Finite Group Databasebeta

← #27539429

Subgroups — #27539429

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285201C211yesmaximal, characteristic
2#20276958C212yesmaximal, complemented
3#20276954C212yesmaximal, complemented
4#20276949C212yesmaximal, complemented
5#20276940C212yesmaximal, complemented
6#20276430C211yesmaximal, characteristic, complemented
7#20276425C211yesmaximal, characteristic, complemented
8#20276420C211yesmaximal, characteristic, complemented
9C2×#16810423C211yesmaximal, characteristic, complemented
10#20276312C211yesmaximal, characteristic, complemented
11C53×#3164C211yesmaximal, characteristic, complemented
12C2×#13133245—21no—
13C2×#13133239C2211yescharacteristic
14C22×D424C2211yescharacteristic
15#16811192—21no—
16C2×#13133157C2211yescharacteristic
17C22×#11719908C2211yescharacteristic
18#16811012—22no—
19#16811007C2212yes—
20#16811000C2212yes—
21#16810426C2211yescharacteristic, complemented
22#16810426C2211yescharacteristic, complemented
23#16810426C2212yescomplemented
24#16810423C2212yes—
25#16810423C2212yes—
26#16810423C2212yes—
27#16810423C2212yes—
28#16810423C2211yescharacteristic, complemented
29#16810423C2211yescharacteristic, complemented
30#16810423C2212yescomplemented
31#16810399C2212yescomplemented
32C4×#11719908C2212yescomplemented
33C53×#544—21no—
34C53×#501—22no—
35C53×#500C2212yescomplemented
36C53×#499C2212yescomplemented
37C53×#428—21no—
38C53×#427C2211yescharacteristic
39C2×#107×C53C2211yescharacteristic
40C53×#405C2211yescharacteristic
41C2×#11719945—21no—
42C22×D212—21no—
43#13133245—44no—
44#13133239—22no—
45#13133239—22no—
46C2×Dic212C2311yescharacteristic
47C2×#11719913—21no—
48C2×#11719913—22no—
49C2×D424—21no—
50C2×D424—22no—
51C2×D424—22no—
52C2×D424—22no—
53C2×D424C2312yes—
54C2×D424C2311yescharacteristic
55#13133214—22no—
56#13133205—22no—
57#13133192—41no—
58#13133192—41no—
59#13133187—42no—
60#13133157—22no—
61C2×#11719908—22no—
62C2×#11719908C2311yescharacteristic
63C2×#11719908C2311yescharacteristic
64C2×#11719908C2312yes—
65C2×M16×C53D411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67C4⋊Q8×C53—22no—
68C4⋊Q8×C53—41no—
69#125×C53—22no—
70#125×C53—41no—
71C22⋊Q8×C53—42no—
72#117×C53—21no—
73#117×C53D411yescharacteristic
74C2×C4⋊C4×C53—21no—
75#107×C53—22no—
76#107×C53—21no—
77#107×C53C2311yescharacteristic
78#107×C53C2312yes—
79#107×C53C2311yescharacteristic
80#106×C53—22no—
81#106×C53—21no—
82#97×C53D412yes—
83C4×C8×C53C2312yes—
84C23×D53—41no—
85C2×Q8×C53—42no—
86C22×C4×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C2×C22⋊D53—41no—
89#11719945—42no—
90#11719945—42no—
91C2×D212—22no—
92C2×D212—21no—
93C2×D212—42no—
94C2×D212—42no—
95C2×D212—41no—
96C2×D212—42no—
97C2×D212—42no—
98C2×C4×D53—41no—
99C2×Dic106—21no—
100M16×C53SD1614yes—
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D412yes—
103C2×C8×C53#4112yes—
104C2×C8×C53#4112yes—
105C2×C8×C53C2411yescharacteristic
106C4⋊C4×C53—21no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—42no—
111C4⋊C4×C53—42no—
112C4⋊C4×C53—42no—
113C4⋊C4×C53D811yescharacteristic
114C4⋊C4×C53D811yescharacteristic
115C22⋊C4×C53—41no—
