Group #16777070 — subgroups

Subgroups — #16777070

Order 3328.

206 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 60 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	#13129877	C₂	1	2	yes	maximal
2	#13115804	C₂	1	1	yes	maximal, characteristic
3	C₂×#88124	C₄	1	1	yes	characteristic
4	C₂×#88106	—	2	2	no	—
5	C₁₃×#384	C₄	1	1	yes	characteristic
6	C₂×#88043	C₂²	1	1	yes	characteristic
7	#11718121	—	2	2	no	—
8	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
9	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
10	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	—	2	1	no	—
12	C₂×#6101	—	4	1	no	—
13	C₂²×#1055	D₄	1	2	yes	—
14	#88124	—	2	1	no	—
15	#88124	—	4	1	no	—
16	#88124	C₈	1	2	yes	—
17	#88106	—	4	2	no	—
18	#88106	—	4	2	no	—
19	C₂×#6092	—	2	4	no	—
20	#88043	—	4	1	no	—
21	#7130	—	13	1	no	maximal, Sylow
22	C₂²×C₄×C₁₃	—	2	1	no	—
23	C₂²×C₄×C₁₃	—	4	1	no	—
24	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×Dic₁₃	—	4	1	no	—
26	C₁₃×C₄⋊C₄	—	4	2	no	—
27	C₁₃×C₄⋊C₄	—	8	2	no	—
28	C₄²×C₁₃	—	2	1	no	—
29	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
30	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
31	C₄⋊Dic₁₃	—	2	2	no	—
32	C₄⋊Dic₁₃	—	4	1	no	—
33	#6101	—	4	2	no	—
34	#6101	—	4	2	no	—
35	C₂×#1055	—	2	2	no	—
36	C₂×#1055	—	2	2	no	—
37	C₂×#1055	SD₁₆	1	2	yes	—
38	C₂×#1055	Q₁₆	1	2	yes	—
39	#6092	—	4	4	no	—
40	#1992	—	13	1	no	—
41	#1375	—	13	2	no	—
42	C₂³×C₁₃	#100	1	1	yes	characteristic
43	C₂×C₄×C₁₃	—	2	1	no	—
44	C₂×C₄×C₁₃	—	2	1	no	—
45	C₂×C₄×C₁₃	—	4	1	no	—
46	C₂×C₄×C₁₃	—	4	1	no	—
47	C₂×C₄×C₁₃	—	8	2	no	—
48	C₂×C₄×C₁₃	—	8	1	no	—
49	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
51	C₂×C₄×C₁₃	#103	1	2	yes	—
52	C₂×C₄×C₁₃	#104	1	2	yes	—
53	C₂×Dic₁₃	—	4	2	no	—
54	C₂×Dic₁₃	—	8	1	no	—
55	C₂×Dic₁₃	—	8	1	no	—
56	#1055	—	2	2	no	—
57	#1055	—	2	2	no	—
58	C₂×#125	—	13	1	no	—
59	C₂×M₃₂	—	26	2	no	—
60	C₂×#97	—	13	1	no	—
61	#384	#261	1	1	yes	characteristic
62	#340	—	26	2	no	—
63	C₂²×C₁₃	—	2	1	no	—
64	C₂²×C₁₃	—	2	1	no	—
65	C₂²×C₁₃	#323	1	1	yes	characteristic
66	C₂²×C₁₃	#329	1	1	yes	characteristic
67	C₂²×C₁₃	#333	1	1	yes	characteristic
68	C₄×C₁₃	—	4	1	no	—
69	C₄×C₁₃	—	4	1	no	—
70	C₄×C₁₃	—	16	1	no	—
71	C₄×C₁₃	Q₈⋊C₈	1	2	yes	—
72	C₄×C₁₃	#328	1	2	yes	—
73	Dic₁₃	—	8	2	no	—
74	C₂²×C₈	—	13	2	no	—
75	#125	—	13	2	no	—
76	#125	—	26	1	no	—
77	#125	—	52	1	no	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	—	4	1	no	—
80	C₂×C₄⋊C₄	—	26	1	no	—
81	C₂×C₄⋊C₄	—	52	1	no	—
82	C₂×C₄²	C₂×#261	1	1	yes	characteristic
83	M₃₂	—	52	2	no	—
84	M₃₂	—	52	2	no	—
85	C₂×C₁₆	—	26	4	no	—
86	#97	—	52	1	no	—
87	C₂×C₁₃	—	2	1	no	—
88	C₂×C₁₃	—	2	1	no	—
89	C₂×C₁₃	#1272	1	1	yes	characteristic
90	C₂×C₁₃	#1355	1	1	yes	characteristic
91	C₂×C₁₃	#1356	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	52	1	no	—
95	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
96	C₂×C₈	—	13	2	no	—
97	C₂×C₈	—	13	2	no	—
98	C₂×C₈	—	26	2	no	—
99	C₂×C₈	—	26	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	8	2	no	—
102	C₄⋊C₄	—	26	2	no	—
103	C₄⋊C₄	—	52	2	no	—
104	C₄⋊C₄	—	52	2	no	—
105	C₄⋊C₄	—	52	1	no	—
106	C₄²	—	2	1	no	—
107	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
108	C₄²	#6118	1	1	yes	characteristic
109	C₁₆	—	52	4	no	—
110	C₁₃	#7130	1	1	yes	characteristic, complemented, Sylow
111	C₂³	#88121	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	52	2	no	—
119	C₂×C₄	—	104	1	no	—
120	C₂×C₄	—	104	1	no	—
121	C₂×C₄	#88107	1	1	yes	characteristic
122	C₂×C₄	#88111	1	1	yes	characteristic
123	C₂×C₄	#88116	1	2	yes	—
124	C₂×C₄	#88117	1	2	yes	—
125	C₈	—	26	2	no	—
126	C₈	—	26	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#11718287	1	1	yes	characteristic
130	C₂²	#11718293	1	1	yes	characteristic
131	C₂²	#11718303	1	1	yes	characteristic
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	16	1	no	—
135	C₄	—	104	2	no	—
136	C₄	#11718241	1	2	yes	—
137	C₄	#11718244	1	2	yes	—
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	#13129657	1	1	yes	characteristic
141	C₂	#13129838	1	1	yes	characteristic
142	C₂	#13129841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129877

