Group #13025111 — subgroups

Subgroups — #13025111

Order 1280.

232 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 96 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	#10603816	C₂	1	2	yes	maximal
2	C₄×#3930	C₂	1	1	yes	maximal, characteristic
3	#10598641	C₂	1	4	yes	maximal, complemented
4	#65681	—	2	2	no	—
5	C₂×#3930	C₄	1	2	yes	—
6	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
7	#64881	—	2	2	no	—
8	#64881	C₂²	1	4	yes	—
9	C₄×C₅×C₁₆	C₄	1	1	yes	characteristic
10	#64722	C₂²	1	2	yes	—
11	C₄×#672	C₄	1	1	yes	characteristic
12	#12020	—	5	1	no	maximal, Sylow
13	C₂×#700	—	4	2	no	—
14	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
16	C₄²×D₅	D₄	1	1	yes	characteristic
17	C₄⋊#170	—	2	4	no	—
18	C₄⋊#170	—	4	2	no	—
19	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
20	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₈×Dic₅	D₄	1	1	yes	characteristic
22	C₈×Dic₅	C₂³	1	1	yes	characteristic
23	C₂×#672	C₂×C₄	1	2	yes	—
24	C₄×#168	Q₈	1	1	yes	characteristic
25	#3930	—	2	4	no	—
26	C₄×M₃₂	—	5	1	no	—
27	#1566	—	5	2	no	—
28	#1368	—	5	4	no	—
29	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×D₅	SD₁₆	1	2	yes	—
31	C₂×#170	—	4	4	no	—
32	#700	—	8	4	no	—
33	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
36	C₄×Dic₅	SD₁₆	1	2	yes	—
37	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
38	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#168	C₂×Q₈	1	1	yes	characteristic
40	C₈×D₅	—	2	2	no	—
41	C₈×D₅	—	2	2	no	—
42	C₅×C₁₆	M₁₆	1	4	yes	complemented
43	#672	—	2	2	no	—
44	C₂×M₃₂	—	5	2	no	—
45	#423	—	10	2	no	—
46	C₂×C₄×C₈	—	5	1	no	—
47	#346	—	5	2	no	—
48	C₄×C₁₆	—	5	1	no	—
49	C₄×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
50	#334	—	5	4	no	—
51	#334	—	10	2	no	—
52	C₂²×D₅	#106	1	1	yes	characteristic
53	C₂×C₄×C₅	#106	1	2	yes	—
54	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₅	#106	1	1	yes	characteristic
56	C₂×Dic₅	C₂×SD₁₆	1	2	yes	—
57	C₄×D₅	—	2	4	no	—
58	C₄×D₅	—	2	1	no	—
59	C₄×D₅	—	2	1	no	—
60	#170	—	8	4	no	—
61	C₅×C₈	#108	1	2	yes	—
62	C₅×C₈	C₂×M₁₆	1	2	yes	—
63	#168	—	2	1	no	—
64	#168	—	2	1	no	—
65	C₂×M₁₆	—	20	2	no	—
66	C₂²×C₈	—	5	1	no	—
67	C₂²×C₈	—	5	1	no	—
68	C₂×C₄²	—	5	1	no	—
69	M₃₂	—	10	4	no	—
70	C₂×C₁₆	—	5	2	no	—
71	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
72	C₄⋊C₈	—	10	4	no	—
73	C₄⋊C₈	—	20	2	no	—
74	C₄×C₈	—	5	1	no	—
75	C₄×C₈	—	5	1	no	—
76	C₄×C₈	—	5	1	no	—
77	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₂²×C₅	C₂×#106	1	1	yes	characteristic
79	C₂×D₅	—	2	1	no	—
80	C₂×D₅	—	2	2	no	—
81	C₄×C₅	#365	1	4	yes	—
82	C₄×C₅	#423	1	1	yes	characteristic
83	C₄×C₅	C₂×#108	1	1	yes	characteristic
84	Dic₅	—	2	1	no	—
85	Dic₅	—	2	2	no	—
86	C₂²×C₄	—	5	1	no	—
87	C₂²×C₄	—	5	2	no	—
88	M₁₆	—	40	4	no	—
89	C₂×C₈	—	5	1	no	—
90	C₂×C₈	—	5	1	no	—
91	C₂×C₈	—	10	2	no	—
92	C₂×C₈	—	10	2	no	—
93	C₂×C₈	—	20	4	no	—
94	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
96	C₄²	—	5	1	no	—
97	C₄²	—	5	2	no	—
98	C₄²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₁₆	—	10	2	no	—
100	C₁₆	#700	1	4	yes	complemented
101	C₂×C₅	#1566	1	1	yes	characteristic
102	C₂×C₅	C₂×#365	1	2	yes	—
103	D₅	—	4	1	no	—
104	C₂³	—	5	1	no	—
105	C₂×C₄	—	5	2	no	—
106	C₂×C₄	—	5	1	no	—
107	C₂×C₄	—	10	4	no	—
108	C₂×C₄	—	10	1	no	—
109	C₂×C₄	—	10	1	no	—
110	C₂×C₄	#3993	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
112	C₈	—	10	1	no	—
113	C₈	—	10	1	no	—
114	C₈	—	40	4	no	—
115	C₈	#3995	1	2	yes	—
116	C₈	C₂×#700	1	2	yes	—
117	C₅	#12020	1	1	yes	characteristic, complemented, Sylow
118	C₂²	—	10	1	no	—
119	C₂²	—	10	2	no	—
120	C₂²	C₂×#3993	1	1	yes	characteristic
121	C₄	—	10	1	no	—
122	C₄	—	10	2	no	—
123	C₄	#64847	1	4	yes	—
124	C₄	#65681	1	1	yes	characteristic
125	C₄	C₂×#3995	1	1	yes	characteristic
126	C₂	—	20	1	no	—
127	C₂	#10603816	1	1	yes	characteristic
128	C₂	C₂×#64847	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10603816

