Group #10589815 — subgroups

Subgroups — #10589815

Order 576.

354 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₉×#127	C₂	1	1	yes	maximal, characteristic
2	#63498	C₂	1	4	yes	maximal, complemented
3	#63497	C₂	1	4	yes	maximal, complemented
4	D₄×Dic₉	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₁₈	C₂	1	2	yes	maximal, complemented
6	#5713	—	3	1	no	maximal
7	C₂×D₄×C₉	C₂²	1	4	yes	—
8	C₂×D₄×C₉	C₂²	1	2	yes	—
9	C₂²×Dic₉	C₂²	1	4	yes	—
10	C₂×Dic₁₈	C₂²	1	2	yes	—
11	C₄²×C₉	C₂²	1	1	yes	characteristic
12	#3676	C₂²	1	8	yes	—
13	#3676	C₂²	1	4	yes	—
14	C₄⋊Dic₉	C₂²	1	2	yes	—
15	Dic₉⋊C₄	C₂²	1	4	yes	—
16	C₄×Dic₉	C₂²	1	4	yes	—
17	C₃×#127	D₃	1	1	yes	characteristic
18	#936	—	3	4	no	—
19	#935	—	3	4	no	—
20	D₄×Dic₃	—	3	4	no	—
21	C₄×Dic₆	—	3	2	no	—
22	C₂³×C₉	C₂³	1	4	yes	—
23	D₄×C₉	—	2	8	no	—
24	D₄×C₉	—	4	2	no	—
25	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₉	C₂³	1	2	yes	—
27	C₂×Dic₉	—	4	4	no	—
28	C₂×Dic₉	C₂³	1	4	yes	—
29	C₂×Dic₉	C₂³	1	4	yes	—
30	Dic₁₈	—	2	4	no	—
31	#559	—	9	1	no	Sylow
32	C₂×C₃×D₄	C₂×D₃	1	4	yes	—
33	C₂×C₃×D₄	C₂×D₃	1	2	yes	—
34	C₂²×Dic₃	—	3	4	no	—
35	C₂×Dic₆	—	3	2	no	—
36	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
37	#217	—	3	8	no	—
38	#217	—	3	4	no	—
39	C₄⋊Dic₃	—	3	2	no	—
40	Dic₃⋊C₄	—	3	4	no	—
41	C₄×Dic₃	—	3	4	no	—
42	C₂²×C₉	—	2	8	no	—
43	C₂²×C₉	—	2	4	no	—
44	C₂²×C₉	C₂⁴	1	1	yes	characteristic
45	C₄×C₉	—	2	1	no	—
46	C₄×C₉	#41	1	4	yes	—
47	Dic₉	—	2	4	no	—
48	Dic₉	—	2	4	no	—
49	#127	D₉	1	1	yes	characteristic
50	#124	—	9	4	no	—
51	#123	—	9	4	no	—
52	C₄×Q₈	—	9	2	no	—
53	C₄×D₄	—	9	4	no	—
54	C₂³×C₃	C₂²×D₃	1	4	yes	—
55	C₃×D₄	—	2	8	no	—
56	C₃×D₄	—	4	2	no	—
57	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
59	C₂×Dic₃	—	3	4	no	—
60	C₂×Dic₃	—	3	4	no	—
61	C₂×Dic₃	—	12	4	no	—
62	Dic₆	—	6	4	no	—
63	C₂×C₉	—	4	4	no	—
64	C₂×C₉	C₂×#41	1	2	yes	—
65	C₂×C₉	H(5,𝔽₂)	1	1	yes	characteristic
66	C₂×Q₈	—	9	2	no	—
67	C₂×D₄	C₂×D₉	1	4	yes	—
68	C₂×D₄	C₂×D₉	1	2	yes	—
69	C₂²×C₄	—	9	4	no	—
70	C₄⋊C₄	—	9	2	no	—
71	C₄⋊C₄	—	9	4	no	—
72	C₂²⋊C₄	—	9	8	no	—
73	C₂²⋊C₄	—	9	4	no	—
74	C₄²	—	9	4	no	—
75	C₄²	C₂×D₉	1	1	yes	characteristic
76	C₂²×C₃	—	2	8	no	—
77	C₂²×C₃	—	2	4	no	—
78	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
79	C₃×C₄	—	2	1	no	—
80	C₃×C₄	#237	1	4	yes	—
81	Dic₃	—	6	4	no	—
82	Dic₃	—	6	4	no	—
83	C₉	#559	1	1	yes	characteristic, complemented, Sylow
84	C₂³	C₂²×D₉	1	4	yes	—
85	Q₈	—	18	4	no	—
86	D₄	—	2	8	no	—
87	D₄	—	4	2	no	—
88	C₂×C₄	—	9	4	no	—
89	C₂×C₄	—	9	4	no	—
90	C₂×C₄	—	36	4	no	—
91	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
92	C₂×C₄	C₂²×D₉	1	2	yes	—
93	C₂×C₃	—	4	4	no	—
94	C₂×C₃	C₂×#237	1	2	yes	—
95	C₂×C₃	#1004	1	1	yes	characteristic
96	C₂²	—	2	8	no	—
97	C₂²	—	2	4	no	—
98	C₂²	C₂³×D₉	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	18	4	no	—
101	C₄	—	18	4	no	—
102	C₄	#3699	1	4	yes	—
103	C₃	#5713	1	1	yes	characteristic
104	C₂	—	4	4	no	—
105	C₂	C₂×#3699	1	2	yes	—
106	C₂	#63710	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#127

