Finite Group Databasebeta

← #27281672

Subgroups — #27281672

Order 13568.

436 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283711C211yesmaximal, characteristic, complemented
2#20273326C211yesmaximal, characteristic, complemented
3#20273275C211yesmaximal, characteristic, complemented
4#20273253C211yesmaximal, characteristic, complemented
5#20272614C211yesmaximal, characteristic, complemented
6#20272597C211yesmaximal, characteristic, complemented
7C53×#3180C211yesmaximal, characteristic
8#16810754—21no—
9#16810695—21no—
10#16810695—21no—
11D4×#88322—21no—
12#16810684—21no—
13#16810680C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810579—21no—
16C2×#13133128C2211yescharacteristic
17#16810489C2211yescharacteristic
18#16810363—21no—
19#16810363—21no—
20#16810360—21no—
21#16810357—21no—
22#16810357—21no—
23#16810357—21no—
24C53×#532C2211yescharacteristic
25C53×#529C2211yescharacteristic
26C53×#501C411yescharacteristic, complemented
27C53×#496C411yescharacteristic, complemented
28C53×#490C411yescharacteristic, complemented
29C53×#489C411yescharacteristic, complemented
30C53×#466—21no—
31C53×#465—21no—
32C53×#442—21no—
33C53×#440—21no—
34C2×#97×C53C2211yescharacteristic
35C2×#11719917—21no—
36C22×#88322—21no—
37C22×#88322—21no—
38C2×C4⋊Dic53—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C22⋊#88322D411yescharacteristic
47C22⋊#88322D411yescharacteristic
48#13133128—21no—
49#13133128—21no—
50#13133128—21no—
51#13133128D411yescharacteristic
52#13133128D411yescharacteristic
53C4×#88322—21no—
54C4×#88322—21no—
55#13133124—21no—
56#13133124—21no—
57C2×#41×C53—21no—
58C2×SD16×C53—21no—
59C2×D8×C53—21no—
60C22×C8×C53—21no—
61C4⋊Q8×C53C2×C411yescharacteristic
62#127×C53C2×C411yescharacteristic
63#126×C53—21no—
64#125×C53C2×C411yescharacteristic
65#124×C53—21no—
66#124×C53C2×C411yescharacteristic
67#123×C53—21no—
68C22⋊Q8×C53D411yescharacteristic
69#121×C53—21no—
70#121×C53—21no—
71#121×C53D411yescharacteristic
72C4×Q8×C53D411yescharacteristic
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53—21no—
76C4×D4×C53D411yescharacteristic
77#117×C53—21no—
78C2×C42×C53C2311yescharacteristic
79#107×C53—21no—
80#106×C53—21no—
81#103×C53—21no—
82#102×C53—21no—
83#102×C53—21no—
84#102×C53—21no—
85#97×C53—21no—
86#97×C53C2×C411yescharacteristic
87#97×C53C2×C411yescharacteristic
88#41×C53—81no—
89#41×C53—81no—
90C2×Q8×C53—21no—
91C2×Q8×C53C22⋊C411yescharacteristic
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—41no—
97C2×D4×C53C22⋊C411yescharacteristic
98C22×C4×C53—21no—
99C22×C4×C53—41no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×Dic53—41no—
104SD16×C53—42no—
105D8×C53—42no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—42no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53C22⋊C411yescharacteristic
116C4⋊C4×C53C22⋊C411yescharacteristic
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C22⋊C4×C53—21no—
121C22⋊C4×C53—21no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C42×C53—21no—
125C42×C53C22×C411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53#4111yescharacteristic
129C4⋊Dic53—81no—
130#11719917—42no—
131C2×#88322—21no—
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—21no—
135C2×#88322—42no—
136C2×#88322—42no—
137C2×#88322—41no—
138C23×C53—21no—
139C23×C53—41no—
140C23×C53#12111yescharacteristic
141Q8×C53—41no—
142Q8×C53—81no—
143D4×C53—41no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—81no—
149D4×C53—81no—
150D4×C53—81no—
151D4×C53—81no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—82no—
163C2×C4×C53C2×C22⋊C411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12311yescharacteristic
169C2×Dic53—81no—
170C2×Dic53—81no—
171C2×Dic53—81no—
172C8×C53—22no—
173C8×C53—41no—
174#88322—22no—
175#88322—41no—
176#88322—41no—
177#88322—41no—
178#88322—81no—
179#9432—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53#38611yescharacteristic
189C22×C53#47111yescharacteristic
190C22×C53#47211yescharacteristic
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—81no—
201Dic53—161no—
202#3180D5311yescharacteristic
203#2968—531no—
204#1530—531no—
205#1517—531no—
206#1512—531no—
207#1302—531no—
208#1295—531no—
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—81no—
212C2×C53—161no—
213C2×C53#188411yescharacteristic
214C2×C53#190611yescharacteristic
