Group #16786693 — subgroups

Subgroups — #16786693

Order 3328.

228 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 100 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	#13131197	C₂	1	2	yes	maximal
2	#13129754	C₂	1	4	yes	maximal, complemented
3	#13113120	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#431	C₄	1	1	yes	characteristic
5	#11718586	C₄	1	1	yes	characteristic
6	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
7	C₂²⋊#6092	C₄	1	4	yes	—
8	#11718231	C₂²	1	4	yes	—
9	C₄⋊#6092	C₄	1	4	yes	—
10	#11718046	C₂²	1	2	yes	—
11	C₈×#1055	C₄	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×#1055	C₂×C₄	1	2	yes	—
14	C₂×#6092	C₂×C₄	1	8	yes	—
15	C₁₃×C₄⋊C₈	—	2	1	no	—
16	C₁₃×C₂²⋊C₈	—	2	1	no	—
17	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂²⋊#1055	—	2	1	no	—
19	C₄⋊#1055	—	2	1	no	—
20	C₄×#1055	C₂×C₄	1	2	yes	—
21	C₄×#1055	C₂×C₄	1	1	yes	characteristic
22	C₄×#1055	C₂×C₄	1	2	yes	—
23	C₄×#1055	C₂³	1	1	yes	characteristic
24	#11416	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
28	C₂×C₈×C₁₃	—	2	2	no	—
29	C₄²×C₁₃	C₄²	1	1	yes	characteristic
30	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
31	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
32	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×#1055	—	2	4	no	—
34	C₂×#1055	—	2	2	no	—
35	C₂×#1055	C₄²	1	2	yes	—
36	C₂×#1055	C₂²×C₄	1	2	yes	—
37	#6092	—	2	8	no	—
38	#2163	—	13	2	no	—
39	#1450	—	13	1	no	—
40	#1313	—	13	4	no	—
41	C₂³×C₁₃	#97	1	1	yes	characteristic
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	—	4	1	no	—
44	C₂×C₄×C₁₃	—	4	1	no	—
45	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
47	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
51	C₈×C₁₃	—	4	2	no	—
52	#1055	—	2	2	no	—
53	#1055	—	4	2	no	—
54	#1055	#131	1	4	yes	—
55	#431	—	13	1	no	—
56	#431	#261	1	1	yes	characteristic
57	C₂×C₄×C₈	—	13	1	no	—
58	C₄⋊C₁₆	—	13	4	no	—
59	C₂²⋊C₁₆	—	13	4	no	—
60	#346	—	13	4	no	—
61	#322	—	13	2	no	—
62	C₈²	—	13	2	no	—
63	C₂²×C₁₃	—	2	1	no	—
64	C₂²×C₁₃	—	2	1	no	—
65	C₂²×C₁₃	—	2	1	no	—
66	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
67	C₄×C₁₃	—	2	1	no	—
68	C₄×C₁₃	—	2	1	no	—
69	C₄×C₁₃	—	2	1	no	—
70	C₄×C₁₃	—	2	1	no	—
71	C₄×C₁₃	—	2	1	no	—
72	C₄×C₁₃	#405	1	2	yes	—
73	C₂²×C₈	—	13	2	no	—
74	C₂×C₄²	C₂×#261	1	1	yes	characteristic
75	C₂×C₁₆	—	13	8	no	—
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	—	26	1	no	—
78	C₂²⋊C₈	—	2	1	no	—
79	C₂²⋊C₈	—	26	1	no	—
80	C₄×C₈	—	13	2	no	—
81	C₄×C₈	—	13	1	no	—
82	C₄×C₈	—	13	1	no	—
83	C₄×C₈	—	13	2	no	—
84	C₄×C₈	C₂×#261	1	2	yes	—
85	C₂×C₁₃	—	4	1	no	—
86	C₂×C₁₃	#1452	1	1	yes	characteristic
87	C₂×C₁₃	#2110	1	1	yes	characteristic
88	C₂×C₁₃	#2110	1	1	yes	characteristic
89	C₂²×C₄	#6118	1	1	yes	characteristic
90	C₂²×C₄	C₄×#261	1	1	yes	characteristic
91	C₂²×C₄	#6122	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	13	2	no	—
94	C₂×C₈	—	13	2	no	—
95	C₂×C₈	—	26	4	no	—
96	C₂×C₈	—	26	2	no	—
97	C₄²	#6118	1	1	yes	characteristic
98	C₄²	C₄×#261	1	1	yes	characteristic
99	C₄²	#6122	1	1	yes	characteristic
100	C₄²	C₂²×#261	1	1	yes	characteristic
101	C₁₆	—	26	8	no	—
102	C₁₃	#11416	1	1	yes	characteristic, complemented, Sylow
103	C₂³	#88110	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	#88110	1	1	yes	characteristic
108	C₂×C₄	#88110	1	1	yes	characteristic
109	C₂×C₄	#88110	1	1	yes	characteristic
110	C₂×C₄	C₂×#6118	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
112	C₂×C₄	C₂×#6122	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	13	4	no	—
115	C₈	—	26	2	no	—
116	C₈	—	52	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×#88110	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	#11719089	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	#13130049	1	1	yes	characteristic
129	C₂	#13131057	1	1	yes	characteristic
130	C₂	#13131059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131197

