Group #16790222 — subgroups

Subgroups — #16790222

Order 3328.

234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 84 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	#13131177	C₂	1	1	yes	maximal, characteristic
2	#13131173	C₂	1	1	yes	maximal, characteristic
3	#13129891	C₂	1	2	yes	maximal
4	#13129887	C₂	1	2	yes	maximal
5	#13114592	C₂	1	1	yes	maximal, characteristic
6	C₂×#88106	C₂²	1	1	yes	characteristic
7	C₂×#88106	C₂²	1	2	yes	—
8	C₂²×#6092	C₂²	1	1	yes	characteristic
9	#11718783	C₂²	1	2	yes	—
10	#11718783	C₂²	1	1	yes	characteristic
11	#11718261	—	2	2	no	—
12	#11718260	—	2	2	no	—
13	C₁₃×#338	C₄	1	1	yes	characteristic
14	#11718149	C₄	1	1	yes	characteristic
15	#11718149	C₄	1	2	yes	—
16	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
18	#88180	C₂×C₄	1	2	yes	—
19	#88180	C₂×C₄	1	1	yes	characteristic
20	C₂²×#1055	C₂³	1	1	yes	characteristic
21	#88106	—	2	2	no	—
22	#88106	—	2	2	no	—
23	#88106	—	2	2	no	—
24	C₂×#6092	—	2	1	no	—
25	C₂×#6092	—	2	1	no	—
26	C₂×#6092	—	2	2	no	—
27	C₂×#6092	—	2	2	no	—
28	#88055	—	2	2	no	—
29	#88055	Q₈	1	2	yes	—
30	C₈×Dic₁₃	—	2	2	no	—
31	C₈×Dic₁₃	D₄	1	2	yes	—
32	#12885	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₁₃×M₁₆	—	2	2	no	—
35	C₁₃×M₁₆	—	2	2	no	—
36	C₁₃×M₁₆	—	2	2	no	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
39	C₂²⋊Dic₁₃	—	2	1	no	—
40	C₂²⋊Dic₁₃	—	2	2	no	—
41	C₄⋊Dic₁₃	—	2	1	no	—
42	C₄⋊Dic₁₃	—	2	2	no	—
43	C₄×Dic₁₃	—	2	2	no	—
44	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
45	C₂×#1055	C₂×D₄	1	1	yes	characteristic
46	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
47	C₂×#1055	#41	1	2	yes	—
48	C₂×#1055	#41	1	2	yes	—
49	#6092	—	4	1	no	—
50	#6092	—	4	1	no	—
51	#6092	—	4	2	no	—
52	#2153	—	13	1	no	—
53	#2151	—	13	1	no	—
54	#1747	—	13	1	no	—
55	#1379	—	13	2	no	—
56	#1378	—	13	2	no	—
57	C₂³×C₁₃	#131	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#117	1	2	yes	—
60	C₂×C₄×C₁₃	#131	1	2	yes	—
61	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
62	C₂×Dic₁₃	—	2	2	no	—
63	C₂×Dic₁₃	—	4	2	no	—
64	C₈×C₁₃	—	2	2	no	—
65	C₈×C₁₃	—	4	2	no	—
66	#1055	#124	1	1	yes	characteristic
67	#1055	#124	1	1	yes	characteristic
68	#1055	#125	1	2	yes	—
69	C₂×M₃₂	—	13	1	no	—
70	C₂×M₃₂	—	13	2	no	—
71	C₂²×C₁₆	—	13	1	no	—
72	#405	—	13	1	no	—
73	#405	—	13	2	no	—
74	#364	—	26	2	no	—
75	C₄⋊C₁₆	—	26	2	no	—
76	#338	#261	1	1	yes	characteristic
77	H(3,ℤ/4)	—	13	2	no	—
78	H(3,ℤ/4)	—	13	1	no	—
79	C₂²×C₁₃	—	4	1	no	—
80	C₂²×C₁₃	#424	1	1	yes	characteristic
81	C₂²×C₁₃	#433	1	2	yes	—
82	C₄×C₁₃	#382	1	1	yes	characteristic
83	C₄×C₁₃	#424	1	1	yes	characteristic
84	C₄×C₁₃	#433	1	2	yes	—
85	Dic₁₃	—	4	2	no	—
86	Dic₁₃	—	8	2	no	—
87	C₂×M₁₆	C₂×#261	1	2	yes	—
88	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
89	C₂²×C₈	—	13	1	no	—
90	#117	—	13	1	no	—
91	#117	—	13	2	no	—
92	M₃₂	—	26	2	no	—
93	M₃₂	—	26	2	no	—
94	M₃₂	—	26	2	no	—
95	C₂×C₁₆	—	26	1	no	—
96	C₂×C₁₆	—	26	1	no	—
97	C₂×C₁₆	—	26	2	no	—
98	C₂×C₁₆	—	26	2	no	—
99	#97	—	13	2	no	—
100	#97	—	26	2	no	—
101	C₄×C₈	—	13	2	no	—
102	C₄×C₈	—	26	2	no	—
103	C₂×C₁₃	—	2	1	no	—
104	C₂×C₁₃	—	2	2	no	—
105	C₂×C₁₃	#1832	1	1	yes	characteristic
106	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	13	1	no	—
112	C₂×C₈	—	13	1	no	—
113	C₂×C₈	—	13	2	no	—
114	C₂×C₈	—	13	2	no	—
115	C₂×C₈	C₄⋊#261	1	2	yes	—
116	C₄⋊C₄	—	26	1	no	—
117	C₄⋊C₄	—	26	2	no	—
118	C₂²⋊C₄	—	26	1	no	—
119	C₂²⋊C₄	—	26	2	no	—
120	C₄²	—	13	2	no	—
121	C₄²	—	26	2	no	—
122	C₁₆	—	52	1	no	—
123	C₁₆	—	52	1	no	—
124	C₁₆	—	52	2	no	—
125	C₁₃	#12885	1	1	yes	characteristic, complemented, Sylow
126	C₂³	#88240	1	1	yes	characteristic
127	C₂×C₄	—	26	2	no	—
128	C₂×C₄	—	52	2	no	—
129	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
130	C₂×C₄	#88237	1	2	yes	—
131	C₂×C₄	#88238	1	2	yes	—
132	C₂×C₄	#88238	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	—	13	1	no	—
136	C₈	—	13	1	no	—
137	C₈	—	13	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	#11719077	1	2	yes	—
140	C₂²	#11719079	1	1	yes	characteristic
141	C₄	—	52	2	no	—
142	C₄	—	104	2	no	—
143	C₄	#11719060	1	1	yes	characteristic
144	C₄	#11719067	1	1	yes	characteristic
145	C₄	#11719068	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#13130754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131177

