Group #36474928 — subgroups

Subgroups — #36474928

Order 22784.

190 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×C₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#26022033	C₂	1	2	yes	maximal
3	#26018453	C₂	1	2	yes	maximal, complemented
4	#26018452	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
7	#19151635	—	2	2	no	—
8	C₂×#15667918	C₂²	1	1	yes	characteristic
9	C₃₂×D₈₉	C₄	1	4	yes	complemented
10	#19151425	—	2	2	no	—
11	#19151424	C₂²	1	2	yes	—
12	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
13	#366×C₈₉	C₂²	1	2	yes	—
14	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
15	#15667970	—	4	2	no	—
16	C₂×#13083549	D₄	1	1	yes	characteristic
17	C₁₆×D₈₉	C₂×C₄	1	2	yes	—
18	C₁₆×D₈₉	Q₈	1	2	yes	—
19	#15667946	—	2	2	no	—
20	#15667943	—	4	2	no	—
21	#15667918	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
23	#107×C₈₉	—	2	2	no	—
24	C₃₂×C₈₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
26	C₂×#10623494	D₈	1	1	yes	characteristic
27	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
28	C₈×D₈₉	Q₁₆	1	2	yes	—
29	C₄⋊Dic₈₉	—	4	2	no	—
30	#13083551	—	8	2	no	—
31	C₄×Dic₈₉	—	8	2	no	—
32	#13083549	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₈₉	—	4	2	no	—
35	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
37	C₂²×D₈₉	D₁₆	1	1	yes	characteristic
38	C₂×C₄×C₈₉	—	8	2	no	—
39	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
40	C₂×Dic₈₉	—	8	2	no	—
41	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
42	C₄×D₈₉	#107	1	2	yes	—
43	C₄×D₈₉	Q₃₂	1	2	yes	—
44	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
46	#10623494	#107	1	2	yes	—
47	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
48	C₂×D₈₉	—	2	1	no	—
49	C₂×D₈₉	—	2	1	no	—
50	C₂×D₈₉	#366	1	2	yes	—
51	C₄×C₈₉	—	16	2	no	—
52	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
53	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
54	Dic₈₉	—	16	2	no	—
55	Dic₈₉	#366	1	2	yes	—
56	#13633	—	89	1	no	maximal, Sylow
57	C₂×C₈₉	C₂×#366	1	1	yes	characteristic
58	C₂×C₈₉	#2262	1	1	yes	characteristic
59	C₂×C₈₉	#2262	1	1	yes	characteristic
60	D₈₉	—	2	2	no	—
61	C₂²×C₃₂	—	89	1	no	—
62	#2158	—	89	2	no	—
63	#1424	—	89	2	no	—
64	#1423	—	89	1	no	—
65	#1423	D₈₉	1	1	yes	characteristic, complemented
66	C₈₉	#13633	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₁₆	—	89	1	no	—
68	#427	—	178	2	no	—
69	C₂×C₃₂	—	89	1	no	—
70	C₂×C₃₂	—	89	4	no	—
71	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
72	#367	—	178	2	no	—
73	#366	—	89	2	no	—
74	#366	C₂×D₈₉	1	2	yes	—
75	C₂²×C₈	—	89	1	no	—
76	#117	—	356	2	no	—
77	C₂×C₁₆	—	89	1	no	—
78	C₂×C₁₆	—	89	2	no	—
79	C₂×C₁₆	—	89	2	no	—
80	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
81	#107	—	2	2	no	—
82	#107	—	178	2	no	—
83	#106	—	356	2	no	—
84	C₃₂	—	89	2	no	—
85	C₃₂	C₄×D₈₉	1	2	yes	—
86	C₂²×C₄	—	89	1	no	—
87	C₂×C₈	—	89	1	no	—
88	C₂×C₈	—	89	2	no	—
89	C₂×C₈	—	89	2	no	—
90	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	356	2	no	—
93	C₂²⋊C₄	—	712	2	no	—
94	C₄²	—	712	2	no	—
95	C₁₆	—	89	2	no	—
96	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
97	C₁₆	Q₈×D₈₉	1	1	yes	characteristic
98	C₂³	—	89	1	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	—	89	1	no	—
101	C₂×C₄	—	89	2	no	—
102	C₂×C₄	—	89	2	no	—
103	C₂×C₄	—	712	2	no	—
104	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
105	C₈	—	89	2	no	—
106	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
107	C₈	Q₁₆×D₈₉	1	1	yes	characteristic
108	C₂²	—	89	2	no	—
109	C₂²	—	178	1	no	—
110	C₂²	—	178	1	no	—
111	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
112	C₄	—	16	2	no	—
113	C₄	—	89	2	no	—
114	C₄	—	1424	2	no	—
115	C₄	#107×D₈₉	1	1	yes	characteristic
116	C₄	Q₃₂×D₈₉	1	1	yes	characteristic
117	C₂	—	178	2	no	—
118	C₂	#366×D₈₉	1	1	yes	characteristic
119	C₂	#26022558	1	1	yes	characteristic
120	C₂	#26022562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₂×D₈₉

