Group #36459113 — subgroups

Subgroups — #36459113

Order 22784.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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	#26019043	C₂	1	3	yes	maximal, complemented
2	#26018921	C₂	1	3	yes	maximal
3	C₈₉×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15668009	C₂²	1	1	yes	characteristic
5	#19151651	C₂²	1	3	yes	—
6	#19151350	—	2	3	no	—
7	#19151348	—	2	3	no	—
8	#19151341	—	2	3	no	—
9	#19151341	—	2	3	no	—
10	#423×C₈₉	C₂²	1	3	yes	—
11	C₂²×D₃₅₆	—	2	3	no	—
12	#15668009	—	2	3	no	—
13	#15668009	Q₈	1	2	yes	—
14	C₄⋊#10623494	D₄	1	3	yes	—
15	C₄⋊#10623494	D₄	1	3	yes	—
16	C₂²⋊#10623494	—	2	3	no	—
17	C₄×#10623494	—	2	3	no	—
18	C₂×M₁₆×C₈₉	—	2	3	no	—
19	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₈₉	—	2	3	no	—
21	C₄⋊C₈×C₈₉	—	2	3	no	—
22	C₂³×D₈₉	—	4	1	no	—
23	C₂×D₃₅₆	—	2	6	no	—
24	C₂×D₃₅₆	—	4	3	no	—
25	C₂×D₃₅₆	—	8	3	no	—
26	C₂×D₃₅₆	—	8	3	no	—
27	C₂×#10623494	—	2	3	no	—
28	C₂×#10623494	—	2	3	no	—
29	C₂²×C₄×C₈₉	#41	1	3	yes	—
30	M₁₆×C₈₉	—	8	3	no	—
31	C₂×C₈×C₈₉	—	4	3	no	—
32	C₄²×C₈₉	C₂×D₄	1	3	yes	—
33	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
34	C₂³×C₈₉	#126	1	1	yes	characteristic
35	C₂²×D₈₉	—	4	2	no	—
36	C₂²×D₈₉	—	8	6	no	—
37	C₂×C₄×C₈₉	—	2	3	no	—
38	C₂×C₄×C₈₉	—	2	3	no	—
39	C₂×C₄×C₈₉	#121	1	3	yes	—
40	C₂×C₄×C₈₉	C₂²⋊Q₈	1	3	yes	—
41	D₃₅₆	—	8	6	no	—
42	D₃₅₆	—	8	3	no	—
43	D₃₅₆	—	8	3	no	—
44	C₈×C₈₉	—	8	3	no	—
45	#10623494	—	4	3	no	—
46	#10623494	—	4	3	no	—
47	C₂²×C₈₉	—	2	3	no	—
48	C₂²×C₈₉	#396	1	1	yes	characteristic
49	C₂×D₈₉	—	8	6	no	—
50	C₂×D₈₉	—	16	4	no	—
51	C₄×C₈₉	—	2	3	no	—
52	C₄×C₈₉	#471	1	3	yes	—
53	C₄×C₈₉	#472	1	3	yes	—
54	#10612	—	89	1	no	maximal, Sylow
55	C₂×C₈₉	—	4	1	no	—
56	C₂×C₈₉	#2063	1	3	yes	—
57	D₈₉	—	16	2	no	—
58	#1574	—	89	3	no	—
59	#1545	—	89	3	no	—
60	#1309	D₈₉	1	1	yes	characteristic, complemented
61	C₈₉	#10612	1	1	yes	characteristic, complemented, Sylow
62	C₂×#127	—	89	1	no	—
63	#431	—	89	3	no	—
64	#423	C₂×D₈₉	1	3	yes	—
65	#335	—	178	3	no	—
66	#334	—	178	3	no	—
67	#331	—	178	3	no	—
68	#331	—	178	3	no	—
69	C₂²×D₄	—	178	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	89	2	no	—
72	#127	—	178	3	no	—
73	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	—	89	3	no	—
77	C₄⋊C₈	—	89	3	no	—
78	C₂²⋊C₈	—	178	3	no	—
79	C₄×C₈	—	178	3	no	—
80	C₂⁴	—	356	1	no	—
81	C₂×D₄	—	356	3	no	—
82	C₂×D₄	—	178	6	no	—
83	C₂×D₄	—	712	3	no	—
84	C₂×D₄	—	712	3	no	—
85	C₂²×C₄	#13083574	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	178	3	no	—
89	C₂×C₈	—	178	3	no	—
90	C₄²	D₄×D₈₉	1	3	yes	—
91	C₄²	Q₈×D₈₉	1	1	yes	characteristic
92	C₂³	—	356	2	no	—
93	C₂³	—	712	6	no	—
94	C₂³	#15668006	1	1	yes	characteristic
95	D₄	—	712	6	no	—
96	D₄	—	712	3	no	—
97	D₄	—	712	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#15667985	1	3	yes	—
101	C₂×C₄	#15667990	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	356	3	no	—
104	C₈	—	356	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	1424	4	no	—
107	C₂²	—	712	6	no	—
108	C₂²	#19151522	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#19151864	1	3	yes	—
111	C₄	#19151867	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	1424	2	no	—
114	C₂	#26021613	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26019043

