Group #36459478 — subgroups

Subgroups — #36459478

Order 22784.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#26020412	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×#15667946	—	2	6	no	—
6	C₂×#15667934	—	2	6	no	—
7	#19151468	C₂²	1	3	yes	—
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	6	no	—
13	C₂×C₄⋊Dic₈₉	—	2	3	no	—
14	#15667946	—	4	12	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	12	no	—
27	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
28	C₂×C₈×C₈₉	—	2	6	no	—
29	C₂×C₈×C₈₉	—	2	6	no	—
30	C₂×C₈×C₈₉	D₈	1	12	yes	—
31	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₈₉	#41	1	3	yes	—
33	C₂³×C₈₉	#127	1	1	yes	characteristic
34	C₂²×D₈₉	—	4	2	no	—
35	C₂²×D₈₉	—	8	3	no	—
36	C₂²×D₈₉	—	8	3	no	—
37	C₂×C₄×C₈₉	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₈₉	#123	1	3	yes	—
39	C₂×C₄×C₈₉	C₂×D₈	1	6	yes	—
40	C₂×C₄×C₈₉	#137	1	6	yes	—
41	C₂×Dic₈₉	—	8	3	no	—
42	C₂×Dic₈₉	—	8	6	no	—
43	D₃₅₆	—	4	12	no	—
44	D₃₅₆	—	8	6	no	—
45	C₈×C₈₉	—	2	12	no	—
46	C₂²×C₈₉	#399	1	1	yes	characteristic
47	C₂²×C₈₉	#493	1	3	yes	—
48	C₂²×C₈₉	#497	1	3	yes	—
49	C₂×D₈₉	—	8	6	no	—
50	C₂×D₈₉	—	16	1	no	—
51	C₂×D₈₉	—	16	3	no	—
52	C₄×C₈₉	#474	1	6	yes	—
53	C₄×C₈₉	#480	1	6	yes	—
54	Dic₈₉	—	16	3	no	—
55	#10673	—	89	1	no	maximal, Sylow
56	C₂×C₈₉	#1712	1	1	yes	characteristic
57	C₂×C₈₉	#1722	1	3	yes	—
58	C₂×C₈₉	#2093	1	3	yes	—
59	D₈₉	—	16	2	no	—
60	#1957	—	89	3	no	—
61	#1845	—	89	3	no	—
62	#1273	D₈₉	1	1	yes	characteristic, complemented
63	C₈₉	#10673	1	1	yes	characteristic, complemented, Sylow
64	C₂×#127	—	89	1	no	—
65	C₂×#107	—	178	6	no	—
66	C₂×#102	—	178	6	no	—
67	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
68	#384	—	89	3	no	—
69	C₂²×D₄	—	178	3	no	—
70	C₂²×C₈	D₃₅₆	1	6	yes	—
71	#127	—	89	2	no	—
72	#127	—	178	3	no	—
73	C₂×C₄⋊C₄	—	178	3	no	—
74	C₂×C₄⋊C₄	—	178	6	no	—
75	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
76	#107	—	356	12	no	—
77	#102	—	356	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	356	1	no	—
80	C₂×D₄	—	356	3	no	—
81	C₂×D₄	—	356	6	no	—
82	C₂×D₄	—	356	6	no	—
83	C₂×D₄	—	178	6	no	—
84	C₂²×C₄	—	356	3	no	—
85	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	D₇₁₂	1	12	yes	—
89	C₄⋊C₄	—	356	6	no	—
90	C₄⋊C₄	—	356	12	no	—
91	C₄²	Q₈×D₈₉	1	1	yes	characteristic
92	C₄²	#13083573	1	3	yes	—
93	C₂³	—	356	2	no	—
94	C₂³	—	712	3	no	—
95	C₂³	—	712	3	no	—
96	C₂³	#15668009	1	1	yes	characteristic
97	D₄	—	356	12	no	—
98	D₄	—	712	6	no	—
99	C₂×C₄	—	712	3	no	—
100	C₂×C₄	—	712	6	no	—
101	C₂×C₄	#15667992	1	3	yes	—
102	C₂×C₄	#15667996	1	3	yes	—
103	C₂×C₄	C₂×D₇₁₂	1	6	yes	—
104	C₂×C₄	#15668045	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	1424	1	no	—
107	C₂²	—	1424	3	no	—
108	C₂²	—	712	6	no	—
109	C₂²	#19151535	1	1	yes	characteristic
110	C₂²	#19151975	1	3	yes	—
111	C₂²	#19151992	1	3	yes	—
112	C₄	—	1424	3	no	—
113	C₄	#19151879	1	6	yes	—
114	C₄	#19151912	1	6	yes	—
115	C₂	—	1424	2	no	—
116	C₂	#26019867	1	1	yes	characteristic
117	C₂	#26019923	1	3	yes	—
118	C₂	#26021779	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021058

C₂

yes

maximal

#26020412

C₂

yes

maximal, complemented

C₈₉×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#15668009

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂²×D₃₅₆

—

#15668009

—

#15668009

Q₈

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667946

—

#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

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

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

C₂×D₈

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

C₂²×C₈₉

#399

yes

characteristic

C₂²×C₈₉

#493

yes

—

C₂²×C₈₉

#497

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#474

yes

—

C₄×C₈₉

#480

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#2093

yes

—

D₈₉

—

#1957

—

#1845

—

#1273

D₈₉

yes

characteristic, complemented

C₈₉

#10673

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#107

—

178

—

C₂×#102

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

—

#384

—

C₂²×D₄

—

178

—

C₂²×C₈

D₃₅₆

yes

—

#127

—

#127

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#107

—

356

—

#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

D₇₁₂

yes

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

Q₈×D₈₉

yes

characteristic

C₄²

#13083573

yes

—

C₂³

—

356

—

C₂³

—

712

—

C₂³

—

712

—

C₂³

#15668009

yes

characteristic

D₄

—

356

—

D₄

—

712

—

C₂×C₄

—

712

—

100

C₂×C₄

—

712

—

101

C₂×C₄

#15667992

yes

—

102

C₂×C₄

#15667996

yes

—

103

C₂×C₄

C₂×D₇₁₂

yes

—

104

C₂×C₄

#15668045

yes

—

105

C₈

—

106

C₂²

—

1424

—

107

C₂²

—

1424

—

108

C₂²

—

712

—

109

C₂²

#19151535

yes

characteristic

110

C₂²

#19151975

yes

—

111

C₂²

#19151992

yes

—

112

C₄

—

1424

—

113

C₄

#19151879

yes

—

114

C₄

#19151912

yes

—

115

C₂

—

1424

—

116

C₂

#26019867

yes

characteristic

117

C₂

#26019923

yes

—

118

C₂

#26021779

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	#10673
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. #26020412	C₂	semidirect
3. C₈₉×#1273	C₂	semidirect
62. #1273	D₈₉	semidirect
63. C₈₉	#10673	semidirect

H ≅ G/N

type

2. #26020412

C₂

semidirect

3. C₈₉×#1273

C₂

semidirect

62. #1273

D₈₉

semidirect

63. C₈₉

#10673

semidirect

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

subgroup	index
#26021058	2
#26020412	2
C₈₉×#1273	2
#10673	89

subgroup

index

#26021058

#26020412

C₈₉×#1273

#10673

Subgroups — #36459478

Conjugacy classes of subgroups

Complements

Maximal subgroups