Group #36466448 — subgroups

Subgroups — #36466448

Order 22784.

198 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 77 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	#26019161	C₂	1	2	yes	maximal
2	#26018318	C₂	1	1	yes	maximal, characteristic
3	#26018222	C₂	1	2	yes	maximal
4	#26018172	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2176	C₂	1	1	yes	maximal, characteristic
6	#19151729	—	2	2	no	—
7	#19151374	C₂²	1	1	yes	characteristic
8	#19151349	C₂²	1	2	yes	—
9	Q₈⋊#10623494	—	2	2	no	—
10	#19151319	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₈₉	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
13	#367×C₈₉	C₄	1	1	yes	characteristic
14	#358×C₈₉	C₄	1	2	yes	—
15	#346×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#13083549	—	2	1	no	—
17	C₄⋊#10623494	—	2	2	no	—
18	C₄⋊#10623494	—	4	2	no	—
19	#15667923	—	4	2	no	—
20	C₄×#10623494	D₄	1	2	yes	—
21	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
22	C₄×Q₈×C₈₉	—	2	2	no	—
23	Q₃₂×C₈₉	—	2	2	no	—
24	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
26	#107×C₈₉	C₂×C₄	1	2	yes	—
27	#103×C₈₉	—	2	2	no	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×#10623494	—	2	2	no	—
30	C₂×#10623494	—	4	2	no	—
31	#13083549	—	4	1	no	—
32	C₂×Q₈×C₈₉	—	2	2	no	—
33	Q₁₆×C₈₉	—	2	2	no	—
34	Q₁₆×C₈₉	—	2	2	no	—
35	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₈₉	—	2	2	no	—
38	C₄⋊C₄×C₈₉	—	4	2	no	—
39	C₄²×C₈₉	—	4	2	no	—
40	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₈₉	—	2	1	no	—
42	C₁₆×C₈₉	M₁₆	1	2	yes	—
43	Q₈×C₈₉	—	4	2	no	—
44	Q₈×C₈₉	—	4	2	no	—
45	C₂×C₄×C₈₉	—	4	2	no	—
46	C₂×C₄×C₈₉	—	4	2	no	—
47	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
49	C₈×C₈₉	—	2	1	no	—
50	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
51	C₈×C₈₉	#131	1	1	yes	characteristic
52	#10623494	—	4	2	no	—
53	#10623494	—	8	2	no	—
54	C₂²×C₈₉	#442	1	1	yes	characteristic
55	C₄×C₈₉	—	2	2	no	—
56	C₄×C₈₉	—	8	2	no	—
57	C₄×C₈₉	—	8	2	no	—
58	C₄×C₈₉	#436	1	1	yes	characteristic
59	C₄×C₈₉	#444	1	1	yes	characteristic
60	#12088	—	89	1	no	maximal, Sylow
61	C₂×C₈₉	#1594	1	1	yes	characteristic
62	C₂×C₈₉	#2180	1	2	yes	—
63	#2176	D₈₉	1	1	yes	characteristic
64	#1593	—	89	2	no	—
65	#1368	—	89	1	no	—
66	#1334	—	89	2	no	—
67	#1313	—	89	1	no	—
68	C₈₉	#12088	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	Dic₈₉	1	1	yes	characteristic
70	#446	—	178	2	no	—
71	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
72	#367	Dic₈₉	1	1	yes	characteristic
73	#358	Dic₈₉	1	2	yes	—
74	#346	—	89	1	no	—
75	#346	C₂×D₈₉	1	1	yes	characteristic
76	#334	—	89	2	no	—
77	Q₈⋊C₈	—	178	2	no	—
78	#322	—	89	1	no	—
79	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
80	C₄×Q₈	—	2	2	no	—
81	Q₃₂	—	2	2	no	—
82	C₂×C₁₆	—	178	1	no	—
83	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
84	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
85	#107	C₂×Dic₈₉	1	2	yes	—
86	C₄⋊C₈	—	356	2	no	—
87	C₄⋊C₈	—	178	2	no	—
88	#103	—	2	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	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	2	2	no	—
95	C₂×C₈	—	356	2	no	—
96	C₂×C₈	—	178	2	no	—
97	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
98	C₂×C₈	#13083573	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	D₄×D₈₉	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	356	1	no	—
105	C₁₆	#13083558	1	2	yes	—
106	Q₈	—	4	2	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
111	C₂×C₄	#15668036	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	356	2	no	—
114	C₈	—	712	2	no	—
115	C₈	C₂×#13083558	1	1	yes	characteristic
116	C₈	#15668019	1	1	yes	characteristic
117	C₂²	#19151710	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	8	2	no	—
120	C₄	—	8	2	no	—
121	C₄	#19151676	1	1	yes	characteristic
122	C₄	#19151720	1	1	yes	characteristic
123	C₂	#26019166	1	1	yes	characteristic
124	C₂	#26022144	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26019161

