Group #35208113 — subgroups

Subgroups — D₃×D₂₃×C₅²⋊C₃ #35208113

Order 20700.

128 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 30 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	C₅²⋊C₃×D₆₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×C₂₃×C₅²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₂₃×C₅²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×D₃×D₂₃	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₃×C₅²⋊C₃	—	3	1	no	maximal
6	C₃×C₂₃×C₅²⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₅²×D₆₉	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅²×D₃×C₂₃	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×C₅²×D₂₃	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₂₃×C₅²⋊C₃	—	3	1	no	—
11	D₂₃×C₅²⋊C₃	—	2	1	no	—
12	D₂₃×C₅²⋊C₃	—	3	1	no	—
13	D₂₃×C₅²⋊C₃	D₃	1	1	yes	characteristic, direct product factor
14	C₂×C₅²×D₂₃	—	3	1	no	Hall
15	C₃×C₅²×C₂₃	C₂²×C₃	1	1	yes	characteristic, complemented
16	C₂₃×C₅²⋊C₃	—	2	1	no	—
17	C₂₃×C₅²⋊C₃	C₂×D₃	1	1	yes	characteristic, complemented
18	C₅×D₃×D₂₃	—	3	2	no	—
19	C₂×C₅²×C₂₃	—	3	1	no	—
20	C₅²×D₂₃	—	3	1	no	—
21	C₅²×D₂₃	C₃×D₃	1	1	yes	characteristic, complemented
22	C₂×D₃×C₅²⋊C₃	—	23	1	no	maximal, Hall
23	C₃×D₃×D₂₃	—	25	1	no	maximal, Hall
24	C₅×D₆₉	—	3	2	no	—
25	C₅×D₃×C₂₃	—	3	2	no	—
26	C₃×C₅×D₂₃	—	3	2	no	—
27	C₅²×C₂₃	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
28	C₂×C₅×D₂₃	—	9	2	no	—
29	C₂×C₃×C₅²⋊C₃	—	23	1	no	—
30	D₃×C₅²⋊C₃	—	23	1	no	—
31	D₃×C₅²⋊C₃	D₂₃	1	1	yes	characteristic, direct product factor
32	C₃×D₆₉	—	25	1	no	—
33	C₃×D₃×C₂₃	—	25	1	no	—
34	C₃²×D₂₃	—	25	1	no	—
35	C₃×C₅×C₂₃	—	3	2	no	—
36	C₂×C₅²×D₃	—	23	1	no	—
37	C₂²×C₅²⋊C₃	—	69	1	no	—
38	C₂×C₃×D₂₃	—	75	1	no	—
39	D₃×D₂₃	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
40	C₂×C₅×C₂₃	—	9	2	no	—
41	C₅×D₂₃	—	3	2	no	—
42	C₅×D₂₃	—	9	2	no	—
43	C₃×C₅²⋊C₃	C₂×D₂₃	1	1	yes	characteristic, complemented, Hall
44	C₃²×C₂₃	—	25	1	no	Hall