215C2×C53#190711yescharacteristic
216#532C2×D5311yescharacteristic
217#529C2×D5311yescharacteristic
218#501Dic5311yescharacteristic, complemented
219#496Dic5311yescharacteristic, complemented
220#490Dic5311yescharacteristic, complemented
221#489Dic5311yescharacteristic, complemented
222#466—21no—
223#465—21no—
224#446—1061no—
225#442—21no—
226#440—21no—
227#435—1061no—
228#435—1061no—
229C8×D4—1061no—
230#433—1061no—
231#432—531no—
232#431—531no—
233#405—1061no—
234C2×#97—531no—
235C2×#97C2×D5311yescharacteristic
236#382—531no—
237#329—1061no—
238#329—1061no—
239#328—1061no—
240#327—1061no—
241#327—1061no—
242#327—1061no—
243C53#943211yescharacteristic, complemented, Sylow
244C2×#41—21no—
245C2×SD16—21no—
246C2×D8—21no—
247C2×M16—1061no—
248C22×C8—21no—
249C22×C8—1061no—
250C22×C8—1061no—
251C4⋊Q8C2×Dic5311yescharacteristic
252#127C2×Dic5311yescharacteristic
253#126—21no—
254#125C2×Dic5311yescharacteristic
255#124—21no—
256#124C2×Dic5311yescharacteristic
257#123—21no—
258C22⋊Q8C22⋊D5311yescharacteristic
259#121—21no—
260#121—21no—
261#121C22⋊D5311yescharacteristic
262C4×Q8C22⋊D5311yescharacteristic
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4C22⋊D5311yescharacteristic
267#117—21no—
268C2×C4⋊C4—1061no—
269C2×C42C22×D5311yescharacteristic
270#107—21no—
271#106—21no—
272C4⋊C8—1061no—
273C4⋊C8—1061no—
274C4⋊C8—1061no—
275C4⋊C8—1061no—
276#103—21no—
277#102—21no—
278#102—21no—
279#102—21no—
280C22⋊C8—531no—
281C22⋊C8—531no—
282C22⋊C8—1061no—
283C22⋊C8—1061no—
284C22⋊C8—1061no—
285#97—21no—
286#97—531no—
287#97—531no—
288#97—1061no—
289#97—1061no—
290#97—1061no—
291#97C2×Dic5311yescharacteristic
292#97C2×Dic5311yescharacteristic
293C4×C8—1061no—
294C4×C8—1061no—
295#95—1061no—
296#95—1061no—
297#41—81no—
298#41—81no—
299C2×Q8—21no—
300C2×Q8C22⋊Dic5311yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4C22⋊Dic5311yescharacteristic
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—2121no—
310C22×C4D4×D5311yescharacteristic
311C22×C4#1171994711yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313SD16—42no—
314D8—42no—
315M16—2122no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—42no—
319C2×C8—1061no—
320C2×C8—1061no—
321C2×C8—1061no—
322C2×C8—1061no—
323C2×C8—2122no—
324C2×C8—2122no—
325C2×C8—2121no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—4241no—
333C4⋊C4C22⋊Dic5311yescharacteristic
334C4⋊C4C22⋊Dic5311yescharacteristic
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C42—21no—
343C42D4×D5311yescharacteristic
344C42#1171994711yescharacteristic
345C42C22×Dic5311yescharacteristic
346C42C2×C22⋊D5311yescharacteristic
347C23—21no—
348C23—41no—
349C23#1313318911yescharacteristic
350Q8—41no—
351Q8—81no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—4241no—
373C2×C4—4241no—
374C2×C4—4241no—
375C2×C4C2×C22⋊Dic5311yescharacteristic
376C2×C4D4×Dic5311yescharacteristic
377C2×C4D4×Dic5311yescharacteristic
378C2×C4#1313318511yescharacteristic
379C2×C4#1313318911yescharacteristic
380C2×C4#1313319811yescharacteristic
381C8—22no—
382C8—41no—
383C8—1062no—
384C8—4241no—
385C8—2121no—
386C8—2121no—
387C8—2121no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#1681050311yescharacteristic
397C22#1681088411yescharacteristic
398C22#1681089111yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—81no—
409C4—8481no—
410C2—21no—
411C2—21no—
412C2—81no—
413C2—161no—
414C2#2027509911yescharacteristic
415C2#2027521811yescharacteristic
416C2#2027522511yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3180C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3180D53
hypocenterC53#9432
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3180

Complements

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

NH ≅ G/Ntype
1. #20283711C2semidirect
2. #20273326C2semidirect
3. #20273275C2semidirect
4. #20273253C2semidirect
5. #20272614C2semidirect
6. #20272597C2semidirect
26. C53×#501C4semidirect
27. C53×#496C4semidirect
28. C53×#490C4semidirect
29. C53×#489C4semidirect
218. #501Dic53semidirect
219. #496Dic53semidirect
220. #490Dic53semidirect
221. #489Dic53semidirect
243. C53#9432semidirect

Maximal subgroups

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

subgroupindex
#202837112
#202733262
#202732752
#202732532
#202726142
#202725972
C53×#31802
#943253