Finite Group Databasebeta

← #27279924

Subgroups — #27279924

Order 13568.

458 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283804C211yesmaximal, characteristic, complemented
2#20274545C211yesmaximal, characteristic, complemented
3#20273326C211yesmaximal, characteristic
4#20273210C211yesmaximal, characteristic
5#20272613C211yesmaximal, characteristic, complemented
6#20272572C211yesmaximal, characteristic, complemented
7C53×#3112C211yesmaximal, characteristic
8#16810754—22no—
9#16810754—21no—
10Q8×#88322—21no—
11#16810680—21no—
12#16810680C2211yescharacteristic
13#16810676—21no—
14#16810676C2211yescharacteristic
15C2×#13133141—21no—
16C2×#13133128C2211yescharacteristic
17C4×C4⋊Dic53C2211yescharacteristic
18#16810385—21no—
19#16810363—22no—
20#16810357—21no—
21#16810357—21no—
22C53×#532C2211yescharacteristic
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#480—21no—
25C53×#477C411yescharacteristic, complemented
26C53×#477C411yescharacteristic, complemented
27C53×#460C411yescharacteristic, complemented
28C53×#460C411yescharacteristic, complemented
29C53×#448—21no—
30C53×#440—21no—
31C53×#440—21no—
32C53×#432C2211yescharacteristic
33C22×#88322—21no—
34C2×C4⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C4⋊#88322—22no—
37C4⋊#88322—22no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40#13133141—24no—
41C22⋊#88322—21no—
42C22⋊#88322D411yescharacteristic
43C22⋊#88322D411yescharacteristic
44#13133128—22no—
45#13133128—21no—
46#13133128—21no—
47#13133128D412yes—
48C4×#88322—22no—
49C4×#88322—21no—
50#13133124—21no—
51C2×#41×C53—21no—
52C22×Q8×C53D411yescharacteristic
53C2×SD16×C53—21no—
54C2×SD16×C53—21no—
55C4⋊Q8×C53C2×C411yescharacteristic
56#127×C53C2×C411yescharacteristic
57#124×C53—21no—
58#121×C53—21no—
59#121×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53C2×C411yescharacteristic
63C4×Q8×C53C2×C411yescharacteristic
64C4×Q8×C53D411yescharacteristic
65C4×Q8×C53D411yescharacteristic
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72#107×C53—21no—
73#107×C53—21no—
74C4⋊C8×C53C2×C411yescharacteristic
75C4⋊C8×C53C2×C411yescharacteristic
76#103×C53—21no—
77#103×C53—21no—
78#102×C53—21no—
79#102×C53—21no—
80C22⋊C8×C53—21no—
81#97×C53—21no—
82#41×C53—81no—
83#41×C53—81no—
84C2×Q8×C53—21no—
85C2×Q8×C53—22no—
86C2×Q8×C53—21no—
87C2×Q8×C53—41no—
88C2×Q8×C53C22⋊C411yescharacteristic
89C2×Q8×C53C22⋊C411yescharacteristic
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—41no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—41no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C22×Dic53—41no—
101SD16×C53—42no—
102SD16×C53—42no—
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53C22⋊C411yescharacteristic
113C4⋊C4×C53C22⋊C411yescharacteristic
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C42×C53—21no—
117C42×C53—21no—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53C22×C411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53#4111yescharacteristic
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126C4×Dic53—81no—
127C2×#88322—22no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—22no—
131C2×#88322—41no—
132C2×#88322—41no—
133C23×C53—41no—
134C23×C53#12111yescharacteristic
135Q8×C53—21no—
136Q8×C53—21no—
137Q8×C53—21no—
138Q8×C53—42no—
139Q8×C53—81no—
140Q8×C53#10412yes—
141D4×C53—41no—
142D4×C53—41no—
143D4×C53—81no—
144D4×C53—81no—
145D4×C53—81no—
146D4×C53—81no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—82no—
161C2×C4×C53C2×C22⋊C411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53C4×D411yescharacteristic
164C2×C4×C53#12011yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12311yescharacteristic
167C2×Dic53—42no—
168C2×Dic53—81no—
169C2×Dic53—81no—
170C8×C53—41no—
171C8×C53—41no—
172#88322—41no—
173#88322—41no—
174#88322—42no—
175#88322—42no—
176#9164—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53#38611yescharacteristic
183C22×C53#47111yescharacteristic
184C22×C53#47211yescharacteristic
185C4×C53—21no—
186C4×C53—22no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53—81no—
195C4×C53C2×#10411yescharacteristic
196C4×C53#42111yescharacteristic
197Dic53—82no—
198#3112D5311yescharacteristic
199#2984—531no—
200#1763—531no—
201#1530—531no—
202#1502—531no—
203#1302—531no—
204#1285—531no—
205C2×C53—21no—
