Group #36469146 — subgroups

Subgroups — #36469146

Order 22784.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#26022508	C₂	1	4	yes	maximal, complemented
2	#26019870	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019867	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15667957	—	2	8	no	—
6	#19152003	—	2	2	no	—
7	#19151978	C₂²	1	2	yes	—
8	#19151975	—	2	2	no	—
9	#19151975	C₂²	1	2	yes	—
10	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
11	C₈²×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₅₆	—	2	4	no	—
13	C₂×D₇₁₂	—	2	4	no	—
14	C₂×D₇₁₂	—	4	4	no	—
15	#15668012	—	2	2	no	—
16	#15668009	—	2	2	no	—
17	#15667957	—	4	16	no	—
18	#15667946	—	4	4	no	—
19	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
20	C₄×C₈×C₈₉	D₄	1	2	yes	—
21	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₇₈	—	4	2	no	—
23	C₂×D₃₅₆	—	4	2	no	—
24	C₂×D₃₅₆	—	4	4	no	—
25	Dic₃₅₆	—	4	8	no	—
26	D₇₁₂	—	4	8	no	—
27	D₇₁₂	—	8	8	no	—
28	C₄⋊Dic₈₉	—	4	4	no	—
29	C₂×C₈×C₈₉	D₈	1	4	yes	—
30	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
31	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₁₆×C₈₉	D₈	1	8	yes	complemented
33	C₂²×D₈₉	—	8	2	no	—
34	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
36	C₂×Dic₈₉	—	8	2	no	—
37	D₃₅₆	—	8	4	no	—
38	D₃₅₆	—	8	8	no	—
39	Dic₁₇₈	—	8	4	no	—
40	C₈×C₈₉	SD₃₂	1	8	yes	complemented
41	C₈×C₈₉	C₂×D₈	1	4	yes	—
42	C₂²×C₈₉	#493	1	1	yes	characteristic
43	C₂×D₈₉	—	16	2	no	—
44	C₂×D₈₉	—	16	4	no	—
45	C₄×C₈₉	#493	1	2	yes	—
46	C₄×C₈₉	C₂×SD₃₂	1	4	yes	—
47	Dic₈₉	—	16	2	no	—
48	#12517	—	89	1	no	maximal, Sylow
49	C₂×C₈₉	#1712	1	1	yes	characteristic
50	C₂×C₈₉	#2248	1	2	yes	—
51	D₈₉	—	32	2	no	—
52	#2248	—	89	4	no	—
53	#1713	—	89	1	no	—
54	#1712	—	89	1	no	—
55	C₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
56	C₈₉	#12517	1	1	yes	characteristic, complemented, Sylow
57	C₂×SD₃₂	—	178	8	no	—
58	#500	—	178	2	no	—
59	#494	—	89	2	no	—
60	#493	—	89	2	no	—
61	#493	—	178	2	no	—
62	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
63	C₈²	C₂×D₈₉	1	1	yes	characteristic
64	C₂×Q₁₆	—	178	4	no	—
65	C₂×D₈	—	356	4	no	—
66	C₂×D₈	—	178	4	no	—
67	C₄⋊Q₈	—	178	2	no	—
68	#127	—	178	2	no	—
69	SD₃₂	—	356	16	no	—
70	C₂×C₁₆	D₃₅₆	1	4	yes	—
71	#107	—	356	4	no	—
72	C₄×C₈	D₃₅₆	1	2	yes	—
73	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
74	C₂×Q₈	—	356	2	no	—
75	C₂×D₄	—	356	2	no	—
76	C₂×D₄	—	356	4	no	—
77	Q₁₆	—	356	8	no	—
78	D₈	—	356	8	no	—
79	D₈	—	712	8	no	—
80	C₂×C₈	D₇₁₂	1	4	yes	—
81	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
82	C₄⋊C₄	—	356	4	no	—
83	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
84	C₁₆	D₇₁₂	1	8	yes	complemented
85	C₂³	—	712	2	no	—
86	Q₈	—	712	4	no	—
87	D₄	—	712	4	no	—
88	D₄	—	712	8	no	—
89	C₂×C₄	—	712	2	no	—
90	C₂×C₄	#15668009	1	1	yes	characteristic
91	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
92	C₈	#15667957	1	8	yes	complemented
93	C₈	C₂×D₇₁₂	1	4	yes	—
94	C₂²	—	1424	2	no	—
95	C₂²	—	1424	4	no	—
96	C₂²	#19151975	1	1	yes	characteristic
97	C₄	—	1424	2	no	—
98	C₄	#19151975	1	2	yes	—
99	C₄	C₂×#15667957	1	4	yes	—
100	C₂	—	2848	2	no	—
101	C₂	#26019867	1	1	yes	characteristic
102	C₂	#26022508	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022508

