Group #36466309 — subgroups

Subgroups — #36466309

Order 22784.

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

normal subgroups only

This group has 214 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	#26022124	C₂	1	1	yes	maximal, characteristic
2	#26019072	C₂	1	2	yes	maximal
3	#26018318	C₂	1	1	yes	maximal, characteristic
4	#26018223	C₂	1	2	yes	maximal
5	C₈₉×#1313	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	C₄	1	1	yes	characteristic
7	C₈×Dic₁₇₈	—	2	2	no	—
8	C₄×Dic₃₅₆	C₂²	1	2	yes	—
9	#19151426	C₄	1	1	yes	characteristic
10	#19151403	C₄	1	2	yes	—
11	#19151349	C₂²	1	2	yes	—
12	#19151330	—	2	2	no	—
13	#346×C₈₉	C₂²	1	1	yes	characteristic
14	#346×C₈₉	C₂²	1	1	yes	characteristic
15	#322×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₇₈	—	2	2	no	—
18	Dic₇₁₂	C₈	1	4	yes	complemented
19	#15667946	C₂×C₄	1	2	yes	—
20	C₄⋊#10623494	—	2	2	no	—
21	#15667941	—	4	2	no	—
22	#15667937	—	2	2	no	—
23	C₈×Dic₈₉	—	4	2	no	—
24	C₂×C₁₆×C₈₉	—	2	1	no	—
25	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₉	D₄	1	2	yes	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₇₈	—	2	2	no	—
30	Dic₃₅₆	—	2	2	no	—
31	Dic₃₅₆	C₂×C₈	1	4	yes	—
32	C₂×#10623494	—	4	2	no	—
33	C₄⋊Dic₈₉	—	2	2	no	—
34	#13083553	—	4	2	no	—
35	C₄×Dic₈₉	—	4	2	no	—
36	C₂×C₈×C₈₉	—	2	2	no	—
37	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
39	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₈₉	—	2	1	no	—
41	C₁₆×C₈₉	—	4	1	no	—
42	C₁₆×C₈₉	C₂×C₈	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	—	2	4	no	—
48	Dic₁₇₈	—	4	2	no	—
49	C₈×C₈₉	—	2	1	no	—
50	C₈×C₈₉	—	4	2	no	—
51	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₈₉	#131	1	1	yes	characteristic
53	#10623494	—	8	2	no	—
54	C₂²×C₈₉	#442	1	1	yes	characteristic
55	C₄×C₈₉	—	2	2	no	—
56	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
57	C₄×C₈₉	#444	1	1	yes	characteristic
58	Dic₈₉	—	4	4	no	—
59	Dic₈₉	—	8	2	no	—
60	#12066	—	89	1	no	maximal, Sylow
61	C₂×C₈₉	#1580	1	1	yes	characteristic
62	C₂×C₈₉	#2180	1	2	yes	—
63	#2176	—	89	1	no	—
64	#1579	—	89	2	no	—
65	#1368	—	89	1	no	—
66	#1334	—	89	2	no	—
67	#1313	D₈₉	1	1	yes	characteristic
68	C₈₉	#12066	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	89	1	no	—
70	C₈×Q₈	—	178	2	no	—
71	C₄×Q₁₆	—	89	2	no	—
72	#367	—	89	1	no	—
73	#358	—	89	2	no	—
74	#346	C₂×D₈₉	1	1	yes	characteristic
75	#346	C₂×D₈₉	1	1	yes	characteristic
76	#334	—	89	2	no	—
77	Q₈⋊C₈	—	178	2	no	—
78	#322	C₂×D₈₉	1	1	yes	characteristic
79	C₂×Q₁₆	—	89	2	no	—
80	C₄×Q₈	—	178	2	no	—
81	Q₃₂	—	89	4	no	—
82	C₂×C₁₆	—	2	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	—	356	2	no	—
87	C₄⋊C₈	—	178	2	no	—
88	#103	—	178	2	no	—
89	C₄×C₈	—	356	2	no	—
90	C₄×C₈	D₃₅₆	1	2	yes	—
91	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
92	C₂×Q₈	—	178	2	no	—
93	Q₁₆	—	89	4	no	—
94	Q₁₆	—	178	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	356	2	no	—
97	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
98	C₂×C₈	#13083574	1	1	yes	characteristic
99	C₄⋊C₄	—	356	2	no	—
100	C₄⋊C₄	—	178	2	no	—
101	C₄²	—	356	2	no	—
102	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	4	1	no	—
105	C₁₆	C₈×D₈₉	1	2	yes	—
106	Q₈	—	356	2	no	—
107	Q₈	—	178	4	no	—
108	C₂×C₄	—	356	2	no	—
109	C₂×C₄	—	356	2	no	—
110	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
111	C₂×C₄	#15668040	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	4	2	no	—
114	C₈	—	712	2	no	—
115	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
116	C₈	#15668018	1	1	yes	characteristic
117	C₂²	#19151711	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	356	4	no	—
120	C₄	—	712	2	no	—
121	C₄	C₈×D₃₅₆	1	1	yes	characteristic
122	C₄	#19151721	1	1	yes	characteristic
123	C₂	#26019077	1	1	yes	characteristic
124	C₂	#26022145	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022124

