Group #36457194 — subgroups

Subgroups — #36457194

Order 22784.

478 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 89 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	#26031646	C₂	1	3	yes	maximal
2	#26021228	C₂	1	3	yes	maximal
3	C₈₉×#1309	C₂	1	1	yes	maximal, characteristic
4	#19152155	C₂²	1	3	yes	—
5	#19151884	—	2	6	no	—
6	#19151856	—	2	6	no	—
7	#19151809	—	2	6	no	—
8	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
9	#19151359	—	2	6	no	—
10	#423×C₈₉	C₂²	1	3	yes	—
11	C₂×Dic₃₅₆	—	2	6	no	—
12	#15668012	D₄	1	6	yes	—
13	#15668004	—	2	3	no	—
14	#15667991	—	2	6	no	—
15	C₄×Dic₁₇₈	—	2	6	no	—
16	#15667971	—	2	3	no	—
17	C₂×C₄×Dic₈₉	—	2	3	no	—
18	#15667944	—	2	6	no	—
19	#15667937	—	2	12	no	—
20	C₂×M₁₆×C₈₉	D₄	1	6	yes	—
21	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	—	2	6	no	—
23	C₂×Dic₁₇₈	—	2	6	no	—
24	C₂²×Dic₈₉	—	4	1	no	—
25	Dic₃₅₆	—	8	6	no	—
26	C₄⋊Dic₈₉	—	2	6	no	—
27	C₄⋊Dic₈₉	—	2	6	no	—
28	#13083553	—	4	6	no	—
29	C₂²⋊Dic₈₉	—	4	3	no	—
30	C₄×Dic₈₉	—	2	6	no	—
31	C₄×Dic₈₉	—	4	3	no	—
32	C₄×Dic₈₉	—	4	3	no	—
33	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
34	M₁₆×C₈₉	—	4	6	no	—
35	C₂×C₈×C₈₉	—	2	6	no	—
36	C₄²×C₈₉	C₂×D₄	1	3	yes	—
37	C₄²×C₈₉	#41	1	1	yes	characteristic
38	C₂³×C₈₉	#127	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	2	3	no	—
40	C₂×C₄×C₈₉	—	2	3	no	—
41	C₂×C₄×C₈₉	#120	1	3	yes	—
42	C₂×C₄×C₈₉	#121	1	3	yes	—
43	C₂×Dic₈₉	—	4	6	no	—
44	C₂×Dic₈₉	—	4	6	no	—
45	Dic₁₇₈	—	4	6	no	—
46	Dic₁₇₈	—	4	6	no	—
47	C₈×C₈₉	—	4	6	no	—
48	C₂²×C₈₉	—	2	3	no	—
49	C₂²×C₈₉	#392	1	1	yes	characteristic
50	C₄×C₈₉	—	2	3	no	—
51	C₄×C₈₉	#455	1	3	yes	—
52	C₄×C₈₉	#456	1	3	yes	—
53	Dic₈₉	—	4	4	no	—
54	Dic₈₉	—	8	6	no	—
55	#10303	—	89	1	no	maximal, Sylow
56	C₂×C₈₉	—	4	1	no	—
57	C₂×C₈₉	#2007	1	3	yes	—
58	#3255	—	89	3	no	—
59	#1994	—	89	3	no	—
60	#1309	D₈₉	1	1	yes	characteristic
61	C₈₉	#10303	1	1	yes	characteristic, complemented, Sylow
62	#533	—	89	3	no	—
63	#475	—	178	6	no	—
64	#470	—	178	6	no	—
65	#462	—	178	6	no	—
66	#423	C₂×D₈₉	1	3	yes	—
67	C₄³	—	89	1	no	—
68	#339	—	178	6	no	—
69	C₂×Q₁₆	—	178	6	no	—
70	C₂×M₁₆	D₃₅₆	1	6	yes	—
71	C₄⋊Q₈	—	89	6	no	—
72	#125	—	178	3	no	—
73	C₂²⋊Q₈	—	178	6	no	—
74	C₄×Q₈	—	178	6	no	—
75	#117	—	178	3	no	—
76	C₂×C₄²	—	178	3	no	—
77	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
78	#106	—	178	6	no	—
79	C₄⋊C₈	—	2	6	no	—
80	#103	—	178	12	no	—
81	C₂×Q₈	—	178	6	no	—
82	C₂²×C₄	—	356	1	no	—
83	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
84	Q₁₆	—	712	6	no	—
85	M₁₆	—	4	6	no	—
86	C₂×C₈	—	2	6	no	—
87	C₄⋊C₄	—	356	6	no	—
88	C₄⋊C₄	—	178	6	no	—
89	C₄⋊C₄	—	178	6	no	—
90	C₂²⋊C₄	—	356	3	no	—
91	C₄²	—	356	3	no	—
92	C₄²	—	356	3	no	—
93	C₄²	—	178	6	no	—
94	C₄²	D₄×D₈₉	1	3	yes	—
95	C₄²	#13083575	1	1	yes	characteristic
96	C₂³	#15668009	1	1	yes	characteristic
97	Q₈	—	356	6	no	—
98	Q₈	—	356	6	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	356	6	no	—
102	C₂×C₄	—	356	6	no	—
103	C₂×C₄	#15667982	1	3	yes	—
104	C₂×C₄	#15667987	1	3	yes	—
105	C₈	—	4	6	no	—
106	C₂²	—	2	3	no	—
107	C₂²	#19151508	1	1	yes	characteristic
108	C₄	—	2	3	no	—
109	C₄	—	356	4	no	—
110	C₄	—	712	6	no	—
111	C₄	#19151778	1	3	yes	—
112	C₄	#19151783	1	3	yes	—
113	C₂	—	4	1	no	—
114	C₂	#26021301	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26031646

