Group #16786692 — subgroups

Subgroups — #16786692

Order 3328.

246 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂×C₄×#1055	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	C₄	1	2	yes	—
11	#11718046	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₃	—	2	1	no	—
13	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×#1055	—	2	1	no	—
15	C₂²×#1055	C₂×C₄	1	2	yes	—
16	C₂×#6092	C₂×C₄	1	8	yes	—
17	C₄×C₈×C₁₃	—	2	1	no	—
18	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
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	#11415	—	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	C₂×C₈×C₁₃	—	4	2	no	—
30	C₄²×C₁₃	C₄²	1	1	yes	characteristic
31	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
32	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
33	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×#1055	—	2	4	no	—
35	C₂×#1055	—	2	2	no	—
36	C₂×#1055	—	4	2	no	—
37	C₂×#1055	C₄²	1	2	yes	—
38	C₂×#1055	C₂²×C₄	1	2	yes	—
39	#6092	—	2	8	no	—
40	#2163	—	13	2	no	—
41	C₂×#322	—	13	1	no	—
42	#1313	—	13	4	no	—
43	C₂³×C₁₃	#97	1	1	yes	characteristic
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
49	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
53	C₈×C₁₃	—	4	2	no	—
54	#1055	—	2	2	no	—
55	#1055	—	4	2	no	—
56	#1055	#131	1	4	yes	—
57	C₂×C₄×C₈	—	13	1	no	—
58	C₂×C₄×C₈	—	13	1	no	—
59	C₂×C₄×C₈	#261	1	1	yes	characteristic
60	C₄⋊C₁₆	—	13	4	no	—
61	C₂²⋊C₁₆	—	13	4	no	—
62	#346	—	13	4	no	—
63	#322	—	13	2	no	—
64	#322	—	13	2	no	—
65	C₂²×C₁₃	—	2	1	no	—
66	C₂²×C₁₃	—	2	1	no	—
67	C₂²×C₁₃	#347	1	2	yes	—
68	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
69	C₄×C₁₃	—	2	1	no	—
70	C₄×C₁₃	—	2	1	no	—
71	C₄×C₁₃	—	2	1	no	—
72	C₄×C₁₃	—	2	1	no	—
73	C₄×C₁₃	#347	1	2	yes	—
74	C₄×C₁₃	#405	1	2	yes	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	13	2	no	—
77	C₂²×C₈	—	26	1	no	—
78	C₂×C₄²	C₂×#261	1	1	yes	characteristic
79	C₂×C₁₆	—	13	8	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	13	2	no	—
82	C₄×C₈	—	13	1	no	—
83	C₄×C₈	—	13	1	no	—
84	C₄×C₈	—	13	2	no	—
85	C₄×C₈	—	26	1	no	—
86	C₄×C₈	C₂×#261	1	2	yes	—
87	C₂×C₁₃	—	2	2	no	—
88	C₂×C₁₃	#1452	1	1	yes	characteristic
89	C₂×C₁₃	C₂×#347	1	1	yes	characteristic
90	C₂×C₁₃	C₂×#347	1	1	yes	characteristic
91	C₂²×C₄	#6118	1	1	yes	characteristic
92	C₂²×C₄	C₄×#261	1	1	yes	characteristic
93	C₂²×C₄	#6122	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	13	2	no	—
97	C₂×C₈	—	13	2	no	—
98	C₂×C₈	—	26	4	no	—
99	C₂×C₈	—	26	2	no	—
100	C₂×C₈	—	52	2	no	—
101	C₄²	#6118	1	1	yes	characteristic
102	C₄²	C₄×#261	1	1	yes	characteristic
103	C₄²	#6122	1	1	yes	characteristic
104	C₄²	C₂²×#261	1	1	yes	characteristic
105	C₁₆	—	26	8	no	—
106	C₁₃	#11415	1	1	yes	characteristic, complemented, Sylow
107	C₂³	#88110	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	#88110	1	1	yes	characteristic
112	C₂×C₄	#88110	1	1	yes	characteristic
113	C₂×C₄	#88110	1	1	yes	characteristic
114	C₂×C₄	C₂×#6118	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
116	C₂×C₄	C₂×#6122	1	1	yes	characteristic
117	C₈	—	4	2	no	—
118	C₈	—	13	4	no	—
119	C₈	—	26	2	no	—
120	C₈	—	52	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#11718262	1	2	yes	—
124	C₂²	C₂×#88110	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	#11718232	1	2	yes	—
130	C₄	#11719089	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	#13130049	1	1	yes	characteristic
133	C₂	C₂×#11718232	1	1	yes	characteristic
134	C₂	C₂×#11718262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131197

C₂

yes

maximal

#13129754

C₂

yes

maximal, complemented

C₂×#11718046

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₄

yes

characteristic

C₂×C₄×#1055

C₄

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

C₂²⋊#6092

C₄

yes

—

#11718231

C₂²

yes

—

C₄⋊#6092

C₄

yes

—

#11718046

C₄

yes

—

#11718046

C₂²

yes

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂²×#1055

—

C₂²×#1055

C₂×C₄

yes

—

C₂×#6092

C₂×C₄

yes

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×#1055

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂³

yes

characteristic

#11415

—

maximal, Sylow

C₂²×C₄×C₁₃

C₄²

yes

characteristic

C₂²×C₄×C₁₃

M₁₆

yes

characteristic

C₂²×C₄×C₁₃

M₁₆

yes

characteristic

C₂×C₈×C₁₃

—

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

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#261

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

#322

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#347

yes

—

C₂²×C₁₃

C₂×#97

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#347

yes

—

C₄×C₁₃

#405

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#261

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×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

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄²

#6118

yes

characteristic

102

C₄²

C₄×#261

yes

characteristic

103

C₄²

#6122

yes

characteristic

104

C₄²

C₂²×#261

yes

characteristic

105

C₁₆

—

106

C₁₃

#11415

yes

characteristic, complemented, Sylow

107

C₂³

#88110

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#88110

yes

characteristic

112

C₂×C₄

#88110

yes

characteristic

113

C₂×C₄

#88110

yes

characteristic

114

C₂×C₄

C₂×#6118

yes

characteristic

115

C₂×C₄

C₂×C₄×#261

yes

characteristic

116

C₂×C₄

C₂×#6122

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₂²

—

122

C₂²

—

123

C₂²

#11718262

yes

—

124

C₂²

C₂×#88110

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#11718232

yes

—

130

C₄

#11719089

yes

—

131

C₂

—

132

C₂

#13130049

yes

characteristic

133

C₂

C₂×#11718232

yes

characteristic

134

C₂

C₂×#11718262

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₄²	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×C₁₃	#97
cosocle	C₄×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#11415
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×C₄×#261

commutator

C₂×C₄×C₁₃

C₂×C₄²

Frattini

C₄²

C₂²×#261

Fitting

C₂×C₄×C₈×C₁₃

C₄

socle

C₂²×C₁₃

C₂×#97

C₂³×C₁₃

C₂³

C₂×C₄×C₈

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #13129754	C₂	semidirect
106. C₁₃	#11415	semidirect

H ≅ G/N

type

2. #13129754

C₂

semidirect

106. C₁₃

#11415

semidirect

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

subgroup	index
#13131197	2
#13129754	2
C₂×#11718046	2
#11415	13

subgroup

index

Subgroups — #16786692

Conjugacy classes of subgroups

Complements

Maximal subgroups