Group #14298919 — subgroups

Subgroups — C₅×#69841 #14298919

Order 1920.

360 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 186 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×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₅×#4924	C₂	1	2	yes	maximal
3	C₅×#4552	C₂	1	4	yes	maximal
4	C₅×#1838	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
6	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₅×#208	C₂²	1	4	yes	—
8	C₃×C₅×#97	C₄	1	4	yes	—
9	C₅×C₄⋊(C₃⋊C₈)	—	2	4	no	—
10	#69841	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₅×#423	—	3	2	no	—
12	C₂×C₅×#97	D₃	1	1	yes	characteristic
13	C₅×#341	—	3	4	no	—
14	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
16	C₂×C₃×C₅×C₈	—	2	4	no	—
17	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
18	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
19	C₃×C₄²×C₅	D₄	1	2	yes	—
20	C₅×#208	—	2	8	no	—
21	C₂×C₅×C₃⋊C₈	—	2	4	no	—
22	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
23	#4924	C₂×C₅	1	2	yes	—
24	#4552	C₂×C₅	1	4	yes	—
25	C₂×C₅×M₁₆	—	3	4	no	—
26	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
27	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
28	C₅×C₄⋊C₈	—	6	4	no	—
29	C₅×#97	Dic₃	1	4	yes	—
30	#1838	—	3	1	no	Sylow
31	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₃×C₄×C₅	—	4	2	no	—
33	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
34	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₅	#41	1	4	yes	—
37	C₃×C₅×C₈	—	2	4	no	—
38	C₅×C₃⋊C₈	—	4	4	no	—
39	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
40	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
41	C₂×#208	C₂²×C₅	1	4	yes	—
42	C₃×#97	C₄×C₅	1	4	yes	—
43	C₄⋊(C₃⋊C₈)	—	2	4	no	—
44	C₂²×C₄×C₅	Dic₆	1	2	yes	—
45	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
46	C₅×M₁₆	—	6	8	no	—
47	C₂×C₅×C₈	—	2	4	no	—
48	C₂×C₅×C₈	—	6	4	no	—
49	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
50	C₄²×C₅	D₁₂	1	2	yes	—
51	C₄²×C₅	C₂×Dic₃	1	2	yes	—
52	#423	—	3	2	no	—
53	C₂×#97	C₅×D₃	1	1	yes	characteristic
54	#341	—	3	4	no	—
55	C₂²×C₃×C₅	—	2	2	no	—
56	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₃×C₅	#125	1	2	yes	—
58	C₃×C₄×C₅	—	2	4	no	—
59	C₃×C₄×C₅	#117	1	2	yes	—
60	C₃×C₄×C₅	#124	1	2	yes	—
61	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
62	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
63	C₂×C₃×C₈	—	2	4	no	—
64	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
65	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
66	C₃×C₄²	C₅×D₄	1	2	yes	—
67	#208	—	2	8	no	—
68	C₂×C₃⋊C₈	—	2	4	no	—
69	C₂³×C₅	C₂×Dic₆	1	1	yes	characteristic
70	C₂×C₄×C₅	—	4	2	no	—
71	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
72	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
73	C₂×C₄×C₅	#235	1	4	yes	—
74	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
75	C₅×C₈	—	2	4	no	—
76	C₅×C₈	—	12	4	no	—
77	C₂×M₁₆	—	3	4	no	—
78	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
79	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
80	C₄⋊C₈	—	6	4	no	—
81	#97	C₅×Dic₃	1	4	yes	—
82	C₂×C₃×C₅	—	2	2	no	—
83	C₂×C₃×C₅	#382	1	1	yes	characteristic
84	C₂×C₃×C₅	#430	1	2	yes	—
85	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
86	C₂×C₃×C₄	—	4	2	no	—
87	C₂×C₃×C₄	C₅×C₄⋊C₄	1	4	yes	—
88	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄	C₅×#41	1	4	yes	—
91	C₃×C₈	—	2	4	no	—
92	C₃⋊C₈	—	4	4	no	—
93	C₂²×C₅	—	2	2	no	—
94	C₂²×C₅	#870	1	2	yes	—
95	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
96	C₄×C₅	—	2	4	no	—
97	C₄×C₅	#875	1	2	yes	—
98	C₄×C₅	#926	1	2	yes	—
99	C₂²×C₄	C₅×Dic₆	1	2	yes	—
100	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
101	M₁₆	—	6	8	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	6	4	no	—
104	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	—
105	C₄²	C₅×D₁₂	1	2	yes	—
106	C₄²	C₂×C₅×Dic₃	1	2	yes	—
107	C₃×C₅	#1838	1	1	yes	characteristic, complemented, Hall
108	C₂²×C₃	—	2	2	no	—
109	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₃	C₅×#125	1	2	yes	—
111	C₃×C₄	—	2	4	no	—
112	C₃×C₄	C₅×#117	1	2	yes	—
113	C₃×C₄	C₅×#124	1	2	yes	—
114	C₂×C₅	—	2	2	no	—
115	C₂×C₅	#4936	1	1	yes	characteristic
116	C₂×C₅	#5125	1	2	yes	—
117	C₂³	C₂×C₅×Dic₆	1	1	yes	characteristic
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₅×C₄⋊Dic₃	1	4	yes	—
120	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
121	C₂×C₄	C₅×#235	1	4	yes	—
122	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
123	C₈	—	2	4	no	—
124	C₈	—	12	4	no	—
125	C₂×C₃	—	2	2	no	—
126	C₂×C₃	C₅×#382	1	1	yes	characteristic
127	C₂×C₃	C₅×#430	1	2	yes	—
128	C₅	#69841	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²	—	2	2	no	—
130	C₂²	C₅×#870	1	2	yes	—
131	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
132	C₄	—	2	4	no	—
133	C₄	C₅×#875	1	2	yes	—
134	C₄	C₅×#926	1	2	yes	—
135	C₃	C₅×#1838	1	1	yes	characteristic, complemented, Sylow
136	C₂	—	2	2	no	—
137	C₂	C₅×#4936	1	1	yes	characteristic
138	C₂	C₅×#5125	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₅×#97