116C22⋊C4×C53D812yes—
117C42×C53—41no—
118C42×C53C2×D412yes—
119#11719921—42no—
120#11719921—42no—
121Dic212—21no—
122Dic212—21no—
123D424—22no—
124D424—22no—
125D424—21no—
126D424—21no—
127D424—42no—
128D424—44no—
129#11719913—42no—
130#11719913—44no—
131#11719908—21no—
132#11719908—21no—
133#11719908—22no—
134C23×C53C2×D811yescharacteristic
135C22×D53—41no—
136C22×D53—42no—
137C22×D53—82no—
138C22×D53—161no—
139C22×D53—161no—
140Q8×C53—162no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—161no—
145C2×C4×C53#12412yes—
146C2×C4×C53C2×D812yes—
147C2×C4×C53C2×D812yes—
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53C2×SD1612yes—
150C2×C4×C53C22×D411yescharacteristic
151C22⋊D53—82no—
152C2×Dic53—41no—
153D212—42no—
154D212—41no—
155D212—42no—
156D212—41no—
157D212—82no—
158D212—162no—
159D212—162no—
160C4×D53—82no—
161Dic106—41no—
162Dic106—41no—
163C8×C53#12412yes—
164C8×C53C2×SD1614yes—
165C8×C53C2×#4111yescharacteristic
166C8×C53C2×#4111yescharacteristic
167#33702—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53#48612yes—
171C22×C53C22×D811yescharacteristic
172C2×D53—81no—
173C2×D53—82no—
174C2×D53—81no—
175C2×D53—82no—
176C2×D53—81no—
177C2×D53—82no—
178C2×D53—164no—
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—82no—
182C4×C53—81no—
183C4×C53—81no—
184C4×C53#48612yes—
185C4×C53C2×#12411yescharacteristic
186C4×C53C22×SD1611yescharacteristic
187Dic53—81no—
188#3164D5311yescharacteristic, complemented
189#3155—531no—
190#2244—532no—
191#2243—532no—
192#2242—532no—
193#2240—532no—
194#2140—531no—
195#2139—531no—
196#2138—531no—
197C2×#357—531no—
198#2108—531no—
199C2×C53—22no—
200C2×C53C2×#48611yescharacteristic
201C2×C53#341511yescharacteristic
202C2×C53#341611yescharacteristic
203D53—161no—
204D53—162no—
205C2×AGL(1,ℤ/8)—1061no—
206C2×#135—531no—
207C22×D8—531no—
208#544—21no—
209#535—1061no—
210C2×M32—531no—
211C22×C16—531no—
212#501—22no—
213#500C2×D5312yescomplemented
214#499C2×D5312yescomplemented
215#496—1062no—
216#495—532no—
217#493—532no—
218#428—21no—
219#427C2×D5311yescharacteristic
220C2×#107C2×D5311yescharacteristic
221#405C2×D5311yescharacteristic
222#358—531no—
223#358—531no—
224#358—532no—
225#357—532no—
226#357—532no—
227#357—531no—
228#357—531no—
229#357—532no—
230#357—532no—
231#357—532no—
232#346—532no—
233C4×C16—532no—
234C53#3370211yescharacteristic, complemented, Sylow
235C2×#41—1061no—
236C22×D4—1061no—
237AGL(1,ℤ/8)—2124no—
238#135—1062no—
239#135—1062no—
240C2×Q16—531no—
241C2×SD16—1061no—
242C2×SD16—1062no—
243C2×D8—532no—
244C2×D8—531no—
245C2×D8—1062no—
246C2×D8—1062no—
247C2×D8—1061no—
248C2×D8—1062no—
249C2×M16C22⋊D5311yescharacteristic
250C22×C8C22×D5311yescharacteristic
251C4⋊Q8—22no—
252C4⋊Q8—41no—
253#127—1062no—
254#125—22no—
255#125—41no—
256#124—1062no—
257C22⋊Q8—42no—
258#121—2121no—
259#121—2121no—
260#120—2122no—
261#117—21no—
262#117C22⋊D5311yescharacteristic
263C2×C4⋊C4—21no—
264M32—1062no—
265C2×C16—531no—
266C2×C16—532no—
267C2×C16—531no—
268C2×C16—1062no—
269#107—21no—
270#107—22no—
271#107C22×D5312yes—
272#107C22×D5311yescharacteristic
273#107C22×D5311yescharacteristic
274#106—21no—
