Group #36466447 — subgroups

Subgroups — #36466447

Order 22784.

198 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 69 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	#26022124	C₂	1	1	yes	maximal, characteristic
2	#26019160	C₂	1	2	yes	maximal
3	#26018221	C₂	1	2	yes	maximal
4	#26018172	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1368	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	C₄	1	1	yes	characteristic
7	#19151725	—	2	2	no	—
8	C₄×Dic₃₅₆	C₂²	1	2	yes	—
9	#19151426	C₄	1	1	yes	characteristic
10	#19151403	C₄	1	2	yes	—
11	#19151374	C₂²	1	1	yes	characteristic
12	#19151328	—	2	2	no	—
13	#19151319	C₂²	1	1	yes	characteristic
14	#346×C₈₉	C₂²	1	1	yes	characteristic
15	#334×C₈₉	C₂²	1	2	yes	—
16	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₇₈	—	2	2	no	—
18	Dic₇₁₂	—	2	2	no	—
19	C₂×#13083549	—	2	1	no	—
20	#15667946	C₂×C₄	1	2	yes	—
21	#15667941	—	4	2	no	—
22	#15667937	—	2	2	no	—
23	#15667922	—	4	2	no	—
24	C₄×#10623494	D₄	1	2	yes	—
25	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	—	2	2	no	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₇₈	—	2	2	no	—
30	Dic₃₅₆	—	2	2	no	—
31	Dic₃₅₆	—	2	2	no	—
32	C₂×#10623494	—	2	2	no	—
33	C₄⋊Dic₈₉	—	2	2	no	—
34	#13083553	—	4	2	no	—
35	C₄×Dic₈₉	—	4	2	no	—
36	#13083549	—	4	1	no	—
37	C₂×C₈×C₈₉	—	4	2	no	—
38	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
40	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₈₉	—	2	1	no	—
42	C₁₆×C₈₉	M₁₆	1	2	yes	—
43	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×Dic₈₉	—	4	2	no	—
46	C₂×Dic₈₉	—	4	2	no	—
47	Dic₁₇₈	—	4	2	no	—
48	Dic₁₇₈	—	4	2	no	—
49	C₈×C₈₉	—	2	1	no	—
50	C₈×C₈₉	—	8	2	no	—
51	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
52	C₈×C₈₉	#131	1	1	yes	characteristic
53	#10623494	—	4	2	no	—
54	C₂²×C₈₉	#442	1	1	yes	characteristic
55	C₄×C₈₉	—	2	2	no	—
56	C₄×C₈₉	#436	1	1	yes	characteristic
57	C₄×C₈₉	#444	1	1	yes	characteristic
58	Dic₈₉	—	8	2	no	—
59	Dic₈₉	—	8	2	no	—
60	#12088	—	89	1	no	maximal, Sylow
61	C₂×C₈₉	#1594	1	1	yes	characteristic
62	C₂×C₈₉	#2180	1	2	yes	—
63	#2176	—	89	1	no	—
64	#1593	—	89	2	no	—
65	#1368	D₈₉	1	1	yes	characteristic
66	#1334	—	89	2	no	—
67	#1313	—	89	1	no	—
68	C₈₉	#12088	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	89	1	no	—
70	#446	—	178	2	no	—
71	C₄×Q₁₆	—	89	2	no	—
72	#367	—	89	1	no	—
73	#358	—	89	2	no	—
74	#346	—	89	1	no	—
75	#346	C₂×D₈₉	1	1	yes	characteristic
76	#334	C₂×D₈₉	1	2	yes	—
77	Q₈⋊C₈	—	178	2	no	—
78	#322	—	89	1	no	—
79	C₂×Q₁₆	—	89	2	no	—
80	C₄×Q₈	—	178	2	no	—
81	Q₃₂	—	178	2	no	—
82	C₂×C₁₆	—	178	1	no	—
83	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
84	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
85	#107	—	89	2	no	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	356	2	no	—
88	#103	—	178	2	no	—
89	#97	—	356	2	no	—
90	C₄×C₈	—	89	2	no	—
91	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
92	C₂×Q₈	—	178	2	no	—
93	Q₁₆	—	178	2	no	—
94	Q₁₆	—	178	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	178	2	no	—
97	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
98	C₂×C₈	#13083575	1	1	yes	characteristic
99	C₄⋊C₄	—	356	2	no	—
100	C₄⋊C₄	—	178	2	no	—
101	C₄²	—	356	2	no	—
102	C₄²	D₄×D₈₉	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	356	1	no	—
105	C₁₆	#13083557	1	2	yes	—
106	Q₈	—	356	2	no	—
107	Q₈	—	356	2	no	—
108	C₂×C₄	—	356	2	no	—
109	C₂×C₄	—	356	2	no	—
110	C₂×C₄	#15667974	1	1	yes	characteristic
111	C₂×C₄	#15668039	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	8	2	no	—
114	C₈	—	356	2	no	—
115	C₈	C₂×#13083557	1	1	yes	characteristic
116	C₈	#15668020	1	1	yes	characteristic
117	C₂²	#19151709	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	712	2	no	—
120	C₄	—	712	2	no	—
121	C₄	#19151675	1	1	yes	characteristic
122	C₄	#19151719	1	1	yes	characteristic
123	C₂	#26019165	1	1	yes	characteristic
124	C₂	#26022142	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022124

