Group #72254992 — subgroups

Subgroups — #72254992

Order 3072.

126 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 65 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×#93878	C₂	1	1	yes	maximal, characteristic
2	#72245357	C₂	1	1	yes	maximal, characteristic
3	#72244930	C₂	1	1	yes	maximal, characteristic
4	#72224648	—	3	1	no	maximal, Sylow
5	#10682994	C₄	1	1	yes	characteristic
6	#10682994	C₄	1	1	yes	characteristic
7	#10681925	—	2	1	no	—
8	C₃×#7456	—	2	1	no	—
9	C₃×#7330	C₂²	1	1	yes	characteristic
10	#94506	—	3	1	no	—
11	#93878	D₃	1	1	yes	characteristic
12	#93877	—	3	1	no	—
13	C₃×#1422	—	4	1	no	—
14	C₃×#1397	C₂×C₄	1	1	yes	characteristic
15	C₃×#1397	Q₈	1	1	yes	characteristic
16	C₃×#1311	D₄	1	1	yes	characteristic
17	#67587	—	2	1	no	—
18	#67587	C₈	1	1	yes	characteristic
19	#67587	C₈	1	1	yes	characteristic
20	#67439	—	4	1	no	—
21	#7511	—	3	1	no	—
22	#7511	—	3	1	no	—
23	#7456	—	2	1	no	—
24	#7330	C₂×D₃	1	1	yes	characteristic
25	#7099	—	6	1	no	—
26	C₃×M₆₄	—	8	1	no	—
27	C₂×C₃×C₃₂	—	2	1	no	—
28	C₂×C₃×C₃₂	C₂×C₈	1	1	yes	characteristic
29	C₂×C₃×C₃₂	M₁₆	1	1	yes	characteristic
30	C₃×C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
31	C₃×C₄×C₁₆	SD₁₆	1	1	yes	characteristic
32	C₃×C₈²	SD₁₆	1	1	yes	characteristic
33	C₄⋊(C₃⋊C₁₆)	—	8	1	no	—
34	C₃⋊C₆₄	—	2	2	no	—
35	C₃⋊C₆₄	—	2	1	no	—
36	C₃⋊C₆₄	C₁₆	1	2	yes	complemented
37	M₁₂₈	—	3	1	no	—
38	M₁₂₈	—	3	1	no	—
39	M₁₂₈	—	6	1	no	—
40	#1422	—	4	1	no	—
41	#1397	Dic₆	1	1	yes	characteristic
42	#1397	C₄×D₃	1	1	yes	characteristic
43	#1367	—	12	1	no	—
44	#1311	#67	1	1	yes	characteristic
45	C₂×C₃×C₁₆	—	2	1	no	—
46	C₂×C₃×C₁₆	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₃×C₁₆	#108	1	1	yes	characteristic
48	C₃×C₄×C₈	—	2	1	no	—
49	C₃×C₄×C₈	#106	1	1	yes	characteristic
50	C₂×C₃⋊C₁₆	—	16	1	no	—
51	C₃×C₃₂	—	2	1	no	—
52	C₃×C₃₂	—	4	1	no	—
53	C₃×C₃₂	—	16	1	no	—
54	C₃×C₃₂	C₂×C₁₆	1	1	yes	characteristic
55	C₃×C₃₂	M₃₂	1	1	yes	characteristic
56	M₆₄	—	8	1	no	—
57	C₂×C₃₂	—	2	1	no	—
58	C₂×C₃₂	D₃×C₈	1	1	yes	characteristic
59	C₂×C₃₂	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
60	C₄⋊C₁₆	—	24	1	no	—
61	C₄×C₁₆	Dic₃⋊C₄	1	1	yes	characteristic
62	C₄×C₁₆	#214	1	1	yes	characteristic
63	C₈²	#215	1	1	yes	characteristic
64	C₆₄	—	3	2	no	—
65	C₆₄	—	6	2	no	—
66	C₆₄	—	6	1	no	—
67	C₂×C₃×C₈	—	4	1	no	—
68	C₂×C₃×C₈	#334	1	1	yes	characteristic
69	C₂×C₃×C₈	#365	1	1	yes	characteristic
70	C₃×C₄²	#365	1	1	yes	characteristic
71	C₃×C₁₆	—	2	1	no	—
72	C₃×C₁₆	—	4	1	no	—
73	C₃×C₁₆	C₄⋊C₁₆	1	1	yes	characteristic
74	C₃×C₁₆	#364	1	1	yes	characteristic
75	C₃⋊C₁₆	—	32	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	48	1	no	—
78	C₂×C₁₆	Dic₃⋊C₈	1	1	yes	characteristic
79	C₂×C₁₆	#822	1	1	yes	characteristic
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	#808	1	1	yes	characteristic
82	C₃₂	—	2	1	no	—
83	C₃₂	—	4	1	no	—
84	C₃₂	—	16	1	no	—
85	C₃₂	D₃×C₁₆	1	1	yes	characteristic
86	C₃₂	#798	1	1	yes	characteristic
87	C₂×C₃×C₄	—	2	1	no	—
88	C₂×C₃×C₄	#1368	1	1	yes	characteristic
89	C₃×C₈	—	2	1	no	—
90	C₃×C₈	—	8	1	no	—
91	C₃×C₈	#1367	1	1	yes	characteristic
92	C₃×C₈	#1369	1	1	yes	characteristic
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	#4480	1	1	yes	characteristic
95	C₂×C₈	#4487	1	1	yes	characteristic
96	C₄²	#4494	1	1	yes	characteristic
97	C₁₆	—	2	1	no	—
98	C₁₆	—	4	1	no	—
99	C₁₆	—	96	1	no	—
100	C₁₆	Dic₃⋊C₁₆	1	1	yes	characteristic
101	C₁₆	#4511	1	1	yes	characteristic
102	C₂²×C₃	#7457	1	1	yes	characteristic
103	C₃×C₄	—	4	1	no	—
104	C₃×C₄	#7099	1	1	yes	characteristic
105	C₃×C₄	#7457	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	#67464	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	8	1	no	—
110	C₈	#67554	1	1	yes	characteristic
111	C₈	#67567	1	1	yes	characteristic
112	C₂×C₃	—	2	1	no	—
113	C₂×C₃	#93877	1	1	yes	characteristic
114	C₂²	#10682858	1	1	yes	characteristic
115	C₄	—	4	1	no	—
116	C₄	#10681924	1	1	yes	characteristic
117	C₄	#10682856	1	1	yes	characteristic
118	C₃	#72224648	1	1	yes	characteristic, complemented, Sylow
119	C₂	—	2	1	no	—
120	C₂	#72244942	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#93878

