Finite Group Databasebeta

← #27294429

Subgroups — #27294429

Order 13568.

470 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 87 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#20277999C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272735C212yesmaximal, complemented
4#20272714C212yesmaximal, complemented
5C4×#99×C53C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic, complemented
7C2×C22⋊#88322—21no—
8#16810561—21no—
9#16810548C411yescharacteristic, complemented
10#16810548C2211yescharacteristic
11#16810537—21no—
12#16810535C411yescharacteristic, complemented
13#16810535C2211yescharacteristic
14C4×C22⋊Dic53—21no—
15C4×C22⋊Dic53—21no—
16#16810469C411yescharacteristic, complemented
17#16810349—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C4×C22⋊C4×C53—21no—
22C53×#342C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×#88322—41no—
25C4×Dic106—41no—
26C4×Dic106—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C22⋊#88322—22no—
36C22⋊#88322—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C4×D4×C53—41no—
48C4×D4×C53—41no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53—41no—
54C2×C42×C53C2×C411yescharacteristic
55#99×C53—24no—
56#95×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—41no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C2×Dic106—41no—
79C2×Dic106—42no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—24no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—42no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—82no—
88C22⋊Dic53—42no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—41no—
93#11719919—42no—
94C4×Dic53—42no—
95C4×Dic53—41no—
96C4×Dic53—41no—
97C2×#88322—42no—
98C2×#88322—82no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#10412yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C23×C53#13512yes—
110D4×C53—44no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—44no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53—82no—
127C2×C4×C53—82no—
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#9911yescharacteristic
130C2×Dic53—22no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142Dic106—42no—
143Dic106—42no—
144#88322—82no—
145#11728—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53C2×#9911yescharacteristic
159C22×C53#41611yescharacteristic
160C22×C53C2×#10411yescharacteristic
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—84no—
167C4×C53#35412yes—
168Dic53—42no—
169Dic53—81no—
170Dic53—81no—
171Dic53—81no—
172#2398—531no—
173C4×#99D5311yescharacteristic
174#1459—531no—
175#1351—532no—
176#1344—532no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—42no—
180C2×C53—42no—
181C2×C53#150111yescharacteristic
182C2×C53#212011yescharacteristic
183C2×C53C2×#35411yescharacteristic
184C2×C4×Q8—531no—
185C2×C4×D4C2×D5311yescharacteristic
186C2×#99—21no—
187C2×C22⋊C8—1061no—
188#400—1061no—
189#397—531no—
190#397—531no—
191#394—1061no—
192#393—531no—
193#393—531no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—1061no—
196C4×C22⋊C4—1061no—
197#376—531no—
198#342C2×D5312yes—
199#324—1061no—
200#323—532no—
201C53#1172811yescharacteristic, complemented, Sylow
202C22×Q8—531no—
203C22×D4C22⋊D5311yescharacteristic
204C23×C4D21211yescharacteristic
205C23×C4C22⋊D5311yescharacteristic
206C22×C8—2121no—
207C4×Q8—2121no—
208C4×Q8—2121no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4D21211yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4C4×D5311yescharacteristic
220C2×C22⋊C4C22×D5311yescharacteristic
221C2×C42—41no—
222C2×C42—1061no—
223C2×C42—1061no—
224C2×C42C4×D5311yescharacteristic
225#99—24no—
226C22⋊C8—1062no—
227C22⋊C8—2121no—
228#95—41no—
229#95—531no—
230#95—531no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235#95—1061no—
236C24#1171992111yescharacteristic
237C24C2×C22⋊D5311yescharacteristic
238C2×Q8—2121no—
239C2×Q8—2122no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992111yescharacteristic
256C22×C4#1171992111yescharacteristic
257C22×C4C2×C4×D5311yescharacteristic
258C22×C4C2×D21211yescharacteristic
259C2×C8—4242no—
260C2×C8—2122no—
261C4⋊C4—41no—
262C4⋊C4—2121no—
263C4⋊C4—2122no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—2122no—
269C22⋊C4—2122no—
270C22⋊C4—2122no—
271C22⋊C4—2122no—
272C42—21no—
273C42—21no—
274C42—82no—
275C42—2122no—
276C42—2121no—
277C42—2121no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1313314512yes—
287C23C2×#1171992111yescharacteristic
288C23#1313323712yes—
289Q8—2122no—
290Q8—2122no—
291D4—44no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—1062no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4#1313313111yescharacteristic
322C2×C4#1313313111yescharacteristic
323C8—4242no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#1313313111yescharacteristic
337C22#1681061711yescharacteristic
338C22C2×#1313314511yescharacteristic
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—84no—
345C4—4241no—
346C4—4241no—
347C4—4241no—
348C4—2122no—
349C4#1681041712yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2#2027320411yescharacteristic
355C2#2027635411yescharacteristic
356C2C2×#1681041711yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenterC4×#99D53
hypocenterC53#11728
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #20272735C2semidirect
4. #20272714C2semidirect
6. C2×C4×Dic106C4semidirect
9. #16810548C4semidirect
12. #16810535C4semidirect
16. #16810469C4semidirect
201. C53#11728semidirect

Maximal subgroups

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

subgroupindex
#202779992
#202730332
#202727352
#202727142
C4×#99×C532
#1172853