Group #36554678 — subgroups

Subgroups — #36554678

Order 22784.

1062 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1062 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	#26020395	C₂	1	8	yes	maximal, complemented
2	#26019946	C₂	1	6	yes	maximal
3	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083558	C₂²	1	4	yes	—
5	#19151353	C₂²	1	24	yes	—
6	C₂²×C₄²×C₈₉	C₄	1	4	yes	—
7	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
8	C₄³×C₈₉	C₄	1	4	yes	—
9	C₄³×C₈₉	C₂²	1	4	yes	—
10	C₂×#13083558	—	2	24	no	—
11	C₂²×#10623494	C₂³	1	8	yes	—
12	C₂³×C₄×C₈₉	C₂×C₄	1	6	yes	—
13	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₈₉	—	2	12	no	—
15	C₂×C₄²×C₈₉	—	2	16	no	—
16	C₂×C₄²×C₈₉	C₂×C₄	1	6	yes	—
17	C₂×C₄²×C₈₉	C₂×C₄	1	12	yes	—
18	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
20	#13083558	—	4	16	no	—
21	C₂×#10623494	—	2	24	no	—
22	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₉	—	2	12	no	—
24	C₂²×C₄×C₈₉	—	2	24	no	—
25	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
26	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	#41	1	12	yes	—
28	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
29	C₄²×C₈₉	—	2	12	no	—
30	C₄²×C₈₉	—	2	24	no	—
31	C₄²×C₈₉	—	2	12	no	—
32	C₄²×C₈₉	M₁₆	1	16	yes	complemented
33	C₂³×C₈₉	—	2	4	no	—
34	C₂³×C₈₉	#117	1	6	yes	—
35	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	—	2	48	no	—
37	C₂×C₄×C₈₉	—	2	12	no	—
38	C₂×C₄×C₈₉	—	2	12	no	—
39	C₂×C₄×C₈₉	#117	1	6	yes	—
40	C₂×C₄×C₈₉	#126	1	16	yes	—
41	C₂×C₄×C₈₉	C₂×M₁₆	1	24	yes	—
42	C₂×C₄×C₈₉	C₂×#41	1	6	yes	—
43	#10623494	—	4	8	no	—
44	C₂²×C₈₉	—	2	12	no	—
45	C₂²×C₈₉	#383	1	4	yes	—
46	C₂²×C₈₉	C₂×#117	1	3	yes	—
47	C₂²×C₈₉	C₂²×M₁₆	1	4	yes	—
48	C₄×C₈₉	—	2	24	no	—
49	C₄×C₈₉	#383	1	4	yes	—
50	C₄×C₈₉	C₂×#126	1	4	yes	—
51	#20772	—	89	1	no	maximal, Sylow
52	C₂×C₈₉	—	2	4	no	—
53	C₂×C₈₉	C₂×#383	1	1	yes	characteristic
54	C₂×C₈₉	#2920	1	6	yes	—
55	C₂×C₄³	D₈₉	1	1	yes	characteristic
56	#1839	—	89	8	no	—
57	#1728	—	89	6	no	—
58	C₈₉	#20772	1	1	yes	characteristic, complemented, Sylow
59	C₂²×M₁₆	—	89	4	no	—
60	C₂²×C₄²	Dic₈₉	1	4	yes	—
61	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
62	C₄³	Dic₈₉	1	4	yes	—
63	C₄³	C₂×D₈₉	1	4	yes	—
64	#336	—	89	24	no	—
65	C₂³×C₄	C₂×Dic₈₉	1	6	yes	—
66	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
67	C₂×M₁₆	—	178	24	no	—
68	C₂²×C₈	—	89	8	no	—
69	C₂×C₄²	—	2	12	no	—
70	C₂×C₄²	—	2	16	no	—
71	C₂×C₄²	C₂×Dic₈₉	1	6	yes	—
72	C₂×C₄²	C₂×Dic₈₉	1	12	yes	—
73	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
74	C₂×C₄²	C₂²×D₈₉	1	6	yes	—
75	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
76	C₂²×C₄	—	2	12	no	—
77	C₂²×C₄	—	2	24	no	—
78	C₂²×C₄	C₂²×Dic₈₉	1	12	yes	—
79	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
80	C₂²×C₄	#13083574	1	12	yes	—
81	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
82	M₁₆	—	356	16	no	—
83	C₂×C₈	—	178	24	no	—
84	C₄²	—	2	12	no	—
85	C₄²	—	2	24	no	—
86	C₄²	—	2	12	no	—
87	C₄²	#13083558	1	16	yes	complemented
88	C₂³	—	2	4	no	—
89	C₂³	#15667971	1	6	yes	—
90	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
91	C₂×C₄	—	2	48	no	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	12	no	—
94	C₂×C₄	#15667971	1	6	yes	—
95	C₂×C₄	#15668006	1	16	yes	—
96	C₂×C₄	C₂×#13083558	1	24	yes	—
97	C₂×C₄	C₂×#13083574	1	6	yes	—
98	C₈	—	356	8	no	—
99	C₂²	—	2	12	no	—
100	C₂²	#19151466	1	4	yes	—
101	C₂²	C₂×#15667971	1	3	yes	—
102	C₂²	C₂²×#13083558	1	4	yes	—
103	C₄	—	2	24	no	—
104	C₄	#19151466	1	4	yes	—
105	C₄	C₂×#15668006	1	4	yes	—
106	C₂	—	2	4	no	—
107	C₂	C₂×#19151466	1	1	yes	characteristic
108	C₂	#26028885	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020395