C₂

yes

maximal

C₄×#3930

C₂

yes

maximal, characteristic

#10598641

C₂

yes

maximal, complemented

#65681

—

C₂×#3930

C₄

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64881

—

#64881

C₂²

yes

—

C₄×C₅×C₁₆

C₄

yes

characteristic

#64722

C₂²

yes

—

C₄×#672

C₄

yes

—

—

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₂×C₈×D₅

Q₈

yes

characteristic

C₄²×D₅

D₄

yes

characteristic

C₄⋊#170

—

C₄⋊#170

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₈×Dic₅

D₄

yes

characteristic

C₈×Dic₅

C₂³

yes

characteristic

C₂×#672

C₂×C₄

yes

—

C₄×#168

Q₈

yes

characteristic

#3930

—

C₄×M₃₂

—

#1566

—

#1368

—

C₂×C₄×D₅

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅

SD₁₆

yes

—

C₂×#170

—

#700

—

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₄²×C₅

C₄⋊C₄

yes

characteristic

C₄×Dic₅

SD₁₆

yes

—

C₄×Dic₅

C₂×D₄

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

C₂×#168

C₂×Q₈

yes

characteristic

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

M₁₆

yes

complemented

#672

—

C₂×M₃₂

—

#423

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

C₂²×D₅

#106

yes

characteristic

C₂×C₄×C₅

#106

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅

#106

yes

characteristic

C₂×Dic₅

C₂×SD₁₆

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

#108

yes

—

C₅×C₈

C₂×M₁₆

yes

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×AGL(1,𝔽₅)

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

C₂²×C₅

yes

C₂×D₅

—

C₂×D₅

—

C₄×C₅

#365

yes

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₄²

—

C₄²

—

C₄²

C₄⋊AGL(1,𝔽₅)

yes

characteristic

C₁₆

—

100

C₁₆

#700

yes

complemented

101

C₂×C₅

#1566

yes

characteristic

102

C₂×C₅

C₂×#365

yes

—

103

D₅

—

104

C₂³

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#3993

yes

—

111

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

#3995

yes

—

116

C₈

C₂×#700

yes

—

117

C₅

#12020

yes

characteristic, complemented, Sylow

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#3993

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#64847

yes

—

124

C₄

#65681

yes

characteristic

125

C₄

C₂×#3995

yes

characteristic

126

C₂

—

127

C₂

#10603816

yes

characteristic

128

C₂

C₂×#64847

yes

—

	subgroup	G/H
center	C₂²	C₂×#3993
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₁₆	C₄
socle	C₂²×C₅	C₂×#106
prime radical	C₂²×D₅	#106
cosocle	C₈×Dic₅	C₂³
hypercenter	C₄×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#12020
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#3993

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

Fitting

C₄×C₅×C₁₆

C₄

C₂²×C₅

C₂²×D₅

C₈×Dic₅

C₂³

C₄×C₁₆

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
3. #10598641	C₂	semidirect
42. C₅×C₁₆	M₁₆	semidirect
100. C₁₆	#700	semidirect
117. C₅	#12020	semidirect

H ≅ G/N

type

3. #10598641

C₂

semidirect

42. C₅×C₁₆

M₁₆

semidirect

100. C₁₆

#700

semidirect

117. C₅

#12020

semidirect

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

subgroup	index
#10603816	2
C₄×#3930	2
#10598641	2
#12020	5

subgroup

index

Subgroups — #13025111

Conjugacy classes of subgroups

Complements

Maximal subgroups