Group #72762 — subgroups

Subgroups — C₃×#2047 #72762

Order 384.

256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₂²×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₃×#419	C₂	1	2	yes	maximal
4	C₂×C₃×#101	C₂	1	2	yes	maximal
5	C₃×#341	C₂	1	1	yes	maximal, characteristic
6	#2047	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×Q₁₆	—	2	4	no	—
9	C₂×C₃×Q₁₆	—	2	2	no	—
10	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
13	C₃×#117	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊C₈	—	2	2	no	—
15	C₃×#103	—	2	4	no	—
16	C₃×#101	—	2	4	no	—
17	C₂²×Q₁₆	C₂×C₃	1	1	yes	characteristic
18	#424	C₂×C₃	1	1	yes	characteristic
19	#419	C₂×C₃	1	2	yes	—
20	C₂×#101	C₂×C₃	1	2	yes	—
21	#341	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×Q₈	—	2	2	no	—
23	C₂×C₃×Q₈	—	2	2	no	—
24	C₂×C₃×Q₈	—	2	2	no	—
25	C₂×C₃×Q₈	—	2	2	no	—
26	C₂×C₃×Q₈	—	2	2	no	—
27	C₂×C₃×Q₈	D₄	1	4	yes	—
28	C₂²×C₃×C₄	—	2	2	no	—
29	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₃×Q₁₆	—	2	8	no	—
31	C₃×M₁₆	—	2	2	no	—
32	C₃×M₁₆	—	2	2	no	—
33	C₃×M₁₆	—	2	2	no	—
34	C₂×C₃×C₈	—	2	2	no	—
35	C₂×C₃×C₈	—	2	2	no	—
36	C₂×C₃×C₈	—	2	1	no	—
37	C₂×C₃×C₈	D₄	1	2	yes	—
38	C₃×C₄⋊C₄	D₄	1	2	yes	—
39	C₃×C₂²⋊C₄	—	2	1	no	—
40	C₃×C₄²	—	2	1	no	—
41	C₂²×Q₈	C₂²×C₃	1	2	yes	—
42	C₂×Q₁₆	—	2	4	no	—
43	C₂×Q₁₆	—	2	2	no	—
44	C₂×M₁₆	C₂²×C₃	1	2	yes	—
45	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
47	#117	C₂²×C₃	1	1	yes	characteristic
48	C₄⋊C₈	—	2	2	no	—
49	#103	—	2	4	no	—
50	#101	—	2	4	no	—
51	C₂³×C₃	#41	1	1	yes	characteristic
52	C₃×Q₈	—	2	4	no	—
53	C₃×Q₈	—	2	4	no	—
54	C₃×Q₈	—	4	2	no	—
55	C₃×Q₈	—	4	2	no	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	—	2	4	no	—
58	C₂×C₃×C₄	—	4	2	no	—
59	C₂×C₃×C₄	—	4	2	no	—
60	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄	#41	1	2	yes	—
64	C₃×C₈	—	2	2	no	—
65	C₃×C₈	—	4	1	no	—
66	C₃×C₈	—	4	2	no	—
67	C₂×Q₈	—	2	2	no	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	—	2	2	no	—
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	—	2	2	no	—
72	C₂×Q₈	C₃×D₄	1	4	yes	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
75	Q₁₆	—	2	8	no	—
76	M₁₆	—	2	2	no	—
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	C₃×D₄	1	2	yes	—
83	C₄⋊C₄	C₃×D₄	1	2	yes	—
84	C₂²⋊C₄	—	2	1	no	—
85	C₄²	—	2	1	no	—
86	C₂²×C₃	—	2	2	no	—
87	C₂²×C₃	#121	1	1	yes	characteristic
88	C₂²×C₃	#123	1	2	yes	—
89	C₃×C₄	—	4	4	no	—
90	C₃×C₄	—	4	2	no	—
91	C₃×C₄	#120	1	1	yes	characteristic
92	C₃×C₄	#121	1	2	yes	—
93	C₃×C₄	#124	1	1	yes	characteristic
94	C₂³	C₃×#41	1	1	yes	characteristic
95	Q₈	—	2	4	no	—
96	Q₈	—	2	4	no	—
97	Q₈	—	4	2	no	—
98	Q₈	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
104	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
105	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
106	C₂×C₄	C₃×#41	1	2	yes	—
107	C₈	—	2	2	no	—
108	C₈	—	4	1	no	—
109	C₈	—	4	2	no	—
110	C₂×C₃	—	2	2	no	—
111	C₂×C₃	#394	1	1	yes	characteristic
112	C₂×C₃	#473	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	C₃×#121	1	1	yes	characteristic
115	C₂²	C₃×#123	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	4	2	no	—
118	C₄	C₃×#120	1	1	yes	characteristic
119	C₄	C₃×#121	1	2	yes	—
120	C₄	C₃×#124	1	1	yes	characteristic
121	C₃	#2047	1	1	yes	characteristic, direct product factor, Sylow
122	C₂	—	2	2	no	—
123	C₂	C₃×#394	1	1	yes	characteristic
124	C₂	C₃×#473	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃×Q₁₆

