Group #26022566 — subgroups

Subgroups — #26022566

Order 11392.

112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 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	#19152031	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151435	C₂	1	1	yes	maximal, characteristic, complemented
4	D₂₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151433	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151431	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#15668035	—	2	1	no	—
9	D₈×D₈₉	—	2	1	no	—
10	#15667955	—	2	1	no	—
11	D₁₄₂₄	C₂²	1	1	yes	characteristic
12	D₁₄₂₄	C₂²	1	1	yes	characteristic, complemented
13	#15667953	—	2	1	no	—
14	C₁₆×D₈₉	C₂²	1	1	yes	characteristic
15	#15667918	C₂²	1	1	yes	characteristic, complemented
16	Q₃₂×C₈₉	C₂²	1	1	yes	characteristic, complemented
17	D₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₃₂×C₈₉	C₂²	1	1	yes	characteristic, complemented
19	#13083575	—	4	1	no	—
20	D₄×D₈₉	—	4	1	no	—
21	#13083561	—	4	1	no	—
22	D₇₁₂	—	2	1	no	—
23	D₇₁₂	—	2	1	no	—
24	#13083559	—	4	1	no	—
25	C₈×D₈₉	D₄	1	1	yes	characteristic
26	#13083549	D₄	1	1	yes	characteristic
27	Q₁₆×C₈₉	—	2	1	no	—
28	D₈×C₈₉	—	2	1	no	—
29	C₁₆×C₈₉	C₂³	1	1	yes	characteristic
30	C₂²×D₈₉	—	8	1	no	—
31	Q₈×C₈₉	—	4	1	no	—
32	D₄×C₈₉	—	4	1	no	—
33	C₂²⋊D₈₉	—	8	1	no	—
34	D₃₅₆	—	4	1	no	—
35	D₃₅₆	—	4	1	no	—
36	D₃₅₆	—	8	1	no	—
37	C₄×D₈₉	—	8	1	no	—
38	C₄×D₈₉	D₈	1	1	yes	characteristic
39	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	#10623494	D₈	1	1	yes	characteristic
41	C₂²×C₈₉	—	8	1	no	—
42	C₂×D₈₉	—	8	1	no	—
43	C₂×D₈₉	—	8	1	no	—
44	C₂×D₈₉	—	16	2	no	—
45	C₂×D₈₉	D₁₆	1	1	yes	characteristic
46	C₄×C₈₉	—	8	1	no	—
47	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
48	Dic₈₉	D₁₆	1	1	yes	characteristic
49	C₂×C₈₉	—	16	1	no	—
50	C₂×C₈₉	C₂×D₁₆	1	1	yes	characteristic
51	D₈₉	—	2	1	no	—
52	D₈₉	—	16	1	no	—
53	D₈₉	—	16	1	no	—
54	#2263	—	89	1	no	maximal, Sylow
55	C₈₉	#2263	1	1	yes	characteristic, complemented, Sylow
56	#508	—	89	1	no	—
57	C₂×D₁₆	—	89	1	no	—
58	SD₆₄	—	89	1	no	—
59	SD₆₄	D₈₉	1	1	yes	characteristic, complemented
60	D₃₂	—	89	1	no	—
61	D₃₂	—	89	1	no	—
62	M₆₄	—	89	1	no	—
63	#135	—	178	1	no	—
64	C₂×D₈	—	178	1	no	—
65	Q₃₂	C₂×D₈₉	1	1	yes	characteristic, complemented
66	SD₃₂	—	178	1	no	—
67	D₁₆	—	89	1	no	—
68	D₁₆	—	89	1	no	—
69	D₁₆	—	178	1	no	—
70	D₁₆	C₂×D₈₉	1	1	yes	characteristic
71	C₂×C₁₆	—	89	1	no	—
72	C₃₂	—	89	1	no	—
73	C₃₂	C₂×D₈₉	1	1	yes	characteristic, complemented
74	#41	—	356	1	no	—
75	C₂×D₄	—	356	1	no	—
76	Q₁₆	—	2	1	no	—
77	SD₁₆	—	356	1	no	—
78	D₈	—	2	1	no	—
79	D₈	—	356	1	no	—
80	D₈	—	178	1	no	—
81	D₈	—	178	1	no	—
82	C₂×C₈	—	89	1	no	—
83	C₁₆	—	89	1	no	—
84	C₁₆	C₂²×D₈₉	1	1	yes	characteristic
85	C₂³	—	712	1	no	—
86	Q₈	—	4	1	no	—
87	D₄	—	4	1	no	—
88	D₄	—	356	1	no	—
89	D₄	—	356	1	no	—
90	D₄	—	712	1	no	—
91	D₄	—	712	1	no	—
92	C₂×C₄	—	89	1	no	—
93	C₂×C₄	—	712	1	no	—
94	C₈	—	89	1	no	—
95	C₈	D₄×D₈₉	1	1	yes	characteristic
96	C₂²	—	8	1	no	—
97	C₂²	—	89	1	no	—
98	C₂²	—	1424	2	no	—
99	C₂²	—	712	1	no	—
100	C₂²	—	712	1	no	—
101	C₄	—	8	1	no	—
102	C₄	—	89	1	no	—
103	C₄	D₈×D₈₉	1	1	yes	characteristic
104	C₂	—	16	1	no	—
105	C₂	—	1424	1	no	—
106	C₂	—	1424	1	no	—
107	C₂	—	178	1	no	—
108	C₂	D₁₆×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152031

