Group #13025647 — subgroups

Subgroups — #13025647

Order 1280.

354 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 156 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	C₄×#3998	C₂	1	2	yes	maximal
2	C₄×#3935	C₂	1	1	yes	maximal, characteristic
3	#10598699	C₂	1	4	yes	maximal
4	C₂×#3998	C₂²	1	4	yes	—
5	C₄³×C₅	C₄	1	1	yes	characteristic
6	C₂×#3935	C₄	1	2	yes	—
7	C₂×#3935	C₂²	1	2	yes	—
8	C₂×C₄×#168	C₄	1	1	yes	characteristic
9	C₂×C₄×#168	C₂²	1	1	yes	characteristic
10	#64859	C₄	1	4	yes	complemented
11	C₄×#674	C₄	1	4	yes	complemented
12	#12594	—	5	1	no	maximal, Sylow
13	C₂×C₄²×C₅	—	2	2	no	—
14	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂²×#168	C₂×C₄	1	2	yes	—
17	C₂²×#168	C₂×C₄	1	1	yes	characteristic
18	C₂²×#168	C₂³	1	1	yes	characteristic
19	#3998	—	2	8	no	—
20	C₂×#674	C₂×C₄	1	8	yes	—
21	#3935	—	2	4	no	—
22	#3935	C₈	1	8	yes	—
23	C₄×#168	—	2	2	no	—
24	C₄×#168	C₂×C₄	1	2	yes	—
25	C₄×#168	C₂×C₄	1	2	yes	—
26	C₄×M₃₂	—	5	2	no	—
27	C₄×#97	—	5	1	no	—
28	#1375	—	5	4	no	—
29	C₂²×C₄×C₅	—	2	2	no	—
30	C₂²×C₄×C₅	M₁₆	1	2	yes	—
31	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₅	—	2	2	no	—
33	C₄²×C₅	—	4	4	no	—
34	C₄²×C₅	C₄²	1	1	yes	characteristic
35	C₄²×C₅	C₄²	1	1	yes	characteristic
36	C₄²×C₅	C₂×C₈	1	4	yes	—
37	C₄²×C₅	M₁₆	1	2	yes	—
38	C₂×#168	—	2	2	no	—
39	C₂×#168	—	2	2	no	—
40	C₂×#168	C₄²	1	2	yes	—
41	C₂×#168	C₂×C₈	1	4	yes	—
42	C₂×#168	C₂×C₈	1	4	yes	—
43	C₂×#168	C₂²×C₄	1	2	yes	—
44	C₂×#168	#41	1	4	yes	—
45	#674	—	2	8	no	—
46	C₂×M₃₂	—	5	4	no	—
47	C₂×#97	—	5	2	no	—
48	C₂×#97	—	5	2	no	—
49	C₂×C₄×C₈	—	5	1	no	—
50	C₂×C₄×C₈	—	5	1	no	—
51	C₄³	AGL(1,𝔽₅)	1	1	yes	characteristic
52	#346	—	5	4	no	—
53	C₄×C₁₆	—	5	4	no	—
54	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅	—	2	4	no	—
56	C₂×C₄×C₅	—	2	4	no	—
57	C₂×C₄×C₅	—	4	4	no	—
58	C₂×C₄×C₅	#97	1	2	yes	—
59	C₂×C₄×C₅	#97	1	2	yes	—
60	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₅	#117	1	2	yes	—
62	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
63	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
64	#168	—	2	4	no	—
65	#168	C₄×C₈	1	4	yes	—
66	#168	#117	1	4	yes	—
67	C₂²×C₈	—	5	1	no	—
68	C₂²×C₈	—	5	1	no	—
69	C₂²×C₈	—	5	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	—
72	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
73	M₃₂	—	10	8	no	—
74	C₂×C₁₆	—	5	8	no	—
75	#97	—	5	8	no	—
76	#97	—	10	4	no	—
77	C₄×C₈	—	5	2	no	—
78	C₄×C₈	—	5	2	no	—
79	C₄×C₈	—	10	2	no	—
80	C₂²×C₅	—	2	2	no	—
81	C₂²×C₅	C₂×#97	1	1	yes	characteristic
82	C₂²×C₅	#431	1	2	yes	—
83	C₄×C₅	—	2	2	no	—
84	C₄×C₅	—	4	4	no	—
85	C₄×C₅	#347	1	4	yes	—
86	C₄×C₅	#376	1	1	yes	characteristic
87	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₅	#431	1	2	yes	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	#704	1	2	yes	—
91	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₂×C₈	—	5	4	no	—
93	C₂×C₈	—	5	2	no	—
94	C₂×C₈	—	5	2	no	—
95	C₂×C₈	—	5	4	no	—
96	C₂×C₈	—	5	4	no	—
97	C₂×C₈	—	10	2	no	—
98	C₂×C₈	—	10	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	4	4	no	—
101	C₄²	D₅⋊C₈	1	4	yes	—
102	C₄²	#700	1	2	yes	—
103	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
104	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₁₆	—	10	8	no	—
106	C₂×C₅	—	2	2	no	—
107	C₂×C₅	#1744	1	1	yes	characteristic
108	C₂×C₅	C₂×#347	1	2	yes	—
109	C₂³	C₂×#704	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	#4002	1	2	yes	—
114	C₂×C₄	#4002	1	2	yes	—
115	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
116	C₂×C₄	C₂×#700	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₂×C₄	#4130	1	2	yes	—
119	C₈	—	5	4	no	—
120	C₈	—	5	4	no	—
121	C₈	—	10	4	no	—
122	C₅	#12594	1	1	yes	characteristic, complemented, Sylow
123	C₂²	—	2	2	no	—
124	C₂²	C₂×#4002	1	1	yes	characteristic
125	C₂²	#65754	1	2	yes	—
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	#64860	1	4	yes	—
129	C₄	C₄×D₅⋊C₈	1	1	yes	characteristic
130	C₄	#65680	1	2	yes	—
131	C₄	#65686	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#64860	1	2	yes	—
134	C₂	#10604174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#3998

