Group #11998693 — subgroups

Subgroups — #11998693

Order 1280.

370 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 119 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×#3160	C₂	1	1	yes	maximal, characteristic
2	#10602631	C₂	1	4	yes	maximal
3	#10600556	C₂	1	1	yes	maximal, characteristic
4	#10600519	C₂	1	1	yes	maximal, characteristic
5	C₅×#533	C₂²	1	2	yes	—
6	C₅×#500	C₄	1	2	yes	complemented
7	C₅×#499	—	2	2	no	—
8	C₅×#498	C₄	1	2	yes	complemented
9	C₅×#427	—	2	2	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	#65225	C₂²	1	2	yes	—
12	#65214	C₂²	1	2	yes	—
13	#64771	—	2	2	no	—
14	#64758	—	2	8	no	—
15	#64674	—	2	2	no	—
16	#9759	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	D₄	1	2	yes	—
18	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
19	C₅×#125	—	2	2	no	—
20	C₅×C₂²⋊Q₈	—	2	4	no	—
21	C₄×C₅×Q₈	—	2	4	no	—
22	C₅×#117	—	2	2	no	—
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₅×#107	—	2	4	no	—
26	C₅×#106	—	2	4	no	—
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₅×C₈	D₄	1	2	yes	—
30	#3963	—	2	4	no	—
31	C₂²⋊#168	D₄	1	4	yes	—
32	C₄⋊#168	—	2	2	no	—
33	C₄×#168	—	2	2	no	—
34	#3160	D₅	1	1	yes	characteristic
35	#1835	—	5	1	no	—
36	#1518	—	5	4	no	—
37	#1452	—	5	1	no	—
38	C₂×C₅×Q₈	—	2	4	no	—
39	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₅	—	4	2	no	—
42	C₂×C₅×C₈	—	2	4	no	—
43	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
44	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
45	C₅×C₄⋊C₄	—	2	4	no	—
46	C₅×C₄⋊C₄	—	2	4	no	—
47	C₅×C₄⋊C₄	—	4	4	no	—
48	C₅×C₂²⋊C₄	—	4	2	no	—
49	C₄²×C₅	—	4	2	no	—
50	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅	#41	1	2	yes	—
53	C₄⋊Dic₅	—	4	4	no	—
54	C₂×#168	—	2	4	no	—
55	#533	C₂×D₅	1	2	yes	—
56	#500	Dic₅	1	2	yes	complemented
57	#499	—	2	2	no	—
58	#498	Dic₅	1	2	yes	complemented
59	#431	—	5	2	no	—
60	#427	—	2	2	no	—
61	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
62	#382	—	5	2	no	—
63	#340	—	10	2	no	—
64	#328	—	10	8	no	—
65	#322	—	10	2	no	—
66	C₂³×C₅	#127	1	1	yes	characteristic
67	C₅×Q₈	—	8	4	no	—
68	C₂×C₄×C₅	—	4	2	no	—
69	C₂×C₄×C₅	—	4	4	no	—
70	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₅	#121	1	2	yes	—
73	C₂×C₄×C₅	#124	1	1	yes	characteristic
74	C₂×C₄×C₅	#137	1	4	yes	—
75	C₂×Dic₅	—	8	4	no	—
76	C₂×Dic₅	—	8	2	no	—
77	C₅×C₈	—	2	2	no	—
78	C₅×C₈	—	2	2	no	—
79	#168	—	4	4	no	—
80	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
81	C₄⋊Q₈	C₂×Dic₅	1	4	yes	—
82	#125	—	2	2	no	—
83	C₂²⋊Q₈	—	2	4	no	—
84	C₄×Q₈	—	2	4	no	—
85	#117	—	2	2	no	—
86	C₂×C₄⋊C₄	—	10	2	no	—
87	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
88	#107	—	2	4	no	—
89	#106	—	2	4	no	—
90	C₄⋊C₈	—	10	2	no	—
91	C₂²⋊C₈	—	5	4	no	—
92	C₄×C₈	—	10	2	no	—
93	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
94	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
95	C₄×C₈	C₂²⋊D₅	1	2	yes	—
96	#95	—	10	4	no	—
97	C₂²×C₅	—	2	2	no	—
98	C₂²×C₅	#390	1	1	yes	characteristic
99	C₂²×C₅	#497	1	2	yes	—
100	C₄×C₅	—	2	2	no	—
101	C₄×C₅	—	2	2	no	—
102	C₄×C₅	—	8	4	no	—
103	C₄×C₅	#419	1	2	yes	—
104	C₄×C₅	#491	1	2	yes	—
105	Dic₅	—	16	2	no	—
106	C₂×Q₈	—	2	4	no	—
107	C₂²×C₄	—	20	2	no	—
108	C₂²×C₄	D₄×D₅	1	2	yes	—
109	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	10	4	no	—
112	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
113	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	4	4	no	—
117	C₄⋊C₄	—	20	4	no	—
118	C₂²⋊C₄	—	4	2	no	—
119	C₄²	—	4	2	no	—
120	C₄²	#711	1	2	yes	—
121	C₄²	C₂²×Dic₅	1	1	yes	characteristic
122	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
123	C₂×C₅	—	2	2	no	—
124	C₂×C₅	#1960	1	1	yes	characteristic
125	C₂×C₅	#1976	1	2	yes	—
126	C₂³	#4086	1	1	yes	characteristic
127	Q₈	—	8	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	40	4	no	—
131	C₂×C₄	—	40	2	no	—
132	C₂×C₄	D₄×Dic₅	1	2	yes	—
133	C₂×C₄	#4082	1	1	yes	characteristic
134	C₂×C₄	#4084	1	2	yes	—
135	C₂×C₄	#4088	1	4	yes	—
136	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	—	20	4	no	—
140	C₅	#9759	1	1	yes	characteristic, complemented, Sylow
141	C₂²	—	2	2	no	—
142	C₂²	#65375	1	2	yes	—
143	C₂²	#65510	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	8	4	no	—
147	C₄	—	80	2	no	—
148	C₄	#65371	1	2	yes	—
149	C₄	#65514	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#10602579	1	1	yes	characteristic
152	C₂	#10602879	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#3160

