Group #12015698 — subgroups

Subgroups — #12015698

Order 1280.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#10602212	C₂	1	1	yes	maximal, characteristic
2	#10601491	C₂	1	2	yes	maximal
3	C₅×#1385	C₂	1	1	yes	maximal, characteristic
4	#10598309	C₂	1	1	yes	maximal, characteristic
5	#10598220	C₂	1	2	yes	maximal
6	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₅×#108	C₂²	1	2	yes	—
8	#65395	C₂²	1	2	yes	—
9	#65390	C₂²	1	1	yes	characteristic
10	C₂²×#672	C₂²	1	1	yes	characteristic
11	#64779	—	2	2	no	—
12	#64777	—	2	2	no	—
13	#64722	C₄	1	2	yes	complemented
14	C₁₆×Dic₅	C₄	1	2	yes	complemented
15	#12893	—	5	1	no	maximal, Sylow
16	C₂×C₅×M₁₆	—	2	2	no	—
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	#4072	—	2	2	no	—
19	#4072	D₄	1	1	yes	characteristic
20	C₂×#681	D₄	1	1	yes	characteristic
21	C₅×M₃₂	—	2	2	no	—
22	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
23	C₅×#108	—	2	2	no	—
24	C₅×#108	—	2	2	no	—
25	#3950	—	2	2	no	—
26	#3949	—	2	2	no	—
27	#3947	Q₈	1	2	yes	—
28	C₈×Dic₅	C₂×C₄	1	2	yes	—
29	C₂×#672	—	2	2	no	—
30	C₂×#672	C₂×C₄	1	1	yes	characteristic
31	C₂×#672	C₂×C₄	1	1	yes	characteristic
32	#2108	—	5	1	no	—
33	#1833	—	5	2	no	—
34	#1385	D₅	1	1	yes	characteristic
35	#1384	—	5	1	no	—
36	#1382	—	5	2	no	—
37	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
38	C₅×M₁₆	—	4	2	no	—
39	C₅×M₁₆	—	4	2	no	—
40	C₂×C₅×C₈	—	4	2	no	—
41	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₅×C₈	#41	1	2	yes	—
44	C₂×C₅×C₈	#41	1	2	yes	—
45	C₂²⋊Dic₅	—	2	1	no	—
46	C₂²⋊Dic₅	—	4	2	no	—
47	C₄⋊Dic₅	—	2	1	no	—
48	C₄⋊Dic₅	—	4	2	no	—
49	C₄×Dic₅	—	4	2	no	—
50	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
51	#681	—	2	2	no	—
52	C₂×#168	C₄⋊C₄	1	2	yes	—
53	C₅×C₁₆	—	2	2	no	—
54	#672	—	2	1	no	—
55	#672	—	2	1	no	—
56	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
57	C₂²×C₁₆	—	5	1	no	—
58	C₂×#108	C₂×D₅	1	2	yes	—
59	#427	—	5	2	no	—
60	#405	—	5	1	no	—
61	#346	—	5	2	no	—
62	C₄×C₁₆	—	5	2	no	—
63	#344	—	10	2	no	—
64	H(3,ℤ/4)	—	10	2	no	—
65	C₂³×C₅	#135	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅	#124	1	2	yes	—
68	C₂×C₄×C₅	#135	1	1	yes	characteristic
69	C₂×C₄×C₅	#135	1	2	yes	—
70	C₂×Dic₅	—	2	2	no	—
71	C₂×Dic₅	—	8	2	no	—
72	C₅×C₈	—	8	2	no	—
73	C₅×C₈	#117	1	1	yes	characteristic
74	C₅×C₈	#117	1	1	yes	characteristic
75	C₅×C₈	#125	1	2	yes	—
76	#168	—	2	2	no	—
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	5	1	no	—
79	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
80	#117	—	5	1	no	—
81	#117	—	10	2	no	—
82	M₃₂	—	2	2	no	—
83	C₂×C₁₆	—	5	1	no	—
84	C₂×C₁₆	—	5	1	no	—
85	C₂×C₁₆	—	10	2	no	—
86	C₂×C₁₆	C₄×D₅	1	2	yes	—
87	#108	—	2	2	no	—
88	#108	—	2	2	no	—
89	#107	—	10	2	no	—
90	#106	—	10	2	no	—
91	#97	—	5	2	no	—
92	C₄×C₈	—	5	2	no	—
93	C₂²×C₅	—	4	1	no	—
94	C₂²×C₅	#427	1	1	yes	characteristic
95	C₂²×C₅	#488	1	2	yes	—
96	C₄×C₅	#382	1	1	yes	characteristic
97	C₄×C₅	#427	1	1	yes	characteristic
98	C₄×C₅	#488	1	2	yes	—
99	Dic₅	—	4	2	no	—
100	Dic₅	—	16	2	no	—
101	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
102	M₁₆	—	4	2	no	—
103	M₁₆	—	4	2	no	—
104	M₁₆	—	10	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	5	2	no	—
107	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
108	C₂×C₈	#709	1	2	yes	—
109	C₂×C₈	#711	1	2	yes	—
110	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
111	C₄⋊C₄	—	10	1	no	—
112	C₄⋊C₄	—	20	2	no	—
113	C₂²⋊C₄	—	10	1	no	—
114	C₂²⋊C₄	—	20	2	no	—
115	C₄²	—	5	2	no	—
116	C₄²	—	20	2	no	—
117	C₁₆	—	2	2	no	—
118	C₁₆	—	10	1	no	—
119	C₁₆	—	10	1	no	—
120	C₂×C₅	—	2	1	no	—
121	C₂×C₅	—	2	2	no	—
122	C₂×C₅	#1835	1	1	yes	characteristic
123	C₂³	#4065	1	1	yes	characteristic
124	C₂×C₄	—	10	2	no	—
125	C₂×C₄	—	40	2	no	—
126	C₂×C₄	#4031	1	2	yes	—
127	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄	#4058	1	1	yes	characteristic
129	C₂×C₄	#4062	1	2	yes	—
130	C₈	—	8	2	no	—
131	C₈	—	10	2	no	—
132	C₈	#4018	1	1	yes	characteristic
133	C₈	#4023	1	1	yes	characteristic
134	C₈	#4035	1	2	yes	—
135	C₅	#12893	1	1	yes	characteristic, complemented, Sylow
136	C₂²	—	4	1	no	—
137	C₂²	#65089	1	2	yes	—
138	C₂²	#65169	1	1	yes	characteristic
139	C₄	—	20	2	no	—
140	C₄	—	80	2	no	—
141	C₄	#64939	1	1	yes	characteristic
142	C₄	#65057	1	2	yes	—
143	C₄	#65149	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	#10599641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10602212