C₂

yes

maximal, characteristic, complemented

D₁₆×D₈₉

C₂

yes

maximal, characteristic, complemented

#19151435

C₂

yes

maximal, characteristic, complemented

D₂₈₄₈

C₂

yes

maximal, characteristic, complemented

#19151433

C₂

yes

maximal, characteristic, complemented

#19151431

C₂

yes

maximal, characteristic, complemented

SD₆₄×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668035

—

D₈×D₈₉

—

#15667955

—

D₁₄₂₄

C₂²

yes

characteristic

D₁₄₂₄

C₂²

yes

characteristic, complemented

#15667953

—

C₁₆×D₈₉

C₂²

yes

characteristic

#15667918

C₂²

yes

characteristic, complemented

Q₃₂×C₈₉

C₂²

yes

characteristic, complemented

D₁₆×C₈₉

C₂²

yes

characteristic

C₃₂×C₈₉

C₂²

yes

characteristic, complemented

#13083575

—

D₄×D₈₉

—

#13083561

—

D₇₁₂

—

D₇₁₂

—

#13083559

—

C₈×D₈₉

D₄

yes

characteristic

#13083549

D₄

yes

characteristic

Q₁₆×C₈₉

—

D₈×C₈₉

—

C₁₆×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

C₂²⋊D₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

D₈

yes

characteristic

C₈×C₈₉

C₂×D₄

yes

characteristic

#10623494

D₈

yes

characteristic

C₂²×C₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×D₈

yes

characteristic

Dic₈₉

D₁₆

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

C₂×D₁₆

yes

characteristic

D₈₉

—

D₈₉

—

D₈₉

—

#2263

—

maximal, Sylow

C₈₉

#2263

yes

characteristic, complemented, Sylow

#508

—

C₂×D₁₆

—

SD₆₄

—

SD₆₄

D₈₉

yes

characteristic, complemented

D₃₂

—

D₃₂

—

M₆₄

—

#135

—

178

—

C₂×D₈

—

178

—

Q₃₂

C₂×D₈₉

yes

characteristic, complemented

SD₃₂

—

178

—

D₁₆

—

D₁₆

—

D₁₆

—

178

—

D₁₆

C₂×D₈₉

yes

characteristic

C₂×C₁₆

—

C₃₂

—

C₃₂

C₂×D₈₉

yes

characteristic, complemented

#41

—

356

—

C₂×D₄

—

356

—

Q₁₆

—

SD₁₆

—

356

—

D₈

—

D₈

—

356

—

D₈

—

178

—

D₈

—

178

—

C₂×C₈

—

C₁₆

—

C₁₆

C₂²×D₈₉

yes

characteristic

C₂³

—

712

—

Q₈

—

D₄

—

D₄

—

356

—

D₄

—

356

—

D₄

