Finite Group Databasebeta

← #27300396

Subgroups — #27300396

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#427×D53C211yesmaximal, characteristic
2#20276325C211yesmaximal, characteristic
3#20274737C211yesmaximal, characteristic
4#20272822C211yesmaximal, characteristic, complemented
5#20272821C211yesmaximal, characteristic, complemented
6#20272820C211yesmaximal, characteristic, complemented
7C53×#1386C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9#117×D53—21no—
10C2×#13133157C2211yescharacteristic
11#16810661C2211yescharacteristic
12#16810660C411yescharacteristic, complemented
13#16810660C411yescharacteristic, complemented
14#16810659—21no—
15#16810658—21no—
16#107×D53C411yescharacteristic, complemented
17#107×D53C411yescharacteristic, complemented
18C2×#13133149C2211yescharacteristic
19#106×D53—21no—
20#16810480—21no—
21#16810402—21no—
22#16810402—21no—
23#16810385—21no—
24C2×M32×C53C2211yescharacteristic
25C53×#427C2211yescharacteristic
26C2×#106×C53C2211yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×#88322D411yescharacteristic
29C2×C8×D53—21no—
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53C2×C411yescharacteristic
34C2×C8×D53C2×C411yescharacteristic
35C2×C8×D53D411yescharacteristic
36C2×C8×D53D411yescharacteristic
37#13133176—21no—
38#13133175—21no—
39#13133175—21no—
40#13133174—41no—
41#13133173—41no—
42C2×C4⋊Dic53—21no—
43C4⋊C4×D53—21no—
44C4⋊C4×D53—21no—
45C2×#11719921—41no—
46C22⋊C4×D53—41no—
47C42×D53—41no—
48#13133157—21no—
49#13133157—21no—
50C2×#11719908—21no—
51#13133151C2×C411yescharacteristic
52#13133151C2×C411yescharacteristic
53#13133150—21no—
54#13133150—21no—
55#13133149—21no—
56#13133149—21no—
57#13133149—21no—
58#13133148—21no—
59#13133148—21no—
60#13133124—41no—
61C22×C8×C53C2311yescharacteristic
62#117×C53—21no—
63C2×C4⋊C4×C53—21no—
64M32×C53—21no—
65M32×C53—21no—
66C2×C16×C53—21no—
67#107×C53C2×C411yescharacteristic
68#107×C53C2×C411yescharacteristic
69#106×C53—21no—
70#106×C53—21no—
71#106×C53—21no—
72C23×D53D811yescharacteristic
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—41no—
76C22×Dic53D811yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53C22⋊C411yescharacteristic
84C2×C4×D53C22⋊C411yescharacteristic
85C2×C4×D53SD1611yescharacteristic
86C2×C4×D53SD1611yescharacteristic
87C2×C8×C53C22×C411yescharacteristic
88C2×C8×C53C2×D411yescharacteristic
89C2×C8×C53#4111yescharacteristic
90C2×C8×C53#4111yescharacteristic
91C2×C8×C53#4111yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C42×C53—41no—
99C22⋊Dic53—41no—
100#11719921—41no—
101#11719921—41no—
102#11719921—82no—
103C4⋊Dic53—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107#11719919—41no—
108#11719919—41no—
109C4×Dic53—41no—
110C4×Dic53—41no—
111C4×Dic53—41no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322C22⋊C411yescharacteristic
115C2×#88322C22⋊C411yescharacteristic
116C8×D53—21no—
117C8×D53—21no—
118C8×D53—21no—
119C8×D53—21no—
120C8×D53—21no—
121C8×D53—21no—
122C8×D53—21no—
123C8×D53—21no—
124C16×C53—41no—
125#11719908—41no—
126C23×C53C2×D811yescharacteristic
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—22no—
130C22×D53—41no—
131C22×D53#10211yescharacteristic
132C22×D53#10211yescharacteristic
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53C2×C22⋊C411yescharacteristic
139C2×C4×C53#12311yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53C2×SD1611yescharacteristic
142C2×C4×C53#13711yescharacteristic
143C2×C4×C53#13711yescharacteristic
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150C2×Dic53—81no—
151C2×Dic53#10211yescharacteristic
152C2×Dic53#10211yescharacteristic
153C4×D53—21no—
154C4×D53—21no—
155C4×D53—21no—
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—21no—
159C4×D53—21no—
160C4×D53—21no—
161C4×D53—82no—
162C4×D53—82no—
163C8×C53#11711yescharacteristic
164C8×C53#11711yescharacteristic
165C8×C53#12311yescharacteristic
166C8×C53#12311yescharacteristic
167#88322—21no—
168#88322—21no—
169#12870—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53C2×#10211yescharacteristic
173C22×C53#48011yescharacteristic
174C22×C53#48011yescharacteristic
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—42no—
181C2×D53—42no—
182C2×D53#36012yes—
183C4×C53—81no—
184C4×C53—81no—
185C4×C53—161no—
186C4×C53#38111yescharacteristic
187C4×C53#41911yescharacteristic
188C4×C53#48311yescharacteristic
189C4×C53#48311yescharacteristic
190Dic53—21no—
191Dic53—21no—
192Dic53—81no—
