Group #13025137 — subgroups

Subgroups — #13025137

Order 1280.

180 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 68 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	#10603835	C₂	1	2	yes	maximal
2	#10599865	C₂	1	1	yes	maximal, characteristic
3	#10598651	C₂	1	2	yes	maximal, complemented
4	#10598649	C₂	1	2	yes	maximal, complemented
5	C₂×#3930	C₄	1	1	yes	characteristic
6	C₂×D₅×C₁₆	C₄	1	1	yes	characteristic
7	#64978	C₂²	1	1	yes	characteristic
8	#64891	—	2	2	no	—
9	#64889	C₂²	1	4	yes	—
10	#64860	—	2	2	no	—
11	C₅×#346	C₄	1	1	yes	characteristic
12	#64722	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
14	C₄×#672	C₄	1	1	yes	characteristic
15	#12027	—	5	1	no	maximal, Sylow
16	C₂×#676	D₄	1	1	yes	characteristic
17	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
18	#4018	Q₈	1	1	yes	characteristic
19	#3998	—	2	4	no	—
20	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	#3947	Q₈	1	1	yes	characteristic
24	C₈×Dic₅	C₂³	1	1	yes	characteristic
25	C₂×#672	C₂×C₄	1	1	yes	characteristic
26	C₂×#672	C₂×C₄	1	1	yes	characteristic
27	#3935	D₄	1	1	yes	characteristic
28	#3930	—	2	2	no	—
29	D₅×C₁₆	—	2	2	no	—
30	#2166	—	5	2	no	—
31	#2108	—	5	1	no	—
32	#1373	—	5	2	no	—
33	#1369	—	5	2	no	—
34	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
38	#685	—	2	1	no	—
39	#683	—	2	1	no	—
40	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
41	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
42	C₂×#168	C₂×D₄	1	1	yes	characteristic
43	#676	—	2	2	no	—
44	C₈×D₅	—	2	1	no	—
45	C₈×D₅	—	2	1	no	—
46	#674	—	4	4	no	—
47	C₅×C₁₆	—	2	1	no	—
48	C₅×C₁₆	—	2	1	no	—
49	#672	M₁₆	1	2	yes	—
50	#672	M₁₆	1	2	yes	—
51	C₂×M₃₂	—	5	1	no	—
52	C₂²×C₁₆	—	5	1	no	—
53	#405	—	5	1	no	—
54	#364	—	5	4	no	—
55	#349	—	10	2	no	—
56	#347	—	10	2	no	—
57	#346	—	5	1	no	—
58	#346	AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₄×C₁₆	—	5	1	no	—
60	C₄×C₁₆	—	5	1	no	—
61	C₂²×D₅	#108	1	1	yes	characteristic
62	C₂×C₄×C₅	—	2	1	no	—
63	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×Dic₅	—	2	1	no	—
65	C₂×Dic₅	#108	1	1	yes	characteristic
66	C₄×D₅	—	2	1	no	—
67	C₄×D₅	—	2	1	no	—
68	C₅×C₈	—	2	1	no	—
69	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
70	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
71	#168	—	2	1	no	—
72	#168	#135	1	2	yes	—
73	C₂×M₁₆	—	5	1	no	—
74	C₂²×C₈	—	5	1	no	—
75	#117	—	5	1	no	—
76	M₃₂	—	10	2	no	—
77	M₃₂	—	10	4	no	—
78	C₂×C₁₆	—	5	1	no	—
79	C₂×C₁₆	—	5	1	no	—
80	C₂×C₁₆	—	10	2	no	—
81	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
83	#97	—	5	1	no	—
84	#97	—	5	1	no	—
85	C₄×C₈	—	5	1	no	—
86	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂²×C₅	C₂×#108	1	1	yes	characteristic
88	C₂×D₅	—	2	1	no	—
89	C₂×D₅	—	4	1	no	—
90	C₄×C₅	—	4	1	no	—
91	C₄×C₅	#423	1	1	yes	characteristic
92	C₄×C₅	#427	1	1	yes	characteristic
93	Dic₅	—	2	1	no	—
94	Dic₅	—	4	1	no	—
95	C₂²×C₄	—	5	1	no	—
96	M₁₆	—	10	2	no	—
97	C₂×C₈	—	5	1	no	—
98	C₂×C₈	—	5	1	no	—
99	C₂×C₈	—	10	1	no	—
100	C₂×C₈	—	10	1	no	—
101	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₄⋊C₄	—	10	1	no	—
104	C₂²⋊C₄	—	10	1	no	—
105	C₄²	—	5	1	no	—
106	C₄²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	—	2	1	no	—
109	C₁₆	—	5	2	no	—
110	C₁₆	—	5	2	no	—
111	C₁₆	—	20	4	no	—
112	C₂×C₅	—	2	1	no	—
113	C₂×C₅	#1568	1	1	yes	characteristic
114	D₅	—	4	1	no	—
115	C₂³	—	5	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	5	1	no	—
118	C₂×C₄	—	10	1	no	—
119	C₂×C₄	—	10	1	no	—
120	C₂×C₄	—	10	1	no	—
121	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	5	2	no	—
124	C₈	—	10	1	no	—
125	C₈	D₅⋊M₁₆	1	1	yes	characteristic
126	C₈	D₅⋊M₁₆	1	1	yes	characteristic
127	C₅	#12027	1	1	yes	characteristic, complemented, Sylow
128	C₂²	—	10	1	no	—
129	C₂²	—	20	1	no	—
130	C₂²	D₅⋊#108	1	1	yes	characteristic
131	C₄	—	4	1	no	—
132	C₄	—	10	1	no	—
133	C₄	—	20	1	no	—
134	C₄	#65682	1	1	yes	characteristic
135	C₄	#65721	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	20	1	no	—
138	C₂	#10603821	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10603835

