Group #12014067 — subgroups

Subgroups — #12014067

Order 1280.

274 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₅×#1707	C₂	1	1	yes	maximal, characteristic
2	#10602175	C₂	1	2	yes	maximal
3	#10602105	C₂	1	2	yes	maximal
4	#10598858	C₂	1	1	yes	maximal, characteristic
5	#10597929	C₂	1	1	yes	maximal, characteristic
6	C₅×#500	C₂²	1	1	yes	characteristic
7	C₅×#498	—	2	1	no	—
8	C₅×#494	C₂²	1	1	yes	characteristic
9	C₅×#487	—	2	1	no	—
10	C₂×#3961	—	2	4	no	—
11	#64988	C₂²	1	2	yes	—
12	#64971	—	2	1	no	—
13	#64971	—	2	1	no	—
14	C₅×C₈²	C₂²	1	1	yes	characteristic
15	#64712	—	2	4	no	—
16	C₄×#672	C₂²	1	2	yes	—
17	#12528	—	5	1	no	maximal, Sylow
18	C₂×C₅×Q₁₆	—	2	2	no	—
19	C₅×C₄⋊Q₈	—	2	1	no	—
20	C₅×C₄⋊Q₈	—	2	1	no	—
21	C₂×Dic₂₀	—	2	2	no	—
22	C₂×Dic₂₀	—	2	2	no	—
23	#4015	—	2	2	no	—
24	C₅×#107	—	2	2	no	—
25	C₅×#106	—	4	2	no	—
26	C₅×#103	—	4	2	no	—
27	C₄×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
30	#3961	—	4	8	no	—
31	#3950	—	4	2	no	—
32	#3950	—	4	2	no	—
33	C₂×#672	D₄	1	4	yes	—
34	#2247	—	5	2	no	—
35	#2241	—	5	2	no	—
36	#1713	—	5	1	no	—
37	#1707	D₅	1	1	yes	characteristic
38	#1312	—	5	1	no	—
39	C₂×C₅×Q₈	—	4	1	no	—
40	C₂×C₅×Q₈	—	4	1	no	—
41	C₂×Dic₁₀	—	4	2	no	—
42	C₅×Q₁₆	—	4	4	no	—
43	C₂×C₅×C₈	D₈	1	2	yes	—
44	C₂×C₅×C₈	D₈	1	2	yes	—
45	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
46	C₅×C₄⋊C₄	—	4	1	no	—
47	C₅×C₄⋊C₄	—	4	1	no	—
48	C₅×C₄⋊C₄	—	4	1	no	—
49	C₅×C₄⋊C₄	—	4	1	no	—
50	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
51	C₄⋊Dic₅	—	4	2	no	—
52	C₄⋊Dic₅	—	4	2	no	—
53	Dic₂₀	—	4	4	no	—
54	Dic₂₀	—	4	4	no	—
55	#672	—	2	4	no	—
56	C₂×Q₃₂	—	10	4	no	—
57	#500	—	10	1	no	—
58	#500	—	10	1	no	—
59	#500	C₂×D₅	1	1	yes	characteristic
60	#498	—	2	1	no	—
61	#494	—	5	2	no	—
62	#494	C₂×D₅	1	1	yes	characteristic
63	#487	—	2	1	no	—
64	#358	—	10	4	no	—
65	C₄×C₁₆	—	5	2	no	—
66	C₈²	C₂×D₅	1	1	yes	characteristic
67	C₅×Q₈	—	8	2	no	—
68	C₅×Q₈	—	8	2	no	—
69	C₂×C₄×C₅	—	8	1	no	—
70	C₂×C₄×C₅	—	8	1	no	—
71	C₂×C₄×C₅	#127	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
74	C₂×Dic₅	—	8	2	no	—
75	Dic₁₀	—	8	2	no	—
76	Dic₁₀	—	8	2	no	—
77	C₅×C₈	—	2	2	no	—
78	C₅×C₈	Q₃₂	1	4	yes	complemented
79	C₅×C₈	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	10	2	no	—
82	C₂×Q₁₆	—	10	2	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄⋊Q₈	—	10	2	no	—
86	Q₃₂	—	20	8	no	—
87	C₂×C₁₆	—	5	4	no	—
88	#107	—	2	2	no	—
89	#107	—	20	2	no	—
90	#107	—	20	2	no	—
91	#106	—	4	2	no	—
92	#103	—	4	2	no	—
93	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
94	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
95	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
96	C₂²×C₅	#493	1	1	yes	characteristic
97	C₄×C₅	—	16	1	no	—
98	C₄×C₅	—	16	1	no	—
99	C₄×C₅	#496	1	2	yes	—
100	C₄×C₅	C₂×Q₃₂	1	2	yes	—
101	C₄×C₅	#510	1	2	yes	—
102	Dic₅	—	16	2	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×Q₈	—	4	1	no	—
105	C₂×Q₈	—	20	2	no	—
106	Q₁₆	—	4	4	no	—
107	Q₁₆	—	20	4	no	—
108	Q₁₆	—	20	4	no	—
109	C₂×C₈	#686	1	2	yes	—
110	C₂×C₈	#686	1	2	yes	—
111	C₂×C₈	D₄×D₅	1	2	yes	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	20	2	no	—
117	C₄⋊C₄	—	20	2	no	—
118	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
119	C₁₆	—	10	4	no	—
120	C₂×C₅	#1721	1	1	yes	characteristic
121	C₂×C₅	#2247	1	1	yes	characteristic
122	C₂×C₅	#2251	1	1	yes	characteristic
123	Q₈	—	8	2	no	—
124	Q₈	—	8	2	no	—
125	Q₈	—	40	2	no	—
126	Q₈	—	40	2	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	40	2	no	—
130	C₂×C₄	C₂×#686	1	1	yes	characteristic
131	C₂×C₄	C₂×#686	1	1	yes	characteristic
132	C₂×C₄	#4086	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	#3961	1	4	yes	complemented
135	C₈	#4060	1	4	yes	—
136	C₅	#12528	1	1	yes	characteristic, complemented, Sylow
137	C₂²	#65362	1	1	yes	characteristic
138	C₄	—	16	1	no	—
139	C₄	—	16	1	no	—
140	C₄	—	80	2	no	—
141	C₄	#65465	1	2	yes	—
142	C₄	C₂×#3961	1	2	yes	—
143	C₄	#65484	1	2	yes	—
144	C₂	#10602055	1	1	yes	characteristic
145	C₂	#10602105	1	1	yes	characteristic
146	C₂	#10602115	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1707