C₂

yes

maximal, characteristic

C₅×#4924

C₂

yes

maximal

C₅×#4552

C₂

yes

maximal

C₅×#1838

—

maximal, Hall

C₂²×C₃×C₅×C₈

C₂²

yes

—

C₂×C₃×C₄²×C₅

C₂²

yes

characteristic

C₂×C₅×#208

C₂²

yes

—

C₃×C₅×#97

C₄

yes

—

C₅×C₄⋊(C₃⋊C₈)

—

#69841

C₅

yes

maximal, characteristic, direct product factor, Hall

C₅×#423

—

C₂×C₅×#97

D₃

yes

characteristic

C₅×#341

—

C₂²×C₃×C₄×C₅

Q₈

yes

—

C₂²×C₃×C₄×C₅

C₂³

yes

characteristic

C₂×C₃×C₅×C₈

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

—

C₃×C₄²×C₅

C₂×C₄

yes

—

C₃×C₄²×C₅

D₄

yes

—

C₅×#208

—

C₂×C₅×C₃⋊C₈

—

C₂×C₃×#97

C₂×C₅

yes

characteristic

#4924

C₂×C₅

yes

—

#4552

C₂×C₅

yes

—

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

C₂×D₃

yes

—

C₂×C₄²×C₅

C₂×D₃

yes

characteristic

C₅×C₄⋊C₈

—

C₅×#97

Dic₃

yes

—

#1838

—

Sylow

C₂³×C₃×C₅

C₂×Q₈

yes

characteristic

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

#41

yes

—

C₃×C₅×C₈

—

C₅×C₃⋊C₈

—

C₂²×C₃×C₈

C₂²×C₅

yes

—

C₂×C₃×C₄²

C₂²×C₅

yes

characteristic

C₂×#208

C₂²×C₅

yes

—

C₃×#97

C₄×C₅

yes

—

C₄⋊(C₃⋊C₈)

