Group #36469147 — subgroups

Subgroups — #36469147

Order 22784.

354 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 103 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	#26022509	C₂	1	4	yes	maximal, complemented
2	#26019867	C₂	1	1	yes	maximal, characteristic
3	C₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15667955	—	2	8	no	—
6	#19151975	—	2	2	no	—
7	#19151975	C₂²	1	2	yes	—
8	C₄×#13083549	C₂²	1	2	yes	—
9	#500×C₈₉	—	2	2	no	—
10	#494×C₈₉	C₂²	1	2	yes	—
11	C₈²×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×D₇₁₂	—	2	4	no	—
13	C₂×D₇₁₂	—	4	4	no	—
14	#15668009	—	2	2	no	—
15	#15667955	—	4	16	no	—
16	C₂×#13083549	D₄	1	4	yes	—
17	C₂×Q₁₆×C₈₉	—	2	4	no	—
18	C₄⋊Q₈×C₈₉	—	2	2	no	—
19	#107×C₈₉	—	4	4	no	—
20	C₄×C₈×C₈₉	D₄	1	2	yes	—
21	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
22	C₂×D₃₅₆	—	4	2	no	—
23	C₂×D₃₅₆	—	4	4	no	—
24	D₇₁₂	—	4	8	no	—
25	D₇₁₂	—	8	8	no	—
26	#13083549	D₈	1	8	yes	complemented
27	C₂×Q₈×C₈₉	—	4	2	no	—
28	Q₁₆×C₈₉	—	4	8	no	—
29	C₂×C₈×C₈₉	D₈	1	4	yes	—
30	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
31	C₄⋊C₄×C₈₉	—	4	4	no	—
32	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×D₈₉	—	8	2	no	—
34	Q₈×C₈₉	—	8	4	no	—
35	C₂×C₄×C₈₉	—	8	2	no	—
36	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
38	D₃₅₆	—	8	4	no	—
39	D₃₅₆	—	8	8	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	—	16	2	no	—
46	C₄×C₈₉	#493	1	2	yes	—
47	C₄×C₈₉	C₂×SD₃₂	1	4	yes	—
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	D₈₉	1	1	yes	characteristic, complemented
54	#1712	—	89	1	no	—
55	C₈×C₁₆	—	89	1	no	—
56	C₈₉	#12517	1	1	yes	characteristic, complemented, Sylow
57	C₂×SD₃₂	—	178	8	no	—
58	#500	—	2	2	no	—
59	#494	C₂×D₈₉	1	2	yes	—
60	#493	—	89	2	no	—
61	#493	—	178	2	no	—
62	C₄×C₁₆	—	89	2	no	—
63	C₈²	C₂×D₈₉	1	1	yes	characteristic
64	C₂×Q₁₆	—	2	4	no	—
65	C₂×D₈	—	356	4	no	—
66	C₂×D₈	—	178	4	no	—
67	C₄⋊Q₈	—	2	2	no	—
68	#127	—	178	2	no	—
69	SD₃₂	—	356	16	no	—
70	C₂×C₁₆	—	89	4	no	—
71	#107	—	4	4	no	—
72	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
73	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
74	C₂×Q₈	—	4	2	no	—
75	C₂×D₄	—	356	2	no	—
76	C₂×D₄	—	356	4	no	—
77	Q₁₆	—	4	8	no	—
78	D₈	—	356	8	no	—
79	D₈	—	712	8	no	—
80	C₂×C₈	#13083559	1	4	yes	—
81	C₂×C₈	D₄×D₈₉	1	2	yes	—
82	C₄⋊C₄	—	4	4	no	—
83	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
84	C₁₆	—	89	8	no	—
85	C₂³	—	712	2	no	—
86	Q₈	—	8	4	no	—
87	D₄	—	712	4	no	—
88	D₄	—	712	8	no	—
89	C₂×C₄	—	8	2	no	—
90	C₂×C₄	#15668008	1	1	yes	characteristic
91	C₂×C₄	C₂×#13083559	1	2	yes	—
92	C₈	#15667955	1	8	yes	complemented
93	C₈	D₈×D₈₉	1	4	yes	—
94	C₂²	—	1424	2	no	—
95	C₂²	—	1424	4	no	—
96	C₂²	#19151974	1	1	yes	characteristic
97	C₄	—	16	2	no	—
98	C₄	#19151973	1	2	yes	—
99	C₄	C₂×#15667955	1	4	yes	—
100	C₂	—	2848	2	no	—
101	C₂	#26019866	1	1	yes	characteristic
102	C₂	#26022509	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022509

C₂

yes

maximal, complemented

#26019867

C₂

yes

maximal, characteristic

C₈×#13083549

C₂

yes

maximal, characteristic, complemented

C₈₉×#1713

C₂

yes

maximal, characteristic, complemented

C₂×#15667955

—

#19151975

—

#19151975

C₂²

yes

—

C₄×#13083549

C₂²

yes

—

#500×C₈₉

—

#494×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

—

#15667955

—

C₂×#13083549

D₄

yes

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

#13083549

D₈

yes

complemented

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

SD₃₂

yes

complemented

C₈×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#493

yes

—

C₄×C₈₉

C₂×SD₃₂

yes

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

#2248

—

#1713

D₈₉

yes

characteristic, complemented

#1712

—

C₈×C₁₆

—

C₈₉

#12517

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

178

—

#500

—

#494

C₂×D₈₉

yes

—

#493

—

#493

—

178

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

356

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

#127

—

178

—

SD₃₂

—

356

—

C₂×C₁₆

—

#107

—

C₄×C₈

C₂²⋊D₈₉

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

Q₁₆

—

D₈

—

356

—

D₈

—

712

—

C₂×C₈

#13083559

yes

—

C₂×C₈

D₄×D₈₉

yes

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

C₁₆

—

C₂³

—

712

—

Q₈

—

D₄

—

712

—

D₄

—

712

—

C₂×C₄

—

C₂×C₄

#15668008

yes

characteristic

C₂×C₄

C₂×#13083559

yes

—

C₈

#15667955

yes

complemented

C₈

D₈×D₈₉

yes

—

C₂²

—

1424

—

C₂²

—

1424

—

C₂²

#19151974

yes

characteristic

C₄

—

C₄

#19151973

yes

—

C₄

C₂×#15667955

yes

—

100

C₂

—

2848

—

101

C₂

#26019866

yes

characteristic

102

C₂

#26022509

yes

—

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1713	C₂
socle	C₂²×C₈₉	#493
prime radical	#26019867	C₂
hypercenter	#1713	D₈₉
hypocenter	C₈₉	#12517
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1713

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. #26022509	C₂	semidirect
3. C₈×#13083549	C₂	semidirect
4. C₈₉×#1713	C₂	semidirect
26. #13083549	D₈	semidirect
40. C₈×C₈₉	SD₃₂	semidirect
53. #1713	D₈₉	semidirect
56. C₈₉	#12517	semidirect
92. C₈	#15667955	semidirect

H ≅ G/N

type

1. #26022509

C₂

semidirect

3. C₈×#13083549

C₂

semidirect

4. C₈₉×#1713

C₂

semidirect

26. #13083549

D₈

semidirect

40. C₈×C₈₉

SD₃₂

semidirect

53. #1713

D₈₉

semidirect

56. C₈₉

#12517

semidirect

92. C₈

#15667955

semidirect

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

subgroup	index
#26022509	2
#26019867	2
C₈×#13083549	2
C₈₉×#1713	2
#12517	89

subgroup

index

C₈₉×#1713

Subgroups — #36469147

Conjugacy classes of subgroups

Complements

Maximal subgroups