Finite Group Databasebeta

← #27539605

Subgroups — #27539605

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285252C211yesmaximal, characteristic
2#20285248C211yesmaximal, characteristic
3#20277001C211yesmaximal, characteristic, complemented
4#20277000C211yesmaximal, characteristic, complemented
5#20276999C211yesmaximal, characteristic, complemented
6#20276996C211yesmaximal, characteristic, complemented
7#20276995C211yesmaximal, characteristic, complemented
8#20276992C211yesmaximal, characteristic, complemented
9#20276991C211yesmaximal, characteristic, complemented
10#20276502C211yesmaximal, characteristic
11#20276500C211yesmaximal, characteristic
12#20276496C211yesmaximal, characteristic
13#367×D53C211yesmaximal, characteristic
14#20276313C211yesmaximal, characteristic
15C53×#2255C211yesmaximal, characteristic, complemented
16#16811253—21no—
17#16811250—21no—
18C2×#13133156C2211yescharacteristic
19C2×C16×D53C2211yescharacteristic
20#16811033—21no—
21#16811031—21no—
22#16811030—21no—
23#16811029—21no—
24#16811028C2211yescharacteristic
25#16811027C2211yescharacteristic
26#16811027C2211yescharacteristic
27#16811025C2211yescharacteristic
28#16811024C2211yescharacteristic
29#16811024C2211yescharacteristic
30#16810664—21no—
31#16810662—21no—
32#16810660C2211yescharacteristic
33#16810660C2211yescharacteristic
34#107×D53C2211yescharacteristic
35#107×D53C2211yescharacteristic
36#16810577C2211yescharacteristic
37#16810451C2211yescharacteristic
38#16810451C2211yescharacteristic
39#16810450C2211yescharacteristic
40#16810450C2211yescharacteristic
41#16810450C2211yescharacteristic
42#16810450C2211yescharacteristic
43#16810450C2211yescharacteristic
44#16810450C2211yescharacteristic
45#16810449C2211yescharacteristic
46#16810449C2211yescharacteristic
47#16810448C2211yescharacteristic
48#16810448C2211yescharacteristic
49#16810398C2211yescharacteristic
50C4×#11719908C2211yescharacteristic
51C16×Dic53C2211yescharacteristic
52C53×#500C2211yescharacteristic
53C53×#499C2211yescharacteristic
54C53×#367C2211yescharacteristic
55C53×#367C2211yescharacteristic
56C53×#366C2211yescharacteristic
57C53×#366C2211yescharacteristic
58C53×#346C2211yescharacteristic
59C2×#11719912D411yescharacteristic
60C2×C8×D53C2311yescharacteristic
61#13133217—21no—
62#13133216—21no—
63#13133216—21no—
64#13133216—41no—
65#13133215—41no—
66#13133209—21no—
67#13133208—21no—
68#13133208—21no—
69#13133207—41no—
70#13133207—41no—
71#13133197—41no—
72#13133195—41no—
73#13133195—41no—
74#13133193—41no—
75#13133175—21no—
76#13133175—21no—
77#13133174D411yescharacteristic
78C4⋊C4×D53—21no—
79C4⋊C4×D53—21no—
80#13133156Q814yescomplemented
81C2×#11719908C2311yescharacteristic
82C2×#11719908C2311yescharacteristic
83C16×D53Q814yescomplemented
84#13133151—21no—
85#13133151C2311yescharacteristic
86#13133151C2311yescharacteristic
87#13133151C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133150—21no—
90#13133150—21no—
91#13133150—21no—
92#13133150—21no—
93#13133149—21no—
94#13133148—21no—
95#13133148—21no—
96#13133148—21no—
97#13133148—21no—
98#13133128D411yescharacteristic
99#13133127D411yescharacteristic
100C8×Dic53C2311yescharacteristic
101C4⋊Q8×C53—21no—
102#125×C53—21no—
103C2×C16×C53C2311yescharacteristic
104C2×C16×C53C2311yescharacteristic
105#107×C53—21no—
106#107×C53C2311yescharacteristic
107#107×C53C2311yescharacteristic
108#107×C53C2311yescharacteristic
109#107×C53C2311yescharacteristic
110#106×C53—21no—