C₂

yes

maximal, complemented

#26019946

C₂

yes

maximal

C₂×C₄³×C₈₉

C₂

yes

maximal, characteristic

C₂²×#13083558

C₂²

yes

—

#19151353

C₂²

yes

—

C₂²×C₄²×C₈₉

C₄

yes

—

C₂²×C₄²×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₄

yes

—

C₄³×C₈₉

C₂²

yes

—

C₂×#13083558

—

C₂²×#10623494

C₂³

yes

—

C₂³×C₄×C₈₉

C₂×C₄

yes

—

C₂³×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

—

#13083558

—

C₂×#10623494

—

C₂⁴×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂²×C₄×C₈₉

C₂⁴

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

M₁₆

yes

complemented

C₂³×C₈₉

—

C₂³×C₈₉

#117

yes

—

C₂³×C₈₉

C₂³×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

#126

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

C₂×#41

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#383

yes

—

C₂²×C₈₉

C₂×#117

yes

—

C₂²×C₈₉

C₂²×M₁₆

yes

—

C₄×C₈₉

—

C₄×C₈₉

#383

yes

—

C₄×C₈₉

C₂×#126

yes

—

#20772

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×#383

yes

characteristic

C₂×C₈₉

#2920

yes

—

C₂×C₄³

D₈₉

yes

characteristic

#1839

—

#1728

—

C₈₉

#20772

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×C₄²

Dic₈₉

yes

—

C₂²×C₄²

C₂×D₈₉

yes

—

C₄³

Dic₈₉

yes

—

C₄³

C₂×D₈₉

yes

—

#336

—

C₂³×C₄

C₂×Dic₈₉

yes

—

C₂³×C₄

C₂²×D₈₉

yes

characteristic

C₂×M₁₆

—

178

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₈₉

yes

—

C₂×C₄²

C₂×Dic₈₉

yes

—

C₂×C₄²

C₂×Dic₈₉

yes

—

C₂×C₄²

C₂²×D₈₉

yes

—

C₂⁴

C₂²×Dic₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×Dic₈₉

yes

—

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

C₂²×C₄

#13083574

yes

—

C₂²×C₄

C₂³×D₈₉

yes

characteristic

M₁₆

—

356

—

C₂×C₈

—

178

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#13083558

yes

complemented

C₂³

—

C₂³

#15667971

yes

—

C₂³

C₂³×Dic₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#15667971

yes

—

C₂×C₄

#15668006

yes

—

C₂×C₄

C₂×#13083558

yes

—

C₂×C₄

C₂×#13083574

yes

—

C₈

—

356

—

C₂²

—

100

C₂²

#19151466

yes

—

101

C₂²

C₂×#15667971

yes

—

102

C₂²

C₂²×#13083558

yes

—

103

C₄

—

104

C₄

#19151466

yes

—

105

C₄

C₂×#15668006

yes

—

106

C₂

—

107

C₂

C₂×#19151466

yes

characteristic

108

C₂

#26028885

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle	C₂³×C₈₉	C₂³×C₄
Fitting	C₂×C₄³×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	#20772
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₈₉

commutator, socle

C₂³×C₈₉

C₂³×C₄

Fitting

C₂×C₄³×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂²×C₄

cosocle

C₂²×C₄×C₈₉

C₂⁴

hypercenter

C₂×C₄³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

N	H ≅ G/N	type
1. #26020395	C₂	semidirect
32. C₄²×C₈₉	M₁₆	semidirect
58. C₈₉	#20772	semidirect
87. C₄²	#13083558	semidirect

H ≅ G/N

type

1. #26020395

C₂

semidirect

32. C₄²×C₈₉

M₁₆

semidirect

58. C₈₉

#20772

semidirect

87. C₄²

#13083558

semidirect

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

subgroup	index
#26020395	2
#26019946	2
C₂×C₄³×C₈₉	2
#20772	89

subgroup

index

#26020395

#26019946

C₂×C₄³×C₈₉

#20772

Subgroups — #36554678

Conjugacy classes of subgroups

Complements

Maximal subgroups