C₂

yes

maximal

#13115804

C₂

yes

maximal, characteristic

C₂×#88124

C₄

yes

characteristic

C₂×#88106

—

C₁₃×#384

C₄

yes

characteristic

C₂×#88043

C₂²

yes

characteristic

#11718121

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×#6101

—

C₂²×#1055

D₄

yes

—

#88124

—

#88124

—

#88124

C₈

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

SD₁₆

yes

—

C₂×#1055

Q₁₆

yes

—

#6092

—

#1992

—

#1375

—

C₂³×C₁₃

#100

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#104

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

#1055

—

C₂×#125

—

C₂×M₃₂

—

C₂×#97

—

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

Q₈⋊C₈

yes

—

C₄×C₁₃

#328

yes

—

Dic₁₃

—

C₂²×C₈

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×#261

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#97

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊#261

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₄²

D₁₃⋊C₈

yes

characteristic

108

C₄²

#6118

yes

characteristic

109

C₁₆

—

110

C₁₃

#7130

yes

characteristic, complemented, Sylow

111

C₂³

#88121

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

104

—

120

C₂×C₄

—

104

—

121

C₂×C₄

#88107

yes

characteristic

122

C₂×C₄

#88111

yes

characteristic

123

C₂×C₄

#88116

yes

—

124

C₂×C₄

#88117

yes

—

125

C₈

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

#11718287

yes

characteristic

130

C₂²

#11718293

yes

characteristic

131

C₂²

#11718303

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

104

—

136

C₄

#11718241

yes

—

137

C₄

#11718244

yes

—

138

C₂

—

139

C₂

—

140

C₂

#13129657

yes

characteristic

141

C₂

#13129838

yes

characteristic

142

C₂

#13129841

yes

characteristic

	subgroup	G/H
center	C₂²	#11718293
commutator	C₄²×C₁₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#261
Fitting	C₁₃×#384	C₄
socle	C₂²×C₁₃	#329
prime radical	C₂³×C₁₃	#100
cosocle	C₂×#88043	C₂²
hypercenter	#384	#261
hypocenter	C₁₃	#7130
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂²

#11718293

commutator

C₄²×C₁₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#261

Fitting

C₁₃×#384

C₄

C₂²×C₁₃

C₂³×C₁₃

C₂²

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
110. C₁₃	#7130	semidirect

H ≅ G/N

type

110. C₁₃

#7130

semidirect

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

subgroup	index
#13129877	2
#13115804	2
#7130	13

subgroup

index

#13129877

#13115804

#7130

Subgroups — #16777070

Conjugacy classes of subgroups

Complements

Maximal subgroups