Group #13163494 — subgroups

Subgroups — #13163494

Order 1728.

1372 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1372 conjugacy classes of subgroups — too many to draw. Only its 411 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	#11723929	C₂	1	2	yes	maximal
2	C₃³×#106	C₂	1	1	yes	maximal, characteristic
3	#11722187	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
5	#11722178	C₂	1	2	yes	maximal
6	#10592554	—	3	13	no	maximal
7	C₂×#6308	—	2	2	no	—
8	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
9	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
10	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
11	#88857	—	2	2	no	—
12	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
14	#64097	—	3	26	no	—
15	C₃²×#106	D₃	1	13	yes	—
16	#63644	—	3	13	no	—
17	C₈×C₃⋊Dic₃	—	3	13	no	—
18	#63635	—	3	26	no	—
19	C₂×C₃³×C₄	—	2	2	no	—
20	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
21	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
22	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	2	yes	—
23	#6308	—	4	4	no	—
24	C₃³×C₈	Q₈	1	2	yes	complemented
25	C₃⋊(C₃⋊(C₃⋊C₈))	Q₈	1	2	yes	complemented
26	#4855	—	9	13	no	—
27	C₂×#634	—	6	26	no	—
28	C₂×C₃²×C₈	C₂×D₃	1	13	yes	—
29	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
30	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
31	#3750	—	6	26	no	—
32	C₄×C₃⋊Dic₃	—	3	13	no	—
33	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
34	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
35	C₃³×C₄	—	4	2	no	—
36	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
37	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
38	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
39	C₃⋊(C₃⋊Dic₃)	SD₁₆	1	4	yes	—
40	#888	—	9	26	no	—
41	C₃×#106	C₃⋊D₃	1	13	yes	—
42	#817	—	9	13	no	—
43	C₈×Dic₃	—	9	13	no	—
44	#808	—	9	26	no	—
45	C₂×C₃²×C₄	—	2	26	no	—
46	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
47	C₂×C₃⋊Dic₃	—	3	26	no	—
48	C₂×C₃⋊Dic₃	—	6	26	no	—
49	#634	—	12	52	no	—
50	C₃²×C₈	Dic₆	1	26	yes	complemented
51	C₃⋊(C₃⋊C₈)	—	3	26	no	—
52	#498	—	27	1	no	Sylow
53	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₃³	C₂×SD₁₆	1	2	yes	—
55	C₂×Dic₆	—	18	26	no	—
56	C₂×C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
57	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
58	C₄⋊Dic₃	—	9	26	no	—
59	Dic₃⋊C₄	—	18	26	no	—
60	C₄×Dic₃	—	9	13	no	—
61	C₂×C₃⋊C₈	—	9	13	no	—
62	C₂²×C₃²	D₃×D₄	1	13	yes	—
63	C₃²×C₄	—	4	26	no	—
64	C₃²×C₄	C₂×Dic₆	1	13	yes	—
65	C₃²×C₄	D₃×Q₈	1	13	yes	—
66	C₃⋊Dic₃	—	3	52	no	—
67	C₃⋊Dic₃	—	12	26	no	—
68	C₄⋊Q₈	—	27	2	no	—
69	#106	—	27	1	no	—
70	#106	—	27	2	no	—
71	#106	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
72	C₄×C₈	—	27	1	no	—
73	C₃³	#498	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃×C₄	—	2	26	no	—
75	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
76	C₂×Dic₃	—	9	26	no	—
77	C₂×Dic₃	—	18	26	no	—
78	Dic₆	—	36	52	no	—
79	C₃×C₈	#634	1	26	yes	complemented
80	C₃⋊C₈	—	9	26	no	—
81	C₂×C₃²	#888	1	13	yes	—
82	C₂×C₃²	D₃×SD₁₆	1	26	yes	—
83	C₂×Q₈	—	54	2	no	—
84	C₂×C₈	—	27	1	no	—
85	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
86	C₄⋊C₄	—	27	2	no	—
87	C₄⋊C₄	—	54	2	no	—
88	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
89	C₄²	—	27	1	no	—
90	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
91	C₃×C₄	—	4	26	no	—
92	C₃×C₄	C₂×#634	1	13	yes	—
93	C₃×C₄	Q₈×C₃⋊D₃	1	13	yes	—
94	Dic₃	—	9	52	no	—
95	Dic₃	—	36	26	no	—
96	C₃²	#4855	1	13	yes	complemented
97	Q₈	—	108	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	27	2	no	—
100	C₂×C₄	—	54	2	no	—
101	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
102	C₈	—	27	2	no	—
103	C₈	#6308	1	2	yes	complemented
104	C₂×C₃	#64097	1	13	yes	—
105	C₂×C₃	SD₁₆×C₃⋊D₃	1	26	yes	—
106	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
107	C₄	—	4	2	no	—
108	C₄	—	27	4	no	—
109	C₄	—	108	2	no	—
110	C₄	C₂×#6308	1	1	yes	characteristic
111	C₄	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
112	C₃	#10592554	1	13	yes	complemented
113	C₂	#11723929	1	1	yes	characteristic
114	C₂	SD₁₆×C₃⋊(C₃⋊D₃)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11723929

