Group #14278564 — subgroups

Subgroups — #14278564

Order 1920.

148 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 66 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	C₃×#64972	C₂	1	1	yes	maximal, characteristic
2	#11746284	C₂	1	1	yes	maximal, characteristic
3	C₅×#4463	C₂	1	1	yes	maximal, characteristic
4	#10598065	—	3	1	no	maximal, Hall
5	C₃×#4016	C₄	1	2	yes	—
6	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	#91096	—	2	1	no	—
8	C₅×#812	—	2	1	no	—
9	C₃×#3950	—	2	1	no	—
10	C₃×#3949	—	2	1	no	—
11	#67800	—	5	1	no	maximal, Hall
12	#64972	D₃	1	1	yes	characteristic
13	C₅×#364	—	3	1	no	—
14	#64683	—	3	1	no	—
15	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
16	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
17	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₃×C₄⋊Dic₅	—	2	2	no	—
19	C₃×#683	—	4	2	no	—
20	C₃⋊#672	—	4	1	no	—
21	C₅×C₃⋊C₁₆	—	4	1	no	—
22	C₃×#499	—	5	1	no	—
23	#4463	—	5	1	no	—
24	#4463	D₅	1	1	yes	characteristic
25	#4016	Dic₃	1	2	yes	—
26	C₅×M₃₂	—	6	1	no	—
27	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
28	#3950	—	2	1	no	—
29	#3949	—	2	1	no	—
30	#3944	—	6	1	no	—
31	#1366	—	15	1	no	Sylow
32	C₂×C₃×C₄×C₅	—	2	1	no	—
33	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₃×Dic₅	—	4	2	no	—
35	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
36	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
37	C₃×C₅×C₈	Q₁₆	1	1	yes	characteristic
38	C₃×C₅×C₈	Q₁₆	1	1	yes	characteristic
39	C₃×#125	—	5	2	no	—
40	C₃×#107	—	10	1	no	—
41	C₃×#106	—	10	1	no	—
42	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
43	#812	—	2	1	no	—
44	#812	—	10	1	no	—
45	C₂×C₅×C₈	#67	1	1	yes	characteristic
46	C₂×C₅×C₈	#67	1	1	yes	characteristic
47	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
48	C₄⋊Dic₅	—	2	2	no	—
49	#683	—	4	2	no	—
50	C₅×C₁₆	—	12	1	no	—
51	#672	—	12	1	no	—
52	#499	—	5	1	no	—
53	#364	—	3	1	no	—
54	#364	—	15	1	no	—
55	C₂²×C₃×C₅	#100	1	1	yes	characteristic
56	C₃×C₄×C₅	—	4	1	no	—
57	C₃×C₄×C₅	#103	1	1	yes	characteristic
58	C₃×C₄×C₅	#103	1	1	yes	characteristic
59	C₃×Dic₅	—	8	2	no	—
60	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
62	C₃×C₄⋊C₄	—	10	2	no	—
63	C₃×C₄⋊C₄	—	20	2	no	—
64	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
65	C₃⋊C₁₆	—	4	1	no	—
66	C₃⋊C₁₆	—	20	1	no	—
67	C₂×C₄×C₅	—	2	1	no	—
68	C₂×C₄×C₅	#217	1	1	yes	characteristic
69	C₂×Dic₅	—	4	2	no	—
70	C₅×C₈	#215	1	1	yes	characteristic
71	C₅×C₈	#215	1	1	yes	characteristic
72	C₅×C₈	#216	1	1	yes	characteristic
73	C₅×C₈	#216	1	1	yes	characteristic
74	#125	—	5	2	no	—
75	M₃₂	—	6	1	no	—
76	M₃₂	—	30	1	no	—
77	#107	—	10	1	no	—
78	#106	—	10	1	no	—
79	C₄×C₈	D₃×D₅	1	1	yes	characteristic
80	C₂×C₃×C₅	—	2	1	no	—
81	C₂×C₃×C₅	#333	1	1	yes	characteristic
82	C₂×C₃×C₄	—	2	1	no	—
83	C₂×C₃×C₄	—	20	2	no	—
84	C₂×C₃×C₄	#685	1	1	yes	characteristic
85	C₃×C₈	#677	1	1	yes	characteristic
86	C₃×C₈	Dic₂₀	1	1	yes	characteristic
87	C₃×C₈	#687	1	1	yes	characteristic
88	C₃×C₈	#689	1	1	yes	characteristic
89	C₂²×C₅	#833	1	1	yes	characteristic
90	C₄×C₅	—	4	1	no	—
91	C₄×C₅	#835	1	1	yes	characteristic
92	C₄×C₅	#835	1	1	yes	characteristic
93	Dic₅	—	8	2	no	—
94	C₂×C₈	#1201	1	1	yes	characteristic
95	C₂×C₈	#1202	1	1	yes	characteristic
96	C₄⋊C₄	—	10	2	no	—
97	C₄⋊C₄	—	20	2	no	—
98	C₄²	D₅×Dic₃	1	1	yes	characteristic
99	C₁₆	—	12	1	no	—
100	C₁₆	—	60	1	no	—
101	C₃×C₅	#1366	1	1	yes	characteristic, complemented, Hall
102	C₂²×C₃	#3955	1	1	yes	characteristic
103	C₃×C₄	—	4	1	no	—
104	C₃×C₄	—	40	2	no	—
105	C₃×C₄	#3942	1	1	yes	characteristic
106	C₃×C₄	#3948	1	1	yes	characteristic
107	C₂×C₅	—	2	1	no	—
108	C₂×C₅	#4546	1	1	yes	characteristic
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	20	2	no	—
111	C₂×C₄	#6665	1	1	yes	characteristic
112	C₈	#6656	1	1	yes	characteristic
113	C₈	#6658	1	1	yes	characteristic
114	C₈	#6661	1	1	yes	characteristic
115	C₈	#6662	1	1	yes	characteristic
116	C₂×C₃	—	2	1	no	—
117	C₂×C₃	#64701	1	1	yes	characteristic
118	C₅	#67800	1	1	yes	characteristic, complemented, Sylow
119	C₂²	#90958	1	1	yes	characteristic
120	C₄	—	4	1	no	—
121	C₄	—	40	2	no	—
122	C₄	#90970	1	1	yes	characteristic
123	C₄	#90971	1	1	yes	characteristic
124	C₃	#10598065	1	1	yes	characteristic, complemented, Sylow
125	C₂	—	2	1	no	—
126	C₂	#11745861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#64972