C₂

yes

maximal, complemented

#26018921

C₂

yes

maximal

C₈₉×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#15668009

C₂²

yes

characteristic

#19151651

C₂²

yes

—

—

—

—

—

#423×C₈₉

C₂²

yes

—

C₂²×D₃₅₆

—

#15668009

—

#15668009

Q₈

yes

—

C₄⋊#10623494

D₄

yes

—

C₄⋊#10623494

D₄

yes

—

C₂²⋊#10623494

—

C₄×#10623494

—

C₂×M₁₆×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×#10623494

—

C₂×#10623494

—

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

#126

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#396

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#471

yes

—

C₄×C₈₉

#472

yes

—

#10612

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2063

yes

—

D₈₉

—

#1574

—

#1545

—

#1309

D₈₉

yes

characteristic, complemented

C₈₉

#10612

yes

characteristic, complemented, Sylow

C₂×#127

—

#431

—

#423

C₂×D₈₉

yes

—

#335

—

178

—

#334

—

178

—

#331

—

178

—

#331

—

178

—

C₂²×D₄

—

178

—

C₂×M₁₆

—

#127

—

#127

—

178

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

178

—

C₄×C₈

—

178

—

C₂⁴

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

178

—

C₂×D₄

—

712

—

C₂×D₄

—

712

—

C₂²×C₄

#13083574

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄²

D₄×D₈₉

yes

—

C₄²

Q₈×D₈₉

yes

characteristic

C₂³

—

356

—

C₂³

—

712

—

C₂³

#15668006

yes

characteristic

D₄

—

712

—

D₄

—

712

—

D₄

—

712

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#15667985

yes

—

101

C₂×C₄

#15667990

yes

—

102

C₈

—

103

C₈

—

356

—

104

C₈

—

356

—

105

C₂²

—

106

C₂²

—

1424

—

107

C₂²

—

712

—

108

C₂²

#19151522

yes

characteristic

109

C₄

—

110

C₄

#19151864

yes

—

111

C₄

#19151867

yes

—

112

C₂

—

113

C₂

—

1424

—

114

C₂

#26021613

yes

—

	subgroup	G/H
center	C₂²	#19151522
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1309	C₂
socle	C₂²×C₈₉	#396
prime radical	C₂×#15668009	C₂²
hypercenter	#1309	D₈₉
hypocenter	C₈₉	#10612
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#19151522

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1309

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #26019043	C₂	semidirect
3. C₈₉×#1309	C₂	semidirect
60. #1309	D₈₉	semidirect
61. C₈₉	#10612	semidirect

H ≅ G/N

type

1. #26019043

C₂

semidirect

3. C₈₉×#1309

C₂

semidirect

60. #1309

D₈₉

semidirect

61. C₈₉

#10612

semidirect

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

subgroup	index
#26019043	2
#26018921	2
C₈₉×#1309	2
#10612	89

subgroup

index

#26019043

#26018921

C₈₉×#1309

#10612

Subgroups — #36459113

Conjugacy classes of subgroups

Complements

Maximal subgroups