C₂

yes

maximal, characteristic

#63498

C₂

yes

maximal, complemented

#63497

C₂

yes

maximal, complemented

D₄×Dic₉

C₂

yes

maximal, complemented

C₄×Dic₁₈

C₂

yes

maximal, complemented

#5713

—

maximal

C₂×D₄×C₉

C₂²

yes

—

C₂×D₄×C₉

C₂²

yes

—

C₂²×Dic₉

C₂²

yes

—

C₂×Dic₁₈

C₂²

yes

—

C₄²×C₉

C₂²

yes

characteristic

#3676

C₂²

yes

—

#3676

C₂²

yes

—

C₄⋊Dic₉

C₂²

yes

—

Dic₉⋊C₄

C₂²

yes

—

C₄×Dic₉

C₂²

yes

—

C₃×#127

D₃

yes

characteristic

#936

—

#935

—

D₄×Dic₃

—

C₄×Dic₆

—

C₂³×C₉

C₂³

yes

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

C₂³

yes

characteristic

C₂×C₄×C₉

C₂³

yes

—

C₂×Dic₉

—

C₂×Dic₉

C₂³

yes

—

C₂×Dic₉

C₂³

yes

—

Dic₁₈

—

#559

—

Sylow

C₂×C₃×D₄

C₂×D₃

yes

—

C₂×C₃×D₄

C₂×D₃

yes

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₃×C₄²

C₂×D₃

yes

characteristic

#217

—

#217

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂⁴

yes

characteristic

C₄×C₉

—

C₄×C₉

#41

yes

—

Dic₉

—

Dic₉

—

#127

D₉

yes

characteristic

#124

—

#123

—

C₄×Q₈

—

C₄×D₄

—

C₂³×C₃

C₂²×D₃

yes

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₂×C₉

—

C₂×C₉

C₂×#41

yes

—

C₂×C₉

H(5,𝔽₂)

yes

characteristic

C₂×Q₈

—

C₂×D₄

C₂×D₉

yes

—

C₂×D₄

C₂×D₉

yes

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₉

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂³×D₃

yes

characteristic

C₃×C₄

—

C₃×C₄

#237

yes

—

Dic₃

—

Dic₃

—

C₉

#559

yes

characteristic, complemented, Sylow

C₂³

C₂²×D₉

yes

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₉

yes

characteristic

C₂×C₄

C₂²×D₉

yes

—

C₂×C₃

—

C₂×C₃

C₂×#237

yes

—

C₂×C₃

#1004

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂³×D₉

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#3699

yes

—

103

C₃

#5713

yes

characteristic

104

C₂

—

105

C₂

C₂×#3699

yes

—

106

C₂

#63710

yes

characteristic

	subgroup	G/H
center	C₂²	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴
Frattini, socle	C₂²×C₃	C₂³×D₃
Fitting	C₉×#127	C₂
prime radical	C₃×#127	D₃
hypercenter	#127	D₉
hypocenter	C₉	#559
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center

C₂²

C₂³×D₉

commutator, cosocle

C₂²×C₉

C₂⁴

Frattini, socle

C₂²×C₃

C₂³×D₃

C₂

D₃

D₉

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
2. #63498	C₂	semidirect
3. #63497	C₂	semidirect
4. D₄×Dic₉	C₂	semidirect
5. C₄×Dic₁₈	C₂	semidirect
83. C₉	#559	semidirect

H ≅ G/N

type

2. #63498

C₂

semidirect

3. #63497

C₂

semidirect

4. D₄×Dic₉

C₂

semidirect

5. C₄×Dic₁₈

C₂

semidirect

83. C₉

#559

semidirect

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

subgroup	index
C₉×#127	2
#63498	2
#63497	2
D₄×Dic₉	2
C₄×Dic₁₈	2
#5713	3

subgroup

index

C₉×#127

#63498

#63497

D₄×Dic₉

C₄×Dic₁₈

#5713

Subgroups — #10589815

Conjugacy classes of subgroups

Complements

Maximal subgroups