Group #26028438 — subgroups

Subgroups — #26028438

Order 11392.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₈₉⋊#401	C₂	1	2	yes	maximal, complemented
2	#19151531	C₂	1	6	yes	maximal, complemented
3	#19151523	C₂	1	3	yes	maximal
4	#19151470	C₂	1	3	yes	maximal
5	#399×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₈₉	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	—
8	C₂×#13083553	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₈₉	C₂²	1	3	yes	—
10	#15667919	C₂²	1	6	yes	—
11	#15667919	C₂²	1	3	yes	—
12	#15667919	C₂²	1	6	yes	—
13	C₂×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
14	C₂×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
15	#95×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₈₉	—	4	3	no	—
17	C₂²×Dic₈₉	C₂³	1	2	yes	—
18	C₂²×Dic₈₉	C₂³	1	6	yes	—
19	C₄⋊Dic₈₉	—	4	3	no	—
20	#13083553	—	4	6	no	—
21	C₂²⋊Dic₈₉	—	2	6	no	—
22	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
24	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
25	C₄⋊C₄×C₈₉	—	4	3	no	—
26	C₂²⋊C₄×C₈₉	—	4	3	no	—
27	C₂³×C₈₉	—	2	3	no	—
28	C₂³×C₈₉	—	2	3	no	—
29	C₂³×C₈₉	#41	1	2	yes	—
30	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₈₉	—	2	3	no	—
32	C₂×C₄×C₈₉	—	2	3	no	—
33	C₂×C₄×C₈₉	—	2	6	no	—
34	C₂×C₄×C₈₉	—	2	6	no	—
35	C₂×Dic₈₉	—	2	6	no	—
36	C₂×Dic₈₉	—	2	6	no	—
37	C₂×Dic₈₉	—	2	6	no	—
38	C₂×Dic₈₉	—	2	6	no	—
39	C₂×Dic₈₉	—	4	4	no	—
40	C₂²×C₈₉	—	2	6	no	—
41	C₂²×C₈₉	—	4	1	no	—
42	C₂²×C₈₉	—	4	3	no	—
43	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
44	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₈₉	2₋¹⁺⁴	1	3	yes	—
46	C₄×C₈₉	—	4	3	no	—
47	C₄×C₈₉	—	4	3	no	—
48	Dic₈₉	—	4	6	no	—
49	Dic₈₉	—	4	2	no	—
50	C₂×C₈₉	—	4	2	no	—
51	C₂×C₈₉	#539	1	3	yes	—
52	C₂×C₈₉	#560	1	1	yes	characteristic
53	C₂×C₈₉	#563	1	3	yes	—
54	#2834	—	89	1	no	maximal, Sylow
55	C₈₉	#2834	1	1	yes	characteristic, complemented, Sylow
56	#401	—	89	2	no	—
57	#399	D₈₉	1	1	yes	characteristic
58	#398	—	89	6	no	—
59	#397	—	89	3	no	—
60	#385	—	89	3	no	—
61	C₂³×C₄	—	89	1	no	—
62	C₂×C₄⋊C₄	—	89	6	no	—
63	C₂×C₄⋊C₄	—	89	3	no	—
64	C₂×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
65	C₂×C₂²⋊C₄	—	89	3	no	—
66	C₂×C₂²⋊C₄	C₂×D₈₉	1	3	yes	—
67	#95	—	89	6	no	—
68	#95	—	89	3	no	—
69	#95	—	89	6	no	—
70	#95	C₂×D₈₉	1	1	yes	characteristic
71	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
72	C₂²×C₄	—	89	6	no	—
73	C₂²×C₄	—	89	2	no	—
74	C₂²×C₄	—	356	3	no	—
75	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
76	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
77	C₄⋊C₄	—	4	3	no	—
78	C₄⋊C₄	—	356	6	no	—
79	C₄⋊C₄	—	356	3	no	—
80	C₂²⋊C₄	—	4	3	no	—
81	C₂²⋊C₄	—	178	6	no	—
82	C₂³	—	2	3	no	—
83	C₂³	—	2	3	no	—
84	C₂³	#13083575	1	2	yes	—
85	C₂³	C₂³×D₈₉	1	1	yes	characteristic
86	C₂×C₄	—	2	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	6	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	356	4	no	—
91	C₂×C₄	—	178	6	no	—
92	C₂×C₄	—	178	6	no	—
93	C₂×C₄	—	178	6	no	—
94	C₂×C₄	—	178	6	no	—
95	C₂²	—	2	6	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	3	no	—
98	C₂²	C₂×#13083575	1	1	yes	characteristic
99	C₂²	#15668057	1	3	yes	—
100	C₂²	#15668059	1	3	yes	—
101	C₄	—	4	3	no	—
102	C₄	—	4	3	no	—
103	C₄	—	356	6	no	—
104	C₄	—	356	2	no	—
105	C₂	—	4	2	no	—
106	C₂	#19152197	1	3	yes	—
107	C₂	#19152335	1	1	yes	characteristic
108	C₂	#19152351	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉⋊#401

C₂

yes

maximal, complemented

#19151531

C₂

yes

maximal, complemented

#19151523

C₂

yes

maximal

#19151470

C₂

yes

maximal

#399×C₈₉

C₂

yes

maximal, characteristic

C₂³×Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#41

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#539

yes

—

C₂×C₈₉

#560

yes

characteristic

C₂×C₈₉

#563

yes

—

#2834

—

maximal, Sylow

C₈₉

#2834

yes

characteristic, complemented, Sylow

#401

—

#399

D₈₉

yes

—

—

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

—

—

—

C₂×D₈₉

yes

characteristic

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

356

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₂²⋊C₄

—

C₂²⋊C₄

—

178

—

C₂³

—

C₂³

—

C₂³

#13083575

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×#13083575

yes

characteristic

C₂²

#15668057

yes

—

100

C₂²

#15668059

yes

—

101

C₄

—

102

C₄

—

103

C₄

—

356

—

104

C₄

—

356

—

105

C₂

—

106

C₂

#19152197

yes

—

107

C₂

#19152335

yes

characteristic

108

C₂

#19152351

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#399×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂³
hypercenter	#399	D₈₉
hypocenter	C₈₉	#2834
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#399×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
1. C₈₉⋊#401	C₂	semidirect
2. #19151531	C₂	semidirect
55. C₈₉	#2834	semidirect

H ≅ G/N

type

1. C₈₉⋊#401

C₂

semidirect

2. #19151531

C₂

semidirect

55. C₈₉

#2834

semidirect

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

subgroup	index
C₈₉⋊#401	2
#19151531	2
#19151523	2
#19151470	2
#399×C₈₉	2
#2834	89

subgroup

index

C₈₉⋊#401

#399×C₈₉

Subgroups — #26028438

Conjugacy classes of subgroups

Complements

Maximal subgroups