C₂

yes

maximal

#26018318

C₂

yes

maximal, characteristic

#26018222

C₂

yes

maximal

#26018172

C₂

yes

maximal, characteristic

C₈₉×#2176

C₂

yes

maximal, characteristic

#19151729

—

#19151374

C₂²

yes

characteristic

#19151349

C₂²

yes

—

Q₈⋊#10623494

—

#19151319

C₂²

yes

characteristic

C₂×Q₃₂×C₈₉

C₄

yes

characteristic

C₄×Q₁₆×C₈₉

C₂²

yes

—

#367×C₈₉

C₄

yes

characteristic

#358×C₈₉

C₄

yes

—

#346×C₈₉

C₂²

yes

—

—

—

—

D₄

yes

—

C₂×Q₁₆×C₈₉

C₂×C₄

yes

—

C₄×Q₈×C₈₉

—

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

#107×C₈₉

C₂×C₄

yes

—

#103×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

—

—

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

M₁₆

yes

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₈×C₈₉

—

C₈×C₈₉

C₂×M₁₆

yes

C₈×C₈₉

yes

—

—

C₂²×C₈₉

#442

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#2176

D₈₉

yes

—

—

—

—

C₈₉

#12088

yes

characteristic, complemented, Sylow

C₂×Q₃₂

Dic₈₉

yes

characteristic

#446

—

178

—

C₄×Q₁₆

C₂×D₈₉

yes

—

#367

Dic₈₉

yes

characteristic

#358

Dic₈₉

yes

—

#346

—

#346

C₂×D₈₉

yes

characteristic

#334

—

Q₈⋊C₈

—

178

—

#322

—

C₂×Q₁₆

C₂×Dic₈₉

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₂×Dic₈₉

yes

characteristic

C₂×C₁₆

C₂×Dic₈₉

yes

characteristic

#107

C₂×Dic₈₉

yes

—

C₄⋊C₈

—

356

—

C₄⋊C₈

—

178

—

#103

—

#97

—

356

—

C₄×C₈

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

356

—

C₂×C₈

—

178

—

C₂×C₈

C₂²×Dic₈₉

yes

characteristic

C₂×C₈

#13083573

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

D₄×D₈₉

yes

characteristic

103

C₁₆

—

104

C₁₆

—

356

—

105

C₁₆

#13083558

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×Dic₈₉

yes

characteristic

111

C₂×C₄

#15668036

yes

—

112

C₈

—

113

C₈

—

356

—

114

C₈

—

712

—

115

C₈

C₂×#13083558

yes

characteristic

116

C₈

#15668019

yes

characteristic

117

C₂²

#19151710

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#19151676

yes

characteristic

122

C₄

#19151720

yes

characteristic

123

C₂

#26019166

yes

characteristic

124

C₂

#26022144

yes

—

	subgroup	G/H
center	C₂²	#19151710
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2176	C₂
socle, prime radical	C₂²×C₈₉	#442
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#2176	D₈₉
hypocenter	C₈₉	#12088
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#19151710

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2176

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 ◁ #2176

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
#26019161	2
#26018318	2
#26018222	2
#26018172	2
C₈₉×#2176	2
#12088	89

subgroup

index

C₈₉×#2176

Subgroups — #36466448

Conjugacy classes of subgroups

Complements

Maximal subgroups