Group #36475401 — subgroups

Subgroups — #36475401

Order 22784.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#26022527	C₂	1	1	yes	maximal, characteristic
2	#26022505	C₂	1	1	yes	maximal, characteristic
3	#26018434	C₂	1	4	yes	maximal
4	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₇₁₂	—	2	1	no	—
6	C₂×Dic₇₁₂	C₂²	1	2	yes	—
7	#19152003	—	2	1	no	—
8	#19151978	—	2	1	no	—
9	#19151424	—	2	1	no	—
10	#19151424	C₂²	1	2	yes	—
11	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
12	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₅₆	—	2	2	no	—
14	C₂×Dic₃₅₆	—	4	1	no	—
15	#15668012	—	4	1	no	—
16	#15668012	—	4	1	no	—
17	Dic₇₁₂	—	2	4	no	—
18	Dic₇₁₂	—	4	2	no	—
19	#15667946	—	2	2	no	—
20	#15667946	—	4	1	no	—
21	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₃₂×C₈₉	—	2	2	no	—
25	C₂×Dic₁₇₈	—	4	2	no	—
26	C₂×Dic₁₇₈	—	8	1	no	—
27	Dic₃₅₆	—	4	2	no	—
28	Dic₃₅₆	—	4	2	no	—
29	Dic₃₅₆	—	8	2	no	—
30	C₄⋊Dic₈₉	—	4	2	no	—
31	C₄⋊Dic₈₉	—	8	1	no	—
32	C₄⋊Dic₈₉	—	8	1	no	—
33	C₄⋊Dic₈₉	—	8	1	no	—
34	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₈₉	D₈	1	1	yes	characteristic
37	C₁₆×C₈₉	SD₁₆	1	2	yes	—
38	C₁₆×C₈₉	#41	1	2	yes	—
39	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
42	C₂×Dic₈₉	—	8	2	no	—
43	C₂×Dic₈₉	—	8	2	no	—
44	Dic₁₇₈	—	8	2	no	—
45	Dic₁₇₈	—	8	2	no	—
46	Dic₁₇₈	—	16	2	no	—
47	C₈×C₈₉	SD₃₂	1	2	yes	—
48	C₈×C₈₉	#124	1	1	yes	characteristic
49	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
50	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
51	C₄×C₈₉	SD₆₄	1	2	yes	—
52	C₄×C₈₉	Q₆₄	1	2	yes	—
53	C₄×C₈₉	#486	1	1	yes	characteristic
54	C₄×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
55	Dic₈₉	—	16	2	no	—
56	Dic₈₉	—	16	2	no	—
57	#13720	—	89	1	no	maximal, Sylow
58	C₂×C₈₉	#2240	1	1	yes	characteristic
59	C₂×C₈₉	C₂×SD₆₄	1	1	yes	characteristic
60	C₂×C₈₉	C₂×Q₆₄	1	1	yes	characteristic
61	#2252	—	89	1	no	—
62	#2247	—	89	1	no	—
63	#1416	—	89	4	no	—
64	C₄×C₃₂	D₈₉	1	1	yes	characteristic
65	C₈₉	#13720	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₃₂	—	89	2	no	—
67	C₂×Q₃₂	—	178	1	no	—
68	#500	—	178	1	no	—
69	#494	—	178	1	no	—
70	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
71	#366	—	89	2	no	—
72	#366	—	178	1	no	—
73	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
74	C₂×Q₁₆	—	356	1	no	—
75	C₂×Q₁₆	—	178	2	no	—
76	C₄⋊Q₈	—	356	1	no	—
77	C₄⋊Q₈	—	356	1	no	—
78	Q₃₂	—	356	2	no	—
79	Q₃₂	—	178	4	no	—
80	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
82	#107	—	356	1	no	—
83	#107	—	178	2	no	—
84	C₄×C₈	D₃₅₆	1	1	yes	characteristic
85	C₃₂	—	2	2	no	—
86	C₂×Q₈	—	356	2	no	—
87	C₂×Q₈	—	712	1	no	—
88	Q₁₆	—	356	2	no	—
89	Q₁₆	—	356	2	no	—
90	Q₁₆	—	712	2	no	—
91	C₂×C₈	D₇₁₂	1	1	yes	characteristic
92	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
93	C₄⋊C₄	—	356	2	no	—
94	C₄⋊C₄	—	712	1	no	—
95	C₄⋊C₄	—	712	1	no	—
96	C₄⋊C₄	—	712	1	no	—
97	C₄²	D₇₁₂	1	1	yes	characteristic
98	C₁₆	#13083563	1	2	yes	—
99	C₁₆	#13083574	1	2	yes	—
100	Q₈	—	1424	2	no	—
101	Q₈	—	712	2	no	—
102	Q₈	—	712	2	no	—
103	C₂×C₄	—	712	2	no	—
104	C₂×C₄	—	712	2	no	—
105	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
106	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
107	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
108	C₈	#15667957	1	2	yes	—
109	C₈	#15668000	1	1	yes	characteristic
110	C₈	C₂×#13083563	1	1	yes	characteristic
111	C₂²	C₂×D₁₄₂₄	1	1	yes	characteristic
112	C₄	—	1424	2	no	—
113	C₄	—	1424	2	no	—
114	C₄	#19151437	1	2	yes	—
115	C₄	Dic₁₄₂₄	1	2	yes	—
116	C₄	#19151945	1	1	yes	characteristic
117	C₄	C₂×#15667957	1	1	yes	characteristic
118	C₂	#26022476	1	1	yes	characteristic
119	C₂	C₂×#19151437	1	1	yes	characteristic
120	C₂	C₂×Dic₁₄₂₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022527