206C2×C53—21no—
207C2×C53—161no—
208C2×C53#187111yescharacteristic
209C2×C53#190511yescharacteristic
210C2×C53#190711yescharacteristic
211#532C2×D5311yescharacteristic
212C2×C4×Q8C2×D5311yescharacteristic
213#480—21no—
214#477Dic5311yescharacteristic, complemented
215#477Dic5311yescharacteristic, complemented
216#460Dic5311yescharacteristic, complemented
217#460Dic5311yescharacteristic, complemented
218#448—21no—
219#446—1062no—
220#446—1061no—
221C8×Q8—1061no—
222#440—21no—
223#440—21no—
224#432—531no—
225#432—1061no—
226#432C2×D5311yescharacteristic
227#431—531no—
228#431—1061no—
229C2×#103—1061no—
230C2×#97—531no—
231C4×C4⋊C4—531no—
232#340—1061no—
233#329—1062no—
234#327—1061no—
235#327—1061no—
236C53#916411yescharacteristic, complemented, Sylow
237C2×#41—21no—
238C22×Q8C22⋊D5311yescharacteristic
239C2×SD16—21no—
240C2×SD16—21no—
241C22×C8—1061no—
242C4⋊Q8C2×Dic5311yescharacteristic
243#127C2×Dic5311yescharacteristic
244#124—21no—
245#121—21no—
246#121—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C2×Dic5311yescharacteristic
250C4×Q8C2×Dic5311yescharacteristic
251C4×Q8C22⋊D5311yescharacteristic
252C4×Q8C22⋊D5311yescharacteristic
253C4×D4—21no—
254C4×D4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4C22⋊D5311yescharacteristic
258C2×C42—21no—
259C2×C42—1061no—
260C2×C42C22×D5311yescharacteristic
261#107—21no—
262#107—21no—
263C4⋊C8—1062no—
264C4⋊C8—1062no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8C2×Dic5311yescharacteristic
268C4⋊C8C2×Dic5311yescharacteristic
269#103—21no—
270#103—21no—
271#103—1064no—
272#102—21no—
273#102—21no—
274C22⋊C8—21no—
275C22⋊C8—531no—
276C22⋊C8—531no—
277C22⋊C8—1061no—
278#97—21no—
279#97—532no—
280#97—1062no—
281#97—1061no—
282#97—1061no—
283C4×C8—1062no—
284C4×C8—1061no—
285#95—1061no—
286#41—81no—
287#41—81no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—21no—
291C2×Q8—41no—
292C2×Q8C22⋊Dic5311yescharacteristic
293C2×Q8C22⋊Dic5311yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—2121no—
302C22×C4D4×D5311yescharacteristic
303C22×C4#1171994711yescharacteristic
304C22×C4C2×C22⋊D5311yescharacteristic
305SD16—42no—
306SD16—42no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—1062no—
310C2×C8—1061no—
311C2×C8—1062no—
312C2×C8—1061no—
313C2×C8—2121no—
314C2×C8—2121no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—1062no—
323C4⋊C4—2121no—
324C4⋊C4C22⋊Dic5311yescharacteristic
325C4⋊C4C22⋊Dic5311yescharacteristic
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—4241no—
333C42D4×D5311yescharacteristic
334C42#1171994711yescharacteristic
335C42C22×Dic5311yescharacteristic
336C42C2×C22⋊D5311yescharacteristic
337C23—41no—
338C23#1313318911yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—42no—
343Q8—81no—
344Q8#1313314412yes—
345D4—41no—
346D4—41no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—4241no—
366C2×C4—4241no—
367C2×C4—2122no—
368C2×C4C2×C22⋊Dic5311yescharacteristic
369C2×C4D4×Dic5311yescharacteristic
370C2×C4D4×Dic5311yescharacteristic
371C2×C4#1313318511yescharacteristic
372C2×C4#1313318911yescharacteristic
373C2×C4#1313319811yescharacteristic
374C8—41no—
375C8—41no—
376C8—2121no—
377C8—2122no—
378C8—2121no—
379C8—2122no—
380C22—21no—
381C22—21no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#1681050311yescharacteristic
386C22#1681088411yescharacteristic
387C22#1681089111yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—4242no—
399C4C2×#1313314411yescharacteristic
400C4#1681064011yescharacteristic
401C2—21no—
402C2—21no—
403C2—161no—
404C2#2027501811yescharacteristic
405C2#2027521111yescharacteristic
406C2#2027522511yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3112C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#3112D53
hypocenterC53#9164
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
1. #20283804C2semidirect
2. #20274545C2semidirect
5. #20272613C2semidirect
6. #20272572C2semidirect
25. C53×#477C4semidirect
26. C53×#477C4semidirect
27. C53×#460C4semidirect
28. C53×#460C4semidirect
214. #477Dic53semidirect
215. #477Dic53semidirect
216. #460Dic53semidirect
217. #460Dic53semidirect
236. C53#9164semidirect

Maximal subgroups

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

subgroupindex
#202838042
#202745452
#202733262
#202732102
#202726132
#202725722
C53×#31122
#916453