C₂

yes

maximal

#13129754

C₂

yes

maximal, complemented

#13113120

C₂

yes

maximal, characteristic

C₁₃×#431

C₄

yes

characteristic

#11718586

C₄

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

C₂²⋊#6092

C₄

yes

—

#11718231

C₂²

yes

—

C₄⋊#6092

C₄

yes

—

#11718046

C₂²

yes

—

C₈×#1055

C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

—

C₂×#6092

C₂×C₄

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₂²⋊#1055

—

C₄⋊#1055

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂³

yes

characteristic

#11416

—

maximal, Sylow

C₂²×C₄×C₁₃

C₄²

yes

characteristic

C₂²×C₄×C₁₃

M₁₆

yes

characteristic

C₂²×C₄×C₁₃

M₁₆

yes

characteristic

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₄²

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

—

—

C₄²

yes

—

C₂×#1055

C₂²×C₄

yes

—

—

—

—

—

C₂³×C₁₃

#97

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₈×C₁₃

—

—

—

yes

—

—

yes

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₈²

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#97

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#405

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×#261

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#261

yes

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

yes

C₄²

yes

C₄²

yes

100

C₄²

C₂²×#261

yes

characteristic

101

C₁₆

—

102

C₁₃

#11416

yes

characteristic, complemented, Sylow

103

C₂³

#88110

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#88110

yes

characteristic

108

C₂×C₄

#88110

yes

characteristic

109

C₂×C₄

#88110

yes

characteristic

110

C₂×C₄

C₂×#6118

yes

characteristic

111

C₂×C₄

C₂×C₄×#261

yes

characteristic

112

C₂×C₄

C₂×#6122

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#88110

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#11719089

yes

—

127

C₂

—

128

C₂

#13130049

yes

characteristic

129

C₂

#13131057

yes

characteristic

130

C₂

#13131059

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#431	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×C₁₃	#97
cosocle	C₄×#1055	C₂³
hypercenter	#431	#261
hypocenter	C₁₃	#11416
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₄×#261

commutator

C₂×C₄×C₁₃

C₂×C₄²

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#431

C₄

socle

C₂²×C₁₃

C₂×#97

C₂³×C₁₃

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #13129754	C₂	semidirect
102. C₁₃	#11416	semidirect

H ≅ G/N

type

2. #13129754

C₂

semidirect

102. C₁₃

#11416

semidirect

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

subgroup	index
#13131197	2
#13129754	2
#13113120	2
#11416	13

subgroup

index

Subgroups — #16786693

Conjugacy classes of subgroups

Complements

Maximal subgroups