Group #13144909 — subgroups

Subgroups — 3₋¹⁺²×#383 #13144909

Order 1728.

552 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 287 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×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×#95	C₂	1	6	yes	maximal
3	C₃²×#383	C₃	1	1	yes	maximal, characteristic
4	C₉×#383	C₃	1	3	yes	maximal, complemented
5	C₂²×C₄×3₋¹⁺²	C₂²	1	4	yes	—
6	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
7	C₄²×3₋¹⁺²	C₄	1	4	yes	complemented
8	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
9	C₃²×#95	C₂×C₃	1	6	yes	—
10	C₂×C₄²×C₉	C₂×C₃	1	3	yes	—
11	C₉×#95	C₂×C₃	1	18	yes	—
12	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
13	C₂×C₄×3₋¹⁺²	—	2	6	no	—
14	C₂×C₄×3₋¹⁺²	—	2	12	no	—
15	C₂×C₄×3₋¹⁺²	C₂×C₄	1	6	yes	—
16	C₃×#383	—	3	1	no	—
17	C₃×#383	C₃²	1	1	yes	characteristic
18	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
19	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
20	C₃²×C₄²	C₃×C₄	1	4	yes	—
21	C₂²×C₄×C₉	C₂²×C₃	1	12	yes	—
22	C₂²×C₄×C₉	C₂²×C₃	1	9	yes	—
23	C₄²×C₉	C₃×C₄	1	12	yes	complemented
24	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
25	C₂²×3₋¹⁺²	#41	1	6	yes	—
26	C₄×3₋¹⁺²	—	2	6	no	—
27	C₄×3₋¹⁺²	—	4	4	no	—
28	C₂×C₃×C₄²	—	3	1	no	—
29	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
30	C₃×#95	—	3	6	no	—
31	C₃×#95	C₂×C₃²	1	6	yes	—
32	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
33	C₂×C₃²×C₄	—	2	6	no	—
34	C₂×C₃²×C₄	—	2	12	no	—
35	C₂×C₃²×C₄	C₂×C₃×C₄	1	6	yes	—
36	C₂³×C₉	C₂³×C₃	1	3	yes	—
37	C₂×C₄×C₉	—	2	18	no	—
38	C₂×C₄×C₉	—	2	36	no	—
39	C₂×C₄×C₉	C₂×C₃×C₄	1	18	yes	—
40	#383	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×3₋¹⁺²	#117	1	3	yes	—
42	C₂×3₋¹⁺²	#126	1	4	yes	—
43	C₂²×C₃×C₄	—	3	3	no	—
44	C₂²×C₃×C₄	—	3	4	no	—
45	C₂²×C₃×C₄	C₂²×C₃²	1	4	yes	—
46	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
47	C₃×C₄²	—	3	4	no	—
48	C₃×C₄²	C₃²×C₄	1	4	yes	—
49	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
50	C₂²×C₃²	C₃×#41	1	6	yes	—
51	C₃²×C₄	—	2	6	no	—
52	C₃²×C₄	—	4	4	no	—
53	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
54	C₂²×C₉	C₃×#41	1	18	yes	—
55	C₄×C₉	—	2	18	no	—
56	C₄×C₉	—	4	12	no	—
57	C₂×C₄²	C₂×3₋¹⁺²	1	1	yes	characteristic
58	#95	C₂×3₋¹⁺²	1	6	yes	—
59	3₋¹⁺²	#383	1	1	yes	characteristic, direct product factor, Sylow
60	C₂³×C₃	—	3	1	no	—
61	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
62	C₂×C₃×C₄	—	2	6	no	—
63	C₂×C₃×C₄	—	2	12	no	—
64	C₂×C₃×C₄	—	3	6	no	—
65	C₂×C₃×C₄	—	6	6	no	—
66	C₂×C₃×C₄	—	6	12	no	—
67	C₂×C₃×C₄	C₂×C₃²×C₄	1	6	yes	—
68	C₂×C₃²	C₃×#117	1	3	yes	—
69	C₂×C₃²	C₃×#126	1	4	yes	—
70	C₂×C₉	C₃×#117	1	9	yes	—
71	C₂×C₉	C₃×#126	1	12	yes	—
72	C₂²×C₄	C₂²×3₋¹⁺²	1	4	yes	—
73	C₂²×C₄	C₂²×3₋¹⁺²	1	3	yes	—
74	C₄²	C₄×3₋¹⁺²	1	4	yes	complemented
75	C₂²×C₃	—	3	1	no	—
76	C₂²×C₃	—	3	6	no	—
77	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
78	C₂²×C₃	C₃²×#41	1	6	yes	—
79	C₃×C₄	—	2	6	no	—
80	C₃×C₄	—	4	4	no	—
81	C₃×C₄	—	6	6	no	—
82	C₃×C₄	—	12	4	no	—
83	C₃²	C₃×#383	1	1	yes	characteristic
84	C₉	C₃×#383	1	3	yes	complemented
85	C₂³	C₂³×3₋¹⁺²	1	1	yes	characteristic
86	C₂×C₄	—	2	6	no	—
87	C₂×C₄	—	2	12	no	—
88	C₂×C₄	C₂×C₄×3₋¹⁺²	1	6	yes	—
89	C₂×C₃	—	3	3	no	—
90	C₂×C₃	—	3	4	no	—
91	C₂×C₃	C₃²×#117	1	3	yes	—
92	C₂×C₃	C₃²×#126	1	4	yes	—
93	C₂²	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
94	C₂²	#41×3₋¹⁺²	1	6	yes	—
95	C₄	—	2	6	no	—
96	C₄	—	4	4	no	—
97	C₃	—	3	1	no	—
98	C₃	C₃²×#383	1	1	yes	characteristic
99	C₂	3₋¹⁺²×#117	1	3	yes	—
100	C₂	3₋¹⁺²×#126	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×3₋¹⁺²

