Finite Group Databasebeta

← #16790153

Subgroups — #16790153

Order 3328.

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

normal subgroups only

This group has 670 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#13130665C212yesmaximal
2#13129931C214yesmaximal, complemented
3#13117324C211yesmaximal, characteristic
4C22×#6127C2211yescharacteristic
5C2×C13×#117C411yescharacteristic, complemented
6C22×C4×Dic13C411yescharacteristic, complemented
7C2×#88119—22no—
8#11719135—22no—
9C2×#88107C2214yes—
10#11718649C2212yes—
11#11718635C412yescomplemented
12#11718619—22no—
13C4×#6103—22no—
14C23×C4×C13C2×C411yescharacteristic
15C22×C22⋊D13D412yes—
16C23×Dic13—21no—
17C2×#6127—24no—
18C2×#6127—42no—
19C2×#6127—42no—
20C22×D52C2311yescharacteristic
21C22×C4×D13D412yes—
22C22×Dic26C2×C411yescharacteristic
23C2×C22⋊#261—24no—
24C13×#117—42no—
25C2×C13×C4⋊C4C2×C412yes—
26C2×C13×C22⋊C4—21no—
27C2×C42×C13—21no—
28C2×#6103—24no—
29C2×C4×Dic13—21no—
30C2×C4×Dic13—42no—
31C2×C4×Dic13C2×C412yes—
32#88119—44no—
33(C2×#261)⋊C4—44no—
34#88107—28no—
35#88071—24no—
36#12857—131nomaximal, Sylow
37C24×C13C22⋊C411yescharacteristic
38C23×D13C2×D412yes—
39C22×#261—44no—
40C22×C4×C13—22no—
41C22×C4×C13—22no—
42C22×C4×C13—22no—
43C22×C4×C13—22no—
44C22×C4×C13C22⋊C411yescharacteristic
45C22×C4×C13C22×C411yescharacteristic
46C2×C22⋊D13—24no—
47C2×C22⋊D13—24no—
48C2×C22⋊D13—42no—
49C22×Dic13—22no—
50C22×Dic13—44no—
51C22×Dic13—42no—
52C22×Dic13C22⋊C412yes—
53#6127—416no—
54C2×D52—41no—
55C2×D52—41no—
56C2×D52#4114yes—
57C2×C4×D13—24no—
58C2×C4×D13—24no—
59C2×C4×D13—42no—
60C2×Dic26—22no—
61C2×Dic26—41no—
62C2×Dic26—41no—
63C22⋊#261—44no—
64C22⋊#261—44no—
65C13×C4⋊C4—42no—
66C13×C22⋊C4—42no—
67C42×C13—42no—
68#6103—48no—
69C4×Dic13—42no—
70C4×Dic13—42no—
71#2297—131no—
72#1782—132no—
73#1393—134no—
74C23×C13—21no—
75C23×C13—21no—
76C23×C13—42no—
77C23×C13#9912yes—
78C23×C13C2×C22⋊C411yescharacteristic
79C22×D13—22no—
80C22×D13—22no—
81C22×D13—44no—
82C22×D13#12314yes—
83C2×#261—84no—
84C2×#261—88no—
85C2×C4×C13—28no—
86C2×C4×C13—21no—
87C2×C4×C13—21no—
88C2×C4×C13—21no—
89C2×C4×C13—21no—
90C2×C4×C13—44no—
91C2×C4×C13—42no—
92C2×C4×C13#9912yes—
93C2×C4×C13#11712yes—
94C22⋊D13—48no—
95C2×Dic13—24no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—42no—
100C2×Dic13—48no—
101C2×Dic13—42no—
102D52—44no—
103C4×D13—48no—
104Dic26—44no—
105C22×#41—131no—
106C2×#117#26111yescharacteristic, complemented
107C22×C42—131no—
108C2×#99—134no—
109C2×#99—262no—
110#391—132no—
111#390—132no—
112#386—262no—
113#381—262no—
114C4×C22⋊C4—262no—
115C22×C13—22no—
116C22×C13—22no—
117C22×C13—42no—
118C22×C13—42no—
119C22×C13—42no—
120C22×C13#38111yescharacteristic
121C22×C13C2×#9911yescharacteristic
122C22×C13C2×#9911yescharacteristic
