Group #15774558 — subgroups

Subgroups — #15774558

Order 3328.

216 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 115 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	#13113599	C₂	1	2	yes	maximal
2	#13112902	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×M₃₂	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	#11718587	—	2	2	no	—
7	C₁₃×#346	C₂²	1	2	yes	—
8	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
10	#11718050	—	2	2	no	—
11	#11718049	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
15	C₁₃×M₃₂	—	2	4	no	—
16	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²⋊#1055	—	4	2	no	—
23	C₄⋊#1055	—	2	4	no	—
24	#88043	—	4	2	no	—
25	#12021	—	13	1	no	maximal, Sylow
26	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₁₃	—	2	2	no	—
29	C₂×C₈×C₁₃	—	2	2	no	—
30	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₄²×C₁₃	SD₁₆	1	2	yes	—
36	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×#1055	—	4	4	no	—
38	C₁₃×C₁₆	—	2	2	no	—
39	C₁₃×C₁₆	M₁₆	1	4	yes	—
40	C₄×M₃₂	D₁₃	1	1	yes	characteristic
41	#1568	—	13	2	no	—
42	#1368	—	13	4	no	—
43	C₂³×C₁₃	#106	1	1	yes	characteristic
44	C₂×C₄×C₁₃	—	2	1	no	—
45	C₂×C₄×C₁₃	—	2	1	no	—
46	C₂×C₄×C₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	#106	1	2	yes	—
48	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
51	C₈×C₁₃	—	2	1	no	—
52	C₈×C₁₃	—	2	1	no	—
53	C₈×C₁₃	#108	1	2	yes	—
54	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
55	#1055	—	8	4	no	—
56	C₂×M₃₂	Dic₁₃	1	2	yes	—
57	#432	—	26	2	no	—
58	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
59	#346	C₂×D₁₃	1	2	yes	—
60	C₄×C₁₆	Dic₁₃	1	1	yes	characteristic
61	C₄×C₁₆	Dic₁₃	1	1	yes	characteristic
62	#335	—	26	2	no	—
63	#334	—	13	4	no	—
64	C₂²×C₁₃	—	2	1	no	—
65	C₂²×C₁₃	—	2	2	no	—
66	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
67	C₄×C₁₃	—	2	1	no	—
68	C₄×C₁₃	—	2	2	no	—
69	C₄×C₁₃	—	2	2	no	—
70	C₄×C₁₃	#423	1	1	yes	characteristic
71	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
72	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
73	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
74	C₂×C₄²	D₅₂	1	1	yes	characteristic
75	M₃₂	—	2	4	no	—
76	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
77	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
78	C₄⋊C₈	—	26	4	no	—
79	C₂²⋊C₈	—	52	2	no	—
80	#97	—	52	2	no	—
81	C₄×C₈	Dic₂₆	1	1	yes	characteristic
82	C₄×C₈	D₅₂	1	1	yes	characteristic
83	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
85	C₂×C₁₃	—	4	1	no	—
86	C₂×C₁₃	#1566	1	1	yes	characteristic
87	C₂×C₁₃	#2155	1	2	yes	—
88	C₂²×C₄	#6095	1	2	yes	—
89	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	52	4	no	—
93	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
95	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
96	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
97	C₄²	#6095	1	2	yes	—
98	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
99	C₄²	C₂×D₅₂	1	1	yes	characteristic
100	C₁₆	—	2	2	no	—
101	C₁₆	#6099	1	4	yes	—
102	C₁₃	#12021	1	1	yes	characteristic, complemented, Sylow
103	C₂³	#88057	1	1	yes	characteristic
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	#88057	1	1	yes	characteristic
108	C₂×C₄	#88057	1	2	yes	—
109	C₂×C₄	C₂×#6095	1	2	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	104	4	no	—
114	C₈	#88059	1	2	yes	—
115	C₈	C₂×#6099	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	C₂×#88057	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	2	2	no	—
121	C₄	—	2	1	no	—
122	C₄	#11718585	1	1	yes	characteristic
123	C₄	C₂×#88059	1	1	yes	characteristic
124	C₂	—	4	1	no	—
125	C₂	#13113592	1	1	yes	characteristic
126	C₂	#13116609	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13113599

C₂

yes

maximal

#13112902

C₂

yes

maximal, complemented

C₄×C₁₃×M₃₂

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₄

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718587

—

C₁₃×#346

C₂²

yes

—

C₄×C₁₃×C₁₆

C₄

yes

characteristic

C₄×C₁₃×C₁₆

C₄

yes

characteristic

#11718050

—

#11718049

C₂²

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

Q₈

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

Q₈

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

C₂²⋊#1055

—

—

—

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₂×#1055

—

C₁₃×C₁₆

—

C₁₃×C₁₆

M₁₆

yes

—

C₄×M₃₂

D₁₃

yes

characteristic

#1568

—

#1368

—

C₂³×C₁₃

#106

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#106

yes

—

C₂×C₄×C₁₃

#106

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#108

yes

—

C₈×C₁₃

C₂×M₁₆

yes

—

#1055

—

C₂×M₃₂

Dic₁₃

yes

—

#432

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#346

C₂×D₁₃

yes

—

C₄×C₁₆

Dic₁₃

yes

characteristic

C₄×C₁₆

Dic₁₃

yes

characteristic

#335

—

#334

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#106

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂²×C₈

Dic₂₆

yes

characteristic

C₂²×C₈

C₂×Dic₁₃

yes

characteristic

C₂×C₄²

D₅₂

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×Dic₁₃

yes

—

C₂×C₁₆

C₂×Dic₁₃

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

Dic₂₆

yes

characteristic

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#1566

yes

characteristic

C₂×C₁₃

#2155

yes

—

C₂²×C₄

#6095

yes

—

C₂²×C₄

C₄⋊Dic₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

C₂×C₈

C₂×Dic₂₆

yes

characteristic

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

C₄²

#6095

yes

—

C₄²

C₄⋊Dic₁₃

yes

characteristic

C₄²

C₂×D₅₂

yes

characteristic

100

C₁₆

—

101

C₁₆

#6099

yes

—

102

C₁₃

#12021

yes

characteristic, complemented, Sylow

103

C₂³

#88057

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#88057

yes

characteristic

108

C₂×C₄

#88057

yes

—

109

C₂×C₄

C₂×#6095

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

104

—

114

C₈

#88059

yes

—

115

C₈

C₂×#6099

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×#88057

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#11718585

yes

characteristic

123

C₄

C₂×#88059

yes

characteristic

124

C₂

—

125

C₂

#13113592

yes

characteristic

126

C₂

#13116609

yes

—

	subgroup	G/H
center	C₂²	C₂×#88057
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂³×C₁₃	#106
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#12021
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂²

C₂×#88057

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₄×C₁₃×M₃₂

C₂

C₂²×C₁₃

C₂³×C₁₃

C₄×C₈×C₁₃

C₂³

hypercenter

C₄×M₃₂

D₁₃

hypocenter

C₁₃

#12021

LCS

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
2. #13112902	C₂	semidirect
102. C₁₃	#12021	semidirect

H ≅ G/N

type

2. #13112902

C₂

semidirect

102. C₁₃

#12021

semidirect

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

subgroup	index
#13113599	2
#13112902	2
C₄×C₁₃×M₃₂	2
#12021	13

subgroup

index

#13113599

#13112902

C₄×C₁₃×M₃₂

#12021

Subgroups — #15774558

Conjugacy classes of subgroups

Complements

Maximal subgroups