Group #15774474 — subgroups

Subgroups — #15774474

Order 3328.

234 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 147 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	#13113584	C₂	1	2	yes	maximal
2	#13112902	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	#11718586	—	2	2	no	—
8	C₁₃×#346	C₄	1	2	yes	—
9	C₁₃×#346	C₂²	1	2	yes	—
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	C₂×C₁₃×C₁₆	—	2	2	no	—
16	C₂×C₁₃×C₁₆	C₈	1	4	yes	complemented
17	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²⋊#1055	—	4	2	no	—
24	C₄⋊#1055	—	2	4	no	—
25	C₄×#1055	—	4	2	no	—
26	#11999	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
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	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
34	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₁₃	SD₁₆	1	2	yes	—
38	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×#1055	—	4	4	no	—
40	C₁₃×C₁₆	—	2	2	no	—
41	C₁₃×C₁₆	C₂×C₈	1	4	yes	—
42	C₂×#346	D₁₃	1	1	yes	characteristic
43	#1564	—	13	2	no	—
44	#1368	—	13	4	no	—
45	C₂³×C₁₃	#106	1	1	yes	characteristic
46	C₂×C₄×C₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
48	C₂×C₄×C₁₃	#106	1	2	yes	—
49	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
50	C₂×C₄×C₁₃	#108	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
53	C₈×C₁₃	—	2	1	no	—
54	C₈×C₁₃	—	2	1	no	—
55	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
56	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
58	#1055	—	8	4	no	—
59	C₂²×C₁₆	Dic₁₃	1	1	yes	characteristic
60	C₂²×C₁₆	Dic₁₃	1	1	yes	characteristic
61	#431	—	26	2	no	—
62	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
63	#346	Dic₁₃	1	2	yes	—
64	#346	C₂×D₁₃	1	2	yes	—
65	#335	—	26	2	no	—
66	#334	—	13	4	no	—
67	C₂²×C₁₃	—	2	2	no	—
68	C₂²×C₁₃	#334	1	2	yes	—
69	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
70	C₄×C₁₃	—	2	2	no	—
71	C₄×C₁₃	—	2	2	no	—
72	C₄×C₁₃	#334	1	2	yes	—
73	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
75	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
76	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
77	C₂×C₄²	D₅₂	1	1	yes	characteristic
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	#1055	1	4	yes	complemented
80	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
81	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
82	C₄⋊C₈	—	26	4	no	—
83	C₂²⋊C₈	—	52	2	no	—
84	C₄×C₈	—	52	2	no	—
85	C₄×C₈	Dic₂₆	1	1	yes	characteristic
86	C₄×C₈	D₅₂	1	1	yes	characteristic
87	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
88	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
89	C₂×C₁₃	—	2	2	no	—
90	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
91	C₂×C₁₃	#2155	1	2	yes	—
92	C₂²×C₄	#6095	1	2	yes	—
93	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	52	4	no	—
96	C₂×C₈	C₂×#1055	1	2	yes	—
97	C₂×C₈	#6099	1	2	yes	—
98	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
99	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
100	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
101	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
102	C₄²	#6095	1	2	yes	—
103	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
104	C₄²	C₂×D₅₂	1	1	yes	characteristic
105	C₁₆	—	2	2	no	—
106	C₁₆	C₂×#1055	1	4	yes	—
107	C₁₃	#11999	1	1	yes	characteristic, complemented, Sylow
108	C₂³	#88057	1	1	yes	characteristic
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₄⋊#1055	1	2	yes	—
111	C₂×C₄	#88057	1	2	yes	—
112	C₂×C₄	#88057	1	1	yes	characteristic
113	C₂×C₄	#88059	1	2	yes	—
114	C₂×C₄	C₂×#6095	1	2	yes	—
115	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	104	4	no	—
119	C₈	C₄⋊#1055	1	2	yes	—
120	C₈	C₂²×#1055	1	1	yes	characteristic
121	C₈	C₂×#6099	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	#11718049	1	2	yes	—
124	C₂²	C₂×#88057	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	#11718049	1	2	yes	—
128	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
129	C₄	C₂×#88059	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×#11718049	1	1	yes	characteristic
132	C₂	#13116609	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13113584

