Group #36448127 — subgroups

Subgroups — #36448127

Order 22784.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#26020383	C₂	1	1	yes	maximal, characteristic
2	#26018544	C₂	1	1	yes	maximal, characteristic
3	#26018112	C₂	1	4	yes	maximal
4	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	2	yes	—
6	#19151464	C₂²	1	2	yes	—
7	#19151361	—	2	2	no	—
8	#19151320	C₄	1	4	yes	complemented
9	C₂×#108×C₈₉	—	2	2	no	—
10	#423×C₈₉	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₈₉	—	2	2	no	—
13	C₄⋊#10623494	—	2	2	no	—
14	C₂²⋊#10623494	—	2	2	no	—
15	C₄×#10623494	C₂×C₄	1	4	yes	—
16	#15667919	—	2	4	no	—
17	C₂×M₁₆×C₈₉	—	2	2	no	—
18	C₂²×C₈×C₈₉	D₄	1	2	yes	—
19	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
20	#108×C₈₉	—	4	4	no	—
21	C₄⋊C₈×C₈₉	—	2	4	no	—
22	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₈₉	Q₈	1	2	yes	—
24	C₂²×Dic₈₉	—	4	2	no	—
25	C₂×#10623494	—	2	4	no	—
26	C₄⋊Dic₈₉	—	4	2	no	—
27	C₄⋊Dic₈₉	—	4	2	no	—
28	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₉	#41	1	2	yes	—
30	M₁₆×C₈₉	—	4	4	no	—
31	C₂×C₈×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	—	2	2	no	—
33	C₂×C₈×C₈₉	—	4	2	no	—
34	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
35	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₈₉	#41	1	2	yes	—
38	C₂³×C₈₉	#124	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	4	2	no	—
40	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₉	#117	1	2	yes	—
42	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
43	C₂×C₄×C₈₉	#125	1	2	yes	—
44	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×C₄×C₈₉	#137	1	2	yes	—
46	C₂×Dic₈₉	—	8	4	no	—
47	C₂×Dic₈₉	—	8	2	no	—
48	C₈×C₈₉	—	2	4	no	—
49	C₈×C₈₉	—	8	2	no	—
50	#10623494	—	4	4	no	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	#382	1	1	yes	characteristic
53	C₂²×C₈₉	#489	1	2	yes	—
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	—	2	2	no	—
56	C₄×C₈₉	#428	1	2	yes	—
57	C₄×C₈₉	#490	1	1	yes	characteristic
58	C₄×C₈₉	#491	1	1	yes	characteristic
59	Dic₈₉	—	16	2	no	—
60	#8794	—	89	1	no	maximal, Sylow
61	C₂×C₈₉	—	2	2	no	—
62	C₂×C₈₉	#1833	1	2	yes	—
63	C₂×C₈₉	#1836	1	1	yes	characteristic
64	#1848	D₈₉	1	1	yes	characteristic
65	#1835	—	89	1	no	—
66	#1452	—	89	1	no	—
67	#1286	—	89	4	no	—
68	C₈₉	#8794	1	1	yes	characteristic, complemented, Sylow
69	#431	—	89	2	no	—
70	C₂×#108	—	2	2	no	—
71	#423	C₂×D₈₉	1	2	yes	—
72	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
73	#382	—	89	2	no	—
74	#340	—	178	2	no	—
75	#322	—	89	4	no	—
76	C₂×M₁₆	—	2	2	no	—
77	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
78	C₂×C₄⋊C₄	—	178	2	no	—
79	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
80	#108	—	4	4	no	—
81	C₄⋊C₈	—	2	4	no	—
82	C₄⋊C₈	—	178	2	no	—
83	C₂²⋊C₈	—	178	2	no	—
84	C₄×C₈	—	89	4	no	—
85	C₄×C₈	Dic₁₇₈	1	2	yes	—
86	C₄×C₈	C₄×D₈₉	1	2	yes	—
87	#95	—	178	4	no	—
88	C₂²×C₄	—	356	2	no	—
89	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
90	C₂²×C₄	#13083575	1	2	yes	—
91	M₁₆	—	4	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	178	4	no	—
96	C₂×C₈	#13083553	1	4	yes	—
97	C₄⋊C₄	—	356	2	no	—
98	C₄⋊C₄	—	356	2	no	—
99	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
100	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
101	C₄²	#13083573	1	2	yes	—
102	C₂³	#15668001	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	712	4	no	—
105	C₂×C₄	—	712	2	no	—
106	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
107	C₂×C₄	#15667970	1	2	yes	—
108	C₂×C₄	#15667999	1	1	yes	characteristic
109	C₂×C₄	#15668003	1	2	yes	—
110	C₂×C₄	#15668038	1	2	yes	—
111	C₂×C₄	#15668043	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	8	2	no	—
114	C₈	—	356	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#19151463	1	1	yes	characteristic
117	C₂²	#19151960	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	1424	2	no	—
121	C₄	#19151638	1	2	yes	—
122	C₄	#19151962	1	1	yes	characteristic
123	C₄	#19151966	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#26020377	1	2	yes	—
126	C₂	#26020387	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020383