C₂

yes

maximal, characteristic

C₃×#424

C₂

yes

maximal, characteristic

C₃×#419

C₂

yes

maximal

C₂×C₃×#101

C₂

yes

maximal

C₃×#341

C₂

yes

maximal, characteristic

#2047

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

—

C₂²×C₃×C₈

C₂²

yes

characteristic

C₃×#117

C₂²

yes

characteristic

C₃×C₄⋊C₈

—

C₃×#103

—

C₃×#101

—

C₂²×Q₁₆

C₂×C₃

yes

characteristic

#424

C₂×C₃

yes

characteristic

#419

C₂×C₃

yes

—

C₂×#101

C₂×C₃

yes

—

#341

C₂×C₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

D₄

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×Q₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄⋊C₄

D₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₂²×Q₈

C₂²×C₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂²×C₃

yes

—

C₂×M₁₆

C₂²×C₃

yes

characteristic

C₂²×C₈

C₂²×C₃

yes

characteristic

#117

C₂²×C₃

yes

characteristic

C₄⋊C₈

—

#103

—

#101

—

C₂³×C₃

#41

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₃×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₃

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₃×D₄

yes

—

C₄⋊C₄

C₃×D₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

#121

yes

characteristic

C₂²×C₃

#123

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#120

yes

characteristic

C₃×C₄

#121

yes

—

C₃×C₄

#124

yes

characteristic

C₂³

C₃×#41

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₃×D₄

yes

—

104

C₂×C₄

C₂×C₃×D₄

yes

characteristic

105

C₂×C₄

C₂×C₃×D₄

yes

characteristic

106

C₂×C₄

C₃×#41

yes

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₂×C₃

—

111

C₂×C₃

#394

yes

characteristic

112

C₂×C₃

#473

yes

—

113

C₂²

—

114

C₂²

C₃×#121

yes

characteristic

115

C₂²

C₃×#123

yes

—

116

C₄

—

117

C₄

—

118

C₄

C₃×#120

yes

characteristic

119

C₄

C₃×#121

yes

—

120

C₄

C₃×#124

yes

characteristic

121

C₃

#2047

yes

characteristic, direct product factor, Sylow

122

C₂

—

123

C₂

C₃×#394

yes

characteristic

124

C₂

C₃×#473

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
prime radical	C₂³×C₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ G

subgroup

G/H

center, socle

C₂²×C₃

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₃

C₂³×C₃

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ G

N	H ≅ G/N	type
6. #2047	C₃	direct (N × H)
121. C₃	#2047	direct (N × H)

H ≅ G/N

type

6. #2047

C₃

direct (N × H)

121. C₃

#2047

direct (N × H)

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

subgroup	index
C₂²×C₃×Q₁₆	2
C₃×#424	2
C₃×#419	2
C₂×C₃×#101	2
C₃×#341	2
#2047	3

subgroup

index

C₂²×C₃×Q₁₆

C₂×C₃×#101

Subgroups — C₃×#2047 #72762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#2047 #72762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2047 #72762