C₂

yes

maximal, characteristic

#26022033

C₂

yes

maximal

#26018453

C₂

yes

maximal, complemented

#26018452

C₂

yes

maximal, characteristic, complemented

C₈₉×#1423

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₈₉

C₂²

yes

characteristic

#19151635

—

C₂×#15667918

C₂²

yes

characteristic

C₃₂×D₈₉

C₄

yes

complemented

#19151425

—

#19151424

C₂²

yes

—

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

#366×C₈₉

C₂²

yes

—

C₂×C₈×D₈₉

D₄

yes

characteristic

#15667970

—

C₂×#13083549

D₄

yes

characteristic

C₁₆×D₈₉

C₂×C₄

yes

—

C₁₆×D₈₉

Q₈

yes

—

#15667946

—

#15667943

—

#15667918

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

C₃₂×C₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

D₈

yes

characteristic

C₂×#10623494

D₈

yes

characteristic

C₈×D₈₉

C₄⋊C₄

yes

—

C₈×D₈₉

Q₁₆

yes

—

C₄⋊Dic₈₉

—

#13083551

—

C₄×Dic₈₉

—

#13083549

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

C₂×Q₈

yes

characteristic

C₂²×D₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

D₁₆

yes

characteristic

C₄×D₈₉

#107

yes

—

C₄×D₈₉

Q₃₂

yes

—

C₈×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₂×Q₁₆

yes

characteristic

#10623494

#107

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

#366

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#107

yes

characteristic

C₄×C₈₉

C₂×Q₃₂

yes

characteristic

Dic₈₉

—

Dic₈₉

#366

yes

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

C₂²×C₃₂

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#13633

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#427

—

178

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#367

—

178

—

#366

—

#366

C₂×D₈₉

yes

—

C₂²×C₈

—

#117

—

356

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#107

—

#107

—

178

—

#106

—

356

—

C₃₂

—

C₃₂

C₄×D₈₉

yes

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₂²⋊C₄

—

712

—

C₄²

—

712

—

C₁₆

—

C₁₆

C₂×C₄×D₈₉

yes

characteristic

C₁₆

Q₈×D₈₉

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

712

—

104

C₂×C₄

D₈×D₈₉

yes

characteristic

105

C₈

—

106

C₈

C₄⋊C₄×D₈₉

yes

characteristic

107

C₈

Q₁₆×D₈₉

yes

characteristic

108

C₂²

—

109

C₂²

—

178

—

110

C₂²

—

178

—

111

C₂²

D₁₆×D₈₉

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

1424

—

115

C₄

#107×D₈₉

yes

characteristic

116

C₄

Q₃₂×D₈₉

yes

characteristic

117

C₂

—

178

—

118

C₂

#366×D₈₉

yes

characteristic

119

C₂

#26022558

yes

characteristic

120

C₂

#26022562

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₈₉
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1423	C₂
socle	C₂²×C₈₉	C₂×D₁₆
prime radical	C₂²×D₈₉	D₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1423	D₈₉
hypocenter	C₈₉	#13633
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

subgroup

G/H

center

C₂²

D₁₆×D₈₉

commutator

C₁₆×C₈₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1423

C₂

socle

C₂²×C₈₉

C₂×D₁₆

prime radical

C₂²×D₈₉

D₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

N	H ≅ G/N	type
3. #26018453	C₂	semidirect
4. #26018452	C₂	semidirect
5. C₈₉×#1423	C₂	semidirect
9. C₃₂×D₈₉	C₄	semidirect
65. #1423	D₈₉	semidirect
66. C₈₉	#13633	semidirect

H ≅ G/N

type

3. #26018453

C₂

semidirect

4. #26018452

C₂

semidirect

5. C₈₉×#1423

C₂

semidirect

9. C₃₂×D₈₉

C₄

semidirect

65. #1423

D₈₉

semidirect

66. C₈₉

#13633

semidirect

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

subgroup	index
C₂×C₃₂×D₈₉	2
#26022033	2
#26018453	2
#26018452	2
C₈₉×#1423	2
#13633	89

subgroup

index

C₂×C₃₂×D₈₉

C₈₉×#1423

Subgroups — #36474928

Conjugacy classes of subgroups

Complements

Maximal subgroups