45	C₂×C₃×C₅²	—	23	1	no	—
46	C₅²×D₃	—	23	1	no	—
47	C₅²×D₃	C₃×D₂₃	1	1	yes	characteristic, complemented
48	C₂×C₅²⋊C₃	—	3	1	no	—
49	C₂×C₅²⋊C₃	—	23	1	no	—
50	C₂×C₅²⋊C₃	—	46	1	no	—
51	C₂×C₅²⋊C₃	—	69	1	no	—
52	C₂×C₃×C₂₃	—	75	1	no	—
53	D₆₉	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
54	C₃×D₂₃	—	25	1	no	—
55	C₃×D₂₃	—	50	1	no	—
56	C₃×D₂₃	—	75	1	no	—
57	C₃×D₂₃	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
58	D₃×C₂₃	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
59	C₅×C₂₃	—	3	2	no	—
60	C₂²×C₅²	—	69	1	no	Hall
61	C₂×D₂₃	—	3	1	no	Hall
62	C₃×C₅²	C₂×C₃×D₂₃	1	1	yes	characteristic, complemented
63	C₅²⋊C₃	—	2	1	no	—
64	C₅²⋊C₃	D₃×D₂₃	1	1	yes	characteristic, direct product factor
65	C₃×C₂₃	—	25	1	no	—
66	C₃×C₂₃	—	50	1	no	—
67	C₃×C₂₃	C₂²×C₅²⋊C₃	1	1	yes	characteristic, complemented
68	C₂×C₅×D₃	—	69	2	no	—
69	C₂×C₅²	—	3	1	no	—
70	C₂×C₅²	—	23	1	no	—
71	C₂×C₅²	—	69	1	no	—
72	C₂×C₂₃	—	3	1	no	—
73	D₂₃	—	3	1	no	—
74	D₂₃	D₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
75	C₂×C₃×D₃	—	575	1	no	Hall
76	C₂×C₃×C₅	—	69	2	no	—
77	C₅×D₃	—	3	2	no	—
78	C₅×D₃	—	69	2	no	—
79	C₅²	C₃×D₃×D₂₃	1	1	yes	characteristic, complemented, Sylow
80	C₂₃	C₂×D₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	—	207	2	no	—
82	C₂×C₃²	—	575	1	no	—
83	C₃×D₃	—	25	1	no	—
84	C₃×D₃	—	575	1	no	—
85	C₃×C₅	—	3	2	no	—
86	C₂²×C₃	—	1725	1	no	—
87	C₂×D₃	—	23	1	no	—
88	C₂×C₅	—	9	2	no	—
89	C₂×C₅	—	69	2	no	—
90	C₂×C₅	—	207	2	no	—
91	C₃²	—	25	1	no	Sylow
92	C₂×C₃	—	23	1	no	—
93	C₂×C₃	—	575	1	no	—
94	C₂×C₃	—	75	1	no	—
95	C₂×C₃	—	1150	1	no	—
96	C₂×C₃	—	1725	1	no	—
97	D₃	—	23	1	no	—
98	D₃	D₂₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
99	C₅	—	3	2	no	—
100	C₂²	—	69	1	no	Sylow
101	C₃	—	25	1	no	—
102	C₃	—	50	1	no	—
103	C₃	C₂×D₂₃×C₅²⋊C₃	1	1	yes	characteristic, complemented
104	C₂	—	3	1	no	—
105	C₂	—	23	1	no	—
106	C₂	—	69	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²⋊C₃×D₆₉

