Finite Group Databasebeta

← #16790168

Subgroups — #16790168

Order 3328.

566 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13130670C212yesmaximal
2#13129934C214yesmaximal, complemented
3#13117420C211yesmaximal, characteristic
4C23×Dic26C2211yescharacteristic
5C2×C13×#117C411yescharacteristic
6C2×#88180C411yescharacteristic
7#11719169—22no—
8C2×#88113—22no—
9C2×#88113C2214yes—
10#11718635C412yes—
11#11718635C2212yes—
12#11718604—22no—
13#11718590—22no—
14C23×C4×C13C2×C411yescharacteristic
15C23×Dic13D412yes—
16C22×Dic26—24no—
17C22×Dic26—42no—
18C22×Dic26—42no—
19C22×Dic26C2×C411yescharacteristic
20C22×Dic26D412yes—
21C22×Dic26C2311yescharacteristic
22C2×#6122—24no—
23C13×#117—42no—
24C2×C13×C4⋊C4C2×C412yes—
25C2×C13×C22⋊C4—21no—
26C2×C42×C13—21no—
27C2×C22⋊Dic13—21no—
28#88180—42no—
29C2×C4⋊Dic13—21no—
30C2×C4×Dic13C2×C412yes—
31C22⋊#1057—44no—
32#88113—28no—
33#88113—44no—
34#88071—24no—
35#88071—24no—
36#12862—131nomaximal, Sylow
37C24×C13C22⋊C411yescharacteristic
38C22×C4×C13—22no—
39C22×C4×C13—22no—
40C22×C4×C13—22no—
41C22×C4×C13—22no—
42C22×C4×C13C22⋊C411yescharacteristic
43C22×C4×C13C22×C411yescharacteristic
44C22×Dic13—24no—
45C22×Dic13—24no—
46C22×Dic13—22no—
47C22×Dic13—42no—
48C22×Dic13C22⋊C412yes—
49C22×Dic13C2×D412yes—
50C2×Dic26—24no—
51C2×Dic26—22no—
52C2×Dic26—24no—
53C2×Dic26—41no—
54C2×Dic26—41no—
55C2×Dic26—42no—
56C2×Dic26—41no—
57C2×Dic26—41no—
58C2×Dic26—416no—
59C2×Dic26#4114yes—
60#6122—44no—
61#6122—44no—
62C2×#1057—44no—
63C13×C4⋊C4—42no—
64C13×C22⋊C4—42no—
65C42×C13—42no—
66C22⋊Dic13—42no—
67C4⋊Dic13—42no—
68C4×Dic13—42no—
69#2310—131no—
70#1784—132no—
71#1394—134no—
72C23×C13—21no—
73C23×C13—21no—
74C23×C13—42no—
75C23×C13#9912yes—
76C23×C13C2×C22⋊C411yescharacteristic
77C2×C4×C13—28no—
78C2×C4×C13—21no—
79C2×C4×C13—21no—
80C2×C4×C13—21no—
81C2×C4×C13—21no—
82C2×C4×C13—44no—
83C2×C4×C13—42no—
84C2×C4×C13#9912yes—
85C2×C4×C13#11712yes—
86C2×Dic13—22no—
87C2×Dic13—22no—
88C2×Dic13—22no—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—48no—
92C2×Dic13—44no—
93C2×Dic13—42no—
94C2×Dic13#12314yes—
95Dic26—44no—
96Dic26—48no—
97Dic26—44no—
98#1057—84no—
99C23×Q8—131no—
100C2×#117—131no—
101C2×#117#26111yescharacteristic
102C2×#101—134no—
103C2×#101—262no—
104#407—262no—
105#391—132no—
106#391—132no—
107#391—262no—
108#381—262no—
109C22×C13—22no—
110C22×C13—22no—
111C22×C13—24no—
112C22×C13—42no—
113C22×C13—42no—
114C22×C13#38111yescharacteristic
115C22×C13C2×#9911yescharacteristic
116C22×C13C2×#9911yescharacteristic
117C4×C13—22no—
118C4×C13—22no—
119C4×C13—82no—
120Dic13—24no—
121Dic13—42no—
122Dic13—82no—
123C22×Q8—132no—
124C22×Q8—131no—
125C22×Q8—131no—
126C22×Q8—264no—
127C22×Q8—522no—
128C22×Q8—522no—
129C23×C4—132no—
130C23×C4C2×#26111yescharacteristic
131C2×M16—264no—
132#117—42no—
133#117—522no—
134C2×C4⋊C4—261no—
135C2×C4⋊C4C2×#26112yes—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—261no—
138C2×C42—21no—
139C2×C42—132no—
140#101—268no—
141#101—524no—
142C22⋊C8—524no—
143#95—264no—
144#95—264no—
145C2×C13—21no—
146C2×C13—21no—
147C2×C13—42no—
148C2×C13#181911yescharacteristic
149C2×C13#213312yes—
150C24C22⋊#26111yescharacteristic
151C2×Q8—134no—
152C2×Q8—264no—
153C2×Q8—264no—
154C2×Q8—262no—
155C2×Q8—521no—
156C2×Q8—521no—
157C2×Q8—522no—
158C2×Q8—521no—
159C2×Q8—521no—
160C2×Q8—5216no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—132no—
166C22×C4—132no—
167C22×C4—264no—
168C22×C4—264no—
169C22×C4—262no—
170C22×C4—522no—
171C22×C4C22⋊#26111yescharacteristic
172C22×C4C22×#26111yescharacteristic
173M16—524no—
174M16—524no—
175C2×C8—524no—
176C4⋊C4—42no—
177C4⋊C4—522no—
178C22⋊C4—42no—
179C22⋊C4—522no—
180C42—42no—
181C42—522no—
182C13#1286211yescharacteristic, complemented, Sylow
183C23—21no—
184C23—21no—
185C23—42no—
186C23#8811912yes—
187C23C2×C22⋊#26111yescharacteristic
188Q8—524no—
189Q8—528no—
190Q8—524no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—134no—
199C2×C4—262no—
200C2×C4—262no—
201C2×C4—262no—
202C2×C4—262no—
203C2×C4—262no—
204C2×C4—528no—
205C2×C4—524no—
206C2×C4—522no—
207C2×C4#8810712yes—
208C2×C4#8823712yes—
209C8—1044no—
210C22—22no—
211C22—24no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22C2×#8810711yescharacteristic
216C22#1171913511yescharacteristic
217C22C2×#8811911yescharacteristic
218C4—22no—
219C4—22no—
220C4—82no—
221C4—264no—
222C4—522no—
223C4—1042no—
224C2—21no—
225C2—21no—
226C2—42no—
227C2#1313073111yescharacteristic
228C2#1313113112yes—
subgroupG/H
centerC22C2×#88119
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC2×C13×#117C4
socleC22×C13C2×#99
prime radicalC24×C13C22⋊C4
cosocleC22×Dic26C23
hypercenterC2×#117#261
hypocenterC13#12862
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #13129934C2semidirect
182. C13#12862semidirect

Maximal subgroups

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

subgroupindex
#131306702
#131299342
#131174202
#1286213