Group #36469166 — subgroups

Subgroups — #36469166

Order 22784.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26022532	C₂	1	2	yes	maximal
2	#26022525	C₂	1	2	yes	maximal
3	#26019862	C₂	1	1	yes	maximal, characteristic
4	C₈×#13083549	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1710	C₂	1	1	yes	maximal, characteristic
6	#19152003	C₂²	1	2	yes	—
7	#19151949	—	2	2	no	—
8	#19151425	—	2	4	no	—
9	#19151423	—	2	4	no	—
10	C₄×#13083549	C₂²	1	1	yes	characteristic
11	C₄×#13083549	C₂²	1	1	yes	characteristic
12	#500×C₈₉	C₂²	1	2	yes	—
13	#487×C₈₉	—	2	2	no	—
14	C₈²×C₈₉	C₂²	1	1	yes	characteristic
15	#15668012	—	2	2	no	—
16	C₂×#13083549	D₄	1	2	yes	—
17	C₂×#13083549	D₄	1	2	yes	—
18	#15667946	—	2	2	no	—
19	#15667946	—	2	2	no	—
20	#15667937	—	4	4	no	—
21	C₄⋊Q₈×C₈₉	—	2	2	no	—
22	#107×C₈₉	—	2	2	no	—
23	#107×C₈₉	—	2	2	no	—
24	#103×C₈₉	—	4	4	no	—
25	C₄×C₈×C₈₉	D₄	1	2	yes	—
26	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₇₈	—	4	2	no	—
28	C₄⋊Dic₈₉	—	4	2	no	—
29	C₄⋊Dic₈₉	—	4	2	no	—
30	#13083549	Q₁₆	1	4	yes	—
31	#13083549	Q₁₆	1	4	yes	—
32	C₂×Q₈×C₈₉	—	4	2	no	—
33	C₂×C₈×C₈₉	D₈	1	4	yes	—
34	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₈₉	—	4	2	no	—
37	C₄⋊C₄×C₈₉	—	4	2	no	—
38	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₈₉	—	8	4	no	—
40	C₂×C₄×C₈₉	—	8	2	no	—
41	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
43	C₂×Dic₈₉	—	8	2	no	—
44	Dic₁₇₈	—	8	4	no	—
45	C₈×C₈₉	—	2	4	no	—
46	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
47	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
48	C₂²×C₈₉	#493	1	1	yes	characteristic
49	C₄×C₈₉	—	16	2	no	—
50	C₄×C₈₉	#494	1	1	yes	characteristic
51	C₄×C₈₉	#494	1	1	yes	characteristic
52	C₄×C₈₉	#508	1	4	yes	—
53	Dic₈₉	—	16	2	no	—
54	#12522	—	89	1	no	maximal, Sylow
55	C₂×C₈₉	#1713	1	1	yes	characteristic
56	C₂×C₈₉	#2249	1	2	yes	—
57	#2254	—	89	2	no	—
58	#2252	—	89	2	no	—
59	#1710	—	89	1	no	—
60	#1710	D₈₉	1	1	yes	characteristic
61	C₈×C₁₆	—	89	1	no	—
62	C₈₉	#12522	1	1	yes	characteristic, complemented, Sylow
63	#500	—	89	2	no	—
64	#500	C₂×D₈₉	1	2	yes	—
65	#487	—	2	2	no	—
66	#487	—	178	2	no	—
67	#367	—	178	4	no	—
68	#366	—	178	4	no	—
69	C₄×C₁₆	—	89	1	no	—
70	C₄×C₁₆	—	89	1	no	—
71	C₈²	C₂×D₈₉	1	1	yes	characteristic
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	—	178	2	no	—
74	C₂×C₁₆	—	89	2	no	—
75	C₂×C₁₆	—	89	2	no	—
76	#107	—	2	2	no	—
77	#107	—	2	2	no	—
78	#107	—	178	2	no	—
79	#107	—	178	2	no	—
80	#103	—	4	4	no	—
81	#103	—	356	4	no	—
82	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
83	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
84	C₂×Q₈	—	4	2	no	—
85	C₂×Q₈	—	356	2	no	—
86	C₂×C₈	#13083559	1	4	yes	—
87	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
88	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	356	2	no	—
92	C₄⋊C₄	—	356	2	no	—
93	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
94	C₁₆	—	89	4	no	—
95	C₁₆	—	89	4	no	—
96	Q₈	—	8	4	no	—
97	Q₈	—	712	4	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	—	712	2	no	—
100	C₂×C₄	#15668008	1	1	yes	characteristic
101	C₂×C₄	C₂×#13083559	1	2	yes	—
102	C₈	—	2	4	no	—
103	C₈	Q₁₆×D₈₉	1	2	yes	—
104	C₈	Q₁₆×D₈₉	1	2	yes	—
105	C₂²	#19151974	1	1	yes	characteristic
106	C₄	—	16	2	no	—
107	C₄	—	1424	2	no	—
108	C₄	#19151976	1	1	yes	characteristic
109	C₄	#19151976	1	1	yes	characteristic
110	C₄	#19152028	1	4	yes	—
111	C₂	#26019868	1	1	yes	characteristic
112	C₂	#26022511	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022532

C₂

yes

maximal

#26022525

C₂

yes

maximal

#26019862

C₂

yes

maximal, characteristic

C₈×#13083549

C₂

yes

maximal, characteristic

C₈₉×#1710

C₂

yes

maximal, characteristic

#19152003

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#13083549

C₂²

yes

characteristic

#500×C₈₉

C₂²

yes

—

#487×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

#15668012

—

C₂×#13083549

D₄

yes

—

C₂×#13083549

D₄

yes

—

#15667946

—

#15667946

—

#15667937

—

C₄⋊Q₈×C₈₉

—

#107×C₈₉

—

#107×C₈₉

—

#103×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083549

Q₁₆

yes

—

#13083549

Q₁₆

yes

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₁₆

yes

—

C₈×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

—

D₈₉

yes

characteristic

C₈×C₁₆

—

C₈₉

#12522

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₈₉

yes

—

#487

—

#487

—

178

—

#367

—

178

—

#366

—

178

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

178

—

#107

—

178

—

#103

—

#103

—

356

—

C₄×C₈

C₂²⋊D₈₉

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂×C₈

#13083559

yes

—

C₂×C₈

D₄×D₈₉

yes

characteristic

C₂×C₈

D₄×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

C₁₆

—

C₁₆

—

Q₈

—

Q₈

—

712

—

C₂×C₄

—

C₂×C₄

—

712

—

100

C₂×C₄

#15668008

yes

characteristic

101

C₂×C₄

C₂×#13083559

yes

—

102

C₈

—

103

C₈

Q₁₆×D₈₉

yes

—

104

C₈

Q₁₆×D₈₉

yes

—

105

C₂²

#19151974

yes

characteristic

106

C₄

—

107

C₄

—

1424

—

108

C₄

#19151976

yes

characteristic

109

C₄

#19151976

yes

characteristic

110

C₄

#19152028

yes

—

111

C₂

#26019868

yes

characteristic

112

C₂

#26022511

yes

—

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1710	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	#1710	D₈₉
hypocenter	C₈₉	#12522
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1710

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
62. C₈₉	#12522	semidirect

H ≅ G/N

type

62. C₈₉

#12522

semidirect

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

subgroup	index
#26022532	2
#26022525	2
#26019862	2
C₈×#13083549	2
C₈₉×#1710	2
#12522	89

subgroup

index

C₈₉×#1710

Subgroups — #36469166

Conjugacy classes of subgroups

Complements

Maximal subgroups