—

C₂²×C₄×C₅

Dic₆

yes

—

C₂²×C₄×C₅

C₂²×D₃

yes

characteristic

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×Dic₃

yes

—

C₄²×C₅

D₁₂

yes

—

C₄²×C₅

C₂×Dic₃

yes

—

#423

—

C₂×#97

C₅×D₃

yes

characteristic

#341

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃×C₅

#125

yes

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#117

yes

—

C₃×C₄×C₅

#124

yes

—

C₂²×C₃×C₄

C₅×Q₈

yes

—

C₂²×C₃×C₄

C₂³×C₅

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₄×C₅

yes

—

C₃×C₄²

C₂×C₄×C₅

yes

—

C₃×C₄²

C₅×D₄

yes

—

#208

—

C₂×C₃⋊C₈

—

C₂³×C₅

C₂×Dic₆

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄⋊Dic₃

yes

—

C₂×C₄×C₅

C₂×D₁₂

yes

characteristic

C₂×C₄×C₅

#235

yes

—

C₂×C₄×C₅

C₂²×Dic₃

yes

characteristic

C₅×C₈

—

C₅×C₈

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₅×D₃

yes

—

C₂×C₄²

C₂×C₅×D₃

yes

characteristic

C₄⋊C₈

—

#97

C₅×Dic₃

yes

—

C₂×C₃×C₅

—

C₂×C₃×C₅

#382

yes

characteristic

C₂×C₃×C₅

#430

yes

—

C₂³×C₃

C₂×C₅×Q₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₅×C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄×C₅

yes

characteristic

C₂×C₃×C₄

C₂×C₅×D₄

yes

characteristic

C₂×C₃×C₄

C₅×#41

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₅

—

C₂²×C₅

#870

yes

—

C₂²×C₅

C₂×C₄⋊Dic₃

yes

characteristic

C₄×C₅

—

C₄×C₅

#875

yes

—

C₄×C₅

#926

yes

—

C₂²×C₄

C₅×Dic₆

yes

—

100

C₂²×C₄

C₂²×C₅×D₃

yes

characteristic

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂×C₅×Dic₃

yes

—

105

C₄²

C₅×D₁₂

yes

—

106

C₄²

C₂×C₅×Dic₃

yes

—

107

C₃×C₅

#1838

yes

characteristic, complemented, Hall

108

C₂²×C₃

—

109

C₂²×C₃

C₂×C₅×C₄⋊C₄

yes

characteristic

110

C₂²×C₃

C₅×#125

yes

—

111

C₃×C₄

—

112

C₃×C₄

C₅×#117

yes

—

113

C₃×C₄

C₅×#124

yes

—

114

C₂×C₅

—

115

C₂×C₅

#4936

yes

characteristic

116

C₂×C₅

#5125

yes

—

117

C₂³

C₂×C₅×Dic₆

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

C₅×C₄⋊Dic₃

yes

—

120

C₂×C₄

C₂×C₅×D₁₂

yes

characteristic

121

C₂×C₄

C₅×#235

yes

—

122

C₂×C₄

C₂²×C₅×Dic₃

yes

characteristic

123

C₈

—

124

C₈

—

125

C₂×C₃

—

126

C₂×C₃

C₅×#382

yes

characteristic

127

C₂×C₃

C₅×#430

yes

—

128

C₅

#69841

yes

characteristic, direct product factor, Sylow

129

C₂²

—

130

C₂²

C₅×#870

yes

—

131

C₂²

C₂×C₅×C₄⋊Dic₃

yes

characteristic

132

C₄

—

133

C₄

C₅×#875

yes

—

134

C₄

C₅×#926

yes

—

135

C₃

C₅×#1838

yes

characteristic, complemented, Sylow

136

C₂

—

137

C₂

C₅×#4936

yes

characteristic

138

C₂

C₅×#5125

yes

—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#97	D₃
hypocenter	C₃	C₅×#1838
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#97

subgroup

G/H

center

C₂×C₄×C₅

C₂×D₁₂

commutator

C₂×C₃×C₄

C₂²×C₄×C₅

Frattini

C₂²×C₄

C₂²×C₅×D₃

Fitting

C₂×C₃×C₅×#97

C₂

socle

C₂²×C₃×C₅

C₂×C₄⋊C₄

prime radical

C₂³×C₃×C₅

C₂×Q₈

cosocle

C₂²×C₃×C₄

C₂³×C₅

hypercenter

C₂×C₅×#97

D₃

hypocenter

C₃

C₅×#1838

LCS

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#97

N	H ≅ G/N	type
10. #69841	C₅	direct (N × H)
107. C₃×C₅	#1838	semidirect
128. C₅	#69841	direct (N × H)
135. C₃	C₅×#1838	semidirect

H ≅ G/N

type

10. #69841

C₅

direct (N × H)

107. C₃×C₅

#1838

semidirect

128. C₅

#69841

direct (N × H)

135. C₃

C₅×#1838

semidirect

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

subgroup	index
C₂×C₃×C₅×#97	2
C₅×#4924	2
C₅×#4552	2
C₅×#1838	3
#69841	5

subgroup

index

C₂×C₃×C₅×#97

Subgroups — C₅×#69841 #14298919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#69841 #14298919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69841 #14298919