Group #12014715 — subgroups

Subgroups — #12014715

Order 1280.

306 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 135 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×#64722	C₂	1	1	yes	maximal, characteristic
2	Dic₅×#108	C₂	1	2	yes	maximal
3	C₂×#64713	C₂	1	2	yes	maximal
4	C₅×#1381	C₂	1	1	yes	maximal, characteristic
5	#10598299	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₅×#108	C₂²	1	2	yes	—
8	M₁₆×Dic₅	—	2	2	no	—
9	C₂×#3951	C₂²	1	2	yes	—
10	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
11	C₂²×#672	C₂²	1	1	yes	characteristic
12	#64778	—	2	2	no	—
13	#64722	C₄	1	4	yes	—
14	#64713	C₄	1	8	yes	complemented
15	#12676	—	5	1	no	maximal, Sylow
16	C₂×C₅×M₁₆	—	2	2	no	—
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
19	C₂×#681	—	2	2	no	—
20	C₂²×#168	D₄	1	1	yes	characteristic
21	C₂×C₅×C₁₆	—	2	2	no	—
22	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
23	C₅×#108	—	2	2	no	—
24	C₅×#108	C₂×C₄	1	4	yes	—
25	#3951	—	2	2	no	—
26	#3951	C₂×C₄	1	4	yes	—
27	#3947	—	4	2	no	—
28	C₈×Dic₅	—	4	2	no	—
29	C₈×Dic₅	C₂×C₄	1	2	yes	—
30	C₈×Dic₅	Q₈	1	2	yes	—
31	C₂×#672	C₂×C₄	1	4	yes	—
32	C₂×#672	C₂×C₄	1	2	yes	—
33	C₂×#368	—	5	2	no	—
34	C₂×#346	—	5	1	no	—
35	C₄×#108	—	5	2	no	—
36	#1381	—	5	1	no	—
37	#1381	D₅	1	1	yes	characteristic
38	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅	Q₁₆	1	2	yes	—
40	C₅×M₁₆	—	2	4	no	—
41	C₅×M₁₆	—	4	2	no	—
42	C₂×C₅×C₈	—	4	2	no	—
43	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
45	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₅×C₈	#41	1	2	yes	—
47	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
48	C₄×Dic₅	D₈	1	2	yes	—
49	#681	—	2	4	no	—
50	#681	—	4	2	no	—
51	C₂×#168	—	2	2	no	—
52	C₂×#168	—	4	2	no	—
53	C₂×#168	C₄⋊C₄	1	2	yes	—
54	C₅×C₁₆	—	2	2	no	—
55	#672	C₄²	1	4	yes	—
56	C₂²×C₁₆	—	5	1	no	—
57	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
58	C₂×#108	—	5	2	no	—
59	C₂×#108	C₂×D₅	1	2	yes	—
60	C₄×M₁₆	—	10	2	no	—
61	C₂×C₄×C₈	—	5	1	no	—
62	#368	—	5	8	no	—
63	#346	—	5	4	no	—
64	#341	—	10	2	no	—
65	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₅	#135	1	2	yes	—
70	C₂×Dic₅	—	4	2	no	—
71	C₂×Dic₅	#107	1	4	yes	—
72	C₅×C₈	—	4	4	no	—
73	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
74	C₅×C₈	#117	1	1	yes	characteristic
75	C₅×C₈	C₄×Q₈	1	2	yes	—
76	#168	—	2	2	no	—
77	#168	—	4	4	no	—
78	C₂×M₁₆	—	2	2	no	—
79	C₂×M₁₆	—	10	2	no	—
80	C₂²×C₈	—	5	1	no	—
81	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
82	C₂×C₄²	—	5	1	no	—
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	—	5	2	no	—
85	C₂×C₁₆	—	5	4	no	—
86	C₂×C₁₆	C₄×D₅	1	2	yes	—
87	#108	—	2	2	no	—
88	#108	—	5	4	no	—
89	#108	—	10	2	no	—
90	#108	C₄×D₅	1	4	yes	—
91	#97	—	20	2	no	—
92	C₄×C₈	—	5	2	no	—
93	C₄×C₈	—	5	2	no	—
94	C₄×C₈	—	20	2	no	—
95	C₂²×C₅	—	2	2	no	—
96	C₂²×C₅	C₂×#107	1	1	yes	characteristic
97	C₂²×C₅	C₄×Q₁₆	1	2	yes	—
98	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₄×C₅	#427	1	1	yes	characteristic
100	C₄×C₅	C₄×D₈	1	2	yes	—
101	Dic₅	—	2	4	no	—
102	C₂²×C₄	—	5	2	no	—
103	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
104	M₁₆	—	2	4	no	—
105	M₁₆	—	4	2	no	—
106	M₁₆	—	10	4	no	—
107	M₁₆	—	20	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	5	2	no	—
110	C₂×C₈	—	10	2	no	—
111	C₂×C₈	—	20	2	no	—
112	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
113	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
114	C₂×C₈	#711	1	2	yes	—
115	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
116	C₄²	—	5	2	no	—
117	C₄²	—	5	2	no	—
118	C₁₆	—	2	2	no	—
119	C₁₆	—	5	4	no	—
120	C₂×C₅	—	2	2	no	—
121	C₂×C₅	C₄×#107	1	1	yes	characteristic
122	C₂×C₅	#2161	1	2	yes	—
123	C₂³	D₅×Q₁₆	1	1	yes	characteristic
124	C₂×C₄	—	5	4	no	—
125	C₂×C₄	—	20	2	no	—
126	C₂×C₄	#4027	1	2	yes	—
127	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄	D₅×D₈	1	1	yes	characteristic
129	C₂×C₄	#4062	1	2	yes	—
130	C₈	—	4	4	no	—
131	C₈	—	10	2	no	—
132	C₈	—	20	4	no	—
133	C₈	C₄²×D₅	1	1	yes	characteristic
134	C₈	#4023	1	1	yes	characteristic
135	C₈	#4033	1	2	yes	—
136	C₅	#12676	1	1	yes	characteristic, complemented, Sylow
137	C₂²	—	2	2	no	—
138	C₂²	#65079	1	2	yes	—
139	C₂²	D₅×#107	1	1	yes	characteristic
140	C₄	—	10	4	no	—
141	C₄	#64938	1	1	yes	characteristic
142	C₄	#65045	1	2	yes	—
143	C₄	#65149	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	#10599578	1	1	yes	characteristic
146	C₂	#10600153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64722

