Group #10627140 — subgroups

Subgroups — C₃×#8533 #10627140

Order 768.

656 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 150 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×#3556	C₂	1	1	yes	maximal, characteristic
2	C₃×#1994	C₂	1	3	yes	maximal
3	C₃×#1561	C₂	1	3	yes	maximal
4	#8533	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₃×#564	C₄	1	4	yes	complemented
6	C₃×#557	—	2	6	no	—
7	C₃×#541	—	2	6	no	—
8	C₃×#533	C₂²	1	3	yes	—
9	C₃×#423	C₂²	1	3	yes	—
10	C₃×C₄³	C₂²	1	1	yes	characteristic
11	C₃×#339	—	2	6	no	—
12	C₃×#332	—	2	6	no	—
13	#3556	C₂×C₃	1	1	yes	characteristic
14	#1994	C₂×C₃	1	3	yes	—
15	#1561	C₂×C₃	1	3	yes	—
16	C₂×C₃×M₁₆	—	2	3	no	—
17	C₃×C₄⋊Q₈	—	2	6	no	—
18	C₃×C₄⋊Q₈	D₄	1	6	yes	—
19	C₃×#126	—	2	8	no	—
20	C₃×#125	—	2	6	no	—
21	C₃×#125	—	2	24	no	—
22	C₃×#125	C₂×C₄	1	6	yes	—
23	C₃×C₂²⋊Q₈	—	2	6	no	—
24	C₃×C₄×Q₈	—	2	12	no	—
25	C₃×C₄×Q₈	—	2	6	no	—
26	C₃×#117	D₄	1	6	yes	—
27	C₂×C₃×C₄²	—	2	3	no	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₃×C₄⋊C₈	—	2	6	no	—
30	#564	C₃×C₄	1	4	yes	complemented
31	#557	—	2	6	no	—
32	#541	—	2	6	no	—
33	#533	C₂²×C₃	1	3	yes	—
34	#423	C₂²×C₃	1	3	yes	—
35	C₄³	C₂²×C₃	1	1	yes	characteristic
36	#339	—	2	6	no	—
37	#332	—	2	6	no	—
38	C₂×C₃×Q₈	—	2	6	no	—
39	C₂²×C₃×C₄	—	4	1	no	—
40	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
41	C₃×M₁₆	—	4	6	no	—
42	C₂×C₃×C₈	—	4	3	no	—
43	C₃×C₄⋊C₄	—	2	24	no	—
44	C₃×C₄⋊C₄	—	2	12	no	—
45	C₃×C₄⋊C₄	—	2	6	no	—
46	C₃×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
47	C₃×C₂²⋊C₄	—	2	6	no	—
48	C₃×C₄²	—	2	8	no	—
49	C₃×C₄²	—	2	6	no	—
50	C₃×C₄²	—	2	6	no	—
51	C₃×C₄²	—	4	3	no	—
52	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
53	C₃×C₄²	C₂×D₄	1	3	yes	—
54	C₂×M₁₆	—	2	3	no	—
55	C₄⋊Q₈	—	2	6	no	—
56	C₄⋊Q₈	C₃×D₄	1	6	yes	—
57	#126	—	2	8	no	—
58	#125	—	2	6	no	—
59	#125	—	2	24	no	—
60	#125	C₂×C₃×C₄	1	6	yes	—
61	C₂²⋊Q₈	—	2	6	no	—
62	C₄×Q₈	—	2	12	no	—
63	C₄×Q₈	—	2	6	no	—
64	#117	C₃×D₄	1	6	yes	—
65	C₂×C₄²	—	2	3	no	—
66	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
67	C₄⋊C₈	—	2	6	no	—
68	C₂³×C₃	#120	1	1	yes	characteristic
69	C₃×Q₈	—	4	6	no	—
70	C₃×Q₈	—	4	6	no	—
71	C₂×C₃×C₄	—	2	3	no	—
72	C₂×C₃×C₄	—	2	3	no	—
73	C₂×C₃×C₄	—	2	12	no	—
74	C₂×C₃×C₄	—	4	6	no	—
75	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	3	yes	—
76	C₂×C₃×C₄	#120	1	3	yes	—
77	C₃×C₈	—	8	3	no	—
78	C₂×Q₈	—	2	6	no	—
79	C₂²×C₄	—	4	1	no	—
80	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
81	M₁₆	—	4	6	no	—
82	C₂×C₈	—	4	3	no	—
83	C₄⋊C₄	—	2	24	no	—
84	C₄⋊C₄	—	2	12	no	—
85	C₄⋊C₄	—	2	6	no	—
86	C₄⋊C₄	C₃×C₂²⋊C₄	1	12	yes	—
87	C₂²⋊C₄	—	2	6	no	—
88	C₄²	—	2	8	no	—
89	C₄²	—	2	6	no	—
90	C₄²	—	2	6	no	—
91	C₄²	—	4	3	no	—
92	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
93	C₄²	C₂×C₃×D₄	1	3	yes	—
94	C₂²×C₃	—	2	3	no	—
95	C₂²×C₃	#379	1	1	yes	characteristic
96	C₃×C₄	—	2	3	no	—
97	C₃×C₄	—	4	4	no	—
98	C₃×C₄	—	4	12	no	—
99	C₃×C₄	#454	1	3	yes	—
100	C₃×C₄	#456	1	3	yes	—
101	C₂³	C₃×#120	1	1	yes	characteristic
102	Q₈	—	4	6	no	—
103	Q₈	—	4	6	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	3	no	—
106	C₂×C₄	—	2	12	no	—
107	C₂×C₄	—	4	6	no	—
108	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	3	yes	—
109	C₂×C₄	C₃×#120	1	3	yes	—
110	C₈	—	8	3	no	—
111	C₂×C₃	—	4	1	no	—
112	C₂×C₃	#1793	1	3	yes	—
113	C₂²	—	2	3	no	—
114	C₂²	C₃×#379	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	4	4	no	—
117	C₄	—	4	12	no	—
118	C₄	C₃×#454	1	3	yes	—
119	C₄	C₃×#456	1	3	yes	—
120	C₃	#8533	1	1	yes	characteristic, direct product factor, Sylow
121	C₂	—	4	1	no	—
122	C₂	C₃×#1793	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#3556

