Finite Group Databasebeta

← #16778719

Subgroups — #16778719

Order 3328.

440 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 90 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#13130751C211yesmaximal, characteristic, complemented
2#13130621C211yesmaximal, characteristic, complemented
3#13129657C212yesmaximal, complemented
4#13129652C212yesmaximal
5#13123207C211yesmaximal, characteristic
6C2×#88190C411yescharacteristic
7C2×#88144C411yescharacteristic
8C2×#88112—21no—
9C2×#88110—21no—
10C2×C4⋊#1057—21no—
11C13×#390C411yescharacteristic
12#11718882C2211yescharacteristic
13#11718635C2211yescharacteristic
14#11718612—21no—
15#11718612—21no—
16#11718605—21no—
17#11718605—21no—
18C42×Dic13C2211yescharacteristic
19#11718590C411yescharacteristic
20#11718290—21no—
21#11718290C2212yes—
22#11718290C2212yes—
23C22×D4×C13C2×C411yescharacteristic
24C22×Dic26C2×C411yescharacteristic
25C22×#1057—22no—
26C22×#1057—22no—
27C2×C13×C4⋊C4C2×C411yescharacteristic
28C2×C13×C22⋊C4—21no—
29C2×C13×C22⋊C4—21no—
30C2×C42×C13C2×C411yescharacteristic
31C2×C22⋊Dic13—21no—
32C2×C22⋊Dic13—21no—
33C2×C22⋊Dic13—21no—
34C2×C22⋊Dic13—21no—
35#88190—42no—
36C2×C4⋊Dic13—21no—
37C2×C4⋊Dic13C2×C411yescharacteristic
38C2×#6101—21no—
39C2×C4×Dic13C2×C411yescharacteristic
40C2×C4×Dic13D411yescharacteristic
41C2×C4×Dic13D411yescharacteristic
42C2×C4×Dic13D411yescharacteristic
43C2×C4×Dic13D411yescharacteristic
44C2×C4×Dic13C2311yescharacteristic
45#88144—42no—
46#88112—24no—
47#88110—42no—
48C4⋊#1057—42no—
49#88071—21no—
50#88071—21no—
51#88071—21no—
52#88071—21no—
53#7696—131nomaximal, Sylow
54C24×C13—21no—
55C2×D4×C13—22no—
56C2×D4×C13—41no—
57C2×D4×C13—41no—
58C22×C4×C13—21no—
59C22×C4×C13C22⋊C411yescharacteristic
60C22×C4×C13C22⋊C411yescharacteristic
61C22×C4×C13C22×C411yescharacteristic
62C22×Dic13—21no—
63C22×Dic13—21no—
64C22×Dic13C22⋊C411yescharacteristic
65C22×Dic13C22⋊C411yescharacteristic
66C22×Dic13C2×D411yescharacteristic
67C22×Dic13C2×D411yescharacteristic
68C2×Dic26—22no—
69C2×Dic26—41no—
70C2×Dic26—41no—
71C2×#1057—24no—
72C2×#1057—42no—
73C2×#1057—42no—
74C2×#1057—42no—
75C2×#1057—42no—
76C2×#1057—42no—
77C13×C4⋊C4—41no—
78C13×C22⋊C4—42no—
79C13×C22⋊C4—42no—
80C42×C13—22no—
81C22⋊Dic13—42no—
82C22⋊Dic13—42no—
83C22⋊Dic13—41no—
84C22⋊Dic13—41no—
85C22⋊Dic13—41no—
86C22⋊Dic13—41no—
87C4⋊Dic13—42no—
88C4⋊Dic13—41no—
89#6101—42no—
90C4×Dic13—22no—
91C4×Dic13—22no—
92C4×Dic13—21no—
93C4×Dic13—21no—
94C4×Dic13—21no—
95C4×Dic13—21no—
96C4×Dic13—41no—
97C4×Dic13—41no—
98#2861—131no—
99#1831—131no—
100#1767—131no—
101#1272—132no—
102#1270—132no—
103C23×C13—41no—
104C23×C13—41no—
105C23×C13—41no—
106C23×C13—41no—
107C23×C13—41no—
108C23×C13—41no—
109C23×C13—41no—
110C23×C13C2×C22⋊C411yescharacteristic
111D4×C13—82no—
112C2×C4×C13—21no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—21no—
116C2×C4×C13—21no—
117C2×C4×C13—21no—
118C2×C4×C13—21no—
119C2×C4×C13—41no—
120C2×C4×C13—41no—
121C2×C4×C13—41no—
122C2×C4×C13#10412yes—
123C2×C4×C13#13112yes—
124C2×Dic13—21no—
125C2×Dic13—21no—
126C2×Dic13—21no—
127C2×Dic13—21no—
128C2×Dic13—21no—
129C2×Dic13—21no—
130C2×Dic13—21no—
131C2×Dic13—21no—
132C2×Dic13—21no—
133C2×Dic13—41no—
134C2×Dic13—41no—
135C2×Dic13—41no—
136C2×Dic13—41no—
137C2×Dic13—41no—
138C2×Dic13—41no—
139C2×Dic13#10012yes—
140C2×Dic13AGL(1,ℤ/8)11yescharacteristic
141C2×Dic13AGL(1,ℤ/8)11yescharacteristic
142C2×Dic13#13711yescharacteristic
143C2×Dic13#13711yescharacteristic
144Dic26—82no—
145#1057—44no—
146#1057—82no—
147C2×#125—131no—
148C2×#124—131no—
149C2×C4⋊C8—261no—
150C2×C4⋊C8—261no—
151C2×#97—261no—
152#391—131no—
153#391—131no—
154#390—131no—
155#390#26111yescharacteristic
156#388—261no—
157#388—261no—
158#388—261no—
159#388—261no—
160C43—131no—
161#336—132no—
162#336—132no—
163#336—261no—
164C22×C13—81no—
165C22×C13—81no—
166C22×C13—81no—