C₂

yes

maximal

#10599865

C₂

yes

maximal, characteristic

#10598651

C₂

yes

maximal, complemented

#10598649

C₂

yes

maximal, complemented

C₂×#3930

C₄

yes

characteristic

C₂×D₅×C₁₆

C₄

yes

characteristic

#64978

C₂²

yes

characteristic

#64891

—

#64889

C₂²

yes

—

#64860

—

C₅×#346

C₄

yes

characteristic

#64722

C₂²

yes

characteristic

C₁₆×Dic₅

C₂²

yes

characteristic

C₄×#672

C₄

yes

—

D₄

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

characteristic

#4018

Q₈

yes

characteristic

#3998

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3947

Q₈

yes

characteristic

C₈×Dic₅

C₂³

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

#3935

D₄

yes

characteristic

#3930

—

D₅×C₁₆

—

—

—

—

—

C₂×C₄×D₅

C₄⋊C₄

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₄²×C₅

C₄⋊C₄

yes

characteristic

#685

—

#683

—

C₄×Dic₅

C₂×Q₈

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

C₂×#168

C₂×D₄

yes

characteristic

#676

—

C₈×D₅

—

C₈×D₅

—

#674

—

C₅×C₁₆

—

C₅×C₁₆

—

#672

M₁₆

yes

—

#672

M₁₆

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

—

—

—

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×D₅

#108

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

#108

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

C₂×M₁₆

yes

characteristic

C₅×C₈

C₂×M₁₆

yes

—

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

yes

C₂²×C₅

yes

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

102

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₄⋊AGL(1,𝔽₅)

yes

characteristic

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₂×C₅

—

113

C₂×C₅

#1568

yes

characteristic

114

D₅

—

115

C₂³

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

D₅⋊M₁₆

yes

characteristic

126

C₈

D₅⋊M₁₆

yes

characteristic

127

C₅

#12027

yes

characteristic, complemented, Sylow

128

C₂²

—

129

C₂²

—

130

C₂²

D₅⋊#108

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#65682

yes

characteristic

135

C₄

#65721

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#10603821

yes

characteristic

	subgroup	G/H
center	C₄	#65721
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#346	C₄
socle	C₂×C₅	#1568
prime radical	C₂²×D₅	#108
cosocle	C₈×Dic₅	C₂³
hypercenter	#346	AGL(1,𝔽₅)
hypocenter	C₅	#12027
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₄

#65721

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

C₄

C₂×C₅

C₂²×D₅

C₈×Dic₅

C₂³

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
3. #10598651	C₂	semidirect
4. #10598649	C₂	semidirect
127. C₅	#12027	semidirect

H ≅ G/N

type

3. #10598651

C₂

semidirect

4. #10598649

C₂

semidirect

127. C₅

#12027

semidirect

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

subgroup	index
#10603835	2
#10599865	2
#10598651	2
#10598649	2
#12027	5

subgroup

index

Subgroups — #13025137

Conjugacy classes of subgroups

Complements

Maximal subgroups