Group #12013678 — subgroups

Subgroups — #12013678

Order 1280.

188 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 67 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×#2252	C₂	1	1	yes	maximal, characteristic
2	#10602171	C₂	1	2	yes	maximal
3	#10599841	C₂	1	1	yes	maximal, characteristic
4	#10598293	C₂	1	2	yes	maximal
5	#10597992	C₂	1	1	yes	maximal, characteristic
6	C₅×#500	C₂²	1	2	yes	—
7	#65380	—	2	2	no	—
8	C₈×Dic₁₀	C₂²	1	1	yes	characteristic
9	C₅×#366	—	2	1	no	—
10	C₅×#366	—	2	1	no	—
11	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	#64766	—	2	2	no	—
13	#64722	—	2	1	no	—
14	#64721	—	2	1	no	—
15	C₄⋊#672	C₂²	1	1	yes	characteristic
16	#64678	C₂²	1	2	yes	—
17	#12461	—	5	1	no	maximal, Sylow
18	C₅×C₄⋊Q₈	—	2	2	no	—
19	C₄×Dic₁₀	D₄	1	1	yes	characteristic
20	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
22	C₅×#107	—	2	2	no	—
23	C₅×#107	—	2	2	no	—
24	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
25	#3946	—	2	1	no	—
26	C₈×Dic₅	—	2	1	no	—
27	C₂×#672	—	2	1	no	—
28	#3942	—	4	2	no	—
29	#3939	—	4	2	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄⋊#168	D₄	1	1	yes	characteristic
32	#2252	D₅	1	1	yes	characteristic
33	#2183	—	5	1	no	—
34	#1676	—	5	2	no	—
35	#1372	—	5	1	no	—
36	#1365	—	5	2	no	—
37	C₂×C₅×Q₈	—	4	2	no	—
38	C₂×Dic₁₀	D₈	1	1	yes	characteristic
39	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₅×C₈	#41	1	1	yes	characteristic
41	C₅×C₄⋊C₄	—	4	2	no	—
42	C₅×C₄⋊C₄	—	4	2	no	—
43	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
44	C₄⋊Dic₅	D₈	1	1	yes	characteristic
45	#683	—	2	1	no	—
46	C₄×Dic₅	—	2	1	no	—
47	C₂×#168	—	2	1	no	—
48	C₂×#168	—	4	2	no	—
49	C₅×C₁₆	—	2	1	no	—
50	C₅×C₁₆	Q₁₆	1	2	yes	—
51	#672	—	4	1	no	—
52	#500	C₂×D₅	1	2	yes	—
53	#477	—	10	2	no	—
54	C₈×Q₈	—	5	1	no	—
55	#366	—	2	1	no	—
56	#366	—	2	1	no	—
57	C₄⋊C₁₆	—	5	1	no	—
58	C₄⋊C₁₆	—	10	1	no	—
59	#346	—	10	1	no	—
60	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
61	#335	—	5	2	no	—
62	#333	—	10	2	no	—
63	C₅×Q₈	—	16	2	no	—
64	C₂×C₄×C₅	—	8	2	no	—
65	C₂×C₄×C₅	#121	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅	#135	1	1	yes	characteristic
68	C₂×Dic₅	—	2	1	no	—
69	C₂×Dic₅	—	2	1	no	—
70	Dic₁₀	—	2	2	no	—
71	C₅×C₈	—	2	1	no	—
72	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
73	C₅×C₈	#137	1	1	yes	characteristic
74	#168	—	4	1	no	—
75	#168	—	8	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄×Q₈	—	5	1	no	—
78	C₂×C₁₆	—	10	1	no	—
79	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
80	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
81	#107	—	2	2	no	—
82	#107	—	2	2	no	—
83	#107	—	20	2	no	—
84	C₄⋊C₈	—	5	1	no	—
85	C₄⋊C₈	—	10	1	no	—
86	C₄⋊C₈	—	10	2	no	—
87	#103	—	20	2	no	—
88	C₄×C₈	—	10	1	no	—
89	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
90	C₂²×C₅	#466	1	1	yes	characteristic
91	C₄×C₅	—	2	1	no	—
92	C₄×C₅	—	16	2	no	—
93	C₄×C₅	#462	1	1	yes	characteristic
94	C₄×C₅	#473	1	1	yes	characteristic
95	C₄×C₅	#510	1	2	yes	—
96	Dic₅	—	4	1	no	—
97	Dic₅	—	4	1	no	—
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	5	1	no	—
100	C₂×C₈	—	10	1	no	—
101	C₂×C₈	—	20	2	no	—
102	C₂×C₈	#711	1	1	yes	characteristic
103	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	5	1	no	—
107	C₄⋊C₄	—	10	1	no	—
108	C₄²	—	10	1	no	—
109	C₄²	D₄×D₅	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	20	1	no	—
112	C₁₆	#689	1	2	yes	—
113	C₂×C₅	#1674	1	1	yes	characteristic
114	C₂×C₅	#2219	1	1	yes	characteristic
115	C₂×C₅	#2221	1	1	yes	characteristic
116	Q₈	—	10	2	no	—
117	Q₈	—	16	2	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	—	10	1	no	—
120	C₂×C₄	—	10	1	no	—
121	C₂×C₄	D₅×D₈	1	1	yes	characteristic
122	C₂×C₄	#4058	1	1	yes	characteristic
123	C₂×C₄	#4084	1	1	yes	characteristic
124	C₈	—	2	1	no	—
125	C₈	—	20	1	no	—
126	C₈	—	40	2	no	—
127	C₈	#4088	1	1	yes	characteristic
128	C₈	C₂×#689	1	1	yes	characteristic
129	C₅	#12461	1	1	yes	characteristic, complemented, Sylow
130	C₂²	#65446	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	16	2	no	—
133	C₄	—	20	1	no	—
134	C₄	—	20	1	no	—
135	C₄	#65207	1	2	yes	—
136	C₄	#65379	1	1	yes	characteristic
137	C₄	#65481	1	1	yes	characteristic
138	C₂	#10602170	1	1	yes	characteristic
139	C₂	#10602265	1	1	yes	characteristic
140	C₂	#10602298	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2252