C₂

yes

maximal, characteristic

3₋¹⁺²×#95

C₂

yes

maximal

C₃²×#383

C₃

yes

maximal, characteristic

C₉×#383

C₃

yes

maximal, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₄²×3₋¹⁺²

C₄

yes

complemented

C₂×C₃²×C₄²

C₂×C₃

yes

characteristic

C₃²×#95

C₂×C₃

yes

—

C₂×C₄²×C₉

C₂×C₃

yes

—

C₉×#95

C₂×C₃

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₃×#383

—

C₃×#383

C₃²

yes

characteristic

C₂²×C₃²×C₄

C₂²×C₃

yes

—

C₂²×C₃²×C₄

C₂²×C₃

yes

—

C₃²×C₄²

C₃×C₄

yes

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₄²×C₉

C₃×C₄

yes

complemented

C₂²×3₋¹⁺²

C₂²×C₄

yes

characteristic

C₂²×3₋¹⁺²

#41

yes

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂×C₃²

yes

characteristic

C₃×#95

—

C₃×#95

C₂×C₃²

yes

—

C₂³×C₃²

C₂³×C₃

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×C₃×C₄

yes

—

C₂³×C₉

C₂³×C₃

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

#383

3₋¹⁺²

yes

characteristic, direct product factor, Sylow

C₂×3₋¹⁺²

#117

yes

—

C₂×3₋¹⁺²

#126

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂²×C₃²

yes

—

C₂²×C₃×C₄

C₂²×C₃²

yes

—

C₃×C₄²

—

C₃×C₄²

C₃²×C₄

yes

—

C₂²×C₃²

C₂²×C₃×C₄

yes

characteristic

C₂²×C₃²

C₃×#41

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₂²×C₉

C₂²×C₃×C₄

yes

—

C₂²×C₉

C₃×#41

yes

—

C₄×C₉

—

C₄×C₉

—

C₂×C₄²

C₂×3₋¹⁺²

yes

characteristic

#95

C₂×3₋¹⁺²

yes

—

3₋¹⁺²

#383

yes

characteristic, direct product factor, Sylow

C₂³×C₃

—

C₂³×C₃

C₂³×C₃²

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃²×C₄

yes

—

C₂×C₃²

C₃×#117

yes

—

C₂×C₃²

C₃×#126

yes

—

C₂×C₉

C₃×#117

yes

—

C₂×C₉

C₃×#126

yes

—

C₂²×C₄

C₂²×3₋¹⁺²

yes

—

C₂²×C₄

C₂²×3₋¹⁺²

yes

—

C₄²

C₄×3₋¹⁺²

yes

complemented

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²×C₃²×C₄

yes

characteristic

C₂²×C₃

C₃²×#41

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃²

yes

C₉

yes

C₂³

C₂³×3₋¹⁺²

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×3₋¹⁺²

yes

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₃²×#117

yes

—

C₂×C₃

C₃²×#126

yes

—

C₂²

C₂²×C₄×3₋¹⁺²

yes

characteristic

C₂²

#41×3₋¹⁺²

yes

—

C₄

—

C₄

—

C₃

—

C₃

C₃²×#383

yes

characteristic

C₂

3₋¹⁺²×#117

yes

—

100

C₂

3₋¹⁺²×#126

yes

—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³×C₃	C₂³×C₃²
commutator	C₂²×C₃	C₂²×C₃²×C₄
prime radical	C₂³×C₃²	C₂³×C₃
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

subgroup

G/H

center, Frattini, socle, cosocle

C₂³×C₃

C₂³×C₃²

commutator

C₂²×C₃

C₂²×C₃²×C₄

prime radical

C₂³×C₃²

C₂³×C₃

LCS

G ▷ C₂²×C₃ ▷ C₁

UCS

C₁ ◁ C₂³×C₃ ◁ G

N	H ≅ G/N	type
4. C₉×#383	C₃	semidirect
7. C₄²×3₋¹⁺²	C₄	semidirect
23. C₄²×C₉	C₃×C₄	semidirect
40. #383	3₋¹⁺²	direct (N × H)
59. 3₋¹⁺²	#383	direct (N × H)
74. C₄²	C₄×3₋¹⁺²	semidirect
84. C₉	C₃×#383	semidirect

H ≅ G/N

type

4. C₉×#383

C₃

semidirect

7. C₄²×3₋¹⁺²

C₄

semidirect

23. C₄²×C₉

C₃×C₄

semidirect

40. #383

3₋¹⁺²

direct (N × H)

59. 3₋¹⁺²

#383

direct (N × H)

74. C₄²

C₄×3₋¹⁺²

semidirect

84. C₉

C₃×#383

semidirect

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

subgroup	index
C₂×C₄²×3₋¹⁺²	2
3₋¹⁺²×#95	2
C₃²×#383	3
C₉×#383	3

subgroup

index

C₂×C₄²×3₋¹⁺²

3₋¹⁺²×#95

C₃²×#383

C₉×#383

Subgroups — 3₋¹⁺²×#383 #13144909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×#383 #13144909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#383 #13144909