Group #26019881 — subgroups

Subgroups — #26019881

Order 11392.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19151975	C₂	1	1	yes	maximal, characteristic
2	#19151972	C₂	1	2	yes	maximal, complemented
3	#19151946	C₂	1	2	yes	maximal, complemented
4	#19151319	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083561	—	2	4	no	—
7	C₂×D₇₁₂	—	2	2	no	—
8	#15668009	C₂²	1	1	yes	characteristic
9	#15668009	C₂²	1	1	yes	characteristic
10	#15667932	—	2	4	no	—
11	C₄×#10623494	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
13	#106×C₈₉	—	2	2	no	—
14	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×D₃₅₆	—	2	1	no	—
16	C₂×D₃₅₆	—	2	1	no	—
17	C₂×D₃₅₆	—	2	2	no	—
18	C₂×D₃₅₆	—	2	2	no	—
19	#13083561	—	4	8	no	—
20	D₇₁₂	—	4	4	no	—
21	C₂×#10623494	D₄	1	4	yes	—
22	C₂×Q₈×C₈₉	—	2	2	no	—
23	C₂×C₈×C₈₉	D₄	1	2	yes	—
24	C₄⋊C₄×C₈₉	—	2	2	no	—
25	C₄⋊C₄×C₈₉	—	2	2	no	—
26	C₄²×C₈₉	C₂³	1	1	yes	characteristic
27	C₂²×D₈₉	—	4	1	no	—
28	C₂²×D₈₉	—	4	1	no	—
29	Q₈×C₈₉	—	4	4	no	—
30	C₂×C₄×C₈₉	—	4	2	no	—
31	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
33	D₃₅₆	—	4	2	no	—
34	D₃₅₆	—	4	2	no	—
35	D₃₅₆	—	4	4	no	—
36	D₃₅₆	—	4	4	no	—
37	C₈×C₈₉	SD₁₆	1	4	yes	complemented
38	#10623494	—	2	4	no	—
39	C₂²×C₈₉	#127	1	1	yes	characteristic
40	C₂×D₈₉	—	8	1	no	—
41	C₂×D₈₉	—	8	1	no	—
42	C₂×D₈₉	—	8	2	no	—
43	C₂×D₈₉	—	8	2	no	—
44	C₄×C₈₉	—	8	2	no	—
45	C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
46	C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
47	C₂×C₈₉	#492	1	1	yes	characteristic
48	C₂×C₈₉	#496	1	2	yes	—
49	D₈₉	—	16	1	no	—
50	D₈₉	—	16	1	no	—
51	#1715	—	89	1	no	maximal, Sylow
52	C₈₉	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	D₈₉	1	1	yes	characteristic, complemented
54	#493	—	89	1	no	—
55	#492	—	89	2	no	—
56	#486	—	89	2	no	—
57	#322	—	89	1	no	—
58	C₂×SD₁₆	—	178	4	no	—
59	C₂×D₈	—	178	2	no	—
60	C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
61	#127	—	89	1	no	—
62	#127	—	89	1	no	—
63	#106	—	2	2	no	—
64	#102	—	178	4	no	—
65	C₄×C₈	—	89	2	no	—
66	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	178	2	no	—
69	C₂×D₄	—	178	2	no	—
70	C₂×D₄	—	178	1	no	—
71	C₂×D₄	—	178	1	no	—
72	SD₁₆	—	356	8	no	—
73	D₈	—	356	4	no	—
74	C₂×C₈	—	89	4	no	—
75	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₈₉	1	1	yes	characteristic
79	C₂³	—	356	1	no	—
80	C₂³	—	356	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	356	2	no	—
83	D₄	—	356	2	no	—
84	D₄	—	356	4	no	—
85	D₄	—	356	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₈₉	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
89	C₈	—	178	4	no	—
90	C₈	#13083561	1	4	yes	complemented
91	C₂²	—	712	1	no	—
92	C₂²	—	712	1	no	—
93	C₂²	—	712	2	no	—
94	C₂²	—	712	2	no	—
95	C₂²	#15668008	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#13083561	1	2	yes	—
98	C₄	#15668039	1	4	yes	—
99	C₂	—	1424	1	no	—
100	C₂	—	1424	1	no	—
101	C₂	#19151972	1	1	yes	characteristic
102	C₂	#19151984	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151975

C₂

yes

maximal, characteristic

#19151972

C₂

yes

maximal, complemented

#19151946

C₂

yes

maximal, complemented

#19151319

C₂

yes

maximal, characteristic, complemented

#498×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083561

—

C₂×D₇₁₂

—

#15668009

C₂²

yes

characteristic

#15668009

C₂²

yes

characteristic

#15667932

—

C₄×#10623494

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

#106×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083561

—

D₇₁₂

—

C₂×#10623494

D₄

yes

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

SD₁₆

yes

complemented

#10623494

—

C₂²×C₈₉

#127

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₈₉

#492

yes

characteristic

C₂×C₈₉

#496

yes

—

D₈₉

—

D₈₉

—

#1715

—

maximal, Sylow

C₈₉

#1715

yes

characteristic, complemented, Sylow

#498

D₈₉

yes

characteristic, complemented

—

—

—

—

C₂×SD₁₆

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

—

—

—

—

—

178

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

SD₁₆

—

356

—

D₈

—

356

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

356

—

C₂³

—

356

—

Q₈

—

D₄

—

356

—

D₄

—

356

—

D₄

—

356

—

D₄

—

356

—

C₂×C₄

—

C₂×C₄

D₄×D₈₉

yes

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

C₈

—

178

—

C₈

#13083561

yes

complemented

C₂²

—

712

—

C₂²

—

712

—

C₂²

—

712

—

C₂²

—

712

—

C₂²

#15668008

yes

characteristic

C₄

—

C₄

C₂×#13083561

yes

—

C₄

#15668039

yes

—

C₂

—

1424

—

100

C₂

—

1424

—

101

C₂

#19151972

yes

characteristic

102

C₂

#19151984

yes

—

	subgroup	G/H
center	C₂²	#15668008
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#498×C₈₉	C₂
socle	C₂²×C₈₉	#127
prime radical	#19151975	C₂
hypercenter	#498	D₈₉
hypocenter	C₈₉	#1715
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#15668008

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#498×C₈₉

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #19151972	C₂	semidirect
3. #19151946	C₂	semidirect
4. #19151319	C₂	semidirect
5. #498×C₈₉	C₂	semidirect
37. C₈×C₈₉	SD₁₆	semidirect
52. C₈₉	#1715	semidirect
53. #498	D₈₉	semidirect
90. C₈	#13083561	semidirect

H ≅ G/N

type

2. #19151972

C₂

semidirect

3. #19151946

C₂

semidirect

4. #19151319

C₂

semidirect

5. #498×C₈₉

C₂

semidirect

37. C₈×C₈₉

SD₁₆

semidirect

52. C₈₉

#1715

semidirect

53. #498

D₈₉

semidirect

90. C₈

#13083561

semidirect

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

subgroup	index
#19151975	2
#19151972	2
#19151946	2
#19151319	2
#498×C₈₉	2
#1715	89

subgroup

index

#498×C₈₉

Subgroups — #26019881

Conjugacy classes of subgroups

Complements

Maximal subgroups