C₂

yes

maximal

#13112902

C₂

yes

maximal, complemented

C₂×C₁₃×#346

C₂

yes

maximal, characteristic

C₂²×C₁₃×C₁₆

C₄

yes

characteristic

C₂²×C₁₃×C₁₆

C₄

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718586

—

C₁₃×#346

C₄

yes

—

C₁₃×#346

C₂²

yes

—

#11718050

—

#11718049

C₂²

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

Q₈

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

C₈

yes

complemented

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

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

—

C₂×C₈×C₁₃

M₁₆

yes

—

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

C₂×C₈

yes

—

C₂×#346

D₁₃

yes

characteristic

#1564

—

#1368

—

C₂³×C₁₃

#106

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

#106

yes

—

C₂×C₄×C₁₃

#106

yes

characteristic

C₂×C₄×C₁₃

#108

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₄⋊C₈

yes

—

C₈×C₁₃

C₂²×C₈

yes

characteristic

C₈×C₁₃

C₂×M₁₆

yes

characteristic

#1055

—

C₂²×C₁₆

Dic₁₃

yes

characteristic

C₂²×C₁₆

Dic₁₃

yes

characteristic

#431

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#346

Dic₁₃

yes

—

#346

C₂×D₁₃

yes

—

#335

—

#334

—

C₂²×C₁₃

—

C₂²×C₁₃

#334

yes

—

C₂²×C₁₃

C₂×#106

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#334

yes

—

C₄×C₁₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₃

C₂×#108

yes

characteristic

C₂²×C₈

Dic₂₆

yes

characteristic

C₂²×C₈

C₂×Dic₁₃

yes

characteristic

C₂×C₄²

D₅₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

#1055

yes

complemented

C₂×C₁₆

C₂×Dic₁₃

yes

—

C₂×C₁₆

C₂×Dic₁₃

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

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

C₂×#334

yes

characteristic

C₂×C₁₃

#2155

yes

—

C₂²×C₄

#6095

yes

—

C₂²×C₄

C₄⋊Dic₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#1055

yes

—

C₂×C₈

#6099

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

100

C₂×C₈

C₂×Dic₂₆

yes

characteristic

101

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

102

C₄²

#6095

yes

—

103

C₄²

C₄⋊Dic₁₃

yes

characteristic

104

C₄²

C₂×D₅₂

yes

characteristic

105

C₁₆

—

106

C₁₆

C₂×#1055

yes

—

107

C₁₃

#11999

yes

characteristic, complemented, Sylow

108

C₂³

#88057

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

C₄⋊#1055

yes

—

111

C₂×C₄

#88057

yes

—

112

C₂×C₄

#88057

yes

characteristic

113

C₂×C₄

#88059

yes

—

114

C₂×C₄

C₂×#6095

yes

—

115

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

104

—

119

C₈

C₄⋊#1055

yes

—

120

C₈

C₂²×#1055

yes

characteristic

121

C₈

C₂×#6099

yes

characteristic

122

C₂²

—

123

C₂²

#11718049

yes

—

124

C₂²

C₂×#88057

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

#11718049

yes

—

128

C₄

C₂×C₄⋊#1055

yes

characteristic

129

C₄

C₂×#88059

yes

characteristic

130

C₂

—

131

C₂

C₂×#11718049

yes

characteristic

132

C₂

#13116609

yes

—

	subgroup	G/H
center	C₂×C₄	#88057
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂³×C₁₃	#106
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#11999
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

#88057

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₂×C₁₃×#346

C₂

C₂²×C₁₃

C₂³×C₁₃

C₄×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #13112902	C₂	semidirect
16. C₂×C₁₃×C₁₆	C₈	semidirect
79. C₂×C₁₆	#1055	semidirect
107. C₁₃	#11999	semidirect

H ≅ G/N

type

2. #13112902

C₂

semidirect

16. C₂×C₁₃×C₁₆

C₈

semidirect

79. C₂×C₁₆

#1055

semidirect

107. C₁₃

#11999

semidirect

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

subgroup	index
#13113584	2
#13112902	2
C₂×C₁₃×#346	2
#11999	13

subgroup

index

#13113584

#13112902

C₂×C₁₃×#346

#11999

Subgroups — #15774474

Conjugacy classes of subgroups

Complements

Maximal subgroups