C₂

yes

maximal, characteristic

#10602631

C₂

yes

maximal

#10600556

C₂

yes

maximal, characteristic

#10600519

C₂

yes

maximal, characteristic

C₅×#533

C₂²

yes

—

C₅×#500

C₄

yes

complemented

C₅×#499

—

C₅×#498

C₄

yes

complemented

C₅×#427

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

#65225

C₂²

yes

—

#65214

C₂²

yes

—

—

—

—

—

C₂²×C₅×C₈

D₄

yes

—

C₅×C₄⋊Q₈

C₂×C₄

yes

—

C₅×#125

—

C₅×C₂²⋊Q₈

—

C₄×C₅×Q₈

—

C₅×#117

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₅×#107

—

C₅×#106

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

—

#3963

—

C₂²⋊#168

D₄

yes

—

C₄⋊#168

—

C₄×#168

—

#3160

D₅

yes

—

—

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

#41

yes

—

C₄⋊Dic₅

—

C₂×#168

—

#533

C₂×D₅

yes

—

#500

Dic₅

yes

complemented

#499

—

#498

Dic₅

yes

complemented

#431

—

#427

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

—

C₂³×C₅

#127

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

—

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

#124

yes

characteristic

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

—

#168

—

C₂²×C₈

C₂²⋊D₅

yes

—

C₄⋊Q₈

C₂×Dic₅

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₅

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×Dic₅

yes

characteristic

C₄×C₈

C₂×Dic₅

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

—

#95

—

C₂²×C₅

—

C₂²×C₅

#390

yes

characteristic

C₂²×C₅

#497

yes

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

—

103

C₄×C₅

#419

yes

—

104

C₄×C₅

#491

yes

—

105

Dic₅

—

106

C₂×Q₈

—

107

C₂²×C₄

—

108

C₂²×C₄

D₄×D₅

yes

—

109

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²⋊Dic₅

yes

—

113

C₂×C₈

C₂²⋊Dic₅

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

#711

yes

—

121

C₄²

C₂²×Dic₅

yes

characteristic

122

C₄²

C₂×C₂²⋊D₅

yes

characteristic

123

C₂×C₅

—

124

C₂×C₅

#1960

yes

characteristic

125

C₂×C₅

#1976

yes

—

126

C₂³

#4086

yes

characteristic

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

D₄×Dic₅

yes

—

133

C₂×C₄

#4082

yes

characteristic

134

C₂×C₄

#4084

yes

—

135

C₂×C₄

#4088

yes

—

136

C₂×C₄

C₂×C₂²⋊Dic₅

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₅

#9759

yes

characteristic, complemented, Sylow

141

C₂²

—

142

C₂²

#65375

yes

—

143

C₂²

#65510

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#65371

yes

—

149

C₄

#65514

yes

—

150

C₂

—

151

C₂

#10602579

yes

characteristic

152

C₂

#10602879

yes

—

	subgroup	G/H
center	C₂²	#65510
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#3160	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#3160	D₅
hypocenter	C₅	#9759
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂²

#65510

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
6. C₅×#500	C₄	semidirect
8. C₅×#498	C₄	semidirect
56. #500	Dic₅	semidirect
58. #498	Dic₅	semidirect
140. C₅	#9759	semidirect

H ≅ G/N

type

6. C₅×#500

C₄

semidirect

8. C₅×#498

C₄

semidirect

56. #500

Dic₅

semidirect

58. #498

Dic₅

semidirect

140. C₅

#9759

semidirect

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

subgroup	index
C₅×#3160	2
#10602631	2
#10600556	2
#10600519	2
#9759	5

subgroup

index

Subgroups — #11998693

Conjugacy classes of subgroups

Complements

Maximal subgroups