Group #36448242 — subgroups

Subgroups — #36448242

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	#26019011	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	#19151351	—	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	D₈	1	4	yes	—
28	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
29	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₈₉	C₂×D₄	1	2	yes	—
31	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
32	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
33	C₂×C₄×C₈₉	—	2	4	no	—
34	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
38	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
40	C₈×C₈₉	—	2	8	no	—
41	C₈×C₈₉	#107	1	8	yes	—
42	#10623494	—	8	4	no	—
43	C₂²×C₈₉	—	2	2	no	—
44	C₂²×C₈₉	#384	1	1	yes	characteristic
45	C₂²×C₈₉	#500	1	2	yes	—
46	C₄×C₈₉	—	2	2	no	—
47	C₄×C₈₉	C₂×#107	1	2	yes	—
48	C₄×C₈₉	#430	1	4	yes	—
49	C₄×C₈₉	#493	1	1	yes	characteristic
50	C₄×C₈₉	#494	1	1	yes	characteristic
51	#8826	—	89	1	no	maximal, Sylow
52	C₂×C₈₉	—	2	2	no	—
53	C₂×C₈₉	#1845	1	1	yes	characteristic
54	C₂×C₈₉	#1849	1	2	yes	—
55	#1848	—	89	2	no	—
56	#1567	—	89	4	no	—
57	C₂×#322	D₈₉	1	1	yes	characteristic
58	C₈₉	#8826	1	1	yes	characteristic, complemented, Sylow
59	C₂×#108	—	178	4	no	—
60	#423	—	89	4	no	—
61	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
63	#335	—	178	8	no	—
64	#322	Dic₈₉	1	4	yes	—
65	C₂×M₁₆	—	178	4	no	—
66	C₂²×C₈	Dic₁₇₈	1	4	yes	—
67	C₂²×C₈	D₃₅₆	1	2	yes	—
68	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
69	#108	—	356	8	no	—
70	C₄⋊C₈	—	178	8	no	—
71	C₄×C₈	Dic₁₇₈	1	2	yes	—
72	C₄×C₈	D₃₅₆	1	4	yes	—
73	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
74	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
75	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
77	M₁₆	—	356	8	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	—	356	4	no	—
80	C₂×C₈	C₄⋊Dic₈₉	1	8	yes	—
81	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
82	C₂×C₈	D₇₁₂	1	4	yes	—
83	C₂×C₈	Dic₃₅₆	1	4	yes	—
84	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
85	C₄²	C₂×D₃₅₆	1	2	yes	—
86	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
87	C₂³	#15668012	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
91	C₂×C₄	#15668009	1	1	yes	characteristic
92	C₂×C₄	#15668012	1	2	yes	—
93	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
94	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	712	4	no	—
97	C₈	#15667946	1	8	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#19151468	1	1	yes	characteristic
100	C₂²	#19152003	1	2	yes	—
101	C₄	—	2	2	no	—
102	C₄	C₂×#15667946	1	2	yes	—
103	C₄	#19151647	1	4	yes	—
104	C₄	#19151975	1	1	yes	characteristic
105	C₄	#19151978	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	#26020412	1	1	yes	characteristic
108	C₂	#26020427	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020424

C₂

yes

maximal

#26019011

C₂

yes

maximal

C₂×#322×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667948

—

#19151622

C₂²

yes

—

#19151351

—

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

D₈

yes

—

C₂×C₈×C₈₉

Q₁₆

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×D₈

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

#107

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#384

yes

characteristic

C₂²×C₈₉

#500

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#107

yes

—

C₄×C₈₉

#430

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

#1845

yes

characteristic

C₂×C₈₉

#1849

yes

—

#1848

—

#1567

—

C₂×#322

D₈₉

yes

characteristic

C₈₉

#8826

yes

characteristic, complemented, Sylow

C₂×#108

—

178

—

#423

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄×C₈

C₂×D₈₉

yes

—

#335

—

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

D₇₁₂

yes

—

C₂×C₈

Dic₃₅₆

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×D₇₁₂

yes

—

C₂×C₄

C₂×Dic₃₅₆

yes

—

C₈

—

C₈

—

712

—

C₈

#15667946

yes

—

C₂²

—

C₂²

#19151468

yes

characteristic

100

C₂²

#19152003

yes

—

101

C₄

—

102

C₄

C₂×#15667946

yes

—

103

C₄

#19151647

yes

—

104

C₄

#19151975

yes

characteristic

105

C₄

#19151978

yes

characteristic

106

C₂

—

107

C₂

#26020412

yes

characteristic

108

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	#8826
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
58. C₈₉	#8826	semidirect

H ≅ G/N

type

58. C₈₉

#8826

semidirect

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

subgroup	index
#26020424	2
#26019011	2
C₂×#322×C₈₉	2
#8826	89

subgroup

index

#26020424

#26019011

C₂×#322×C₈₉

#8826

Subgroups — #36448242

Conjugacy classes of subgroups

Complements

Maximal subgroups