C₂

yes

maximal, characteristic

#26019072

C₂

yes

maximal

#26018318

C₂

yes

maximal, characteristic

#26018223

C₂

yes

maximal

C₈₉×#1313

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₄

yes

characteristic

C₈×Dic₁₇₈

—

C₄×Dic₃₅₆

C₂²

yes

—

#19151426

C₄

yes

characteristic

#19151403

C₄

yes

—

#19151349

C₂²

yes

—

#19151330

—

#346×C₈₉

C₂²

yes

characteristic

#346×C₈₉

C₂²

yes

characteristic

#322×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

C₂×C₄

yes

—

C₄×Dic₁₇₈

—

Dic₇₁₂

C₈

yes

complemented

#15667946

C₂×C₄

yes

—

C₄⋊#10623494

—

#15667941

—

#15667937

—

C₈×Dic₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

C₂×C₈

yes

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

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

—

C₁₆×C₈₉

C₂×C₈

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

C₂²×C₈

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#442

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₈×D₄

yes

characteristic

C₄×C₈₉

#444

yes

characteristic

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#12066

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₈×Q₈

—

178

—

C₄×Q₁₆

—

#367

—

#358

—

#346

C₂×D₈₉

yes

characteristic

#346

C₂×D₈₉

yes

characteristic

#334

—

Q₈⋊C₈

—

178

—

#322

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

178

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₄×D₈₉

yes

characteristic

#107

—

C₄⋊C₈

—

356

—

C₄⋊C₈

—

178

—

#103

—

178

—

C₄×C₈

—

356

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

178

—

Q₁₆

—

Q₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

C₂×C₈

#13083574

yes

characteristic

C₄⋊C₄

—

356

—

100

C₄⋊C₄

—

178

—

101

C₄²

—

356

—

102

C₄²

C₂×D₃₅₆

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

C₈×D₈₉

yes

—

106

Q₈

—

356

—

107

Q₈

—

178

—

108

C₂×C₄

—

356

—

109

C₂×C₄

—

356

—

110

C₂×C₄

C₄×D₃₅₆

yes

characteristic

111

C₂×C₄

#15668040

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

712

—

115

C₈

C₂×C₈×D₈₉

yes

characteristic

116

C₈

#15668018

yes

characteristic

117

C₂²

#19151711

yes

characteristic

118

C₄

—

119

C₄

—

356

—

120

C₄

—

712

—

121

C₄

C₈×D₃₅₆

yes

characteristic

122

C₄

#19151721

yes

characteristic

123

C₂

#26019077

yes

characteristic

124

C₂

#26022145

yes

—

	subgroup	G/H
center	C₂²	#19151711
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1313	C₂
socle, prime radical	C₂²×C₈₉	#442
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1313	D₈₉
hypocenter	C₈₉	#12066
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#19151711

commutator

C₈×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1313

C₂

socle, prime radical

C₂²×C₈₉

#442

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
18. Dic₇₁₂	C₈	semidirect
68. C₈₉	#12066	semidirect

H ≅ G/N

type

18. Dic₇₁₂

C₈

semidirect

68. C₈₉

#12066

semidirect

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

subgroup	index
#26022124	2
#26019072	2
#26018318	2
#26018223	2
C₈₉×#1313	2
#12066	89

subgroup

index

C₈₉×#1313

Subgroups — #36466309

Conjugacy classes of subgroups

Complements

Maximal subgroups