111C4×C8×C53C2311yescharacteristic
112C2×Q8×C53—41no—
113C2×C4×D53—41no—
114C2×C4×D53—41no—
115C2×C4×D53C2×D411yescharacteristic
116C2×Dic106—41no—
117C2×Dic106—41no—
118C2×Dic106—41no—
119C2×C8×C53C2×D411yescharacteristic
120C2×C8×C53C2411yescharacteristic
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—41no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C42×C53C2×D411yescharacteristic
129#11719921—41no—
130#11719921—41no—
131#11719921D811yescharacteristic
132#11719921D811yescharacteristic
133C4⋊Dic53—21no—
134C4⋊Dic53—21no—
135C4⋊Dic53—21no—
136C4⋊Dic53—21no—
137C4⋊Dic53—41no—
138C4⋊Dic53—41no—
139C4⋊Dic53—41no—
140#11719919—41no—
141#11719919—41no—
142#11719919—41no—
143#11719919—41no—
144#11719919—41no—
145#11719919—41no—
146#11719919—41no—
147#11719919—41no—
148#11719919D811yescharacteristic
149#11719919D811yescharacteristic
150C4×Dic53—41no—
151C4×Dic53—41no—
152C4×Dic53C2×D411yescharacteristic
153C2×#88322C2×D411yescharacteristic
154C2×#88322C2×D411yescharacteristic
155#11719912Q1614yes—
156C8×D53C2×Q812yes—
157C8×D53C2×Q812yes—
158C16×C53C2×Q812yes—
159C16×C53C2×Q812yes—
160#11719908C2×Q812yes—
161#11719908C2×Q812yes—
162C22×D53C2×D811yescharacteristic
163Q8×C53—161no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53C2×D811yescharacteristic
169C2×C4×C53C2×D811yescharacteristic
170C2×C4×C53C22×D411yescharacteristic
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53C2×D811yescharacteristic
176C2×Dic53C2×D811yescharacteristic
177C2×Dic53C2×D811yescharacteristic
178C4×D53—161no—
179C4×D53—161no—
180C4×D53C4⋊Q812yes—
181C4×D53C2×Q1612yes—
182Dic106—161no—
183Dic106—161no—
184Dic106—161no—
185C8×C53C2×Q1612yes—
186C8×C53C22×Q811yescharacteristic
187C8×C53C22×Q811yescharacteristic
188#88322C4⋊Q812yes—
189#88322C2×Q1612yes—
190#33730—531nomaximal, Sylow
191C22×C53C22×D811yescharacteristic
192C2×D53—21no—
193C2×D53—21no—
194C2×D53#50012yes—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—81no—
198C4×C53—81no—
199C4×C53—81no—
200C4×C53—81no—
201C4×C53C2×C4⋊Q811yescharacteristic
202C4×C53C22×Q1611yescharacteristic
203Dic53—21no—
204Dic53—21no—
205Dic53—81no—
206Dic53—81no—
207Dic53—81no—
208Dic53—81no—
209Dic53#50012yes—
210#3164—531no—
211#3164—531no—
212#2255—531no—
213#2255—531no—
214#2255—531no—
215#2255D5311yescharacteristic, complemented
216#2254—531no—
217#2254—531no—
218#2253—531no—
219#2253—531no—
220#2159—531no—
221#2159—531no—
222#2158—531no—
223C2×#367—531no—
224#2108—531no—
225C2×C53C2×#50011yescharacteristic
226C2×C53#341611yescharacteristic
227C2×C53#341611yescharacteristic
228D53—22no—
229#544—1061no—
230#544—1061no—
231C2×M32—531no—
232C22×C16—531no—
233#501—1061no—
234#501—1061no—
235#501—1061no—
236#501—1061no—
237#500—531no—
238#500—531no—
239#500—531no—
240#500C2×D5311yescharacteristic
241#499—531no—
242#499—531no—
243#499—531no—
244#499C2×D5311yescharacteristic
245#428—1061no—
246#428—1061no—
247#427—531no—
248#427—531no—
249C2×#107—531no—
250C2×#107—531no—
251#405—531no—
252#367—531no—
253#367—531no—
254#367—531no—
255#367—531no—
256#367—531no—
257#367—531no—
258#367—531no—
259#367—531no—
260#367C2×D5311yescharacteristic
261#367C2×D5311yescharacteristic
262#366—531no—
263#366—531no—
264#366—531no—
265#366—531no—
266#366C2×D5311yescharacteristic
267#366C2×D5311yescharacteristic
268#346—531no—
269#346C2×D5311yescharacteristic
270C4×C16—531no—
271C4×C16—531no—
272C53#3373011yescharacteristic, complemented, Sylow