C₂

yes

maximal, complemented

#26019870

C₂

yes

maximal, characteristic, complemented

#26019867

C₂

yes

maximal, characteristic

C₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#15667957

—

#19152003

—

#19151978

C₂²

yes

—

#19151975

—

#19151975

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

—

—

—

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

Dic₃₅₆

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

D₈

yes

complemented

C₂²×D₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

SD₃₂

yes

complemented

C₈×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#493

yes

—

C₄×C₈₉

C₂×SD₃₂

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

#2248

—

#1713

—

#1712

—

C₈×C₁₆

D₈₉

yes

characteristic, complemented

C₈₉

#12517

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

178

—

#500

—

178

—

#494

—

#493

—

#493

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

178

—

C₂×D₈

—

356

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

178

—

#127

—

178

—

SD₃₂

—

356

—

C₂×C₁₆

D₃₅₆

yes

—

#107

—

356

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

Q₁₆

—

356

—

D₈

—

356

—

D₈

—

712

—

C₂×C₈

D₇₁₂

yes

—

C₂×C₈

C₂×D₃₅₆

yes

—

C₄⋊C₄

—

356

—

C₄²

C₂×D₃₅₆

yes

characteristic

C₁₆

D₇₁₂

yes

complemented

C₂³

—

712

—

Q₈

—

712

—

D₄

—

712

—

D₄

—

712

—

C₂×C₄

—

712

—

C₂×C₄

#15668009

yes

characteristic

C₂×C₄

C₂×D₇₁₂

yes

—

C₈

#15667957

yes

complemented

C₈

C₂×D₇₁₂

yes

—

C₂²

—

1424

—

C₂²

—

1424

—

C₂²

#19151975

yes

characteristic

C₄

—

1424

—

C₄

#19151975

yes

—

C₄

C₂×#15667957

yes

—

100

C₂

—

2848

—

101

C₂

#26019867

yes

characteristic

102

C₂

#26022508

yes

—

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#493
prime radical	#26019867	C₂
hypercenter	C₈×C₁₆	D₈₉
hypocenter	C₈₉	#12517
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#19151975

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈×C₁₆×C₈₉

C₂

C₂²×C₈₉

C₂

C₈×C₁₆

D₈₉

hypocenter

C₈₉

#12517

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #26022508	C₂	semidirect
2. #26019870	C₂	semidirect
4. C₈×C₁₆×C₈₉	C₂	semidirect
32. C₁₆×C₈₉	D₈	semidirect
40. C₈×C₈₉	SD₃₂	semidirect
55. C₈×C₁₆	D₈₉	semidirect
56. C₈₉	#12517	semidirect
84. C₁₆	D₇₁₂	semidirect
92. C₈	#15667957	semidirect

H ≅ G/N

type

1. #26022508

C₂

semidirect

2. #26019870

C₂

semidirect

4. C₈×C₁₆×C₈₉

C₂

semidirect

32. C₁₆×C₈₉

D₈

semidirect

40. C₈×C₈₉

SD₃₂

semidirect

55. C₈×C₁₆

D₈₉

semidirect

56. C₈₉

#12517

semidirect

84. C₁₆

D₇₁₂

semidirect

92. C₈

#15667957

semidirect

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

subgroup	index
#26022508	2
#26019870	2
#26019867	2
C₈×C₁₆×C₈₉	2
#12517	89

subgroup

index

C₈×C₁₆×C₈₉

Subgroups — #36469146

Conjugacy classes of subgroups

Complements

Maximal subgroups