C₂

yes

maximal, characteristic

#72245357

C₂

yes

maximal, characteristic

#72244930

C₂

yes

maximal, characteristic

#72224648

—

maximal, Sylow

#10682994

C₄

yes

characteristic

#10682994

C₄

yes

—

—

C₂²

yes

characteristic

#94506

—

#93878

D₃

yes

—

—

C₂×C₄

yes

characteristic

C₃×#1397

Q₈

yes

characteristic

C₃×#1311

D₄

yes

characteristic

#67587

—

#67587

C₈

yes

characteristic

#67587

C₈

yes

—

—

—

—

C₂×D₃

yes

characteristic

#7099

—

C₃×M₆₄

—

C₂×C₃×C₃₂

—

C₂×C₃×C₃₂

C₂×C₈

yes

characteristic

C₂×C₃×C₃₂

M₁₆

yes

characteristic

C₃×C₄×C₁₆

C₄⋊C₄

yes

characteristic

C₃×C₄×C₁₆

SD₁₆

yes

characteristic

C₃×C₈²

SD₁₆

yes

characteristic

C₄⋊(C₃⋊C₁₆)

—

C₃⋊C₆₄

—

C₃⋊C₆₄

—

C₃⋊C₆₄

C₁₆

yes

complemented

M₁₂₈

—

M₁₂₈

—

M₁₂₈

—

#1422

—

#1397

Dic₆

yes

characteristic

#1397

C₄×D₃

yes

—

yes

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

C₄⋊C₈

yes

characteristic

C₂×C₃×C₁₆

#108

yes

characteristic

C₃×C₄×C₈

—

C₃×C₄×C₈

#106

yes

characteristic

C₂×C₃⋊C₁₆

—

C₃×C₃₂

—

C₃×C₃₂

—

C₃×C₃₂

—

C₃×C₃₂

C₂×C₁₆

yes

characteristic

C₃×C₃₂

M₃₂

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

D₃×C₈

yes

characteristic

C₂×C₃₂

ΓL(1,𝔽₂₅)

yes

characteristic

C₄⋊C₁₆

—

C₄×C₁₆

Dic₃⋊C₄

yes

C₄×C₁₆

yes

C₈²

yes

C₆₄

—

C₆₄

—

C₆₄

—

C₂×C₃×C₈

—

C₂×C₃×C₈

yes

C₂×C₃×C₈

yes

C₃×C₄²

yes

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

C₄⋊C₁₆

yes

characteristic

C₃×C₁₆

#364

yes

characteristic

C₃⋊C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Dic₃⋊C₈

yes

characteristic

C₂×C₁₆

#822

yes

characteristic

C₄×C₈

—

C₄×C₈

#808

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

C₃₂

D₃×C₁₆

yes

characteristic

C₃₂

#798

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

#1368

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

yes

C₃×C₈

yes

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

100

C₁₆

Dic₃⋊C₁₆

yes

characteristic

101

C₁₆

#4511

yes

characteristic

102

C₂²×C₃

#7457

yes

characteristic

103

C₃×C₄

—

104

C₃×C₄

#7099

yes

characteristic

105

C₃×C₄

#7457

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

#67464

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

#67554

yes

characteristic

111

C₈

#67567

yes

characteristic

112

C₂×C₃

—

113

C₂×C₃

#93877

yes

characteristic

114

C₂²

#10682858

yes

characteristic

115

C₄

—

116

C₄

#10681924

yes

characteristic

117

C₄

#10682856

yes

characteristic

118

C₃

#72224648

yes

characteristic, complemented, Sylow

119

C₂

—

120

C₂

#72244942

yes

characteristic

	subgroup	G/H
center	C₂	#72244942
commutator	C₃×C₃₂	C₂×C₁₆
Frattini	#7330	C₂×D₃
Fitting	C₃×#93878	C₂
socle	C₂×C₃	#93877
prime radical	C₂²×C₃	#7457
cosocle	C₃×#7330	C₂²
hypercenter	#93878	D₃
hypocenter	C₃	#72224648
LCS	G ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1311 ◁ #7330 ◁ #93878

subgroup

G/H

center

C₂

#72244942

commutator

C₃×C₃₂

C₂×C₁₆

C₂×D₃

C₂

C₂×C₃

C₂²×C₃

C₂²

D₃

C₃

G ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1311 ◁ #7330 ◁ #93878

N	H ≅ G/N	type
36. C₃⋊C₆₄	C₁₆	semidirect
118. C₃	#72224648	semidirect

H ≅ G/N

type

36. C₃⋊C₆₄

C₁₆

semidirect

118. C₃

#72224648

semidirect

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

subgroup	index
C₃×#93878	2
#72245357	2
#72244930	2
#72224648	3

subgroup

index

Subgroups — #72254992

Conjugacy classes of subgroups

Complements

Maximal subgroups