Group #36448181 — subgroups

Subgroups — #36448181

Order 22784.

402 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 211 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	#26020404	C₂	1	2	yes	maximal
2	C₂×#19151349	C₂	1	4	yes	maximal
3	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10623494	—	2	4	no	—
5	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
6	#19151349	—	2	8	no	—
7	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
8	C₄³×C₈₉	C₂²	1	1	yes	characteristic
9	#336×C₈₉	C₄	1	4	yes	—
10	C₂²×#10623494	—	2	4	no	—
11	C₄⋊#10623494	—	2	8	no	—
12	C₄⋊#10623494	—	4	8	no	—
13	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₈₉	D₄	1	2	yes	—
16	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
17	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₈₉	D₄	1	4	yes	—
19	C₄×C₈×C₈₉	Q₈	1	4	yes	—
20	C₂×#10623494	—	4	4	no	—
21	C₂×#10623494	—	4	8	no	—
22	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
23	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₈×C₈₉	—	2	8	no	—
27	C₂×C₈×C₈₉	C₄⋊C₄	1	8	yes	—
28	C₄²×C₈₉	—	2	4	no	—
29	C₄²×C₈₉	—	4	2	no	—
30	C₄²×C₈₉	SD₁₆	1	8	yes	—
31	C₄²×C₈₉	C₂×D₄	1	2	yes	—
32	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
33	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	—	2	4	no	—
35	C₂×C₄×C₈₉	—	2	4	no	—
36	C₂×C₄×C₈₉	—	2	2	no	—
37	C₂×C₄×C₈₉	#106	1	8	yes	—
38	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
41	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄×C₈₉	#131	1	4	yes	—
43	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
44	C₈×C₈₉	—	2	8	no	—
45	#10623494	—	8	4	no	—
46	C₂²×C₈₉	#384	1	1	yes	characteristic
47	C₂²×C₈₉	#424	1	2	yes	—
48	C₂²×C₈₉	C₂×#106	1	2	yes	—
49	C₂²×C₈₉	#430	1	2	yes	—
50	C₄×C₈₉	—	2	8	no	—
51	C₄×C₈₉	—	2	2	no	—
52	C₄×C₈₉	#424	1	4	yes	—
53	C₄×C₈₉	#492	1	2	yes	—
54	C₄×C₈₉	#498	1	2	yes	—
55	#8809	—	89	1	no	maximal, Sylow
56	C₂×C₈₉	#1570	1	4	yes	—
57	C₂×C₈₉	#1843	1	1	yes	characteristic
58	C₂×C₈₉	#1844	1	1	yes	characteristic
59	C₂×C₈₉	#1849	1	1	yes	characteristic
60	#1842	—	89	2	no	—
61	#1744	D₈₉	1	1	yes	characteristic
62	C₂×#334	—	89	4	no	—
63	C₈₉	#8809	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₈	—	89	4	no	—
65	C₂×C₄⋊C₈	—	178	4	no	—
66	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
67	C₄³	C₂×D₈₉	1	1	yes	characteristic
68	#336	Dic₈₉	1	4	yes	—
69	#334	—	178	8	no	—
70	C₂²×C₈	—	178	4	no	—
71	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
72	C₂×C₄²	Dic₁₇₈	1	2	yes	—
73	C₂×C₄²	D₃₅₆	1	2	yes	—
74	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
75	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
76	C₄⋊C₈	—	356	8	no	—
77	C₄⋊C₈	—	178	8	no	—
78	C₄×C₈	Dic₁₇₈	1	4	yes	—
79	C₄×C₈	D₃₅₆	1	4	yes	—
80	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
81	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
83	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
84	C₂×C₈	—	2	8	no	—
85	C₂×C₈	—	356	4	no	—
86	C₂×C₈	—	356	8	no	—
87	C₂×C₈	C₄⋊Dic₈₉	1	8	yes	—
88	C₄²	—	2	4	no	—
89	C₄²	—	4	2	no	—
90	C₄²	#13083563	1	8	yes	—
91	C₄²	C₂×D₃₅₆	1	2	yes	—
92	C₄²	C₂×Dic₁₇₈	1	2	yes	—
93	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	#15667944	1	8	yes	—
98	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
99	C₂×C₄	#15668009	1	1	yes	characteristic
100	C₂×C₄	#15668012	1	2	yes	—
101	C₂×C₄	#15668012	1	1	yes	characteristic
102	C₂×C₄	#15668019	1	4	yes	—
103	C₂×C₄	C₂×#13083563	1	4	yes	—
104	C₈	—	2	8	no	—
105	C₈	—	712	4	no	—
106	C₂²	#19151468	1	1	yes	characteristic
107	C₂²	#19151625	1	2	yes	—
108	C₂²	C₂×#15667944	1	2	yes	—
109	C₂²	#19151647	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	8	no	—
112	C₄	#19151625	1	4	yes	—
113	C₄	#19151971	1	2	yes	—
114	C₄	#19151996	1	2	yes	—
115	C₂	#26019023	1	4	yes	—
116	C₂	#26020406	1	1	yes	characteristic
117	C₂	#26020409	1	1	yes	characteristic
118	C₂	#26020427	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020404

