Group #13026586 — subgroups

Subgroups — #13026586

Order 1280.

322 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 140 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	C₂×C₄⋊#674	C₂	1	2	yes	maximal
2	#10600227	C₂	1	1	yes	maximal, characteristic
3	#10598698	C₂	1	4	yes	maximal
4	C₂²×#674	C₂²	1	4	yes	—
5	C₅×#376	C₄	1	1	yes	characteristic
6	C₂×#3935	C₄	1	2	yes	—
7	C₂×#3935	C₄	1	1	yes	characteristic
8	C₂×C₄×#168	C₂²	1	2	yes	—
9	C₂×C₄×#168	C₂²	1	1	yes	characteristic
10	C₄⋊#674	—	2	4	no	—
11	#12880	—	5	1	no	maximal, Sylow
12	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂²×#168	C₂×C₄	1	1	yes	characteristic
15	C₂²×#168	C₂×C₄	1	2	yes	—
16	C₂²×#168	C₂³	1	1	yes	characteristic
17	C₂×#674	—	2	4	no	—
18	C₂×#674	—	2	8	no	—
19	C₅×#95	C₈	1	4	yes	—
20	#3935	—	2	2	no	—
21	#3935	—	2	4	no	—
22	C₄×#168	—	2	4	no	—
23	C₄×#168	D₄	1	2	yes	—
24	C₄×#168	Q₈	1	2	yes	—
25	C₂×C₄⋊C₁₆	—	5	2	no	—
26	#1726	—	5	1	no	—
27	#1374	—	5	4	no	—
28	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
29	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
30	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₅	—	2	2	no	—
32	C₄²×C₅	—	2	2	no	—
33	C₄²×C₅	C₄⋊C₄	1	2	yes	—
34	C₄²×C₅	M₁₆	1	2	yes	—
35	C₂×#168	—	2	2	no	—
36	C₂×#168	—	2	2	no	—
37	C₂×#168	—	2	2	no	—
38	C₂×#168	C₄⋊C₄	1	2	yes	—
39	C₂×#168	M₁₆	1	4	yes	—
40	C₂×#168	C₂×D₄	1	1	yes	characteristic
41	C₂×#168	C₂×Q₈	1	1	yes	characteristic
42	C₂×#168	#41	1	4	yes	—
43	#674	—	4	4	no	—
44	C₂²×C₁₆	—	5	4	no	—
45	C₂×#97	—	5	2	no	—
46	C₂×#97	—	5	1	no	—
47	C₂×C₄×C₈	—	5	1	no	—
48	C₂×C₄×C₈	—	5	2	no	—
49	#376	AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₄⋊C₁₆	—	10	4	no	—
51	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅	—	2	4	no	—
53	C₂×C₄×C₅	—	2	4	no	—
54	C₂×C₄×C₅	—	2	8	no	—
55	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
56	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅	#117	1	2	yes	—
58	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
60	#168	—	2	2	no	—
61	#168	—	2	4	no	—
62	#168	#124	1	2	yes	—
63	#168	#125	1	2	yes	—
64	C₂²×C₈	—	5	1	no	—
65	C₂²×C₈	—	5	2	no	—
66	C₂²×C₈	—	5	1	no	—
67	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	—
68	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂×C₁₆	—	10	4	no	—
70	C₂×C₁₆	—	10	8	no	—
71	#97	—	10	4	no	—
72	#97	—	10	2	no	—
73	C₄×C₈	—	5	2	no	—
74	C₄×C₈	—	5	2	no	—
75	C₄×C₈	—	10	4	no	—
76	#95	#170	1	4	yes	—
77	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₅	#431	1	2	yes	—
79	C₂²×C₅	#504	1	4	yes	—
80	C₄×C₅	—	2	4	no	—
81	C₄×C₅	—	4	4	no	—
82	C₄×C₅	#382	1	1	yes	characteristic
83	C₄×C₅	#423	1	1	yes	characteristic
84	C₄×C₅	#432	1	2	yes	—
85	C₂²×C₄	C₂×#170	1	4	yes	—
86	C₂²×C₄	C₂×#170	1	2	yes	—
87	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₂×C₈	—	5	1	no	—
89	C₂×C₈	—	5	1	no	—
90	C₂×C₈	—	5	4	no	—
91	C₂×C₈	—	5	4	no	—
92	C₂×C₈	—	5	2	no	—
93	C₂×C₈	—	10	2	no	—
94	C₂×C₈	—	10	2	no	—
95	C₂×C₈	—	10	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	#700	1	2	yes	—
99	C₄²	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
100	C₁₆	—	20	4	no	—
101	C₂×C₅	#1831	1	1	yes	characteristic
102	C₂×C₅	#2151	1	2	yes	—
103	C₂×C₅	#2164	1	4	yes	—
104	C₂³	C₂²×#170	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	C₄⋊#170	1	4	yes	—
109	C₂×C₄	C₂×#700	1	1	yes	characteristic
110	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
112	C₂×C₄	#4130	1	2	yes	—
113	C₈	—	5	2	no	—
114	C₈	—	5	2	no	—
115	C₈	—	10	2	no	—
116	C₈	—	10	4	no	—
117	C₅	#12880	1	1	yes	characteristic, complemented, Sylow
118	C₂²	C₂×C₄⋊#170	1	1	yes	characteristic
119	C₂²	#65748	1	2	yes	—
120	C₂²	#65791	1	4	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#65682	1	1	yes	characteristic
124	C₄	#65683	1	2	yes	—
125	C₄	#65689	1	1	yes	characteristic
126	C₂	#10604185	1	1	yes	characteristic
127	C₂	#10604367	1	2	yes	—
128	C₂	#10604368	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#674

