2	1	no	—
85	C₄×C₅	#347	1	2	yes	—
86	C₄×C₅	#405	1	2	yes	—
87	C₂²×C₄	#700	1	1	yes	characteristic
88	C₂²×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
89	C₂²×C₄	#704	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	5	2	no	—
93	C₂×C₈	—	5	2	no	—
94	C₂×C₈	—	10	4	no	—
95	C₂×C₈	—	10	2	no	—
96	C₂×C₈	—	20	2	no	—
97	C₄²	#700	1	1	yes	characteristic
98	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₄²	#704	1	1	yes	characteristic
100	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
101	C₁₆	—	10	8	no	—
102	C₂×C₅	—	2	2	no	—
103	C₂×C₅	#1452	1	1	yes	characteristic
104	C₂×C₅	C₂×#347	1	1	yes	characteristic
105	C₂×C₅	C₂×#347	1	1	yes	characteristic
106	C₂³	#4002	1	1	yes	characteristic
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	#4002	1	1	yes	characteristic
111	C₂×C₄	#4002	1	1	yes	characteristic
112	C₂×C₄	#4002	1	1	yes	characteristic
113	C₂×C₄	C₂×#700	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×C₄	C₂×#704	1	1	yes	characteristic
116	C₈	—	4	2	no	—
117	C₈	—	5	4	no	—
118	C₈	—	10	2	no	—
119	C₈	—	20	2	no	—
120	C₅	#11415	1	1	yes	characteristic, complemented, Sylow
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#64860	1	2	yes	—
124	C₂²	C₂×#4002	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	#64890	1	2	yes	—
130	C₄	#65718	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	C₂×#64860	1	1	yes	characteristic
133	C₂	#10604689	1	1	yes	characteristic
134	C₂	C₂×#64890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604152

C₂

yes

maximal

C₂×#64674

C₂

yes

maximal, characteristic

#10598515

C₂

yes

maximal, complemented

C₂×C₄×C₅×C₈

C₄

yes

characteristic

C₂×C₄×#168

C₄

yes

characteristic

C₂×C₄×#168

C₂²

yes

characteristic

C₂²⋊#674

C₄

yes

—

#64859

C₂²

yes

—

C₄⋊#674

C₄

yes

—

#64674

C₄

yes

—

#64674

C₂²

yes

—

#11415

—

maximal, Sylow

C₂²×C₅×C₈

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂²×#168

—

C₂²×#168

C₂×C₄

yes

—

C₂×#674

C₂×C₄

yes

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

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

—

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

—

#674

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

#322

—

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

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#347

yes

—

C₂²×C₅

C₂×#97

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#347

yes

—

C₄×C₅

#405

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

yes

C₄²

yes

C₄²

yes

100

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

101

C₁₆

—

102

C₂×C₅

—

103

C₂×C₅

#1452

yes

characteristic

104

C₂×C₅

C₂×#347

yes

characteristic

105

C₂×C₅

C₂×#347

yes

characteristic

106

C₂³

#4002

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#4002

yes

characteristic

111

C₂×C₄

#4002

yes

characteristic

112

C₂×C₄

#4002

yes

characteristic

113

C₂×C₄

C₂×#700

yes

characteristic

114

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

115

C₂×C₄

C₂×#704

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₅

#11415

yes

characteristic, complemented, Sylow

121

C₂²

—

122

C₂²

—

123

C₂²

#64860

yes

—

124

C₂²

C₂×#4002

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#64890

yes

—

130

C₄

#65718

yes

—

131

C₂

—

132

C₂

C₂×#64860

yes

characteristic

133

C₂

#10604689

yes

characteristic

134

C₂

C₂×#64890

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	C₂×C₄×C₅×C₈	C₄
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	#97
cosocle	C₄×#168	C₂³
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#11415
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₂×C₄×C₅×C₈

C₄

socle

C₂²×C₅

C₂×#97

C₂³×C₅

C₂³

C₂×C₄×C₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₅

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #10598515	C₂	semidirect
120. C₅	#11415	semidirect

H ≅ G/N

type

3. #10598515

C₂

semidirect

120. C₅

#11415

semidirect

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

subgroup	index
#10604152	2
C₂×#64674	2
#10598515	2
#11415	5

subgroup

index

Subgroups — #13023067

Conjugacy classes of subgroups

Complements

Maximal subgroups