C₂

yes

maximal, characteristic

#26019160

C₂

yes

maximal

#26018221

C₂

yes

maximal

#26018172

C₂

yes

maximal, characteristic

C₈₉×#1368

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₄

yes

characteristic

#19151725

—

C₄×Dic₃₅₆

C₂²

yes

—

#19151426

C₄

yes

characteristic

#19151403

C₄

yes

—

#19151374

C₂²

yes

characteristic

#19151328

—

#19151319

C₂²

yes

characteristic

#346×C₈₉

C₂²

yes

characteristic

#334×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

C₂×C₄

yes

—

C₄×Dic₁₇₈

—

Dic₇₁₂

—

C₂×#13083549

—

#15667946

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

#13083549

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

M₁₆

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×M₁₆

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#442

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#2176

—

#1593

—

#1368

D₈₉

yes

characteristic

#1334

—

#1313

—

C₈₉

#12088

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#446

—

178

—

C₄×Q₁₆

—

—

—

—

C₂×D₈₉

yes

characteristic

#334

C₂×D₈₉

yes

—

Q₈⋊C₈

—

178

—

#322

—

C₂×Q₁₆

—

C₄×Q₈

—

178

—

Q₃₂

—

178

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₄×D₈₉

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

356

—

#103

—

178

—

#97

—

356

—

C₄×C₈

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

178

—

Q₁₆

—

178

—

Q₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

C₂×C₈

#13083575

yes

characteristic

C₄⋊C₄

—

356

—

100

C₄⋊C₄

—

178

—

101

C₄²

—

356

—

102

C₄²

D₄×D₈₉

yes

characteristic

103

C₁₆

—

104

C₁₆

—

356

—

105

C₁₆

#13083557

yes

—

106

Q₈

—

356

—

107

Q₈

—

356

—

108

C₂×C₄

—

356

—

109

C₂×C₄

—

356

—

110

C₂×C₄

#15667974

yes

characteristic

111

C₂×C₄

#15668039

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

356

—

115

C₈

C₂×#13083557

yes

characteristic

116

C₈

#15668020

yes

characteristic

117

C₂²

#19151709

yes

characteristic

118

C₄

—

119

C₄

—

712

—

120

C₄

—

712

—

121

C₄

#19151675

yes

characteristic

122

C₄

#19151719

yes

characteristic

123

C₂

#26019165

yes

characteristic

124

C₂

#26022142

yes

—

	subgroup	G/H
center	C₂²	#19151709
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1368	C₂
socle, prime radical	C₂²×C₈₉	#442
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1368	D₈₉
hypocenter	C₈₉	#12088
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#19151709

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1368

C₂

socle, prime radical

C₂²×C₈₉

#442

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
68. C₈₉	#12088	semidirect

H ≅ G/N

type

68. C₈₉

#12088

semidirect

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

subgroup	index
#26022124	2
#26019160	2
#26018221	2
#26018172	2
C₈₉×#1368	2
#12088	89

subgroup

index

C₈₉×#1368

Subgroups — #36466447

Conjugacy classes of subgroups

Complements

Maximal subgroups