C₂

yes

maximal, characteristic

#26018544

C₂

yes

maximal, characteristic

#26018112

C₂

yes

maximal

C₈₉×#1848

C₂

yes

maximal, characteristic

#19151651

C₂²

yes

—

#19151464

C₂²

yes

—

#19151361

—

#19151320

C₄

yes

complemented

C₂×#108×C₈₉

—

#423×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

—

C₄⋊#10623494

—

C₂²⋊#10623494

—

C₄×#10623494

C₂×C₄

yes

—

#15667919

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

#125

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#382

yes

characteristic

C₂²×C₈₉

#489

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#428

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#8794

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1833

yes

—

C₂×C₈₉

#1836

yes

characteristic

#1848

D₈₉

yes

—

—

—

C₈₉

#8794

yes

characteristic, complemented, Sylow

#431

—

C₂×#108

—

#423

C₂×D₈₉

yes

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#382

—

#340

—

178

—

#322

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₈₉

yes

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₂²⋊C₈

—

178

—

C₄×C₈

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

C₄×D₈₉

yes

—

#95

—

178

—

C₂²×C₄

—

356

—

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

#13083575

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

#13083553

yes

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

C₂×C₄×D₈₉

yes

characteristic

100

C₄²

C₂×Dic₁₇₈

yes

characteristic

101

C₄²

#13083573

yes

—

102

C₂³

#15668001

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

712

—

105

C₂×C₄

—

712

—

106

C₂×C₄

C₂×#13083553

yes

characteristic

107

C₂×C₄

#15667970

yes

—

108

C₂×C₄

#15667999

yes

characteristic

109

C₂×C₄

#15668003

yes

—

110

C₂×C₄

#15668038

yes

—

111

C₂×C₄

#15668043

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

356

—

115

C₂²

—

116

C₂²

#19151463

yes

characteristic

117

C₂²

#19151960

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

1424

—

121

C₄

#19151638

yes

—

122

C₄

#19151962

yes

characteristic

123

C₄

#19151966

yes

characteristic

124

C₂

—

125

C₂

#26020377

yes

—

126

C₂

#26020387

yes

characteristic

	subgroup	G/H
center	C₂²	#19151463
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	#382
prime radical	C₂³×C₈₉	#124
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#8794
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂²

#19151463

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1848

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
8. #19151320	C₄	semidirect
68. C₈₉	#8794	semidirect

H ≅ G/N

type

8. #19151320

C₄

semidirect

68. C₈₉

#8794

semidirect

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

subgroup	index
#26020383	2
#26018544	2
#26018112	2
C₈₉×#1848	2
#8794	89

subgroup

index

C₈₉×#1848

Subgroups — #36448127

Conjugacy classes of subgroups

Complements

Maximal subgroups