C₂

yes

maximal

C₄×#3935

C₂

yes

maximal, characteristic

#10598699

C₂

yes

maximal

C₂×#3998

C₂²

yes

—

C₄³×C₅

C₄

yes

characteristic

C₂×#3935

C₄

yes

—

C₂×#3935

C₂²

yes

—

C₂×C₄×#168

C₄

yes

characteristic

C₂×C₄×#168

C₂²

yes

characteristic

#64859

C₄

yes

complemented

C₄×#674

C₄

yes

complemented

#12594

—

maximal, Sylow

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂²×#168

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

characteristic

C₂²×#168

C₂³

yes

characteristic

#3998

—

C₂×#674

C₂×C₄

yes

—

#3935

—

#3935

C₈

yes

—

C₄×#168

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

C₄×M₃₂

—

C₄×#97

—

#1375

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

M₁₆

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₄²

yes

characteristic

C₄²×C₅

C₄²

yes

characteristic

C₄²×C₅

C₂×C₈

yes

—

C₄²×C₅

M₁₆

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

C₄²

yes

—

C₂×#168

C₂×C₈

yes

—

C₂×#168

C₂×C₈

yes

—

C₂×#168

C₂²×C₄

yes

—

C₂×#168

#41

yes

—

#674

—

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

AGL(1,𝔽₅)

yes

characteristic

#346

—

C₄×C₁₆

—

C₂³×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#97

yes

—

C₂×C₄×C₅

#97

yes

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

#168

—

#168

C₄×C₈

yes

—

#168

#117

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

C₂×#97

yes

characteristic

C₂²×C₅

#431

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#347

yes

—

C₄×C₅

#376

yes

characteristic

C₄×C₅

C₂×C₄×C₈

yes

characteristic

C₄×C₅

#431

yes

—

C₂²×C₄

—

C₂²×C₄

#704

yes

—

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

100

C₄²

—

101

C₄²

D₅⋊C₈

yes

—

102

C₄²

#700

yes

—

103

C₄²

C₄×AGL(1,𝔽₅)

yes

characteristic

104

C₄²

C₄×AGL(1,𝔽₅)

yes

characteristic

105

C₁₆

—

106

C₂×C₅

—

107

C₂×C₅

#1744

yes

characteristic

108

C₂×C₅

C₂×#347

yes

—

109

C₂³

C₂×#704

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#4002

yes

—

114

C₂×C₄

#4002

yes

—

115

C₂×C₄

C₂×D₅⋊C₈

yes

—

116

C₂×C₄

C₂×#700

yes

characteristic

117

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

118

C₂×C₄

#4130

yes

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₅

#12594

yes

characteristic, complemented, Sylow

123

C₂²

—

124

C₂²

C₂×#4002

yes

characteristic

125

C₂²

#65754

yes

—

126

C₄

—

127

C₄

—

128

C₄

#64860

yes

—

129

C₄

C₄×D₅⋊C₈

yes

characteristic

130

C₄

#65680

yes

—

131

C₄

#65686

yes

characteristic

132

C₂

—

133

C₂

C₂×#64860

yes

—

134

C₂

#10604174

yes

characteristic

	subgroup	G/H
center	C₄²	C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₄³×C₅	C₄
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×#168	C₂³
hypercenter	C₄³	AGL(1,𝔽₅)
hypocenter	C₅	#12594
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₄³

subgroup

G/H

center

C₄²

C₄×AGL(1,𝔽₅)

commutator

C₄×C₅

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₄³×C₅

C₄

socle

C₂²×C₅

C₂×#97

prime radical

C₂³×C₅

C₂×M₁₆

cosocle

C₂²×#168

C₂³

C₄³

C₅

G ▷ C₄×C₅ ▷ C₅

C₁ ◁ C₄² ◁ C₄³

N	H ≅ G/N	type
10. #64859	C₄	semidirect
11. C₄×#674	C₄	semidirect
122. C₅	#12594	semidirect

H ≅ G/N

type

10. #64859

C₄

semidirect

11. C₄×#674

C₄

semidirect

122. C₅

#12594

semidirect

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

subgroup	index
C₄×#3998	2
C₄×#3935	2
#10598699	2
#12594	5

subgroup

index

Subgroups — #13025647

Conjugacy classes of subgroups

Complements

Maximal subgroups