C₂

yes

maximal, characteristic

Dic₅×#108

C₂

yes

maximal

C₂×#64713

C₂

yes

maximal

C₅×#1381

C₂

yes

maximal, characteristic

#10598299

C₂

yes

maximal, characteristic

C₂²×C₅×C₁₆

C₂²

yes

characteristic

C₂×C₅×#108

C₂²

yes

—

M₁₆×Dic₅

—

C₂×#3951

C₂²

yes

—

C₂×C₈×Dic₅

C₂²

yes

characteristic

C₂²×#672

C₂²

yes

characteristic

#64778

—

#64722

C₄

yes

—

#64713

C₄

yes

complemented

#12676

—

maximal, Sylow

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄×Dic₅

D₄

yes

characteristic

C₂×#681

—

C₂²×#168

D₄

yes

characteristic

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₅×#108

—

C₅×#108

C₂×C₄

yes

—

#3951

—

#3951

C₂×C₄

yes

—

#3947

—

C₈×Dic₅

—

C₈×Dic₅

C₂×C₄

yes

—

C₈×Dic₅

Q₈

yes

—

C₂×#672

C₂×C₄

yes

—

C₂×#672

C₂×C₄

yes

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

Q₁₆

yes

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

—

C₂×C₅×C₈

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₄×Dic₅

C₄⋊C₄

yes

—

C₄×Dic₅

D₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₅×C₁₆

—

#672

C₄²

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₅

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#368

—

#346

—

#341

—

C₂³×C₅

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

—

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

#135

yes

—

C₂×Dic₅

—

C₂×Dic₅

#107

yes

—

C₅×C₈

—

C₅×C₈

C₂×C₄²

yes

characteristic

C₅×C₈

#117

yes

characteristic

C₅×C₈

C₄×Q₈

yes

—

#168

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₅

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅

yes

—

—

—

—

C₄×D₅

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

C₂×#107

yes

characteristic

C₂²×C₅

C₄×Q₁₆

yes

—

C₄×C₅

C₄×C₄⋊C₄

yes

characteristic

C₄×C₅

#427

yes

characteristic

100

C₄×C₅

C₄×D₈

yes

—

101

Dic₅

—

102

C₂²×C₄

—

103

C₂²×C₄

D₄×D₅

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂×C₄×D₅

yes

characteristic

113

C₂×C₈

C₂×C₄×D₅

yes

—

114

C₂×C₈

#711

yes

—

115

C₂×C₈

Q₈×D₅

yes

characteristic

116

C₄²

—

117

C₄²

—

118

C₁₆

—

119

C₁₆

—

120

C₂×C₅

—

121

C₂×C₅

C₄×#107

yes

characteristic

122

C₂×C₅

#2161

yes

—

123

C₂³

D₅×Q₁₆

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#4027

yes

—

127

C₂×C₄

D₅×C₄⋊C₄

yes

characteristic

128

C₂×C₄

D₅×D₈

yes

characteristic

129

C₂×C₄

#4062

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

C₄²×D₅

yes

characteristic

134

C₈

#4023

yes

characteristic

135

C₈

#4033

yes

—

136

C₅

#12676

yes

characteristic, complemented, Sylow

137

C₂²

—

138

C₂²

#65079

yes

—

139

C₂²

D₅×#107

yes

characteristic

140

C₄

—

141

C₄

#64938

yes

characteristic

142

C₄

#65045

yes

—

143

C₄

#65149

yes

characteristic

144

C₂

—

145

C₂

#10599578

yes

characteristic

146

C₂

#10600153

yes

—

	subgroup	G/H
center	C₂×C₄	D₅×D₈
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1381	C₂
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	C₂×Q₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1381	D₅
hypocenter	C₅	#12676
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₅×D₈

commutator

C₅×C₈

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×Q₁₆

cosocle

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
14. #64713	C₄	semidirect
136. C₅	#12676	semidirect

H ≅ G/N

type

14. #64713

C₄

semidirect

136. C₅

#12676

semidirect

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

subgroup	index
C₂×#64722	2
Dic₅×#108	2
C₂×#64713	2
C₅×#1381	2
#10598299	2
#12676	5

subgroup

index

Subgroups — #12014715

Conjugacy classes of subgroups

Complements

Maximal subgroups