Group #26022093 — subgroups

Subgroups — #26022093

Order 11392.

144 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 63 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	#19152016	C₂	1	2	yes	maximal, complemented
2	#19151716	C₂	1	1	yes	maximal, characteristic
3	#19151383	C₂	1	2	yes	maximal, complemented
4	#19151374	C₂	1	1	yes	maximal, characteristic, complemented
5	#364×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668034	C₄	1	1	yes	characteristic
7	#15668018	C₂²	1	2	yes	—
8	#15667951	—	2	2	no	—
9	C₂×#13083549	C₂²	1	1	yes	characteristic
10	C₂×#13083549	C₂²	1	1	yes	characteristic
11	C₁₆×D₈₉	—	2	2	no	—
12	#15667948	C₄	1	1	yes	characteristic
13	#15667940	C₄	1	2	yes	—
14	M₃₂×C₈₉	C₂²	1	2	yes	—
15	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#13083574	C₂×C₄	1	2	yes	—
17	Dic₃₅₆	C₈	1	1	yes	characteristic
18	#13083563	C₈	1	2	yes	—
19	D₇₁₂	C₈	1	1	yes	characteristic
20	#13083558	C₂×C₄	1	2	yes	—
21	#13083557	—	2	2	no	—
22	C₈×D₈₉	—	2	2	no	—
23	#13083549	—	2	1	no	—
24	#13083549	D₄	1	2	yes	—
25	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₈₉	—	2	2	no	—
29	C₂×C₄×C₈₉	—	2	1	no	—
30	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
31	C₂²⋊D₈₉	—	2	2	no	—
32	D₃₅₆	C₂×C₈	1	2	yes	—
33	C₄×D₈₉	—	2	2	no	—
34	Dic₁₇₈	C₂×C₈	1	2	yes	—
35	C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
36	C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
37	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₈₉	#41	1	1	yes	characteristic
39	#10623494	—	2	2	no	—
40	C₂²×C₈₉	#131	1	1	yes	characteristic
41	C₂×D₈₉	—	2	2	no	—
42	C₄×C₈₉	—	4	1	no	—
43	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
44	C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
45	Dic₈₉	—	2	2	no	—
46	C₂×C₈₉	—	2	1	no	—
47	C₂×C₈₉	C₈×D₄	1	1	yes	characteristic
48	D₈₉	—	4	2	no	—
49	#2171	—	89	1	no	maximal, Sylow
50	C₈₉	#2171	1	1	yes	characteristic, complemented, Sylow
51	#504	—	89	2	no	—
52	#443	—	89	1	no	—
53	#364	D₈₉	1	1	yes	characteristic, complemented
54	#350	—	89	2	no	—
55	#346	—	89	1	no	—
56	#135	—	89	1	no	—
57	#131	—	89	2	no	—
58	M₃₂	—	178	2	no	—
59	M₃₂	C₂×D₈₉	1	2	yes	—
60	C₂×C₁₆	—	89	1	no	—
61	C₂×C₁₆	—	89	1	no	—
62	C₂×C₁₆	—	178	2	no	—
63	#108	—	89	1	no	—
64	#104	—	89	2	no	—
65	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
66	#41	—	89	2	no	—
67	Q₁₆	—	89	1	no	—
68	SD₁₆	—	89	2	no	—
69	D₈	—	89	1	no	—
70	M₁₆	—	89	2	no	—
71	M₁₆	—	178	2	no	—
72	C₂×C₈	—	178	2	no	—
73	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
75	C₄²	C₄×D₈₉	1	1	yes	characteristic
76	C₁₆	—	2	2	no	—
77	C₁₆	—	89	2	no	—
78	C₁₆	—	178	1	no	—
79	Q₈	—	89	2	no	—
80	D₄	—	89	2	no	—
81	D₄	—	178	2	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	178	2	no	—
84	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
85	C₈	—	178	2	no	—
86	C₈	C₈×D₈₉	1	1	yes	characteristic
87	C₈	C₈×D₈₉	1	1	yes	characteristic
88	C₈	D₄×D₈₉	1	1	yes	characteristic
89	C₈	#13083575	1	1	yes	characteristic
90	C₂²	—	178	2	no	—
91	C₂²	#15668020	1	1	yes	characteristic
92	C₄	—	4	1	no	—
93	C₄	—	178	2	no	—
94	C₄	#15667974	1	1	yes	characteristic
95	C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	356	2	no	—
98	C₂	#19151663	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152016