C₂

yes

maximal, characteristic, complemented

D₃×C₂₃×C₅²⋊C₃

C₂

yes

maximal, characteristic, complemented

C₃×D₂₃×C₅²⋊C₃

C₂

yes

maximal, characteristic, complemented

C₅²×D₃×D₂₃

C₃

yes

maximal, characteristic, complemented

C₂×D₂₃×C₅²⋊C₃

—

maximal

C₃×C₂₃×C₅²⋊C₃

C₂²

yes

characteristic, complemented, Hall

C₅²×D₆₉

C₂×C₃

yes

characteristic, complemented

C₅²×D₃×C₂₃

C₂×C₃

yes

characteristic, complemented

C₃×C₅²×D₂₃

C₂×C₃

yes

characteristic, complemented

C₂×C₂₃×C₅²⋊C₃

—

D₂₃×C₅²⋊C₃

—

D₂₃×C₅²⋊C₃

—

D₂₃×C₅²⋊C₃

D₃

yes

characteristic, direct product factor

C₂×C₅²×D₂₃

—

Hall

C₃×C₅²×C₂₃

C₂²×C₃

yes

characteristic, complemented

C₂₃×C₅²⋊C₃

—

C₂₃×C₅²⋊C₃

C₂×D₃

yes

characteristic, complemented

C₅×D₃×D₂₃

—

C₂×C₅²×C₂₃

—

C₅²×D₂₃

—

C₅²×D₂₃

C₃×D₃

yes

characteristic, complemented

C₂×D₃×C₅²⋊C₃

—

maximal, Hall

C₃×D₃×D₂₃

—

maximal, Hall

C₅×D₆₉

—

C₅×D₃×C₂₃

—

C₃×C₅×D₂₃

—

C₅²×C₂₃

C₂×C₃×D₃

yes

characteristic, complemented, Hall

C₂×C₅×D₂₃

—

C₂×C₃×C₅²⋊C₃

—

D₃×C₅²⋊C₃

—

D₃×C₅²⋊C₃

D₂₃

yes

characteristic, direct product factor

C₃×D₆₉

—

C₃×D₃×C₂₃

—

C₃²×D₂₃

—

C₃×C₅×C₂₃

—

C₂×C₅²×D₃

—

C₂²×C₅²⋊C₃

—

C₂×C₃×D₂₃

—

D₃×D₂₃

C₅²⋊C₃

yes

characteristic, direct product factor

C₂×C₅×C₂₃

—

C₅×D₂₃

—

C₅×D₂₃

—

C₃×C₅²⋊C₃

C₂×D₂₃

yes

characteristic, complemented, Hall

C₃²×C₂₃

—

Hall

C₂×C₃×C₅²

—

C₅²×D₃

—

C₅²×D₃

C₃×D₂₃

yes

characteristic, complemented

C₂×C₅²⋊C₃

—

C₂×C₅²⋊C₃

—

C₂×C₅²⋊C₃

—

C₂×C₅²⋊C₃

—

C₂×C₃×C₂₃

—

D₆₉

C₂×C₅²⋊C₃

yes

characteristic, complemented

C₃×D₂₃

—

C₃×D₂₃

—

C₃×D₂₃

—

C₃×D₂₃

C₂×C₅²⋊C₃

yes

characteristic, complemented

D₃×C₂₃

C₂×C₅²⋊C₃

yes

characteristic, complemented

C₅×C₂₃

—

C₂²×C₅²

—

Hall

C₂×D₂₃

—

Hall

C₃×C₅²

C₂×C₃×D₂₃

yes

characteristic, complemented

C₅²⋊C₃

—

C₅²⋊C₃

D₃×D₂₃

yes

characteristic, direct product factor

C₃×C₂₃

—

C₃×C₂₃

—

C₃×C₂₃

C₂²×C₅²⋊C₃

yes

characteristic, complemented

C₂×C₅×D₃

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₂₃

—

D₂₃

—

D₂₃

D₃×C₅²⋊C₃

yes

characteristic, direct product factor

C₂×C₃×D₃

—

575

Hall

C₂×C₃×C₅

—

C₅×D₃

—

C₅×D₃

—

C₅²

C₃×D₃×D₂₃

yes

characteristic, complemented, Sylow

C₂₃

C₂×D₃×C₅²⋊C₃

yes

characteristic, complemented, Sylow

C₂²×C₅

—

207

—

C₂×C₃²

—

575

—

C₃×D₃

—

C₃×D₃

—

575

—

C₃×C₅

—

C₂²×C₃

—

1725

—

C₂×D₃

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

207

—

C₃²

—

Sylow

C₂×C₃

—

C₂×C₃

—

575

—

C₂×C₃

—

C₂×C₃

—

1150

—

C₂×C₃

—

1725

—

D₃

—

D₃

D₂₃×C₅²⋊C₃

yes

characteristic, direct product factor

C₅

—

100

C₂²

—

Sylow

101

C₃

—

102

C₃

—

103

C₃

C₂×D₂₃×C₅²⋊C₃

yes

characteristic, complemented

104

C₂

—

105

C₂

—

106

C₂

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₂₃	C₂²×C₃
LCS	G ▷ C₃×C₅²×C₂₃
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, cosocle, hypocenter