C₂

yes

maximal

C₂×#19151349

C₂

yes

maximal

C₈₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

—

C₂×C₄⋊#10623494

C₂²

yes

—

#19151349

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

C₄

yes

—

C₂²×#10623494

—

C₄⋊#10623494

—

C₄⋊#10623494

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#106

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#384

yes

characteristic

C₂²×C₈₉

#424

yes

—

C₂²×C₈₉

C₂×#106

yes

—

C₂²×C₈₉

#430

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#424

yes

—

C₄×C₈₉

#492

yes

—

C₄×C₈₉

#498

yes

—

#8809

—

maximal, Sylow

C₂×C₈₉

#1570

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

C₂×#334

—

C₈₉

#8809

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

—

C₄³

C₂×D₈₉

yes

characteristic

#336

Dic₈₉

yes

—

#334

—

178

—

C₂²×C₈

—

178

—

C₂²×C₈

C₂×Dic₈₉

yes

—

C₂×C₄²

Dic₁₇₈

yes

—

C₂×C₄²

D₃₅₆

yes

—

C₂×C₄²

C₂×Dic₈₉

yes

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

356

—

C₄⋊C₈

—

178

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

D₃₅₆

yes

—

C₂²×C₄

C₄⋊Dic₈₉

yes

—

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

—

356

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₄²

—

C₄²

—

C₄²

#13083563

yes

—

C₄²

C₂×D₃₅₆

yes

—

C₄²

C₂×Dic₁₇₈

yes

—

C₂³

C₂×C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#15667944

yes

—

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

—

C₂×C₄

#15668009

yes

characteristic

100

C₂×C₄

#15668012

yes

—

101

C₂×C₄

#15668012

yes

characteristic

102

C₂×C₄

#15668019

yes

—

103

C₂×C₄

C₂×#13083563

yes

—

104

C₈

—

105

C₈

—

712

—

106

C₂²

#19151468

yes

characteristic

107

C₂²

#19151625

yes

—

108

C₂²

C₂×#15667944

yes

—

109

C₂²

#19151647

yes

—

110

C₄

—

111

C₄

—

112

C₄

#19151625

yes

—

113

C₄

#19151971

yes

—

114

C₄

#19151996

yes

—

115

C₂

#26019023

yes

—

116

C₂

#26020406

yes

characteristic

117

C₂

#26020409

yes

characteristic

118

C₂

#26020427

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1744	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1744	D₈₉
hypocenter	C₈₉	#8809
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₃₅₆

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1744

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
63. C₈₉	#8809	semidirect

H ≅ G/N

type

63. C₈₉

#8809

semidirect

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

subgroup	index
#26020404	2
C₂×#19151349	2
C₈₉×#1744	2
#8809	89

subgroup

index

#26020404

C₂×#19151349

C₈₉×#1744

#8809

Subgroups — #36448181

Conjugacy classes of subgroups

Complements

Maximal subgroups