C₂

yes

maximal, characteristic

#10602171

C₂

yes

maximal

#10599841

C₂

yes

maximal, characteristic

#10598293

C₂

yes

maximal

#10597992

C₂

yes

maximal, characteristic

C₅×#500

C₂²

yes

—

#65380

—

C₈×Dic₁₀

C₂²

yes

characteristic

C₅×#366

—

C₅×#366

—

C₄×C₅×C₁₆

C₂²

yes

—

—

—

C₂²

yes

characteristic

#64678

C₂²

yes

—

#12461

—

maximal, Sylow

C₅×C₄⋊Q₈

—

C₄×Dic₁₀

D₄

yes

characteristic

C₂×C₅×C₁₆

D₄

yes

characteristic

C₂×C₅×C₁₆

D₄

yes

characteristic

C₅×#107

—

C₅×#107

—

C₄×C₅×C₈

C₂³

yes

characteristic

#3946

—

C₈×Dic₅

—

—

—

—

—

D₄

yes

characteristic

#2252

D₅

yes

—

—

—

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

D₈

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

#41

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

D₈

yes

characteristic

#683

—

C₄×Dic₅

—

C₂×#168

—

C₂×#168

—

C₅×C₁₆

—

C₅×C₁₆

Q₁₆

yes

—

#672

—

#500

C₂×D₅

yes

—

#477

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

C₂×D₅

yes

characteristic

#335

—

#333

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

#135

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

C₂×Q₁₆

yes

C₅×C₈

yes

—

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅

yes

characteristic

C₂×C₁₆

C₂²⋊D₅

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#466

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

#711

yes

characteristic

103

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

D₄×D₅

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

#689

yes

—

113

C₂×C₅

#1674

yes

characteristic

114

C₂×C₅

#2219

yes

characteristic

115

C₂×C₅

#2221

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

D₅×D₈

yes

characteristic

122

C₂×C₄

#4058

yes

characteristic

123

C₂×C₄

#4084

yes

characteristic

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

#4088

yes

characteristic

128

C₈

C₂×#689

yes

characteristic

129

C₅

#12461

yes

characteristic, complemented, Sylow

130

C₂²

#65446

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#65207

yes

—

136

C₄

#65379

yes

characteristic

137

C₄

#65481

yes

characteristic

138

C₂

#10602170

yes

characteristic

139

C₂

#10602265

yes

characteristic

140

C₂

#10602298

yes

characteristic

	subgroup	G/H
center	C₂²	#65446
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2252	C₂
socle, prime radical	C₂²×C₅	#466
hypercenter	#2252	D₅
hypocenter	C₅	#12461
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#65446

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 ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
129. C₅	#12461	semidirect

H ≅ G/N

type

129. C₅

#12461

semidirect

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

subgroup	index
C₅×#2252	2
#10602171	2
#10599841	2
#10598293	2
#10597992	2
#12461	5

subgroup

index

Subgroups — #12013678

Conjugacy classes of subgroups

Complements

Maximal subgroups