193Dic53—81no—
194Dic53—161no—
195C2×#427—531no—
196#2112—531no—
197#1807—531no—
198#1386—531no—
199#1386—531no—
200#1386—531no—
201#1386D5311yescharacteristic, complemented
202C2×C53—21no—
203C2×C53—21no—
204C2×C53#182311yescharacteristic
205C2×C53C2×#36011yescharacteristic
206C2×C53#214511yescharacteristic
207D53—22no—
208D53—41no—
209C23×C8—531no—
210C2×#117—1061no—
211C2×M32—531no—
212C2×M32C2×D5311yescharacteristic
213#427—531no—
214#427—531no—
215#427—531no—
216#427—1061no—
217#427—1061no—
218#427C2×D5311yescharacteristic
219C2×#107—531no—
220C2×#107—531no—
221C2×#106—531no—
222C2×#106—1061no—
223C2×#106C2×D5311yescharacteristic
224#380—1061no—
225C22⋊C16—1061no—
226C22⋊C16—1061no—
227#340—1061no—
228C53#1287011yescharacteristic, complemented, Sylow
229C23×C4—531no—
230C22×C8—531no—
231C22×C8—531no—
232C22×C8—531no—
233C22×C8—531no—
234C22×C8—531no—
235C22×C8—1061no—
236C22×C8—1061no—
237C22×C8—1061no—
238C22×C8—1061no—
239C22×C8C22×D5311yescharacteristic
240#117—21no—
241#117—1061no—
242#117—1061no—
243#117—1061no—
244#117—2121no—
245#117—2121no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—1061no—
250C2×C22⋊C4—2121no—
251C2×C22⋊C4—2121no—
252C2×C42—2121no—
253M32—21no—
254M32—21no—
255M32—1061no—
256M32—1061no—
257C2×C16—21no—
258C2×C16—1061no—
259#107—531no—
260#107—531no—
261#107—1061no—
262#107—1061no—
263#107C4×D5311yescharacteristic
264#107C4×D5311yescharacteristic
265#106—21no—
266#106—21no—
267#106—21no—
268#106—1061no—
269#106—1061no—
270#106—1061no—
271#106—1061no—
272#106—1061no—
273#95—2121no—
274C24—531no—
275C22×C4—41no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4D4×D5311yescharacteristic
289C2×C8—531no—
290C2×C8—531no—
291C2×C8—1061no—
292C2×C8—1061no—
293C2×C8—1061no—
294C2×C8—1061no—
295C2×C8—1061no—
296C2×C8—1061no—
297C2×C8—1061no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1061no—
301C2×C8C2×C4×D5311yescharacteristic
302C2×C8#1171994511yescharacteristic
303C2×C8#1171994511yescharacteristic
304C2×C8D4×D5311yescharacteristic
305C2×C8#1171994711yescharacteristic
306C2×C8#1171994911yescharacteristic
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—2121no—
314C4⋊C4—2121no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C22⋊C4—41no—
318C22⋊C4—4242no—
319C22⋊C4—2121no—
320C22⋊C4—2121no—
321C22⋊C4—2121no—
322C42—41no—
323C42—2121no—
324C42—2121no—
325C42—2121no—
326C16—41no—
327C16—2121no—
328C23—531no—
329C23—531no—
330C23—1062no—
331C23—1061no—
332C23—1061no—
333C23—2121no—
334C23D8×D5311yescharacteristic
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—531no—
341C2×C4—531no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—4242no—
353C2×C4—4242no—
354C2×C4—4241no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4C22⋊C4×D5311yescharacteristic
360C2×C4#1313319911yescharacteristic
361C2×C4#1313320211yescharacteristic
362C2×C4SD16×D5311yescharacteristic
363C2×C4#1313324611yescharacteristic
364C2×C4#1313324911yescharacteristic
365C8—1061no—
366C8—1061no—
367C8#1313317411yescharacteristic
368C8#1313317511yescharacteristic
369C8#1313319911yescharacteristic
370C8#1313320211yescharacteristic
371C22—21no—
372C22—21no—
373C22—532no—
374C22—1061no—
375C22—1061no—
376C22—1061no—
377C22—1061no—
378C22—1061no—
379C22—2122no—
380C22—2122no—
381C22#102×D5311yescharacteristic
382C22#1681093511yescharacteristic
383C22#1681093811yescharacteristic
384C4—81no—
385C4—81no—
386C4—161no—
387C4—8481no—
388C4—1061no—
389C4—1061no—
390C4—4241no—
391C4—4241no—
392C4#1681048511yescharacteristic
393C4#1681063011yescharacteristic
394C4#1681095211yescharacteristic
395C4#1681095511yescharacteristic
396C2—21no—
397C2—21no—
398C2—1062no—
399C2—2121no—
400C2#2027480011yescharacteristic
401C2#360×D5311yescharacteristic
402C2#2027644911yescharacteristic
subgroupG/H
centerC22#102×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1386C2
socleC22×C53C2×#102
prime radicalC23×D53D8
cosocleC22×C8×C53C23
hypercenter#1386D53
hypocenterC53#12870
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
4. #20272822C2semidirect
5. #20272821C2semidirect
6. #20272820C2semidirect
7. C53×#1386C2semidirect
12. #16810660C4semidirect
13. #16810660C4semidirect
16. #107×D53C4semidirect
17. #107×D53C4semidirect
201. #1386D53semidirect
228. C53#12870semidirect

Maximal subgroups

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

subgroupindex
#427×D532
#202763252
#202747372
#202728222
#202728212
#202728202
C53×#13862
#1287053