C₂

yes

maximal, characteristic

#10602175

C₂

yes

maximal

#10602105

C₂

yes

maximal

#10598858

C₂

yes

maximal, characteristic

#10597929

C₂

yes

maximal, characteristic

C₅×#500

C₂²

yes

characteristic

C₅×#498

—

C₅×#494

C₂²

yes

—

—

C₂²

yes

—

#64971

—

#64971

—

C₅×C₈²

C₂²

yes

characteristic

#64712

—

C₄×#672

C₂²

yes

—

#12528

—

maximal, Sylow

C₂×C₅×Q₁₆

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

—

—

—

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

C₂³

yes

—

—

—

D₄

yes

—

—

—

—

D₅

yes

characteristic

#1312

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₅×Q₁₆

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

Dic₂₀

—

Dic₂₀

—

#672

—

C₂×Q₃₂

—

#500

—

#500

—

#500

C₂×D₅

yes

—

—

C₂×D₅

yes

characteristic

#487

—

#358

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

Q₃₂

yes

complemented

C₅×C₈

AGL(1,ℤ/8)

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

—

—

—

—

—

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#496

yes

—

100

C₄×C₅

C₂×Q₃₂

yes

—

101

C₄×C₅

#510

yes

—

102

Dic₅

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

Q₁₆

—

107

Q₁₆

—

108

Q₁₆

—

109

C₂×C₈

#686

yes

—

110

C₂×C₈

#686

yes

—

111

C₂×C₈

D₄×D₅

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

C₂×C₂²⋊D₅

yes

characteristic

119

C₁₆

—

120

C₂×C₅

#1721

yes

characteristic

121

C₂×C₅

#2247

yes

characteristic

122

C₂×C₅

#2251

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×#686

yes

characteristic

131

C₂×C₄

C₂×#686

yes

characteristic

132

C₂×C₄

#4086

yes

characteristic

133

C₈

—

134

C₈

#3961

yes

complemented

135

C₈

#4060

yes

—

136

C₅

#12528

yes

characteristic, complemented, Sylow

137

C₂²

#65362

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#65465

yes

—

142

C₄

C₂×#3961

yes

—

143

C₄

#65484

yes

—

144

C₂

#10602055

yes

characteristic

145

C₂

#10602105

yes

characteristic

146

C₂

#10602115

yes

characteristic

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1707	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#1707	D₅
hypocenter	C₅	#12528
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

subgroup

G/H

center

C₂²

#65362

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

N	H ≅ G/N	type
78. C₅×C₈	Q₃₂	semidirect
134. C₈	#3961	semidirect
136. C₅	#12528	semidirect

H ≅ G/N

type

78. C₅×C₈

Q₃₂

semidirect

134. C₈

#3961

semidirect

136. C₅

#12528

semidirect

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

subgroup	index
C₅×#1707	2
#10602175	2
#10602105	2
#10598858	2
#10597929	2
#12528	5

subgroup

index

Subgroups — #12014067

Conjugacy classes of subgroups

Complements

Maximal subgroups