C₂

yes

maximal, complemented

#19151716

C₂

yes

maximal, characteristic

#19151383

C₂

yes

maximal, complemented

#19151374

C₂

yes

maximal, characteristic, complemented

#364×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668034

C₄

yes

characteristic

#15668018

C₂²

yes

—

#15667951

—

C₂×#13083549

C₂²

yes

characteristic

C₂×#13083549

C₂²

yes

characteristic

C₁₆×D₈₉

—

#15667948

C₄

yes

characteristic

#15667940

C₄

yes

—

M₃₂×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

#13083574

C₂×C₄

yes

—

Dic₃₅₆

C₈

yes

characteristic

#13083563

C₈

yes

—

D₇₁₂

C₈

yes

characteristic

#13083558

C₂×C₄

yes

—

#13083557

—

C₈×D₈₉

—

#13083549

—

#13083549

D₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₁₆×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²⋊D₈₉

—

D₃₅₆

C₂×C₈

yes

—

C₄×D₈₉

—

Dic₁₇₈

C₂×C₈

yes

—

C₈×C₈₉

C₂×C₈

yes

characteristic

C₈×C₈₉

C₂×C₈

yes

characteristic

C₈×C₈₉

C₂×D₄

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#131

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

characteristic

C₄×C₈₉

C₂²×C₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₈×D₄

yes

characteristic

D₈₉

—

#2171

—

maximal, Sylow

C₈₉

#2171

yes

characteristic, complemented, Sylow

#504

—

#443

—

#364

D₈₉

yes

characteristic, complemented

—

—

—

—

M₃₂

—

178

—

M₃₂

C₂×D₈₉

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

#108

—

#104

—

C₄×C₈

C₂×D₈₉

yes

characteristic

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄²

C₄×D₈₉

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

178

—

Q₈

—

D₄

—

D₄

—

178

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₈

—

178

—

C₈

C₈×D₈₉

yes

characteristic

C₈

C₈×D₈₉

yes

characteristic

C₈

D₄×D₈₉

yes

characteristic

C₈

#13083575

yes

characteristic

C₂²

—

178

—

C₂²

#15668020

yes

characteristic

C₄

—

C₄

—

178

—

C₄

#15667974

yes

characteristic

C₄

C₂×C₈×D₈₉

yes

characteristic

C₂

—

C₂

—

356

—

C₂

#19151663

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₈₉
commutator	C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#364×C₈₉	C₂
socle	C₂×C₈₉	C₈×D₄
prime radical	#15668034	C₄
cosocle	C₂×C₈×C₈₉	C₂³
hypercenter	#364	D₈₉
hypocenter	C₈₉	#2171
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×D₈₉

commutator

C₄×C₈₉

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#364×C₈₉

C₂

socle

C₂×C₈₉

C₈×D₄

prime radical

#15668034

C₄

cosocle

C₂×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
1. #19152016	C₂	semidirect
3. #19151383	C₂	semidirect
4. #19151374	C₂	semidirect
5. #364×C₈₉	C₂	semidirect
50. C₈₉	#2171	semidirect
53. #364	D₈₉	semidirect

H ≅ G/N

type

1. #19152016

C₂

semidirect

3. #19151383

C₂

semidirect

4. #19151374

C₂

semidirect

5. #364×C₈₉

C₂

semidirect

50. C₈₉

#2171

semidirect

53. #364

D₈₉

semidirect

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

subgroup	index
#19152016	2
#19151716	2
#19151383	2
#19151374	2
#364×C₈₉	2
#2171	89

subgroup

index

#364×C₈₉

Subgroups — #26022093

Conjugacy classes of subgroups

Complements

Maximal subgroups