C₂

yes

maximal

#26021228

C₂

yes

maximal

C₈₉×#1309

C₂

yes

maximal, characteristic

#19152155

C₂²

yes

—

#19151884

—

#19151856

—

#19151809

—

C₄²×Dic₈₉

C₂²

yes

characteristic

#19151359

—

#423×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

—

#15668012

D₄

yes

—

#15668004

—

#15667991

—

C₄×Dic₁₇₈

—

#15667971

—

C₂×C₄×Dic₈₉

—

#15667944

—

#15667937

—

C₂×M₁₆×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

yes

C₂³×C₈₉

yes

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#392

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#455

yes

—

C₄×C₈₉

#456

yes

—

Dic₈₉

—

Dic₈₉

—

#10303

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2007

yes

—

#3255

—

#1994

—

#1309

D₈₉

yes

characteristic

C₈₉

#10303

yes

characteristic, complemented, Sylow

#533

—

#475

—

178

—

#470

—

178

—

#462

—

178

—

#423

C₂×D₈₉

yes

—

C₄³

—

#339

—

178

—

C₂×Q₁₆

—

178

—

C₂×M₁₆

D₃₅₆

yes

—

C₄⋊Q₈

—

#125

—

178

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

178

—

#117

—

178

—

C₂×C₄²

—

178

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#106

—

178

—

C₄⋊C₈

—

#103

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

356

—

C₂²×C₄

C₂×D₃₅₆

yes

—

Q₁₆

—

712

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

356

—

C₄²

—

356

—

C₄²

—

356

—

C₄²

—

178

—

C₄²

D₄×D₈₉

yes

—

C₄²

yes

C₂³

yes

Q₈

—

356

—

Q₈

—

356

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

356

—

102

C₂×C₄

—

356

—

103

C₂×C₄

#15667982

yes

—

104

C₂×C₄

#15667987

yes

—

105

C₈

—

106

C₂²

—

107

C₂²

#19151508

yes

characteristic

108

C₄

—

109

C₄

—

356

—

110

C₄

—

712

—

111

C₄

#19151778

yes

—

112

C₄

#19151783

yes

—

113

C₂

—

114

C₂

#26021301

yes

—

	subgroup	G/H
center	C₂²	#19151508
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1309	C₂
socle	C₂²×C₈₉	#392
prime radical	C₂³×C₈₉	#127
hypercenter	#1309	D₈₉
hypocenter	C₈₉	#10303
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#19151508

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1309

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
61. C₈₉	#10303	semidirect

H ≅ G/N

type

61. C₈₉

#10303

semidirect

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

subgroup	index
#26031646	2
#26021228	2
C₈₉×#1309	2
#10303	89

subgroup

index

#26031646

#26021228

C₈₉×#1309

#10303

Subgroups — #36457194

Conjugacy classes of subgroups

Complements

Maximal subgroups