273C2×M16—531no—
274C22×C8—531no—
275C4⋊Q8—21no—
276C4⋊Q8—1061no—
277C4⋊Q8—1061no—
278C4⋊Q8—1061no—
279C4⋊Q8—2121no—
280C4⋊Q8—2121no—
281#125—21no—
282#125—1061no—
283#125—1061no—
284#125—1061no—
285#125—2121no—
286#125—2121no—
287C22⋊Q8—2121no—
288C22⋊Q8—2121no—
289C22⋊Q8—2121no—
290C22⋊Q8—2121no—
291#117—531no—
292#117—1061no—
293#117—1061no—
294C2×C4⋊C4—1061no—
295C2×C4⋊C4—1061no—
296M32—534no—
297C2×C16—531no—
298C2×C16—531no—
299C2×C16—534no—
300C2×C16C22×D5311yescharacteristic
301C2×C16C22×D5311yescharacteristic
302#107—21no—
303#107—531no—
304#107—531no—
305#107—531no—
306#107—531no—
307#107—1061no—
308#107—1061no—
309#107—1061no—
310#107—1061no—
311#107—1061no—
312#107C22×D5311yescharacteristic
313#107C22×D5311yescharacteristic
314#107C22×D5311yescharacteristic
315#107C22×D5311yescharacteristic
316#106—21no—
317#106—1061no—
318#106—1061no—
319#106—1061no—
320#106—1061no—
321#106—1061no—
322#97—531no—
323#97—531no—
324C4×C8—531no—
325C4×C8C22×D5311yescharacteristic
326C2×Q8—41no—
327C2×Q8—2121no—
328C2×Q8—2121no—
329C2×Q8—2121no—
330C22×C4—531no—
331C22×C4—2121no—
332C22×C4—2121no—
333M16—534no—
334C2×C8—531no—
335C2×C8—531no—
336C2×C8—532no—
337C2×C8—532no—
338C2×C8D4×D5311yescharacteristic
339C2×C8C23×D5311yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—531no—
348C4⋊C4—531no—
349C4⋊C4—1061no—
350C4⋊C4—1061no—
351C4⋊C4—1061no—
352C4⋊C4—1061no—
353C4⋊C4—2121no—
354C4⋊C4—2121no—
355C4⋊C4—2121no—
356C4⋊C4—2121no—
357C4⋊C4—2121no—
358C4⋊C4—2121no—
359C4⋊C4—2121no—
360C4⋊C4—2121no—
361C4⋊C4—2121no—
362C4⋊C4—2121no—
363C4⋊C4—2121no—
364C22⋊C4—531no—
365C22⋊C4—531no—
366C22⋊C4—2121no—
367C22⋊C4—2121no—
368C42—531no—
369C42—2121no—
370C42—2121no—
371C42D4×D5311yescharacteristic
372C16—532no—
373C16—532no—
374C16Q8×D5312yes—
375C16Q8×D5312yes—
376C23—531no—
377Q8—161no—
378Q8—8481no—
379Q8—8481no—
380Q8—8481no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—531no—
386C2×C4—532no—
387C2×C4—532no—
388C2×C4—531no—
389C2×C4—531no—
390C2×C4—8481no—
391C2×C4—8481no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4D8×D5311yescharacteristic
397C2×C4D8×D5311yescharacteristic
398C2×C4C2×D4×D5311yescharacteristic
399C8—532no—
400C8—532no—
401C8Q16×D5312yes—
402C8C2×Q8×D5311yescharacteristic
403C8C2×Q8×D5311yescharacteristic
404C22—532no—
405C22—1061no—
406C22—1061no—
407C22C2×D8×D5311yescharacteristic
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—532no—
415C4—1061no—
416C4—1061no—
417C4—4241no—
418C4—4241no—
419C4—4241no—
420C4—4241no—
421C4C4⋊Q8×D5311yescharacteristic
422C4C2×Q16×D5311yescharacteristic
423C2—1062no—
424C2#500×D5311yescharacteristic
425C2#2028786011yescharacteristic
426C2#2028786111yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2255C2
socleC22×C53C22×D8
prime radicalC22×D53C2×D8
hypercenter#2255D53
hypocenterC53#33730
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #20277001C2semidirect
4. #20277000C2semidirect
5. #20276999C2semidirect
6. #20276996C2semidirect
7. #20276995C2semidirect
8. #20276992C2semidirect
9. #20276991C2semidirect
15. C53×#2255C2semidirect
80. #13133156Q8semidirect
83. C16×D53Q8semidirect
215. #2255D53semidirect
272. C53#33730semidirect

Maximal subgroups

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

subgroupindex
#202852522
#202852482
#202770012
#202770002
#202769992
#202769962
#202769952
#202769922
#202769912
#202765022
#202765002
#202764962
#367×D532
#202763132
C53×#22552
#3373053