Group #19152410 — subgroups

Subgroups — #19152410

Order 5696.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#15668059	C₂	1	1	yes	maximal, characteristic, complemented
2	#15668058	C₂	1	1	yes	maximal, characteristic, complemented
3	#15668045	C₂	1	3	yes	maximal, complemented
4	#15668040	C₂	1	3	yes	maximal, complemented
5	#15668034	C₂	1	3	yes	maximal, complemented
6	C₂×#13083563	C₂	1	3	yes	maximal, complemented
7	#131×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13083575	—	2	1	no	—
9	#13083574	C₂²	1	3	yes	—
10	#13083574	C₂²	1	3	yes	complemented
11	#13083573	—	2	3	no	—
12	C₂×Dic₁₇₈	C₂²	1	3	yes	complemented
13	Q₈×D₈₉	—	2	1	no	—
14	C₂×D₃₅₆	C₂²	1	3	yes	—
15	D₄×D₈₉	—	2	3	no	—
16	Dic₃₅₆	C₂²	1	3	yes	complemented
17	#13083563	C₂²	1	3	yes	complemented
18	#13083563	C₂²	1	6	yes	complemented
19	#13083563	C₂²	1	1	yes	characteristic, complemented
20	D₇₁₂	C₂²	1	3	yes	complemented
21	#41×C₈₉	C₂²	1	1	yes	characteristic
22	M₁₆×C₈₉	C₂²	1	3	yes	complemented
23	C₂×C₈×C₈₉	C₂²	1	3	yes	complemented
24	C₂²×D₈₉	—	2	3	no	—
25	Q₈×C₈₉	D₄	1	1	yes	characteristic
26	D₄×C₈₉	D₄	1	3	yes	—
27	C₂×C₄×C₈₉	C₂³	1	3	yes	—
28	C₂²⋊D₈₉	—	2	3	no	—
29	C₂²⋊D₈₉	—	2	3	no	—
30	C₂×Dic₈₉	—	2	3	no	—
31	D₃₅₆	—	2	3	no	—
32	D₃₅₆	C₂³	1	1	yes	characteristic
33	D₃₅₆	C₂³	1	3	yes	—
34	C₄×D₈₉	—	2	3	no	—
35	C₄×D₈₉	—	2	3	no	—
36	Dic₁₇₈	—	2	3	no	—
37	Dic₁₇₈	C₂³	1	3	yes	—
38	Dic₁₇₈	C₂³	1	1	yes	characteristic
39	C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₈×C₈₉	C₂³	1	3	yes	—
41	C₂²×C₈₉	C₂×D₄	1	3	yes	—
42	C₂×D₈₉	—	2	1	no	—
43	C₂×D₈₉	—	2	3	no	—
44	C₂×D₈₉	—	8	3	no	—
45	C₄×C₈₉	C₂×D₄	1	3	yes	—
46	C₄×C₈₉	C₂⁴	1	1	yes	characteristic
47	Dic₈₉	—	2	3	no	—
48	Dic₈₉	—	2	1	no	—
49	C₂×C₈₉	—	2	3	no	—
50	C₂×C₈₉	C₂²×D₄	1	1	yes	characteristic
51	D₈₉	—	4	1	no	—
52	D₈₉	—	4	3	no	—
53	C₈₉	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	89	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	89	1	no	—
56	H(5,𝔽₂)	—	89	1	no	—
57	#137	—	89	3	no	—
58	AGL(1,ℤ/8)	—	89	3	no	—
59	#135	—	89	3	no	—
60	C₂×SD₁₆	—	89	3	no	—
61	#131	D₈₉	1	1	yes	characteristic, complemented
62	#41	—	89	3	no	—
63	#41	—	89	3	no	—
64	#41	—	178	1	no	—
65	#41	—	178	3	no	—
66	#41	C₂×D₈₉	1	1	yes	characteristic
67	C₂×Q₈	—	89	3	no	—
68	C₂×Q₈	—	178	1	no	—
69	C₂×D₄	—	89	3	no	—
70	C₂×D₄	—	178	3	no	—
71	Q₁₆	—	89	3	no	—
72	SD₁₆	—	89	3	no	—
73	SD₁₆	—	89	6	no	—
74	SD₁₆	—	89	1	no	—
75	D₈	—	89	3	no	—
76	M₁₆	C₂×D₈₉	1	3	yes	complemented
77	C₂×C₈	C₂×D₈₉	1	3	yes	complemented
78	C₂³	—	178	3	no	—
79	Q₈	—	89	3	no	—
80	Q₈	—	89	1	no	—
81	Q₈	—	178	3	no	—
82	Q₈	D₃₅₆	1	1	yes	characteristic
83	D₄	—	89	1	no	—
84	D₄	—	89	3	no	—
85	D₄	—	178	3	no	—
86	D₄	—	178	3	no	—
87	D₄	—	178	3	no	—
88	D₄	D₃₅₆	1	3	yes	—
89	C₂×C₄	—	178	3	no	—
90	C₂×C₄	—	178	3	no	—
91	C₂×C₄	—	178	3	no	—
92	C₂×C₄	C₂²×D₈₉	1	3	yes	—
93	C₈	C₂²×D₈₉	1	1	yes	characteristic
94	C₈	C₂²×D₈₉	1	3	yes	—
95	C₂²	—	178	1	no	—
96	C₂²	—	178	3	no	—
97	C₂²	—	712	3	no	—
98	C₂²	C₂×D₃₅₆	1	3	yes	—
99	C₄	—	178	1	no	—
100	C₄	—	178	3	no	—
101	C₄	C₂×D₃₅₆	1	3	yes	—
102	C₄	C₂³×D₈₉	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	356	1	no	—
105	C₂	—	356	3	no	—
106	C₂	C₂²×D₃₅₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15668059

