Group #10625762 — subgroups

Subgroups — C₃×#7155 #10625762

Order 768.

222 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 134 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×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#1564	C₂	1	2	yes	maximal
3	#7155	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂²×C₃×C₁₆	C₄	1	2	yes	—
5	C₃×#431	—	2	2	no	—
6	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₁₆	—	2	2	no	—
8	C₃×#335	C₄	1	4	yes	—
9	C₃×#334	—	2	2	no	—
10	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic
11	#1564	C₂×C₃	1	2	yes	—
12	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
14	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
15	C₂×C₃×C₁₆	—	2	2	no	—
16	C₂×C₃×C₁₆	—	2	2	no	—
17	C₂×C₃×C₁₆	C₈	1	4	yes	complemented
18	C₃×C₄⋊C₈	—	2	4	no	—
19	C₃×C₂²⋊C₈	—	4	2	no	—
20	C₃×C₄×C₈	—	4	2	no	—
21	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
22	C₃×C₄×C₈	D₄	1	2	yes	—
23	C₂²×C₁₆	C₃×C₄	1	2	yes	—
24	#431	—	2	2	no	—
25	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
26	C₄×C₁₆	—	2	2	no	—
27	#335	C₃×C₄	1	4	yes	—
28	#334	—	2	2	no	—
29	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄	SD₁₆	1	2	yes	—
31	C₂×C₃×C₈	—	2	2	no	—
32	C₂×C₃×C₈	—	4	4	no	—
33	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
34	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
36	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
37	C₂×C₃×C₈	M₁₆	1	2	yes	—
38	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
39	C₃×C₄²	D₈	1	1	yes	characteristic
40	C₃×C₄²	Q₁₆	1	1	yes	characteristic
41	C₃×C₁₆	—	2	4	no	—
42	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₃×Q₈	1	1	yes	characteristic
44	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
45	C₂×C₁₆	—	2	2	no	—
46	C₂×C₁₆	—	2	2	no	—
47	C₂×C₁₆	C₃×C₈	1	4	yes	complemented
48	C₄⋊C₈	—	2	4	no	—
49	C₂²⋊C₈	—	4	2	no	—
50	C₄×C₈	—	4	2	no	—
51	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
52	C₄×C₈	C₃×D₄	1	2	yes	—
53	C₂³×C₃	#106	1	1	yes	characteristic
54	C₂×C₃×C₄	—	2	4	no	—
55	C₂×C₃×C₄	#95	1	1	yes	characteristic
56	C₂×C₃×C₄	#102	1	2	yes	—
57	C₂×C₃×C₄	#103	1	2	yes	—
58	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
59	C₂×C₃×C₄	#107	1	1	yes	characteristic
60	C₂×C₃×C₄	#108	1	2	yes	—
61	C₃×C₈	—	2	2	no	—
62	C₃×C₈	—	8	4	no	—
63	C₃×C₈	C₄×C₈	1	1	yes	characteristic
64	C₃×C₈	#97	1	1	yes	characteristic
65	C₃×C₈	C₂²⋊C₈	1	2	yes	—
66	C₂²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄	C₃×SD₁₆	1	2	yes	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	4	4	no	—
70	C₂×C₈	C₃×C₄²	1	1	yes	characteristic
71	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈	C₃×C₄⋊C₄	1	2	yes	—
73	C₂×C₈	C₂×C₃×C₈	1	2	yes	—
74	C₂×C₈	C₃×M₁₆	1	2	yes	—
75	C₄²	C₃×C₂²⋊C₄	1	2	yes	—
76	C₄²	C₃×D₈	1	1	yes	characteristic
77	C₄²	C₃×Q₁₆	1	1	yes	characteristic
78	C₁₆	—	2	4	no	—
79	C₂²×C₃	—	2	2	no	—
80	C₂²×C₃	#334	1	2	yes	—
81	C₂²×C₃	#340	1	1	yes	characteristic
82	C₃×C₄	—	2	2	no	—
83	C₃×C₄	#335	1	2	yes	—
84	C₃×C₄	#336	1	1	yes	characteristic
85	C₃×C₄	#341	1	1	yes	characteristic
86	C₃×C₄	#359	1	4	yes	—
87	C₂³	C₃×#106	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₃×#95	1	1	yes	characteristic
90	C₂×C₄	C₃×#102	1	2	yes	—
91	C₂×C₄	C₃×#103	1	2	yes	—
92	C₂×C₄	C₃×C₄⋊C₈	1	2	yes	—
93	C₂×C₄	C₃×#107	1	1	yes	characteristic
94	C₂×C₄	C₃×#108	1	2	yes	—
95	C₈	—	2	2	no	—
96	C₈	—	8	4	no	—
97	C₈	C₃×C₄×C₈	1	1	yes	characteristic
98	C₈	C₃×#97	1	1	yes	characteristic
99	C₈	C₃×C₂²⋊C₈	1	2	yes	—
100	C₂×C₃	—	2	2	no	—
101	C₂×C₃	#1277	1	1	yes	characteristic
102	C₂×C₃	#1382	1	2	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	C₃×#334	1	2	yes	—
105	C₂²	C₃×#340	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	C₃×#335	1	2	yes	—
108	C₄	C₃×#336	1	1	yes	characteristic
109	C₄	C₃×#341	1	1	yes	characteristic
110	C₄	C₃×#359	1	4	yes	—
111	C₃	#7155	1	1	yes	characteristic, direct product factor, Sylow
112	C₂	—	2	2	no	—
113	C₂	C₃×#1277	1	1	yes	characteristic
114	C₂	C₃×#1382	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄×C₁₆

