Group #36469124 — subgroups

Subgroups — #36469124

Order 22784.

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

normal subgroups only

This group has 260 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	#26022481	C₂	1	2	yes	maximal, complemented
2	#26022477	C₂	1	2	yes	maximal
3	#26019858	C₂	1	1	yes	maximal, characteristic
4	C₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151978	C₂²	1	1	yes	characteristic
7	#19151975	C₂²	1	1	yes	characteristic
8	#19151945	—	2	2	no	—
9	#19151401	—	2	4	no	—
10	#19151398	—	2	4	no	—
11	C₄×#13083549	C₂²	1	2	yes	—
12	#500×C₈₉	C₂²	1	2	yes	—
13	#487×C₈₉	—	2	2	no	—
14	C₈²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₅₆	—	2	2	no	—
16	C₂×D₇₁₂	—	2	2	no	—
17	#15668012	—	2	1	no	—
18	#15668009	—	2	1	no	—
19	C₂×#13083549	D₄	1	4	yes	—
20	#15667934	—	4	4	no	—
21	C₄⋊Q₈×C₈₉	—	2	2	no	—
22	#107×C₈₉	—	2	2	no	—
23	#107×C₈₉	—	2	2	no	—
24	#103×C₈₉	—	4	4	no	—
25	C₄×C₈×C₈₉	D₄	1	2	yes	—
26	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₇₈	—	4	1	no	—
28	C₂×D₃₅₆	—	4	1	no	—
29	C₂×D₃₅₆	—	4	2	no	—
30	Dic₃₅₆	—	4	4	no	—
31	D₇₁₂	—	4	4	no	—
32	C₄⋊Dic₈₉	—	4	2	no	—
33	#13083549	—	2	4	no	—
34	C₂×Q₈×C₈₉	—	4	2	no	—
35	C₂×C₈×C₈₉	D₈	1	4	yes	—
36	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
37	C₄⋊C₄×C₈₉	—	4	2	no	—
38	C₄⋊C₄×C₈₉	—	4	2	no	—
39	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₈₉	—	8	1	no	—
41	Q₈×C₈₉	—	8	4	no	—
42	C₂×C₄×C₈₉	—	8	2	no	—
43	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
45	C₂×Dic₈₉	—	8	1	no	—
46	D₃₅₆	—	8	2	no	—
47	D₃₅₆	—	8	4	no	—
48	Dic₁₇₈	—	8	2	no	—
49	C₈×C₈₉	—	2	4	no	—
50	C₈×C₈₉	#135	1	4	yes	—
51	C₂²×C₈₉	#493	1	1	yes	characteristic
52	C₂×D₈₉	—	16	1	no	—
53	C₂×D₈₉	—	16	2	no	—
54	C₄×C₈₉	—	16	2	no	—
55	C₄×C₈₉	#495	1	2	yes	—
56	C₄×C₈₉	#508	1	4	yes	—
57	Dic₈₉	—	16	1	no	—
58	#12511	—	89	1	no	maximal, Sylow
59	C₂×C₈₉	#1709	1	1	yes	characteristic
60	C₂×C₈₉	#2249	1	2	yes	—
61	D₈₉	—	32	1	no	—
62	#2241	—	89	2	no	—
63	#2240	—	89	2	no	—
64	#1710	D₈₉	1	1	yes	characteristic, complemented
65	#1709	—	89	1	no	—
66	C₈×C₁₆	—	89	1	no	—
67	C₈₉	#12511	1	1	yes	characteristic, complemented, Sylow
68	#500	C₂×D₈₉	1	2	yes	—
69	#494	—	89	1	no	—
70	#493	—	89	1	no	—
71	#487	—	2	2	no	—
72	#486	—	178	2	no	—
73	#358	—	178	4	no	—
74	#357	—	178	4	no	—
75	C₄×C₁₆	—	89	2	no	—
76	C₈²	C₂×D₈₉	1	1	yes	characteristic
77	C₂×Q₁₆	—	178	2	no	—
78	C₂×D₈	—	178	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	178	1	no	—
81	#127	—	178	1	no	—
82	C₂×C₁₆	—	89	4	no	—
83	#107	—	2	2	no	—
84	#107	—	2	2	no	—
85	#103	—	4	4	no	—
86	#102	—	356	4	no	—
87	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
88	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	356	1	no	—
91	C₂×D₄	—	356	1	no	—
92	C₂×D₄	—	356	2	no	—
93	Q₁₆	—	356	4	no	—
94	D₈	—	356	4	no	—
95	C₂×C₈	#13083559	1	4	yes	—
96	C₂×C₈	D₄×D₈₉	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	356	2	no	—
100	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
101	C₁₆	—	178	4	no	—
102	C₂³	—	712	1	no	—
103	Q₈	—	8	4	no	—
104	Q₈	—	712	2	no	—
105	D₄	—	712	2	no	—
106	D₄	—	712	4	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	712	1	no	—
109	C₂×C₄	#15668008	1	1	yes	characteristic
110	C₂×C₄	C₂×#13083559	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#15668035	1	4	yes	—
113	C₂²	—	1424	1	no	—
114	C₂²	—	1424	2	no	—
115	C₂²	#19151974	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	1424	1	no	—
118	C₄	#19151983	1	2	yes	—
119	C₄	#19152028	1	4	yes	—
120	C₂	—	2848	1	no	—
121	C₂	#26019859	1	1	yes	characteristic
122	C₂	#26022511	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022481