C₂

yes

maximal

C₃³×#106

C₂

yes

maximal, characteristic

#11722187

C₂

yes

maximal, characteristic

C₈×C₃⋊(C₃⋊Dic₃)

C₂

yes

maximal, characteristic

#11722178

C₂

yes

—

—

C₂×C₃³×C₈

C₂²

yes

characteristic

C₃³×C₄⋊C₄

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

#88857

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

#64097

—

C₃²×#106

D₃

yes

—

#63644

—

C₈×C₃⋊Dic₃

—

#63635

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

D₄

yes

—

#6308

—

C₃³×C₈

Q₈

yes

complemented

C₃⋊(C₃⋊(C₃⋊C₈))

Q₈

yes

complemented

#4855

—

C₂×#634

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

C₂×Q₈

yes

characteristic

C₃³×C₄

C₂×Q₈

yes

characteristic

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

SD₁₆

yes

—

#888

—

C₃×#106

C₃⋊D₃

yes

—

#817

—

C₈×Dic₃

—

#808

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

C₃²×C₈

Dic₆

yes

complemented

C₃⋊(C₃⋊C₈)

—

#498

—

Sylow

C₂×C₃³

C₄⋊Q₈

yes

characteristic

C₂×C₃³

C₂×SD₁₆

yes

—

C₂×Dic₆

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

D₃×D₄

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃²×C₄

D₃×Q₈

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₄⋊Q₈

—

#106

—

#106

—

#106

C₃⋊(C₃⋊D₃)

yes

characteristic

C₄×C₈

—

C₃³

#498

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

#634

yes

complemented

C₃⋊C₈

—

C₂×C₃²

#888

yes

—

C₂×C₃²

D₃×SD₁₆

yes

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

C₄²

—

C₂²×C₃

D₄×C₃⋊D₃

yes

—

C₃×C₄

—

C₃×C₄

C₂×#634

yes

—

C₃×C₄

Q₈×C₃⋊D₃

yes

—

Dic₃

—

Dic₃

—

C₃²

#4855

yes

complemented

Q₈

—

108

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

102

C₈

—

103

C₈

#6308

yes

complemented

104

C₂×C₃

#64097

yes

—

105

C₂×C₃

SD₁₆×C₃⋊D₃

yes

—

106

C₂²

D₄×C₃⋊(C₃⋊D₃)

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

108

—

110

C₄

C₂×#6308

yes

characteristic

111

C₄

Q₈×C₃⋊(C₃⋊D₃)

yes

characteristic

112

C₃

#10592554

yes

complemented

113

C₂

#11723929

yes

characteristic

114

C₂

SD₁₆×C₃⋊(C₃⋊D₃)

yes

—

	subgroup	G/H
center	C₂²	D₄×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃³×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×#106	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	#106	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#498
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×C₃⋊(C₃⋊D₃)

commutator, cosocle

C₂×C₃³×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₃³×#106

C₂

socle, prime radical

C₂²×C₃³

C₂×D₄

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
24. C₃³×C₈	Q₈	semidirect
25. C₃⋊(C₃⋊(C₃⋊C₈))	Q₈	semidirect
50. C₃²×C₈	Dic₆	semidirect
73. C₃³	#498	semidirect
79. C₃×C₈	#634	semidirect
96. C₃²	#4855	semidirect
103. C₈	#6308	semidirect
112. C₃	#10592554	semidirect

H ≅ G/N

type

24. C₃³×C₈

Q₈

semidirect

25. C₃⋊(C₃⋊(C₃⋊C₈))

Q₈

semidirect

50. C₃²×C₈

Dic₆

semidirect

73. C₃³

#498

semidirect

79. C₃×C₈

#634

semidirect

96. C₃²

#4855

semidirect

103. C₈

#6308

semidirect

112. C₃

#10592554

semidirect

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

subgroup	index
#11723929	2
C₃³×#106	2
#11722187	2
C₈×C₃⋊(C₃⋊Dic₃)	2
#11722178	2
#10592554	3

subgroup

index

C₃³×#106

C₈×C₃⋊(C₃⋊Dic₃)

Subgroups — #13163494

Conjugacy classes of subgroups

Complements

Maximal subgroups