167C22×C13—81no—
168C22×C13—81no—
169C22×C13—81no—
170C22×C13—81no—
171C22×C13#40811yescharacteristic
172C22×C13#41011yescharacteristic
173C22×C13C2×#10011yescharacteristic
174C22×C13#41711yescharacteristic
175C22×C13#41711yescharacteristic
176C22×C13C2×#10411yescharacteristic
177C22×C13#42111yescharacteristic
178C4×C13—22no—
179C4×C13—22no—
180C4×C13—41no—
181C4×C13—81no—
182Dic13—21no—
183Dic13—21no—
184Dic13—21no—
185Dic13—21no—
186Dic13—22no—
187Dic13—41no—
188Dic13—81no—
189Dic13—81no—
190C22×Q8—131no—
191C22×D4C2×#26111yescharacteristic
192C22×C8—262no—
193C22×C8—262no—
194#125—522no—
195#124—522no—
196C2×C4⋊C4—131no—
197C2×C4⋊C4—261no—
198C2×C4⋊C4—261no—
199C2×C4⋊C4C2×#26111yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—261no—
203C2×C22⋊C4—261no—
204C2×C22⋊C4—261no—
205C2×C22⋊C4—261no—
206C2×C42—131no—
207C2×C42—131no—
208C2×C42—131no—
209C2×C42—131no—
210C2×C42—131no—
211C2×C42—131no—
212C2×C42C2×#26111yescharacteristic
213C4⋊C8—264no—
214C4⋊C8—522no—
215#97—522no—
216#95—261no—
217#95—261no—
218#95—261no—
219#95—261no—
220C2×C13—161no—
221C2×C13#146911yescharacteristic
222C2×C13#148311yescharacteristic
223C2×C13#149511yescharacteristic
224C2×C13#150911yescharacteristic
225C2×C13#151211yescharacteristic
226C2×C13#153011yescharacteristic
227C2×C13#156011yescharacteristic
228C24—21no—
229C2×Q8—262no—
230C2×Q8—521no—
231C2×Q8—521no—
232C2×D4—22no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—131no—
237C22×C4—131no—
238C22×C4—131no—
239C22×C4—131no—
240C22×C4—261no—
241C22×C4—261no—
242C22×C4C22⋊#26111yescharacteristic
243C22×C4C22⋊#26111yescharacteristic
244C22×C4C22×#26111yescharacteristic
245C2×C8—264no—
246C2×C8—522no—
247C2×C8—522no—
248C2×C8—522no—
249C2×C8—522no—
250C2×C8—522no—
251C4⋊C4—41no—
252C4⋊C4—521no—
253C4⋊C4—522no—
254C4⋊C4—522no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—522no—
258C22⋊C4—521no—
259C22⋊C4—521no—
260C22⋊C4—521no—
261C22⋊C4—521no—
262C22⋊C4—522no—
263C42—22no—
264C42—261no—
265C42—261no—
266C42—262no—
267C42—261no—
268C42—261no—
269C42—262no—
270C42—521no—
271C42—521no—
272C13#769611yescharacteristic, complemented, Sylow
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊#26111yescharacteristic
281Q8—1042no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—132no—
294C2×C4—131no—
295C2×C4—131no—
296C2×C4—131no—
297C2×C4—131no—
298C2×C4—261no—
299C2×C4—261no—
300C2×C4—261no—
301C2×C4—261no—
302C2×C4—261no—
303C2×C4—261no—
304C2×C4—261no—
305C2×C4—261no—
306C2×C4—261no—
307C2×C4—521no—
308C2×C4—521no—
309C2×C4—521no—
310C2×C4—521no—
311C2×C4—521no—
312C2×C4—521no—
313C2×C4#8811612yes—
314C2×C4#8823812yes—
315C8—524no—
316C8—1042no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1171911611yescharacteristic
325C22C2×#8811611yescharacteristic
326C22#1171914011yescharacteristic
327C22#1171914611yescharacteristic
328C22D13⋊#10011yescharacteristic
329C22#1171916411yescharacteristic
330C22#1171916411yescharacteristic
331C4—22no—
332C4—22no—
333C4—41no—
334C4—81no—
335C4—261no—
336C4—261no—
337C4—262no—
338C4—261no—
339C4—261no—
340C4—521no—
341C4—1041no—
342C4—1041no—
343C2—161no—
344C2#1313010511yescharacteristic
345C2#1313015111yescharacteristic
346C2#1313019111yescharacteristic
347C2#1313023811yescharacteristic
348C2#1313024711yescharacteristic
349C2#1313030611yescharacteristic
350C2#1313037411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC13×#390C4
socleC23×C13C2×C22⋊C4
prime radicalC22×D4×C13C2×C4
cosocleC2×C4×Dic13C23
hypercenter#390#261
hypocenterC13#7696
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #13130751C2semidirect
2. #13130621C2semidirect
3. #13129657C2semidirect
272. C13#7696semidirect

Maximal subgroups

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

subgroupindex
#131307512
#131306212
#131296572
#131296522
#131232072
#769613