Group #26022132 — subgroups

Subgroups — #26022132

Order 11392.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19152030	C₂	1	1	yes	maximal, characteristic
2	#19151714	C₂	1	2	yes	maximal, complemented
3	#19151404	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#368×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668034	C₂²	1	2	yes	—
7	#15668020	—	2	2	no	—
8	Dic₇₁₂	C₄	1	1	yes	characteristic, complemented
9	#15667957	C₄	1	2	yes	complemented
10	D₁₄₂₄	C₄	1	1	yes	characteristic, complemented
11	C₂×#13083549	C₂²	1	1	yes	characteristic
12	#15667938	—	2	2	no	—
13	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	#108×C₈₉	C₂²	1	2	yes	—
16	#13083574	—	2	2	no	—
17	Dic₃₅₆	C₂×C₄	1	2	yes	—
18	#13083563	—	2	2	no	—
19	D₇₁₂	C₂×C₄	1	2	yes	—
20	#13083557	—	4	2	no	—
21	C₂×#10623494	D₄	1	1	yes	characteristic
22	C₈×D₈₉	—	4	2	no	—
23	C₄×Dic₈₉	D₄	1	1	yes	characteristic
24	#13083549	—	2	1	no	—
25	M₁₆×C₈₉	—	2	2	no	—
26	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
30	C₂²⋊D₈₉	—	4	2	no	—
31	C₂×Dic₈₉	—	2	1	no	—
32	D₃₅₆	—	2	2	no	—
33	C₄×D₈₉	—	4	2	no	—
34	Dic₁₇₈	—	2	2	no	—
35	C₈×C₈₉	—	4	2	no	—
36	C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₈₉	#41	1	1	yes	characteristic
38	#10623494	D₈	1	1	yes	characteristic
39	#10623494	D₈	1	1	yes	characteristic
40	C₂²×C₈₉	#135	1	1	yes	characteristic
41	C₂×D₈₉	—	4	2	no	—
42	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
43	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
44	Dic₈₉	—	2	2	no	—
45	Dic₈₉	—	4	2	no	—
46	C₂×C₈₉	—	2	1	no	—
47	C₂×C₈₉	C₄×D₈	1	1	yes	characteristic
48	D₈₉	—	8	2	no	—
49	#2178	—	89	1	no	maximal, Sylow
50	C₈₉	#2178	1	1	yes	characteristic, complemented, Sylow
51	#508	—	89	1	no	—
52	#443	—	89	2	no	—
53	#368	D₈₉	1	1	yes	characteristic, complemented
54	#359	—	89	2	no	—
55	C₄×C₁₆	—	89	1	no	—
56	#135	—	89	2	no	—
57	#131	—	178	2	no	—
58	Q₃₂	—	89	1	no	—
59	SD₃₂	—	89	2	no	—
60	D₁₆	—	89	1	no	—
61	C₂×C₁₆	—	89	1	no	—
62	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic
63	#108	C₂×D₈₉	1	2	yes	—
64	#104	—	178	2	no	—
65	C₄×C₈	—	89	1	no	—
66	#41	—	178	2	no	—
67	Q₁₆	—	89	2	no	—
68	SD₁₆	—	178	2	no	—
69	D₈	—	89	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	356	2	no	—
72	C₂×C₈	—	89	1	no	—
73	C₂×C₈	—	356	2	no	—
74	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
75	C₄²	—	89	1	no	—
76	C₁₆	—	178	1	no	—
77	C₁₆	C₄×D₈₉	1	1	yes	characteristic
78	C₁₆	C₄×D₈₉	1	1	yes	characteristic
79	Q₈	—	178	2	no	—
80	D₄	—	356	2	no	—
81	D₄	—	178	2	no	—
82	C₂×C₄	—	356	2	no	—
83	C₂×C₄	—	178	1	no	—
84	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
85	C₈	—	4	2	no	—
86	C₈	—	89	1	no	—
87	C₈	—	89	1	no	—
88	C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
89	C₈	#13083575	1	1	yes	characteristic
90	C₂²	—	356	2	no	—
91	C₂²	#15668035	1	1	yes	characteristic
92	C₄	—	356	2	no	—
93	C₄	—	178	2	no	—
94	C₄	#15667974	1	1	yes	characteristic
95	C₄	D₈×D₈₉	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	712	2	no	—
98	C₂	#19151680	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152030