—

712

—

D₄

—

712

—

C₂×C₄

—

C₂×C₄

—

712

—

C₈

—

C₈

D₄×D₈₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

1424

—

C₂²

—

712

—

100

C₂²

—

712

—

101

C₄

—

102

C₄

—

103

C₄

D₈×D₈₉

yes

characteristic

104

C₂

—

105

C₂

—

1424

—

106

C₂

—

1424

—

107

C₂

—

178

—

108

C₂

D₁₆×D₈₉

yes

characteristic

	subgroup	G/H
center	C₂	D₁₆×D₈₉
commutator, cosocle	C₁₆×C₈₉	C₂³
Frattini	C₁₆	C₂²×D₈₉
Fitting	SD₆₄×C₈₉	C₂
socle	C₂×C₈₉	C₂×D₁₆
hypercenter	SD₆₄	D₈₉
hypocenter	C₈₉	#2263
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ SD₆₄

subgroup

G/H

center

C₂

D₁₆×D₈₉

commutator, cosocle

C₁₆×C₈₉

C₂³

Frattini

C₁₆

C₂²×D₈₉

Fitting

SD₆₄×C₈₉

C₂

socle

C₂×C₈₉

C₂×D₁₆

hypercenter

SD₆₄

D₈₉

hypocenter

C₈₉

#2263

LCS

G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ SD₆₄

N	H ≅ G/N	type
1. #19152031	C₂	semidirect
2. D₁₆×D₈₉	C₂	semidirect
3. #19151435	C₂	semidirect
4. D₂₈₄₈	C₂	semidirect
5. #19151433	C₂	semidirect
6. #19151431	C₂	semidirect
7. SD₆₄×C₈₉	C₂	semidirect
12. D₁₄₂₄	C₂²	semidirect
15. #15667918	C₂²	semidirect
16. Q₃₂×C₈₉	C₂²	semidirect
18. C₃₂×C₈₉	C₂²	semidirect
55. C₈₉	#2263	semidirect
59. SD₆₄	D₈₉	semidirect
65. Q₃₂	C₂×D₈₉	semidirect
73. C₃₂	C₂×D₈₉	semidirect

H ≅ G/N

type

1. #19152031

C₂

semidirect

2. D₁₆×D₈₉

C₂

semidirect

3. #19151435

C₂

semidirect

4. D₂₈₄₈

C₂

semidirect

5. #19151433

C₂

semidirect

6. #19151431

C₂

semidirect

7. SD₆₄×C₈₉

C₂

semidirect

12. D₁₄₂₄

C₂²

semidirect

15. #15667918

C₂²

semidirect

16. Q₃₂×C₈₉

C₂²

semidirect

18. C₃₂×C₈₉

C₂²

semidirect

55. C₈₉

#2263

semidirect

59. SD₆₄

D₈₉

semidirect

65. Q₃₂

C₂×D₈₉

semidirect

73. C₃₂

C₂×D₈₉

semidirect

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

subgroup	index
#19152031	2
D₁₆×D₈₉	2
#19151435	2
D₂₈₄₈	2
#19151433	2
#19151431	2
SD₆₄×C₈₉	2
#2263	89

subgroup

index

D₁₆×D₈₉

D₂₈₄₈

SD₆₄×C₈₉

Subgroups — #26022566

Conjugacy classes of subgroups

Complements

Maximal subgroups