123C2×D13—24no—
124C2×D13—44no—
125C2×D13—44no—
126C2×D13—82no—
127#261—164no—
128C4×C13—22no—
129C4×C13—22no—
130C4×C13—82no—
131Dic13—44no—
132Dic13—42no—
133C2×#41—264no—
134C2×#41—522no—
135C2×#41—522no—
136C22×Q8—131no—
137C22×D4—132no—
138C22×D4—131no—
139C23×C4—132no—
140C23×C4—261no—
141C23×C4C2×#26111yescharacteristic
142#117—42no—
143C2×C4⋊C4C2×#26112yes—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—264no—
146C2×C22⋊C4—264no—
147C2×C42—21no—
148C2×C42—132no—
149C2×C42—261no—
150C2×C42—522no—
151#99—268no—
152#99—524no—
153#95—264no—
154#95—524no—
155C2×C13—21no—
156C2×C13—21no—
157C2×C13—42no—
158C2×C13#181911yescharacteristic
159C2×C13#212612yes—
160D13—44no—
161C24—132no—
162C24C22⋊#26111yescharacteristic
163#41—5216no—
164C2×Q8—262no—
165C2×Q8—521no—
166C2×Q8—521no—
167C2×D4—134no—
168C2×D4—264no—
169C2×D4—264no—
170C2×D4—521no—
171C2×D4—521no—
172C2×D4—522no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—132no—
178C22×C4—264no—
179C22×C4—262no—
180C22×C4—264no—
181C22×C4—522no—
182C22×C4—524no—
183C22×C4—522no—
184C22×C4—524no—
185C22×C4C22⋊#26111yescharacteristic
186C22×C4C22×#26111yescharacteristic
187C4⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—528no—
190C22⋊C4—524no—
191C22⋊C4—524no—
192C42—42no—
193C42—522no—
194C42—522no—
195C13#1285711yescharacteristic, complemented, Sylow
196C23—21no—
197C23—21no—
198C23—42no—
199C23—134no—
200C23—262no—
201C23—262no—
202C23—524no—
203C23#8810712yes—
204C23C2×C22⋊#26111yescharacteristic
205Q8—524no—
206D4—524no—
207D4—528no—
208C2×C4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—264no—
216C2×C4—262no—
217C2×C4—262no—
218C2×C4—262no—
219C2×C4—528no—
220C2×C4—522no—
221C2×C4—528no—
222C2×C4—522no—
223C2×C4—1044no—
224C2×C4—1048no—
225C2×C4#8811912yes—
226C2×C4#8823712yes—
227C22—22no—
228C22—22no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—264no—
233C22—524no—
234C22—524no—
235C22—1042no—
236C22C2×#8810711yescharacteristic
237C22#1171913511yescharacteristic
238C22C2×#8811911yescharacteristic
239C4—22no—
240C4—22no—
241C4—82no—
242C4—524no—
243C4—522no—
244C4—2084no—
245C2—21no—
246C2—21no—
247C2—42no—
248C2—524no—
249C2#1313073111yescharacteristic
250C2#1313110712yes—
subgroupG/H
centerC22C2×#88119
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC2×C13×#117C4
socleC22×C13C2×#99
prime radicalC22×#6127C22
cosocleC22×D52C23
hypercenterC2×#117#261
hypocenterC13#12857
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #13129931C2semidirect
5. C2×C13×#117C4semidirect
6. C22×C4×Dic13C4semidirect
11. #11718635C4semidirect
106. C2×#117#261semidirect
195. C13#12857semidirect

Maximal subgroups

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

subgroupindex
#131306652
#131299312
#131173242
#1285713