Group #11986277 — subgroups

Subgroups — #11986277

Order 1280.

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

normal subgroups only

This group has 210 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	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic
2	#10600538	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
4	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₅×#322	—	2	2	no	—
7	C₂²⋊#672	C₄	1	4	yes	—
8	C₄⋊#672	—	2	2	no	—
9	#64681	—	2	2	no	—
10	#7196	—	5	1	no	maximal, Sylow
11	C₂²×C₅×C₈	—	2	1	no	—
12	C₂²×C₅×C₈	—	2	1	no	—
13	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
15	C₄×C₅×C₈	—	2	1	no	—
16	C₄×C₅×C₈	—	2	1	no	—
17	C₄×C₅×C₈	C₈	1	2	yes	—
18	C₄×C₅×C₈	C₈	1	2	yes	—
19	C₄×C₅×C₈	D₄	1	1	yes	characteristic
20	C₄×C₅×C₈	D₄	1	2	yes	—
21	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
22	C₂×#672	—	2	4	no	—
23	#2163	—	5	2	no	—
24	C₂×#322	D₅	1	1	yes	characteristic
25	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
27	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
28	C₂×C₅×C₈	—	2	4	no	—
29	C₂×C₅×C₈	—	2	2	no	—
30	C₂×C₅×C₈	—	2	2	no	—
31	C₂×C₅×C₈	—	4	2	no	—
32	C₂×C₅×C₈	—	4	2	no	—
33	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
34	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
35	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
36	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
39	#672	—	4	4	no	—
40	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
41	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
43	C₄⋊C₁₆	—	10	2	no	—
44	C₂²⋊C₁₆	—	5	4	no	—
45	#346	—	10	2	no	—
46	#322	—	2	2	no	—
47	C₂³×C₅	#97	1	1	yes	characteristic
48	C₂×C₄×C₅	—	2	2	no	—
49	C₂×C₄×C₅	—	2	2	no	—
50	C₂×C₄×C₅	—	2	2	no	—
51	C₂×C₄×C₅	#95	1	1	yes	characteristic
52	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
57	C₅×C₈	—	2	4	no	—
58	C₅×C₈	—	2	2	no	—
59	C₅×C₈	—	2	2	no	—
60	C₅×C₈	—	4	2	no	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	C₄×D₅	1	2	yes	—
64	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
65	C₂×C₁₆	—	10	4	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	#168	1	2	yes	—
69	C₄×C₈	#168	1	2	yes	—
70	C₄×C₈	Dic₁₀	1	1	yes	characteristic
71	C₄×C₈	D₂₀	1	1	yes	characteristic
72	C₄×C₈	C₂²⋊D₅	1	2	yes	—
73	C₂²×C₅	—	2	1	no	—
74	C₂²×C₅	—	2	1	no	—
75	C₂²×C₅	#336	1	1	yes	characteristic
76	C₂²×C₅	#347	1	2	yes	—
77	C₄×C₅	—	2	1	no	—
78	C₄×C₅	—	2	1	no	—
79	C₄×C₅	—	2	1	no	—
80	C₄×C₅	—	2	1	no	—
81	C₄×C₅	#344	1	2	yes	—
82	C₄×C₅	#364	1	2	yes	—
83	C₂²×C₄	#681	1	1	yes	characteristic
84	C₂²×C₄	#681	1	1	yes	characteristic
85	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	—	4	2	no	—
91	C₂×C₈	#683	1	2	yes	—
92	C₂×C₈	#685	1	2	yes	—
93	C₄²	C₂×#168	1	1	yes	characteristic
94	C₄²	C₂×#168	1	1	yes	characteristic
95	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
96	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
97	C₁₆	—	20	4	no	—
98	C₂×C₅	—	2	2	no	—
99	C₂×C₅	#1281	1	1	yes	characteristic
100	C₂×C₅	#1375	1	1	yes	characteristic
101	C₂×C₅	#1376	1	1	yes	characteristic
102	C₂³	#3935	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	C₄×#168	1	1	yes	characteristic
107	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
108	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
111	C₂×C₄	#3963	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	—	4	2	no	—
116	C₅	#7196	1	1	yes	characteristic, complemented, Sylow
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#64686	1	2	yes	—
120	C₂²	#64744	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	#64683	1	2	yes	—
126	C₄	#64776	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	#10598149	1	1	yes	characteristic
129	C₂	#10598152	1	1	yes	characteristic
130	C₂	#10598154	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#322