C₂

yes

maximal, characteristic, complemented

#15668058

C₂

yes

maximal, characteristic, complemented

#15668045

C₂

yes

maximal, complemented

#15668040

C₂

yes

maximal, complemented

#15668034

C₂

yes

maximal, complemented

C₂×#13083563

C₂

yes

maximal, complemented

#131×C₈₉

C₂

yes

maximal, characteristic, complemented

#13083575

—

#13083574

C₂²

yes

—

#13083574

C₂²

yes

complemented

#13083573

—

C₂×Dic₁₇₈

C₂²

yes

complemented

Q₈×D₈₉

—

C₂×D₃₅₆

C₂²

yes

—

D₄×D₈₉

—

Dic₃₅₆

C₂²

yes

complemented

#13083563

C₂²

yes

complemented

#13083563

C₂²

yes

complemented

#13083563

C₂²

yes

characteristic, complemented

D₇₁₂

C₂²

yes

complemented

#41×C₈₉

C₂²

yes

characteristic

M₁₆×C₈₉

C₂²

yes

complemented

C₂×C₈×C₈₉

C₂²

yes

complemented

C₂²×D₈₉

—

Q₈×C₈₉

D₄

yes

characteristic

D₄×C₈₉

D₄

yes

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

C₂³

yes

characteristic

D₃₅₆

C₂³

yes

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

C₂³

yes

—

Dic₁₇₈

C₂³

yes

characteristic

C₈×C₈₉

C₂³

yes

characteristic

C₈×C₈₉

C₂³

yes

—

C₂²×C₈₉

C₂×D₄

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

C₂×D₄

yes

—

C₄×C₈₉

C₂⁴

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×D₄

yes

characteristic

D₈₉

—

D₈₉

—

C₈₉

#577

yes

characteristic, complemented, Sylow

#577

—

maximal, Sylow

2₋¹⁺⁴

—

—

—

—

—

C₂×SD₁₆

—

#131

D₈₉

yes

characteristic, complemented

#41

—

#41

—

#41

—

178

—

#41

—

178

—

#41

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×D₄

—

C₂×D₄

—

178

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

C₂×D₈₉

yes

complemented

C₂×C₈

C₂×D₈₉

yes

complemented

C₂³

—

178

—

Q₈

—

Q₈

—

Q₈

—

178

—

Q₈

D₃₅₆

yes

characteristic

D₄

—

D₄

—

D₄

—

178

—

D₄

—

178

—

D₄

—

178

—

D₄

D₃₅₆

yes

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂²×D₈₉

yes

—

C₈

C₂²×D₈₉

yes

characteristic

C₈

C₂²×D₈₉

yes

—

C₂²

—

178

—

C₂²

—

178

—

C₂²

—

712

—

C₂²

C₂×D₃₅₆

yes

—

C₄

—

178

—

100

C₄

—

178

—

101

C₄

C₂×D₃₅₆

yes

—

102

C₄

C₂³×D₈₉

yes

characteristic

103

C₂

—

104

C₂

—

356

—

105

C₂

—

356

—

106

C₂

C₂²×D₃₅₆

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₃₅₆
commutator	C₄×C₈₉	C₂⁴
Frattini	C₄	C₂³×D₈₉
Fitting	#131×C₈₉	C₂
socle	C₂×C₈₉	C₂²×D₄
hypercenter	#131	D₈₉
hypocenter	C₈₉	#577
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₃₅₆

commutator

C₄×C₈₉

C₂⁴

Frattini

C₄

C₂³×D₈₉

Fitting

#131×C₈₉

C₂

socle

C₂×C₈₉

C₂²×D₄

D₈₉

C₈₉

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #15668059	C₂	semidirect
2. #15668058	C₂	semidirect
3. #15668045	C₂	semidirect
4. #15668040	C₂	semidirect
5. #15668034	C₂	semidirect
6. C₂×#13083563	C₂	semidirect
7. #131×C₈₉	C₂	semidirect
10. #13083574	C₂²	semidirect
12. C₂×Dic₁₇₈	C₂²	semidirect
16. Dic₃₅₆	C₂²	semidirect
17. #13083563	C₂²	semidirect
18. #13083563	C₂²	semidirect
19. #13083563	C₂²	semidirect
20. D₇₁₂	C₂²	semidirect
22. M₁₆×C₈₉	C₂²	semidirect
23. C₂×C₈×C₈₉	C₂²	semidirect
53. C₈₉	#577	semidirect
61. #131	D₈₉	semidirect
76. M₁₆	C₂×D₈₉	semidirect
77. C₂×C₈	C₂×D₈₉	semidirect

H ≅ G/N

type

1. #15668059

C₂

semidirect

2. #15668058

C₂

semidirect

3. #15668045

C₂

semidirect

4. #15668040

C₂

semidirect

5. #15668034

C₂

semidirect

6. C₂×#13083563

C₂

semidirect

7. #131×C₈₉

C₂

semidirect

10. #13083574

C₂²

semidirect

12. C₂×Dic₁₇₈

C₂²

semidirect

16. Dic₃₅₆

C₂²

semidirect

17. #13083563

C₂²

semidirect

18. #13083563

C₂²

semidirect

19. #13083563

C₂²

semidirect

20. D₇₁₂

C₂²

semidirect

22. M₁₆×C₈₉

C₂²

semidirect

23. C₂×C₈×C₈₉

C₂²

semidirect

53. C₈₉

#577

semidirect

61. #131

D₈₉

semidirect

76. M₁₆

C₂×D₈₉

semidirect

77. C₂×C₈

C₂×D₈₉

semidirect

Subgroups — #19152410

Conjugacy classes of subgroups

Complements