C₂

yes

maximal, characteristic

#26022505

C₂

yes

maximal, characteristic

#26018434

C₂

yes

maximal

C₄×C₃₂×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

—

C₂×Dic₇₁₂

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₃₂×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

#15668012

—

Dic₇₁₂

—

Dic₇₁₂

—

#15667946

—

#15667946

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₃₂×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

SD₁₆

yes

—

C₁₆×C₈₉

#41

yes

—

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

SD₃₂

yes

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

SD₆₄

yes

—

C₄×C₈₉

Q₆₄

yes

—

C₄×C₈₉

#486

yes

characteristic

C₄×C₈₉

C₂×SD₃₂

yes

characteristic

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

C₂×SD₆₄

yes

characteristic

C₂×C₈₉

C₂×Q₆₄

yes

—

—

—

C₄×C₃₂

D₈₉

yes

characteristic

C₈₉

#13720

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

178

—

#500

—

178

—

#494

—

178

—

C₂×C₃₂

C₂×D₈₉

yes

—

#366

—

#366

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

356

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

356

—

C₄⋊Q₈

—

356

—

Q₃₂

—

356

—

Q₃₂

—

178

—

C₂×C₁₆

D₃₅₆

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#107

—

356

—

#107

—

178

—

C₄×C₈

D₃₅₆

yes

characteristic

C₃₂

—

C₂×Q₈

—

356

—

C₂×Q₈

—

712

—

Q₁₆

—

356

—

Q₁₆

—

356

—

Q₁₆

—

712

—

C₂×C₈

D₇₁₂

yes

characteristic

C₂×C₈

C₂×D₃₅₆

yes

characteristic

C₄⋊C₄

—

356

—

C₄⋊C₄

—

712

—

C₄⋊C₄

—

712

—

C₄⋊C₄

—

712

—

C₄²

D₇₁₂

yes

characteristic

C₁₆

#13083563

yes

—

C₁₆

#13083574

yes

—

100

Q₈

—

1424

—

101

Q₈

—

712

—

102

Q₈

—

712

—

103

C₂×C₄

—

712

—

104

C₂×C₄

—

712

—

105

C₂×C₄

D₁₄₂₄

yes

characteristic

106

C₂×C₄

D₁₄₂₄

yes

characteristic

107

C₂×C₄

C₂×D₇₁₂

yes

characteristic

108

C₈

#15667957

yes

—

109

C₈

#15668000

yes

characteristic

110

C₈

C₂×#13083563

yes

characteristic

111

C₂²

C₂×D₁₄₂₄

yes

characteristic

112

C₄

—

1424

—

113

C₄

—

1424

—

114

C₄

#19151437

yes

—

115

C₄

Dic₁₄₂₄

yes

—

116

C₄

#19151945

yes

characteristic

117

C₄

C₂×#15667957

yes

characteristic

118

C₂

#26022476

yes

characteristic

119

C₂

C₂×#19151437

yes

characteristic

120

C₂

C₂×Dic₁₄₂₄

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₄₂₄
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₄×C₃₂×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	C₄×C₃₂	D₈₉
hypocenter	C₈₉	#13720
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂²

C₂×D₁₄₂₄

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₄×C₃₂×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₁₆

hypercenter

C₄×C₃₂

D₈₉

hypocenter

C₈₉

#13720

LCS

G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
65. C₈₉	#13720	semidirect

H ≅ G/N

type

65. C₈₉

#13720

semidirect

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

subgroup	index
#26022527	2
#26022505	2
#26018434	2
C₄×C₃₂×C₈₉	2
#13720	89

subgroup

index

C₄×C₃₂×C₈₉

Subgroups — #36475401

Conjugacy classes of subgroups

Complements

Maximal subgroups