C₂

yes

maximal, characteristic

#10600538

C₂

yes

maximal

C₂×C₄×C₅×C₈

C₄

yes

characteristic

C₂×C₄×C₅×C₈

C₄

yes

characteristic

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#322

—

C₂²⋊#672

C₄

yes

—

—

—

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₄×C₅×C₈

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₈

yes

—

C₄×C₅×C₈

C₈

yes

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

Q₈

yes

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

C₄²

yes

characteristic

C₂²×C₄×C₅

M₁₆

yes

characteristic

C₂²×C₄×C₅

M₁₆

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄²×C₅

C₂²⋊C₄

yes

characteristic

C₄²×C₅

C₄⋊C₄

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

#672

—

C₂×C₄×C₈

Dic₅

yes

characteristic

C₂×C₄×C₈

Dic₅

yes

characteristic

C₂×C₄×C₈

C₂×D₅

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₂³×C₅

#97

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#95

yes

characteristic

C₂×C₄×C₅

C₄×C₈

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

characteristic

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₅

yes

—

C₂×C₄²

C₂×Dic₅

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#168

yes

—

C₄×C₈

#168

yes

—

C₄×C₈

Dic₁₀

yes

characteristic

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#336

yes

characteristic

C₂²×C₅

#347

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#344

yes

—

C₄×C₅

#364

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₄×Dic₅

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#683

yes

—

C₂×C₈

#685

yes

—

C₄²

yes

C₄²

yes

C₄²

C₄⋊Dic₅

yes

characteristic

C₄²

C₂²⋊Dic₅

yes

characteristic

C₁₆

—

C₂×C₅

—

C₂×C₅

#1281

yes

characteristic

100

C₂×C₅

#1375

yes

characteristic

101

C₂×C₅

#1376

yes

characteristic

102

C₂³

#3935

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₄×#168

yes

characteristic

107

C₂×C₄

C₄⋊#168

yes

characteristic

108

C₂×C₄

C₄⋊#168

yes

characteristic

109

C₂×C₄

C₂²⋊#168

yes

characteristic

110

C₂×C₄

C₂²⋊#168

yes

characteristic

111

C₂×C₄

#3963

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₅

#7196

yes

characteristic, complemented, Sylow

117

C₂²

—

118

C₂²

—

119

C₂²

#64686

yes

—

120

C₂²

#64744

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#64683

yes

—

126

C₄

#64776

yes

—

127

C₂

—

128

C₂

#10598149

yes

characteristic

129

C₂

#10598152

yes

characteristic

130

C₂

#10598154

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#3963
commutator	C₂×C₄×C₅	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅
Fitting	C₂×C₅×#322	C₂
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	#97
cosocle	C₂×C₄×C₅×C₈	C₂²
hypercenter	C₂×#322	D₅
hypocenter	C₅	#7196
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#3963

commutator

C₂×C₄×C₅

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅

Fitting

C₂×C₅×#322

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄×C₅×C₈

C₂²

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

N	H ≅ G/N	type
116. C₅	#7196	semidirect

H ≅ G/N

type

116. C₅

#7196

semidirect

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

subgroup	index
C₂×C₅×#322	2
#10600538	2
#7196	5

subgroup

index

C₂×C₅×#322

#10600538

#7196

Subgroups — #11986277

Conjugacy classes of subgroups

Complements

Maximal subgroups