C₂

yes

maximal, complemented

#26022477

C₂

yes

maximal

#26019858

C₂

yes

maximal, characteristic

C₈×#13083549

C₂

yes

maximal, characteristic, complemented

C₈₉×#1710

C₂

yes

maximal, characteristic, complemented

#19151978

C₂²

yes

characteristic

#19151975

C₂²

yes

—

—

—

C₂²

yes

—

#500×C₈₉

C₂²

yes

—

#487×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×D₇₁₂

—

#15668012

—

#15668009

—

C₂×#13083549

D₄

yes

—

#15667934

—

C₄⋊Q₈×C₈₉

—

#107×C₈₉

—

#107×C₈₉

—

#103×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

Dic₃₅₆

—

D₇₁₂

—

C₄⋊Dic₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#495

yes

—

C₄×C₈₉

#508

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

#2241

—

#2240

—

#1710

D₈₉

yes

characteristic, complemented

#1709

—

C₈×C₁₆

—

C₈₉

#12511

yes

characteristic, complemented, Sylow

#500

C₂×D₈₉

yes

—

—

—

—

—

178

—

#358

—

178

—

#357

—

178

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

#127

—

178

—

C₂×C₁₆

—

—

—

—

—

356

—

C₄×C₈

C₂²⋊D₈₉

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

Q₁₆

—

356

—

D₈

—

356

—

C₂×C₈

#13083559

yes

—

C₂×C₈

D₄×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

100

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

101

C₁₆

—

178

—

102

C₂³

—

712

—

103

Q₈

—

104

Q₈

—

712

—

105

D₄

—

712

—

106

D₄

—

712

—

107

C₂×C₄

—

108

C₂×C₄

—

712

—

109

C₂×C₄

#15668008

yes

characteristic

110

C₂×C₄

C₂×#13083559

yes

—

111

C₈

—

112

C₈

#15668035

yes

—

113

C₂²

—

1424

—

114

C₂²

—

1424

—

115

C₂²

#19151974

yes

characteristic

116

C₄

—

117

C₄

—

1424

—

118

C₄

#19151983

yes

—

119

C₄

#19152028

yes

—

120

C₂

—

2848

—

121

C₂

#26019859

yes

characteristic

122

C₂

#26022511

yes

—

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1710	C₂
socle	C₂²×C₈₉	#493
prime radical	#19151975	C₂²
hypercenter	#1710	D₈₉
hypocenter	C₈₉	#12511
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1710

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #26022481	C₂	semidirect
4. C₈×#13083549	C₂	semidirect
5. C₈₉×#1710	C₂	semidirect
64. #1710	D₈₉	semidirect
67. C₈₉	#12511	semidirect

H ≅ G/N

type

1. #26022481

C₂

semidirect

4. C₈×#13083549

C₂

semidirect

5. C₈₉×#1710

C₂

semidirect

64. #1710

D₈₉

semidirect

67. C₈₉

#12511

semidirect

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

subgroup	index
#26022481	2
#26022477	2
#26019858	2
C₈×#13083549	2
C₈₉×#1710	2
#12511	89

subgroup

index

C₈₉×#1710

Subgroups — #36469124

Conjugacy classes of subgroups

Complements

Maximal subgroups