C₂

yes

maximal, characteristic

#19151714

C₂

yes

maximal, complemented

#19151404

C₂

yes

maximal, complemented

C₁₆×Dic₈₉

C₂

yes

maximal, characteristic, complemented

#368×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668034

C₂²

yes

—

#15668020

—

Dic₇₁₂

C₄

yes

characteristic, complemented

#15667957

C₄

yes

complemented

D₁₄₂₄

C₄

yes

characteristic, complemented

C₂×#13083549

C₂²

yes

characteristic

#15667938

—

C₈×Dic₈₉

C₂²

yes

characteristic

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

#108×C₈₉

C₂²

yes

—

#13083574

—

Dic₃₅₆

C₂×C₄

yes

—

#13083563

—

D₇₁₂

C₂×C₄

yes

—

#13083557

—

C₂×#10623494

D₄

yes

characteristic

C₈×D₈₉

—

C₄×Dic₈₉

D₄

yes

characteristic

#13083549

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂²×C₄

yes

C₈×C₈₉

yes

D₈

yes

characteristic

#10623494

D₈

yes

characteristic

C₂²×C₈₉

#135

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

C₄×D₄

yes

characteristic

C₄×C₈₉

C₂×D₈

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₄×D₈

yes

characteristic

D₈₉

—

#2178

—

maximal, Sylow

C₈₉

#2178

yes

characteristic, complemented, Sylow

#508

—

#443

—

#368

D₈₉

yes

characteristic, complemented

#359

—

C₄×C₁₆

—

#135

—

#131

—

178

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

characteristic

#108

C₂×D₈₉

yes

—

#104

—

178

—

C₄×C₈

—

#41

—

178

—

Q₁₆

—

SD₁₆

—

178

—

D₈

—

M₁₆

—

M₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄²

—

C₁₆

—

178

—

C₁₆

C₄×D₈₉

yes

characteristic

C₁₆

C₄×D₈₉

yes

characteristic

Q₈

—

178

—

D₄

—

356

—

D₄

—

178

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

D₄×D₈₉

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

C₂×C₄×D₈₉

yes

characteristic

C₈

#13083575

yes

characteristic

C₂²

—

356

—

C₂²

#15668035

yes

characteristic

C₄

—

356

—

C₄

—

178

—

C₄

#15667974

yes

characteristic

C₄

D₈×D₈₉

yes

characteristic

C₂

—

C₂

—

712

—

C₂

#19151680

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₈₉
commutator	C₈×C₈₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#368×C₈₉	C₂
socle	C₂×C₈₉	C₄×D₈
prime radical	#19152030	C₂
cosocle	C₂×C₈×C₈₉	C₂³
hypercenter	#368	D₈₉
hypocenter	C₈₉	#2178
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

D₈×D₈₉

commutator

C₈×C₈₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#368×C₈₉

C₂

socle

C₂×C₈₉

C₄×D₈

prime radical

#19152030

C₂

cosocle

C₂×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
2. #19151714	C₂	semidirect
3. #19151404	C₂	semidirect
4. C₁₆×Dic₈₉	C₂	semidirect
5. #368×C₈₉	C₂	semidirect
8. Dic₇₁₂	C₄	semidirect
9. #15667957	C₄	semidirect
10. D₁₄₂₄	C₄	semidirect
50. C₈₉	#2178	semidirect
53. #368	D₈₉	semidirect

H ≅ G/N

type

2. #19151714

C₂

semidirect

3. #19151404

C₂

semidirect

4. C₁₆×Dic₈₉

C₂

semidirect

5. #368×C₈₉

C₂

semidirect

8. Dic₇₁₂

C₄

semidirect

9. #15667957

C₄

semidirect

10. D₁₄₂₄

C₄

semidirect

50. C₈₉

#2178

semidirect

53. #368

D₈₉

semidirect

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

subgroup	index
#19152030	2
#19151714	2
#19151404	2
C₁₆×Dic₈₉	2
#368×C₈₉	2
#2178	89

subgroup

index

#19152030

#19151714

#19151404

C₁₆×Dic₈₉

#368×C₈₉

#2178

Subgroups — #26022132

Conjugacy classes of subgroups

Complements

Maximal subgroups