C₂

yes

maximal, characteristic

#11746284

C₂

yes

maximal, characteristic

C₅×#4463

C₂

yes

maximal, characteristic

#10598065

—

maximal, Hall

C₃×#4016

C₄

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic

C₅×#364

—

#64683

—

C₂×C₃×C₅×C₈

D₄

yes

characteristic

C₂×C₃×C₅×C₈

D₄

yes

characteristic

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₃×C₄⋊Dic₅

—

C₃×#683

—

C₃⋊#672

—

C₅×C₃⋊C₁₆

—

C₃×#499

—

#4463

—

#4463

D₅

yes

characteristic

#4016

Dic₃

yes

—

C₅×M₃₂

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

—

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₃×Dic₅

—

C₃×C₅×C₈

SD₁₆

yes

characteristic

C₃×C₅×C₈

SD₁₆

yes

characteristic

C₃×C₅×C₈

Q₁₆

yes

characteristic

C₃×C₅×C₈

Q₁₆

yes

—

—

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#812

—

#812

—

C₂×C₅×C₈

yes

C₂×C₅×C₈

yes

C₄²×C₅

C₂×Dic₃

yes

characteristic

C₄⋊Dic₅

—

#683

—

C₅×C₁₆

—

—

—

—

—

C₂²×C₃×C₅

#100

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

C₃×Dic₅

—

C₂×C₃×C₈

D₂₀

yes

characteristic

C₂×C₃×C₈

C₂²⋊D₅

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₄×D₅

yes

characteristic

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#217

yes

characteristic

C₂×Dic₅

—

C₅×C₈

yes

C₅×C₈

yes

C₅×C₈

yes

C₅×C₈

yes

—

M₃₂

—

M₃₂

—

#107

—

#106

—

C₄×C₈

D₃×D₅

yes

characteristic

C₂×C₃×C₅

—

C₂×C₃×C₅

#333

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

yes

C₃×C₈

yes

C₃×C₈

Dic₂₀

yes

C₃×C₈

yes

C₃×C₈

yes

C₂²×C₅

yes

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₅×Dic₃

yes

characteristic

C₁₆

—

100

C₁₆

—

101

C₃×C₅

#1366

yes

characteristic, complemented, Hall

102

C₂²×C₃

#3955

yes

characteristic

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

#3942

yes

characteristic

106

C₃×C₄

#3948

yes

characteristic

107

C₂×C₅

—

108

C₂×C₅

#4546

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#6665

yes

characteristic

112

C₈

#6656

yes

characteristic

113

C₈

#6658

yes

characteristic

114

C₈

#6661

yes

characteristic

115

C₈

#6662

yes

characteristic

116

C₂×C₃

—

117

C₂×C₃

#64701

yes

characteristic

118

C₅

#67800

yes

characteristic, complemented, Sylow

119

C₂²

#90958

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

#90970

yes

characteristic

123

C₄

#90971

yes

characteristic

124

C₃

#10598065

yes

characteristic, complemented, Sylow

125

C₂

—

126

C₂

#11745861

yes

characteristic

	subgroup	G/H
center	C₂	#11745861
commutator	C₃×C₄²×C₅	C₂×C₄
Frattini, hypercenter	C₄×C₈	D₃×D₅
Fitting, cosocle	C₃×C₄×C₅×C₈	C₂²
socle	C₂×C₃×C₅	#333
prime radical	C₂²×C₃×C₅	#100
hypocenter	C₃×C₅	#1366
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center

C₂

#11745861

commutator

C₃×C₄²×C₅

C₂×C₄

Frattini, hypercenter

C₄×C₈

D₃×D₅

Fitting, cosocle

C₃×C₄×C₅×C₈

C₂²

C₂×C₃×C₅

C₂²×C₃×C₅

C₃×C₅

G ▷ C₃×C₄²×C₅ ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
101. C₃×C₅	#1366	semidirect
118. C₅	#67800	semidirect
124. C₃	#10598065	semidirect

H ≅ G/N

type

101. C₃×C₅

#1366

semidirect

118. C₅

#67800

semidirect

124. C₃

#10598065

semidirect

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

subgroup	index
C₃×#64972	2
#11746284	2
C₅×#4463	2
#10598065	3
#67800	5

subgroup

index

Subgroups — #14278564

Conjugacy classes of subgroups

Complements

Maximal subgroups