C₂

yes

maximal

#10600227

C₂

yes

maximal, characteristic

#10598698

C₂

yes

maximal

C₂²×#674

C₂²

yes

—

C₅×#376

C₄

yes

characteristic

C₂×#3935

C₄

yes

—

C₂×#3935

C₄

yes

characteristic

C₂×C₄×#168

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

—

—

C₂×C₄²×C₅

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂²×#168

C₂×C₄

yes

characteristic

C₂²×#168

C₂×C₄

yes

—

C₂²×#168

C₂³

yes

characteristic

C₂×#674

—

C₂×#674

—

C₅×#95

C₈

yes

—

—

—

—

D₄

yes

—

C₄×#168

Q₈

yes

—

C₂×C₄⋊C₁₆

—

#1726

—

#1374

—

C₂²×C₄×C₅

C₂×C₈

yes

—

C₂²×C₄×C₅

C₂×C₈

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₄⋊C₄

yes

—

C₄²×C₅

M₁₆

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×#168

M₁₆

yes

—

C₂×#168

C₂×D₄

yes

characteristic

C₂×#168

C₂×Q₈

yes

characteristic

C₂×#168

#41

yes

—

#674

—

C₂²×C₁₆

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₁₆

—

C₂³×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

—

—

yes

—

#168

#125

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

#170

yes

—

C₂²×C₅

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅

#431

yes

—

C₂²×C₅

#504

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

C₂²×C₄

C₂×#170

yes

—

C₂²×C₄

C₂×#170

yes

—

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

#700

yes

—

C₄²

C₄⋊AGL(1,𝔽₅)

yes

—

100

C₁₆

—

101

C₂×C₅

#1831

yes

characteristic

102

C₂×C₅

#2151

yes

—

103

C₂×C₅

#2164

yes

—

104

C₂³

C₂²×#170

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₄⋊#170

yes

—

109

C₂×C₄

C₂×#700

yes

characteristic

110

C₂×C₄

D₅⋊M₁₆

yes

—

111

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

112

C₂×C₄

#4130

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₅

#12880

yes

characteristic, complemented, Sylow

118

C₂²

C₂×C₄⋊#170

yes

characteristic

119

C₂²

#65748

yes

—

120

C₂²

#65791

yes

—

121

C₄

—

122

C₄

—

123

C₄

#65682

yes

characteristic

124

C₄

#65683

yes

—

125

C₄

#65689

yes

characteristic

126

C₂

#10604185

yes

characteristic

127

C₂

#10604367

yes

—

128

C₂

#10604368

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
commutator, socle, prime radical	C₂³×C₅	C₂²×C₈
Fitting	C₅×#376	C₄
cosocle	C₂²×#168	C₂³
hypercenter	#376	AGL(1,𝔽₅)
hypocenter	C₅	#12880
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

commutator, socle, prime radical

C₂³×C₅

C₂²×C₈

Fitting

C₅×#376

C₄

cosocle

C₂²×#168

C₂³

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
117. C₅	#12880	semidirect

H ≅ G/N

type

117. C₅

#12880

semidirect

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

subgroup	index
C₂×C₄⋊#674	2
#10600227	2
#10598698	2
#12880	5

subgroup

index

C₂×C₄⋊#674

#10600227

#10598698

#12880

Subgroups — #13026586

Conjugacy classes of subgroups

Complements

Maximal subgroups