C₂

yes

maximal, characteristic

C₃×#1564

C₂

yes

maximal

#7155

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃×C₁₆

C₄

yes

—

C₃×#431

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₃×C₄×C₁₆

—

C₃×#335

C₄

yes

—

C₃×#334

—

C₂×C₄×C₁₆

C₂×C₃

yes

characteristic

#1564

C₂×C₃

yes

—

C₂²×C₃×C₈

C₂×C₄

yes

characteristic

C₂²×C₃×C₈

Q₈

yes

characteristic

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

C₈

yes

complemented

C₃×C₄⋊C₈

—

C₃×C₂²⋊C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

C₂²×C₁₆

C₃×C₄

yes

—

#431

—

C₂×C₄×C₈

C₂²×C₃

yes

characteristic

C₄×C₁₆

—

#335

C₃×C₄

yes

—

#334

—

C₂²×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄²

yes

characteristic

C₂×C₃×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₈

C₄⋊C₄

yes

—

C₂×C₃×C₈

C₂×C₈

yes

—

C₂×C₃×C₈

M₁₆

yes

—

C₃×C₄²

C₂²⋊C₄

yes

—

C₃×C₄²

D₈

yes

characteristic

C₃×C₄²

Q₁₆

yes

characteristic

C₃×C₁₆

—

C₂²×C₈

C₂×C₃×C₄

yes

characteristic

C₂²×C₈

C₃×Q₈

yes

characteristic

C₂×C₄²

C₃×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₃×C₈

yes

complemented

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₃×C₄

yes

—

C₄×C₈

C₃×D₄

yes

—

C₂³×C₃

#106

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

#95

yes

characteristic

C₂×C₃×C₄

#102

yes

—

C₂×C₃×C₄

#103

yes

—

C₂×C₃×C₄

C₄⋊C₈

yes

—

C₂×C₃×C₄

#107

yes

characteristic

C₂×C₃×C₄

#108

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₄×C₈

yes

characteristic

C₃×C₈

#97

yes

characteristic

C₃×C₈

C₂²⋊C₈

yes

—

C₂²×C₄

C₃×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₃×SD₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₃×C₄²

yes

characteristic

C₂×C₈

C₃×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₃×C₄⋊C₄

yes

—

C₂×C₈

C₂×C₃×C₈

yes

—

C₂×C₈

C₃×M₁₆

yes

—

C₄²

C₃×C₂²⋊C₄

yes

—

C₄²

C₃×D₈

yes

characteristic

C₄²

C₃×Q₁₆

yes

characteristic

C₁₆

—

C₂²×C₃

—

C₂²×C₃

#334

yes

—

C₂²×C₃

#340

yes

characteristic

C₃×C₄

—

C₃×C₄

#335

yes

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

C₂³

C₃×#106

yes

characteristic

C₂×C₄

—

C₂×C₄

C₃×#95

yes

characteristic

C₂×C₄

C₃×#102

yes

—

C₂×C₄

C₃×#103

yes

—

C₂×C₄

C₃×C₄⋊C₈

yes

—

C₂×C₄

C₃×#107

yes

characteristic

C₂×C₄

C₃×#108

yes

—

C₈

—

C₈

—

C₈

C₃×C₄×C₈

yes

characteristic

C₈

C₃×#97

yes

characteristic

C₈

C₃×C₂²⋊C₈

yes

—

100

C₂×C₃

—

101

C₂×C₃

#1277

yes

characteristic

102

C₂×C₃

#1382

yes

—

103

C₂²

—

104

C₂²

C₃×#334

yes

—

105

C₂²

C₃×#340

yes

characteristic

106

C₄

—

107

C₄

C₃×#335

yes

—

108

C₄

C₃×#336

yes

characteristic

109

C₄

C₃×#341

yes

characteristic

110

C₄

C₃×#359

yes

—

111

C₃

#7155

yes

characteristic, direct product factor, Sylow

112

C₂

—

113

C₂

C₃×#1277

yes

characteristic

114

C₂

C₃×#1382

yes

—

	subgroup	G/H
center	C₃×C₄²	D₈
commutator	C₈	C₃×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃
socle	C₂²×C₃	#340
prime radical	C₂³×C₃	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄² ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

subgroup

G/H

center

C₃×C₄²

D₈

commutator

C₈

C₃×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₃

C₂²×C₃

C₂³×C₃

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₃×C₄² ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

N	H ≅ G/N	type
3. #7155	C₃	direct (N × H)
17. C₂×C₃×C₁₆	C₈	semidirect
47. C₂×C₁₆	C₃×C₈	semidirect
111. C₃	#7155	direct (N × H)

H ≅ G/N

type

3. #7155

C₃

direct (N × H)

17. C₂×C₃×C₁₆

C₈

semidirect

47. C₂×C₁₆

C₃×C₈

semidirect

111. C₃

#7155

direct (N × H)

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

subgroup	index
C₂×C₃×C₄×C₁₆	2
C₃×#1564	2
#7155	3

subgroup

index

C₂×C₃×C₄×C₁₆

C₃×#1564

#7155

Subgroups — C₃×#7155 #10625762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#7155 #10625762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#7155 #10625762