Group #26022057 — subgroups

Subgroups — #26022057

Order 11392.

138 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 69 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	#19151552	C₂	1	1	yes	maximal, characteristic
2	#19151416	C₂	1	2	yes	maximal, complemented
3	#19151377	C₂	1	2	yes	maximal
4	#19151374	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083557	C₄	1	1	yes	characteristic
7	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
8	#15667969	C₄	1	1	yes	characteristic
9	C₂×#13083549	C₂²	1	2	yes	—
10	#15667923	C₄	1	1	yes	characteristic
11	#15667922	C₄	1	1	yes	characteristic
12	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₈₉	C₂²	1	2	yes	—
14	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
16	#13083557	—	2	2	no	—
17	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
18	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
19	C₈×D₈₉	—	2	2	no	—
20	#13083553	C₈	1	2	yes	—
21	#13083551	C₈	1	2	yes	—
22	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
23	#13083549	—	2	2	no	—
24	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₈₉	—	2	2	no	—
28	C₂²×D₈₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₈₉	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₈₉	C₂×C₈	1	2	yes	—
32	C₂×Dic₈₉	C₂×C₈	1	1	yes	characteristic
33	C₄×D₈₉	—	2	1	no	—
34	C₄×D₈₉	—	2	1	no	—
35	C₈×C₈₉	M₁₆	1	2	yes	—
36	C₈×C₈₉	#41	1	2	yes	—
37	#10623494	—	2	1	no	—
38	#10623494	—	2	1	no	—
39	C₂²×C₈₉	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₈₉	—	2	1	no	—
41	C₂×D₈₉	—	2	2	no	—
42	C₄×C₈₉	—	2	2	no	—
43	C₄×C₈₉	#117	1	1	yes	characteristic
44	C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
45	Dic₈₉	—	2	1	no	—
46	Dic₈₉	—	2	2	no	—
47	C₂×C₈₉	#431	1	1	yes	characteristic
48	C₂×C₈₉	#504	1	2	yes	—
49	D₈₉	—	4	1	no	—
50	#2164	—	89	1	no	maximal, Sylow
51	C₈₉	#2164	1	1	yes	characteristic, complemented, Sylow
52	#405	—	89	1	no	—
53	C₄⋊C₁₆	—	89	2	no	—
54	C₂²⋊C₁₆	—	89	2	no	—
55	#346	—	89	1	no	—
56	C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
57	C₂×M₁₆	—	89	1	no	—
58	C₂²×C₈	—	89	1	no	—
59	#117	—	89	1	no	—
60	C₂×C₁₆	—	89	2	no	—
61	C₂×C₁₆	C₂×D₈₉	1	2	yes	—
62	#97	—	89	1	no	—
63	#97	—	89	1	no	—
64	C₄×C₈	—	89	1	no	—
65	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
66	C₂²×C₄	—	89	1	no	—
67	M₁₆	—	178	2	no	—
68	C₂×C₈	—	89	1	no	—
69	C₂×C₈	—	89	1	no	—
70	C₂×C₈	—	178	2	no	—
71	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
73	C₄⋊C₄	—	89	2	no	—
74	C₂²⋊C₄	—	89	2	no	—
75	C₄²	—	89	1	no	—
76	C₄²	C₄×D₈₉	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	178	2	no	—
79	C₂³	—	89	1	no	—
80	C₂×C₄	—	89	2	no	—
81	C₂×C₄	—	89	1	no	—
82	C₂×C₄	—	178	1	no	—
83	C₂×C₄	—	178	1	no	—
84	C₂×C₄	C₈×D₈₉	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
86	C₈	—	178	1	no	—
87	C₈	—	178	1	no	—
88	C₈	#13083557	1	2	yes	—
89	C₈	#13083574	1	2	yes	—
90	C₂²	—	178	1	no	—
91	C₂²	—	178	2	no	—
92	C₂²	C₂×C₈×D₈₉	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	178	1	no	—
95	C₄	—	178	2	no	—
96	C₄	#15667969	1	1	yes	characteristic
97	C₄	C₂×#13083557	1	1	yes	characteristic
98	C₂	—	356	1	no	—
99	C₂	#19151649	1	1	yes	characteristic
100	C₂	#19152014	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151552

C₂

yes

maximal, characteristic

#19151416

C₂

yes

maximal, complemented

#19151377

C₂

yes

maximal

#19151374

C₂

yes

maximal, characteristic, complemented

C₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083557

C₄

yes

characteristic

C₂×C₈×D₈₉

C₂²

yes

characteristic

#15667969

C₄

yes

characteristic

C₂×#13083549

C₂²

yes

—

#15667923

C₄

yes

characteristic

#15667922

C₄

yes

characteristic

C₈×Dic₈₉

C₂²

yes

characteristic

C₂×C₁₆×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂×C₄

yes

characteristic

#13083557

—

C₂×#10623494

C₂×C₄

yes

characteristic

C₂×#10623494

C₂×C₄

yes

characteristic

C₈×D₈₉

—

#13083553

C₈

yes

—

#13083551

C₈

yes

—

C₄×Dic₈₉

C₂×C₄

yes

characteristic

#13083549

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₁₆×C₈₉

—

C₂²×D₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×C₈

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×Dic₈₉

C₂×C₈

yes

—

C₂×Dic₈₉

C₂×C₈

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

M₁₆

yes

—

C₈×C₈₉

#41

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

C₂²×C₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#117

yes

characteristic

C₄×C₈₉

C₂×M₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#431

yes

characteristic

C₂×C₈₉

#504

yes

—

D₈₉

—

#2164

—

maximal, Sylow

C₈₉

#2164

yes

characteristic, complemented, Sylow

#405

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

D₈₉

yes

characteristic, complemented

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₄

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₈₉

yes

characteristic

C₁₆

—

C₁₆

—

178

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₈×D₈₉

yes

—

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₈

—

178

—

C₈

—

178

—

C₈

#13083557

yes

—

C₈

#13083574

yes

—

C₂²

—

178

—

C₂²

—

178

—

C₂²

C₂×C₈×D₈₉

yes

characteristic

C₄

—

C₄

—

178

—

C₄

—

178

—

C₄

yes

C₄

yes

C₂

—

356

—

C₂

#19151649

yes

characteristic

100

C₂

#19152014

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₈
Fitting	C₄×C₁₆×C₈₉	C₂
prime radical	C₂²×D₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂³
hypercenter	C₄×C₁₆	D₈₉
hypocenter	C₈₉	#2164
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₈₉

commutator, socle

C₂²×C₈₉

C₂²×C₈

Fitting

C₄×C₁₆×C₈₉

C₂

prime radical

C₂²×D₈₉

C₂×C₈

cosocle

C₂×C₈×C₈₉

C₂³

hypercenter

C₄×C₁₆

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #19151416	C₂	semidirect
4. #19151374	C₂	semidirect
5. C₄×C₁₆×C₈₉	C₂	semidirect
51. C₈₉	#2164	semidirect
56. C₄×C₁₆	D₈₉	semidirect

H ≅ G/N

type

2. #19151416

C₂

semidirect

4. #19151374

C₂

semidirect

5. C₄×C₁₆×C₈₉

C₂

semidirect

51. C₈₉

#2164

semidirect

56. C₄×C₁₆

D₈₉

semidirect

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

subgroup	index
#19151552	2
#19151416	2
#19151377	2
#19151374	2
C₄×C₁₆×C₈₉	2
#2164	89

subgroup

index

C₄×C₁₆×C₈₉

Subgroups — #26022057

Conjugacy classes of subgroups

Complements

Maximal subgroups