Group #36448239 — subgroups

Subgroups — #36448239

Order 22784.

338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 187 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	#26020424	C₂	1	2	yes	maximal
2	#26019008	C₂	1	4	yes	maximal
3	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667948	—	2	4	no	—
5	#19151622	C₂²	1	4	yes	—
6	#19151349	—	2	8	no	—
7	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	#322×C₈₉	C₄	1	4	yes	—
10	C₂×#13083558	—	2	4	no	—
11	#15667948	—	4	8	no	—
12	C₄⋊#10623494	—	2	8	no	—
13	C₂²×C₈×C₈₉	D₄	1	2	yes	—
14	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
15	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₈₉	D₄	1	4	yes	—
19	C₄×C₈×C₈₉	Q₈	1	2	yes	—
20	#13083558	—	4	8	no	—
21	C₂×#10623494	—	4	4	no	—
22	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
24	C₂×C₈×C₈₉	—	4	4	no	—
25	C₂×C₈×C₈₉	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₈₉	SD₁₆	1	8	yes	—
28	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₈₉	C₂×D₄	1	2	yes	—
30	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
31	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	2	4	no	—
33	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
38	C₈×C₈₉	—	2	8	no	—
39	C₈×C₈₉	#106	1	8	yes	—
40	#10623494	—	8	4	no	—
41	C₂²×C₈₉	—	2	2	no	—
42	C₂²×C₈₉	#384	1	1	yes	characteristic
43	C₂²×C₈₉	#498	1	2	yes	—
44	C₄×C₈₉	—	2	2	no	—
45	C₄×C₈₉	C₂×#106	1	2	yes	—
46	C₄×C₈₉	#430	1	4	yes	—
47	C₄×C₈₉	#492	1	2	yes	—
48	#8825	—	89	1	no	maximal, Sylow
49	C₂×C₈₉	—	2	2	no	—
50	C₂×C₈₉	#1844	1	1	yes	characteristic
51	C₂×C₈₉	#1849	1	2	yes	—
52	#1848	—	89	2	no	—
53	#1566	—	89	4	no	—
54	C₂×#322	D₈₉	1	1	yes	characteristic
55	C₈₉	#8825	1	1	yes	characteristic, complemented, Sylow
56	C₂×#108	—	178	4	no	—
57	#423	—	89	4	no	—
58	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
60	#334	—	178	8	no	—
61	#322	Dic₈₉	1	4	yes	—
62	C₂×M₁₆	—	178	4	no	—
63	C₂²×C₈	Dic₁₇₈	1	4	yes	—
64	C₂²×C₈	D₃₅₆	1	2	yes	—
65	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
66	#108	—	356	8	no	—
67	C₄⋊C₈	—	178	8	no	—
68	C₄×C₈	Dic₁₇₈	1	2	yes	—
69	C₄×C₈	D₃₅₆	1	4	yes	—
70	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
71	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
72	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
73	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
74	M₁₆	—	356	8	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	356	4	no	—
77	C₂×C₈	C₄⋊Dic₈₉	1	8	yes	—
78	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
79	C₂×C₈	#13083563	1	8	yes	—
80	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
81	C₄²	C₂×D₃₅₆	1	2	yes	—
82	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
83	C₂³	#15668012	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
86	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
87	C₂×C₄	#15668009	1	1	yes	characteristic
88	C₂×C₄	#15668012	1	2	yes	—
89	C₂×C₄	C₂×#13083563	1	4	yes	—
90	C₈	—	2	8	no	—
91	C₈	—	712	4	no	—
92	C₈	#15667944	1	8	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	#19151468	1	1	yes	characteristic
95	C₂²	#19151996	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	C₂×#15667944	1	2	yes	—
98	C₄	#19151647	1	4	yes	—
99	C₄	#19151971	1	2	yes	—
100	C₂	—	2	2	no	—
101	C₂	#26020409	1	1	yes	characteristic
102	C₂	#26020427	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020424

C₂

yes

maximal

#26019008

C₂

yes

maximal

C₂×#322×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667948

—

#19151622

C₂²

yes

—

#19151349

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#322×C₈₉

C₄

yes

—

C₂×#13083558

—

#15667948

—

C₄⋊#10623494

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#13083558

—

C₂×#10623494

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

#106

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#384

yes

characteristic

C₂²×C₈₉

#498

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#106

yes

—

C₄×C₈₉

#430

yes

—

C₄×C₈₉

#492

yes

—

#8825

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1844

yes

characteristic

C₂×C₈₉

#1849

yes

—

#1848

—

#1566

—

C₂×#322

D₈₉

yes

characteristic

C₈₉

#8825

yes

characteristic, complemented, Sylow

C₂×#108

—

178

—

#423

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄×C₈

C₂×D₈₉

yes

—

#334

—

178

—

#322

Dic₈₉

yes

—

C₂×M₁₆

—

178

—

C₂²×C₈

Dic₁₇₈

yes

—

C₂²×C₈

D₃₅₆

yes

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#108

—

356

—

C₄⋊C₈

—

178

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

C₂×Dic₈₉

yes

—

C₄×C₈

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₇₈

yes

—

M₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

#13083563

yes

—

C₄²

C₂²×Dic₈₉

yes

characteristic

C₄²

C₂×D₃₅₆

yes

—

C₄²

C₂×Dic₁₇₈

yes

characteristic

C₂³

#15668012

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

—

C₂×C₄

#15668009

yes

characteristic

C₂×C₄

#15668012

yes

—

C₂×C₄

C₂×#13083563

yes

—

C₈

—

C₈

—

712

—

C₈

#15667944

yes

—

C₂²

—

C₂²

#19151468

yes

characteristic

C₂²

#19151996

yes

—

C₄

—

C₄

C₂×#15667944

yes

—

C₄

#19151647

yes

—

C₄

#19151971

yes

—

100

C₂

—

101

C₂

#26020409

yes

characteristic

102

C₂

#26020427

yes

—

	subgroup	G/H
center	C₂×C₄	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂²×C₈₉	#384
prime radical	C₂³×C₈₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#8825
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#15668009

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₂×#322×C₈₉

C₂

socle

C₂²×C₈₉

#384

prime radical

C₂³×C₈₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
55. C₈₉	#8825	semidirect

H ≅ G/N

type

55. C₈₉

#8825

semidirect

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

subgroup	index
#26020424	2
#26019008	2
C₂×#322×C₈₉	2
#8825	89

subgroup

index

#26020424

#26019008

C₂×#322×C₈₉

#8825

Subgroups — #36448239

Conjugacy classes of subgroups

Complements

Maximal subgroups