Group #36459503 — subgroups

Subgroups — #36459503

Order 22784.

478 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 141 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	#26021058	C₂	1	3	yes	maximal
2	#26020380	C₂	1	3	yes	maximal, complemented
3	C₈₉×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15668009	C₂²	1	1	yes	characteristic
5	C₂×#15667934	—	2	6	no	—
6	#19151468	C₂²	1	3	yes	—
7	#19151361	—	2	6	no	—
8	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
9	C₂²×D₃₅₆	—	2	3	no	—
10	#15668009	—	2	3	no	—
11	#15668009	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₈₉	—	2	3	no	—
13	C₂×C₄⋊Dic₈₉	—	2	3	no	—
14	C₂×C₄⋊Dic₈₉	—	2	3	no	—
15	#15667934	—	4	12	no	—
16	C₂²×C₈×C₈₉	D₄	1	6	yes	—
17	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₈₉	—	2	6	no	—
19	C₂³×D₈₉	—	4	1	no	—
20	C₂×D₃₅₆	—	2	6	no	—
21	C₂×D₃₅₆	—	4	6	no	—
22	C₂×D₃₅₆	—	4	6	no	—
23	C₂×D₃₅₆	—	4	3	no	—
24	C₂²×Dic₈₉	—	4	3	no	—
25	C₄⋊Dic₈₉	—	4	6	no	—
26	C₄⋊Dic₈₉	—	4	6	no	—
27	C₄⋊Dic₈₉	—	4	6	no	—
28	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
29	C₂×C₈×C₈₉	—	2	6	no	—
30	C₂×C₈×C₈₉	—	2	6	no	—
31	C₂×C₈×C₈₉	—	2	6	no	—
32	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₈₉	#41	1	3	yes	—
34	C₂³×C₈₉	#127	1	1	yes	characteristic
35	C₂²×D₈₉	—	4	2	no	—
36	C₂²×D₈₉	—	8	3	no	—
37	C₂²×D₈₉	—	8	3	no	—
38	C₂×C₄×C₈₉	C₂²⋊Q₈	1	3	yes	—
39	C₂×C₄×C₈₉	#123	1	3	yes	—
40	C₂×C₄×C₈₉	#135	1	6	yes	—
41	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	6	yes	—
42	C₂×Dic₈₉	—	8	3	no	—
43	C₂×Dic₈₉	—	8	3	no	—
44	C₂×Dic₈₉	—	8	3	no	—
45	D₃₅₆	—	4	12	no	—
46	D₃₅₆	—	8	6	no	—
47	C₈×C₈₉	—	4	6	no	—
48	C₂²×C₈₉	#399	1	1	yes	characteristic
49	C₂²×C₈₉	#495	1	3	yes	—
50	C₂²×C₈₉	#496	1	3	yes	—
51	C₂×D₈₉	—	8	6	no	—
52	C₂×D₈₉	—	16	1	no	—
53	C₂×D₈₉	—	16	3	no	—
54	C₄×C₈₉	#478	1	6	yes	—
55	C₄×C₈₉	#482	1	6	yes	—
56	Dic₈₉	—	16	3	no	—
57	#10678	—	89	1	no	maximal, Sylow
58	C₂×C₈₉	#1711	1	1	yes	characteristic
59	C₂×C₈₉	#1717	1	3	yes	—
60	C₂×C₈₉	#2092	1	3	yes	—
61	D₈₉	—	16	2	no	—
62	#1957	—	89	3	no	—
63	#1834	—	89	3	no	—
64	#1273	D₈₉	1	1	yes	characteristic, complemented
65	C₈₉	#10678	1	1	yes	characteristic, complemented, Sylow
66	C₂×#127	—	89	1	no	—
67	C₂×#102	—	178	6	no	—
68	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
69	#384	—	89	3	no	—
70	#340	—	178	6	no	—
71	C₂²×D₄	—	178	3	no	—
72	C₂²×C₈	D₃₅₆	1	6	yes	—
73	#127	—	89	2	no	—
74	#127	—	178	3	no	—
75	C₂×C₄⋊C₄	—	178	3	no	—
76	C₂×C₄⋊C₄	—	178	3	no	—
77	C₂×C₄⋊C₄	—	178	3	no	—
78	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
79	#102	—	356	12	no	—
80	C₄×C₈	—	2	6	no	—
81	C₂⁴	—	356	1	no	—
82	C₂×D₄	—	356	3	no	—
83	C₂×D₄	—	356	6	no	—
84	C₂×D₄	—	356	6	no	—
85	C₂×D₄	—	178	6	no	—
86	C₂²×C₄	—	356	3	no	—
87	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	2	6	no	—
91	C₄⋊C₄	—	356	6	no	—
92	C₄⋊C₄	—	356	6	no	—
93	C₄⋊C₄	—	356	6	no	—
94	C₄²	Q₈×D₈₉	1	1	yes	characteristic
95	C₄²	#13083573	1	3	yes	—
96	C₂³	—	356	2	no	—
97	C₂³	—	712	3	no	—
98	C₂³	—	712	3	no	—
99	C₂³	#15668009	1	1	yes	characteristic
100	D₄	—	356	12	no	—
101	D₄	—	712	6	no	—
102	C₂×C₄	—	712	3	no	—
103	C₂×C₄	—	712	3	no	—
104	C₂×C₄	—	712	3	no	—
105	C₂×C₄	#15667992	1	3	yes	—
106	C₂×C₄	#15667996	1	3	yes	—
107	C₂×C₄	#15668034	1	6	yes	—
108	C₂×C₄	#15668040	1	6	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	1424	1	no	—
111	C₂²	—	1424	3	no	—
112	C₂²	—	712	6	no	—
113	C₂²	#19151535	1	1	yes	characteristic
114	C₂²	#19151982	1	3	yes	—
115	C₂²	#19151987	1	3	yes	—
116	C₄	—	1424	3	no	—
117	C₄	#19151899	1	6	yes	—
118	C₄	#19151923	1	6	yes	—
119	C₂	—	1424	2	no	—
120	C₂	#26019864	1	1	yes	characteristic
121	C₂	#26019891	1	3	yes	—
122	C₂	#26021772	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021058