C₃×C₅²×C₂₃

C₂²×C₃

LCS

G ▷ C₃×C₅²×C₂₃

UCS

C₁

N	H ≅ G/N	type
1. C₅²⋊C₃×D₆₉	C₂	semidirect
2. D₃×C₂₃×C₅²⋊C₃	C₂	semidirect
3. C₃×D₂₃×C₅²⋊C₃	C₂	semidirect
4. C₅²×D₃×D₂₃	C₃	semidirect
6. C₃×C₂₃×C₅²⋊C₃	C₂²	semidirect
7. C₅²×D₆₉	C₂×C₃	semidirect
8. C₅²×D₃×C₂₃	C₂×C₃	semidirect
9. C₃×C₅²×D₂₃	C₂×C₃	semidirect
13. D₂₃×C₅²⋊C₃	D₃	direct (N × H)
15. C₃×C₅²×C₂₃	C₂²×C₃	semidirect
17. C₂₃×C₅²⋊C₃	C₂×D₃	semidirect
21. C₅²×D₂₃	C₃×D₃	semidirect
27. C₅²×C₂₃	C₂×C₃×D₃	semidirect
31. D₃×C₅²⋊C₃	D₂₃	direct (N × H)
39. D₃×D₂₃	C₅²⋊C₃	direct (N × H)
43. C₃×C₅²⋊C₃	C₂×D₂₃	semidirect
47. C₅²×D₃	C₃×D₂₃	semidirect
53. D₆₉	C₂×C₅²⋊C₃	semidirect
57. C₃×D₂₃	C₂×C₅²⋊C₃	semidirect
58. D₃×C₂₃	C₂×C₅²⋊C₃	semidirect
62. C₃×C₅²	C₂×C₃×D₂₃	semidirect
64. C₅²⋊C₃	D₃×D₂₃	direct (N × H)
67. C₃×C₂₃	C₂²×C₅²⋊C₃	semidirect
74. D₂₃	D₃×C₅²⋊C₃	direct (N × H)
79. C₅²	C₃×D₃×D₂₃	semidirect
80. C₂₃	C₂×D₃×C₅²⋊C₃	semidirect
98. D₃	D₂₃×C₅²⋊C₃	direct (N × H)
103. C₃	C₂×D₂₃×C₅²⋊C₃	semidirect

H ≅ G/N

type

1. C₅²⋊C₃×D₆₉

C₂

semidirect

2. D₃×C₂₃×C₅²⋊C₃

C₂

semidirect

3. C₃×D₂₃×C₅²⋊C₃

C₂

semidirect

4. C₅²×D₃×D₂₃

C₃

semidirect

6. C₃×C₂₃×C₅²⋊C₃

C₂²

semidirect

7. C₅²×D₆₉

C₂×C₃

semidirect

8. C₅²×D₃×C₂₃

C₂×C₃

semidirect

9. C₃×C₅²×D₂₃

C₂×C₃

semidirect

13. D₂₃×C₅²⋊C₃

D₃

direct (N × H)

15. C₃×C₅²×C₂₃

C₂²×C₃

semidirect

17. C₂₃×C₅²⋊C₃

C₂×D₃

semidirect

21. C₅²×D₂₃

C₃×D₃

semidirect

27. C₅²×C₂₃

C₂×C₃×D₃

semidirect

31. D₃×C₅²⋊C₃

D₂₃

direct (N × H)

39. D₃×D₂₃

C₅²⋊C₃

direct (N × H)

43. C₃×C₅²⋊C₃

C₂×D₂₃

semidirect

47. C₅²×D₃

C₃×D₂₃

semidirect

53. D₆₉

C₂×C₅²⋊C₃

semidirect

57. C₃×D₂₃

C₂×C₅²⋊C₃

semidirect

58. D₃×C₂₃

C₂×C₅²⋊C₃

semidirect

62. C₃×C₅²

C₂×C₃×D₂₃

semidirect

64. C₅²⋊C₃

D₃×D₂₃

direct (N × H)

67. C₃×C₂₃

C₂²×C₅²⋊C₃

semidirect

74. D₂₃

D₃×C₅²⋊C₃

direct (N × H)

79. C₅²

C₃×D₃×D₂₃

semidirect

80. C₂₃

C₂×D₃×C₅²⋊C₃

semidirect

98. D₃

D₂₃×C₅²⋊C₃

direct (N × H)

103. C₃

C₂×D₂₃×C₅²⋊C₃

semidirect

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

subgroup	index
C₅²⋊C₃×D₆₉	2
D₃×C₂₃×C₅²⋊C₃	2
C₃×D₂₃×C₅²⋊C₃	2
C₅²×D₃×D₂₃	3
C₂×D₂₃×C₅²⋊C₃	3
C₂×D₃×C₅²⋊C₃	23
C₃×D₃×D₂₃	25

subgroup

index

C₅²⋊C₃×D₆₉

D₃×C₂₃×C₅²⋊C₃

C₃×D₂₃×C₅²⋊C₃

C₅²×D₃×D₂₃

C₂×D₂₃×C₅²⋊C₃

C₂×D₃×C₅²⋊C₃

C₃×D₃×D₂₃

Subgroups — D₃×D₂₃×C₅²⋊C₃ #35208113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D3×D23×C52⋊C3 #35208113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×D₂₃×C₅²⋊C₃ #35208113