Finite Group Databasebeta

← #27287261

Subgroups — #27287261

Order 13568.

422 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 91 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#20279934C211yesmaximal, characteristic
2#20273287C211yesmaximal, characteristic
3C2×#16810360C211yesmaximal, characteristic
4#20272593C211yesmaximal, characteristic
5#20272582C211yesmaximal, characteristic
6#20272563C211yesmaximal, characteristic
7C53×#1996C211yesmaximal, characteristic
8C2×C22⋊#88322C2211yescharacteristic
9C2×C22⋊#88322C2211yescharacteristic
10#16810565—21no—
11#16810557—21no—
12#16810507—21no—
13#16810507—21no—
14#16810495—21no—
15#16810483C2211yescharacteristic
16#16810481—21no—
17C2×#13133124C2211yescharacteristic
18#16810360—21no—
19#16810360—21no—
20#16810360—21no—
21#16810360—21no—
22C2×C22⋊Q8×C53C2211yescharacteristic
23C53×#380C2211yescharacteristic
24C4×C22⋊C4×C53C2211yescharacteristic
25C53×#342—21no—
26C53×#342—21no—
27C23×Dic53—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C2×C4⋊Dic53—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C22⋊#88322—21no—
37C22⋊#88322—21no—
38C22⋊#88322—21no—
39C22⋊#88322—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124D411yescharacteristic
46#13133124D411yescharacteristic
47#13133124D411yescharacteristic
48#13133124D411yescharacteristic
49C22×Q8×C53Q811yescharacteristic
50C23×C4×C53C2311yescharacteristic
51C22⋊Q8×C53—21no—
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53Q811yescharacteristic
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53D411yescharacteristic
61C2×C22⋊C4×C53D411yescharacteristic
62C2×C42×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65C24×C53C2×D411yescharacteristic
66C2×Q8×C53—21no—
67C2×Q8×C53—21no—
68C2×Q8×C53—81no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—41no—
74C22×C4×C53—81no—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×Q811yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—42no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88C22×Dic53—81no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—81no—
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C42×C53—81no—
103C22⋊Dic53—41no—
104C22⋊Dic53—41no—
105C4⋊Dic53—81no—
106#11719919—42no—
107#11719919—42no—
108#11719919—81no—
109#11719919—81no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—81no—
113C2×#88322—81no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53#12011yescharacteristic
117C23×C53C22⋊Q811yescharacteristic
118C23×C53#12411yescharacteristic
119C23×C53#13511yescharacteristic
120C23×C53#13511yescharacteristic
121C23×C53#13711yescharacteristic
122C23×C53#13711yescharacteristic
123Q8×C53—81no—
124Q8×C53—81no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53—81no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53C22⋊Q811yescharacteristic
145C2×C4×C53C22⋊Q811yescharacteristic
146C2×C4×C53#12411yescharacteristic
147C2×C4×C53#12411yescharacteristic
148C2×Dic53—22no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—82no—
161C2×Dic53—82no—
162#88322—81no—
163#88322—81no—
164#10377—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53#39311yescharacteristic
176C22×C53#45211yescharacteristic
177C22×C53#45211yescharacteristic
178C22×C53#47911yescharacteristic
179C22×C53#47911yescharacteristic
180C22×C53#48811yescharacteristic
181C22×C53#49111yescharacteristic
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—81no—
187C4×C53—81no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53—161no—
191Dic53—42no—
192Dic53—81no—
193Dic53—81no—
194Dic53—81no—
195#2569—531no—
196#1996D5311yescharacteristic
197#1520—531no—
198C2×#328—531no—
199#1294—531no—
200#1289—531no—
201#1282—531no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53#202611yescharacteristic
209C2×C53#203111yescharacteristic
210C2×C53#203211yescharacteristic
211C2×C22⋊Q8C2×D5311yescharacteristic
212C2×C22⋊C8—531no—
213C2×C22⋊C8—531no—
214#400—1061no—
215#398—1061no—
216#387—1061no—
217#387—1061no—
218#385—1061no—
219#380—531no—
220#380—1061no—
221#380C2×D5311yescharacteristic
222C4×C22⋊C4C2×D5311yescharacteristic
223C2×#95—531no—
224#342—21no—
225#342—21no—
226#328—1061no—
227#328—1061no—
228#328—1061no—
229#328—1061no—
230C53#1037711yescharacteristic, complemented, Sylow
231C22×Q8Dic10611yescharacteristic
232C23×C4—1061no—
233C23×C4C22×D5311yescharacteristic
234C22×C8—1061no—
235C22×C8—1061no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4Dic10611yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—1061no—
251C2×C22⋊C4C22⋊D5311yescharacteristic
252C2×C22⋊C4C22⋊D5311yescharacteristic
253C2×C42—21no—
254C22⋊C8—1061no—
255C22⋊C8—1061no—
256C22⋊C8—1061no—
257C22⋊C8—1061no—
258#95—21no—
259#95—21no—
260#95—531no—
261#95—531no—
262#95—531no—
263#95—531no—
264#95—1061no—
265#95—1061no—
266#95—1061no—
267#95—1061no—
268#95—1061no—
269C24C2×C22⋊D5311yescharacteristic
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—81no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—81no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—4241no—
284C22×C4—2122no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4C2×Dic10611yescharacteristic
288C22×C4#1171994511yescharacteristic
289C22×C4D4×D5311yescharacteristic
290C22×C4D4×D5311yescharacteristic
291C22×C4#1171994711yescharacteristic
292C22×C4#1171994711yescharacteristic
293C2×C8—4241no—
294C2×C8—4241no—
295C2×C8—2121no—
296C2×C8—2121no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—81no—
302C4⋊C4—4241no—
303C4⋊C4—4241no—
304C4⋊C4—4241no—
305C4⋊C4—2122no—
306C4⋊C4—2122no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C42—81no—
318C23—41no—
319C23—41no—
320C23#1313318511yescharacteristic
321C23#1313319611yescharacteristic
322C23#1313320411yescharacteristic
323C23#1313323711yescharacteristic
324C23#1313323711yescharacteristic
325C23#1313324811yescharacteristic
326C23#1313324811yescharacteristic
327Q8—81no—
328Q8—81no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—1062no—
349C2×C4—4242no—
350C2×C4—4242no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4#1313319411yescharacteristic
363C2×C4#1313319411yescharacteristic
364C2×C4#1313320311yescharacteristic
365C2×C4#1313320311yescharacteristic
366C8—4241no—
367C8—4241no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22#1681053511yescharacteristic
379C22#1681078311yescharacteristic
380C22#1681078311yescharacteristic
381C22#1681093011yescharacteristic
382C22#1681093011yescharacteristic
383C22#1681097711yescharacteristic
384C22#1681099111yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—161no—
394C4—4241no—
395C4—4241no—
396C4—4241no—
397C4—2122no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2#2027586611yescharacteristic
405C2#2027588911yescharacteristic
406C2#2027589211yescharacteristic
subgroupG/H
centerC22#16810535
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1996C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#1996D53
hypocenterC53#10377
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
230. C53#10377semidirect

Maximal subgroups

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

subgroupindex
#202799342
#202732872
C2×#168103602
#202725932
#202725822
#202725632
C53×#19962
#1037753