C₂

yes

maximal, characteristic

#13131173

C₂

yes

maximal, characteristic

#13129891

C₂

yes

maximal

#13129887

C₂

yes

maximal

#13114592

C₂

yes

maximal, characteristic

C₂×#88106

C₂²

yes

characteristic

C₂×#88106

C₂²

yes

—

C₂²×#6092

C₂²

yes

characteristic

#11718783

C₂²

yes

—

#11718783

C₂²

yes

characteristic

#11718261

—

#11718260

—

C₁₃×#338

C₄

yes

characteristic

#11718149

C₄

yes

characteristic

#11718149

C₄

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

characteristic

#88180

C₂×C₄

yes

—

#88180

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

—

—

—

—

—

—

—

—

Q₈

yes

—

C₈×Dic₁₃

—

C₈×Dic₁₃

D₄

yes

—

#12885

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₄⋊C₄

yes

—

C₂×#1055

C₂×D₄

yes

characteristic

C₂×#1055

C₂×Q₈

yes

characteristic

C₂×#1055

#41

yes

—

C₂×#1055

#41

yes

—

—

—

—

—

—

—

—

—

C₂³×C₁₃

#131

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#131

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

yes

yes

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#405

—

#364

—

C₄⋊C₁₆

—

yes

—

—

C₂²×C₁₃

—

C₂²×C₁₃

#424

yes

characteristic

C₂²×C₁₃

#433

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×M₁₆

C₂×#261

yes

—

C₂×M₁₆

C₂×#261

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₂×C₁₃

—

104

C₂×C₁₃

—

105

C₂×C₁₃

#1832

yes

characteristic

106

C₂²×C₄

C₂²×#261

yes

characteristic

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₄⋊#261

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₁₃

#12885

yes

characteristic, complemented, Sylow

126

C₂³

#88240

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

130

C₂×C₄

#88237

yes

—

131

C₂×C₄

#88238

yes

—

132

C₂×C₄

#88238

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

#11719077

yes

—

140

C₂²

#11719079

yes

characteristic

141

C₄

—

142

C₄

—

104

—

143

C₄

#11719060

yes

characteristic

144

C₄

#11719067

yes

characteristic

145

C₄

#11719068

yes

—

146

C₂

—

147

C₂

—

148

C₂

#13130754

yes

characteristic

	subgroup	G/H
center	C₄	#11719060
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#338	C₄
socle	C₂×C₁₃	#1832
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×#1055	C₂³
hypercenter	#338	#261
hypocenter	C₁₃	#12885
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

subgroup

G/H

center

C₄

#11719060

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₁₃×#338

C₄

C₂×C₁₃

C₂³×C₁₃

C₂²×#1055

C₂³

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

N	H ≅ G/N	type
125. C₁₃	#12885	semidirect

H ≅ G/N

type

125. C₁₃

#12885

semidirect

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

subgroup	index
#13131177	2
#13131173	2
#13129891	2
#13129887	2
#13114592	2
#12885	13

subgroup

index

Subgroups — #16790222

Conjugacy classes of subgroups

Complements

Maximal subgroups