C₂

yes

maximal, characteristic

C₃×#1994

C₂

yes

maximal

C₃×#1561

C₂

yes

maximal

#8533

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₃×#564

C₄

yes

—

—

C₂²

yes

—

C₃×#423

C₂²

yes

—

C₃×C₄³

C₂²

yes

—

—

C₂×C₃

yes

characteristic

#1994

C₂×C₃

yes

—

#1561

C₂×C₃

yes

—

C₂×C₃×M₁₆

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

—

C₃×#117

D₄

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₃×C₄⋊C₈

—

#564

C₃×C₄

yes

—

—

C₂²×C₃

yes

—

#423

C₂²×C₃

yes

—

C₄³

C₂²×C₃

yes

characteristic

#339

—

#332

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂²⋊C₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₃×D₄

yes

—

—

—

—

C₂×C₃×C₄

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₃×D₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³×C₃

yes

characteristic

C₄⋊C₈

—

C₂³×C₃

#120

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

—

C₂×C₃×C₄

#120

yes

—

C₃×C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₃×D₄

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₃×C₂²⋊C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₃×C₄

yes

characteristic

C₄²

C₂×C₃×D₄

yes

—

C₂²×C₃

—

C₂²×C₃

#379

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#454

yes

—

100

C₃×C₄

#456

yes

—

101

C₂³

C₃×#120

yes

characteristic

102

Q₈

—

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₃×C₂²⋊C₄

yes

—

109

C₂×C₄

C₃×#120

yes

—

110

C₈

—

111

C₂×C₃

—

112

C₂×C₃

#1793

yes

—

113

C₂²

—

114

C₂²

C₃×#379

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₃×#454

yes

—

119

C₄

C₃×#456

yes

—

120

C₃

#8533

yes

characteristic, direct product factor, Sylow

121

C₂

—

122

C₂

C₃×#1793

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	#379
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂³×C₃	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

subgroup

G/H

center, socle

C₂²×C₃

#379

commutator

C₄²

C₂²×C₃×C₄

Frattini, cosocle

C₂×C₄²

C₂³×C₃

C₂³×C₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

N	H ≅ G/N	type
4. #8533	C₃	direct (N × H)
5. C₃×#564	C₄	semidirect
30. #564	C₃×C₄	semidirect
120. C₃	#8533	direct (N × H)

H ≅ G/N

type

4. #8533

C₃

direct (N × H)

5. C₃×#564

C₄

semidirect

30. #564

C₃×C₄

semidirect

120. C₃

#8533

direct (N × H)

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

subgroup	index
C₃×#3556	2
C₃×#1994	2
C₃×#1561	2
#8533	3

subgroup

index

Subgroups — C₃×#8533 #10627140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#8533 #10627140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#8533 #10627140