C₂

yes

maximal

#26020380

C₂

yes

maximal, complemented

C₈₉×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#15668009

C₂²

yes

characteristic

C₂×#15667934

—

#19151468

C₂²

yes

—

#19151361

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂²×D₃₅₆

—

#15668009

—

#15668009

Q₈

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667934

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

—

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#127

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

C₂²×C₈₉

#399

yes

characteristic

C₂²×C₈₉

#495

yes

—

C₂²×C₈₉

#496

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#478

yes

—

C₄×C₈₉

#482

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#2092

yes

—

D₈₉

—

#1957

—

#1834

—

#1273

D₈₉

yes

characteristic, complemented

C₈₉

#10678

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#102

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

—

#384

—

#340

—

178

—

C₂²×D₄

—

178

—

C₂²×C₈

D₃₅₆

yes

—

#127

—

#127

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#102

—

356

—

C₄×C₈

—

C₂⁴

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

178

—

C₂²×C₄

—

356

—

C₂²×C₄

C₂×D₃₅₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

Q₈×D₈₉

yes

characteristic

C₄²

#13083573

yes

—

C₂³

—

356

—

C₂³

—

712

—

C₂³

—

712

—

C₂³

#15668009

yes

characteristic

100

D₄

—

356

—

101

D₄

—

712

—

102

C₂×C₄

—

712

—

103

C₂×C₄

—

712

—

104

C₂×C₄

—

712

—

105

C₂×C₄

#15667992

yes

—

106

C₂×C₄

#15667996

yes

—

107

C₂×C₄

#15668034

yes

—

108

C₂×C₄

#15668040

yes

—

109

C₈

—

110

C₂²

—

1424

—

111

C₂²

—

1424

—

112

C₂²

—

712

—

113

C₂²

#19151535

yes

characteristic

114

C₂²

#19151982

yes

—

115

C₂²

#19151987

yes

—

116

C₄

—

1424

—

117

C₄

#19151899

yes

—

118

C₄

#19151923

yes

—

119

C₂

—

1424

—

120

C₂

#26019864

yes

characteristic

121

C₂

#26019891

yes

—

122

C₂

#26021772

yes

—

	subgroup	G/H
center	C₂³	#15668009
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1273	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#15668009	C₂²
hypercenter	#1273	D₈₉
hypocenter	C₈₉	#10678
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#15668009

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1273

C₂

C₂³×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #26020380	C₂	semidirect
3. C₈₉×#1273	C₂	semidirect
64. #1273	D₈₉	semidirect
65. C₈₉	#10678	semidirect

H ≅ G/N

type

2. #26020380

C₂

semidirect

3. C₈₉×#1273

C₂

semidirect

64. #1273

D₈₉

semidirect

65. C₈₉

#10678

semidirect

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

subgroup	index
#26021058	2
#26020380	2
C₈₉×#1273	2
#10678	89

subgroup

index

#26021058

#26020380

C₈₉×#1273

#10678

Subgroups — #36459503

Conjugacy classes of subgroups

Complements

Maximal subgroups