C₂

yes

maximal, characteristic

#10601491

C₂

yes

maximal

C₅×#1385

C₂

yes

maximal, characteristic

#10598309

C₂

yes

maximal, characteristic

#10598220

C₂

yes

maximal

C₂×C₅×M₃₂

C₂²

yes

characteristic

C₂×C₅×#108

C₂²

yes

—

#65395

C₂²

yes

—

#65390

C₂²

yes

characteristic

C₂²×#672

C₂²

yes

—

—

C₄

yes

complemented

C₁₆×Dic₅

C₄

yes

complemented

#12893

—

maximal, Sylow

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

C₂³

yes

characteristic

#4072

—

#4072

D₄

yes

characteristic

C₂×#681

D₄

yes

characteristic

C₅×M₃₂

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

—

—

—

—

Q₈

yes

—

C₈×Dic₅

C₂×C₄

yes

—

C₂×#672

—

C₂×#672

C₂×C₄

yes

characteristic

C₂×#672

C₂×C₄

yes

—

—

D₅

yes

characteristic

#1384

—

#1382

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₂×C₅×C₈

#41

yes

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₄⋊C₄

yes

—

#681

—

C₂×#168

C₄⋊C₄

yes

—

C₅×C₁₆

—

#672

—

#672

—

C₂×M₃₂

C₂×D₅

yes

characteristic

C₂²×C₁₆

—

C₂×#108

C₂×D₅

yes

—

#427

—

#405

—

#346

—

C₄×C₁₆

—

#344

—

H(3,ℤ/4)

—

C₂³×C₅

#135

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#124

yes

—

C₂×C₄×C₅

#135

yes

characteristic

C₂×C₄×C₅

#135

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

yes

C₅×C₈

yes

C₅×C₈

yes

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²×D₅

yes

characteristic

#117

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅

yes

—

—

—

—

—

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

#427

yes

characteristic

C₂²×C₅

#488

yes

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

Dic₅

—

100

Dic₅

—

101

C₂²×C₄

D₄×D₅

yes

characteristic

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂×C₄×D₅

yes

characteristic

108

C₂×C₈

#709

yes

—

109

C₂×C₈

#711

yes

—

110

C₂×C₈

Q₈×D₅

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

120

C₂×C₅

—

121

C₂×C₅

—

122

C₂×C₅

#1835

yes

characteristic

123

C₂³

#4065

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#4031

yes

—

127

C₂×C₄

D₅×C₄⋊C₄

yes

characteristic

128

C₂×C₄

#4058

yes

characteristic

129

C₂×C₄

#4062

yes

—

130

C₈

—

131

C₈

—

132

C₈

#4018

yes

characteristic

133

C₈

#4023

yes

characteristic

134

C₈

#4035

yes

—

135

C₅

#12893

yes

characteristic, complemented, Sylow

136

C₂²

—

137

C₂²

#65089

yes

—

138

C₂²

#65169

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#64939

yes

characteristic

142

C₄

#65057

yes

—

143

C₄

#65149

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#10599641

yes

characteristic

	subgroup	G/H
center	C₄	#65149
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1385	C₂
socle	C₂×C₅	#1835
prime radical	C₂³×C₅	#135
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1385	D₅
hypocenter	C₅	#12893
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#65149

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅

C₂

C₂×C₅

C₂³×C₅

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
13. #64722	C₄	semidirect
14. C₁₆×Dic₅	C₄	semidirect
135. C₅	#12893	semidirect

H ≅ G/N

type

13. #64722

C₄

semidirect

14. C₁₆×Dic₅

C₄

semidirect

135. C₅

#12893

semidirect

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

subgroup	index
#10602212	2
#10601491	2
C₅×#1385	2
#10598309	2
#10598220	2
#12893	5

subgroup

index

Subgroups — #12015698

Conjugacy classes of subgroups

Complements

Maximal subgroups