275#106—22no—
276#97C22⋊D5312yes—
277C4×C8C22×D5312yes—
278C24—2121no—
279#41—2122no—
280#41—2122no—
281C2×Q8—42no—
282C2×Q8—1061no—
283C2×D4—1062no—
284C2×D4—1061no—
285C2×D4—2121no—
286C2×D4—2122no—
287C2×D4—2122no—
288C2×D4—2122no—
289C2×D4—2121no—
290C2×D4—2122no—
291C22×C4—41no—
292C22×C4—2121no—
293C22×C4C2×C22⋊D5311yescharacteristic
294Q16—1061no—
295Q16—1061no—
296SD16—2122no—
297SD16—2124no—
298D8—1062no—
299D8—1062no—
300D8—1061no—
301D8—1061no—
302D8—2122no—
303D8—2124no—
304M16#1171992414yes—
305C2×C8#1171994912yes—
306C2×C8#1171994912yes—
307C2×C8C2×C22⋊D5311yescharacteristic
308C2×C8C2×C22⋊D5312yes—
309C2×C8C23×D5311yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4#1171992211yescharacteristic
318C4⋊C4#1171992211yescharacteristic
319C22⋊C4—41no—
320C22⋊C4—2122no—
321C22⋊C4—2122no—
322C22⋊C4#1171992212yes—
323C42—41no—
324C42C2×C22⋊D5312yes—
325C16—1061no—
326C16—1061no—
327C16—1062no—
328C23—8481no—
329C23—8481no—
330C23—4242no—
331C23—2122no—
332C23—2121no—
333C23C2×#1171992211yescharacteristic
334Q8—162no—
335Q8—2121no—
336Q8—2121no—
337D4—8482no—
338D4—8482no—
339D4—4242no—
340D4—4242no—
341D4—2121no—
342D4—2122no—
343D4—2122no—
344D4—2121no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—161no—
349C2×C4—4242no—
350C2×C4—2121no—
351C2×C4#1313320612yes—
352C2×C4C2×#1171992212yes—
353C2×C4C2×#1171992212yes—
354C2×C4C2×#1171992211yescharacteristic
355C2×C4C2×#1171992412yes—
356C2×C4C22×C22⋊D5311yescharacteristic
357C8#1313320612yes—
358C8C2×#1171992414yes—
359C8C2×#1171994911yescharacteristic
360C8C2×#1171994911yescharacteristic
361C22—21no—
362C22—21no—
363C22—8484no—
364C22—4241no—
365C22—4242no—
366C22—4241no—
367C22—4242no—
368C22—4241no—
369C22—4242no—
370C22#1681097112yes—
371C22C22×#1171992211yescharacteristic
372C4—22no—
373C4—22no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4—4241no—
378C4#1681097112yes—
379C4C2×#1313320611yescharacteristic
380C4C22×#1171992411yescharacteristic
381C2—22no—
382C2—8482no—
383C2—8481no—
384C2C2×#1681097111yescharacteristic
385C2#2028785411yescharacteristic
386C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3164C2
socleC22×C53C22×D8
prime radical#20285201C2
hypercenter#3164D53
hypocenterC53#33702
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #20276958C2semidirect
3. #20276954C2semidirect
4. #20276949C2semidirect
5. #20276940C2semidirect
6. #20276430C2semidirect
7. #20276425C2semidirect
8. #20276420C2semidirect
9. C2×#16810423C2semidirect
10. #20276312C2semidirect
11. C53×#3164C2semidirect
21. #16810426C22semidirect
22. #16810426C22semidirect
23. #16810426C22semidirect
28. #16810423C22semidirect
29. #16810423C22semidirect
30. #16810423C22semidirect
31. #16810399C22semidirect
32. C4×#11719908C22semidirect
35. C53×#500C22semidirect
36. C53×#499C22semidirect
188. #3164D53semidirect
213. #500C2×D53semidirect
214. #499C2×D53semidirect
234. C53#33702semidirect

Maximal subgroups

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

subgroupindex
#202852012
#202769582
#202769542
#202769492
#202769402
#202764302
#202764252
#202764202
C2×#168104232
#202763122
C53×#31642
#3370253