Finite Group Databasebeta

← #27539426

Subgroups — #27539426

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#20285254C211yesmaximal, characteristic, complemented
2#20285201C211yesmaximal, characteristic
3#20276957C212yesmaximal, complemented
4#20276952C212yesmaximal, complemented
5#20276947C212yesmaximal, complemented
6#20276939C212yesmaximal, complemented
7#20276432C211yesmaximal, characteristic, complemented
8#20276428C211yesmaximal, characteristic, complemented
9#20276423C211yesmaximal, characteristic, complemented
10C2×#16810425C211yesmaximal, characteristic, complemented
11C53×#2108C211yesmaximal, characteristic, complemented
12C2×#13133245—21no—
13C2×#13133239C2211yescharacteristic
14C22×D424C2211yescharacteristic
15#16811251—21no—
16#16811192—21no—
17#16811032—22no—
18#16811028C2212yescomplemented
19#16811025C2212yescomplemented
20#16811012—22no—
21#16811007C2212yes—
22#16811000C2212yes—
23#16810665—21no—
24#16810661C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26#16810428C2211yescharacteristic, complemented
27#16810428C2211yescharacteristic, complemented
28#16810428C2212yescomplemented
29#16810425C2212yes—
30#16810425C2212yes—
31#16810425C2212yes—
32#16810425C2212yes—
33#16810425C2211yescharacteristic, complemented
34#16810425C2211yescharacteristic, complemented
35#16810425C2212yescomplemented
36C2×M32×C53C2211yescharacteristic
37C22×C16×C53C2211yescharacteristic
38C53×#405C2211yescharacteristic
39C53×#346C2212yescomplemented
40C4×C16×C53C2212yescomplemented
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#13133217—22no—
56#13133216—41no—
57#13133214—22no—
58#13133209—22no—
59#13133208—41no—
60#13133205—22no—
61#13133194—42no—
62#13133192—41no—
63#13133192—41no—
64#13133187—42no—
65#13133176—21no—
66C2×C4⋊Dic53—21no—
67#13133151—22no—
68#13133151—21no—
69#13133151C2311yescharacteristic
70#13133151C2312yes—
71#13133151C2311yescharacteristic
72#13133149—22no—
73#13133149—21no—
74C2×M16×C53D411yescharacteristic
75C22×C8×C53C2311yescharacteristic
76#117×C53D411yescharacteristic
77M32×C53—22no—
78C2×C16×C53—22no—
79C2×C16×C53C2311yescharacteristic
80C2×C16×C53C2311yescharacteristic
81C2×C16×C53C2312yes—
82#97×C53D412yes—
83C4×C8×C53C2312yes—
84C23×D53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C2×C22⋊D53—41no—
87C22×Dic53—41no—
88#11719945—42no—
89#11719945—42no—
90C2×D212—22no—
91C2×D212—21no—
92C2×D212—42no—
93C2×D212—42no—
94C2×D212—41no—
95C2×D212—42no—
96C2×D212—42no—
97C2×C4×D53—41no—
98C2×Dic106—21no—
99C2×Dic106—42no—
100M16×C53SD1614yes—
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D412yes—
103C2×C8×C53#4112yes—
104C2×C8×C53#4112yes—
105C2×C8×C53C2411yescharacteristic
106C4⋊C4×C53D811yescharacteristic
107C4⋊C4×C53D811yescharacteristic
108C22⋊C4×C53D812yes—
109C42×C53C2×D412yes—
110C22⋊Dic53—41no—
111#11719921—42no—
112#11719921—42no—
113C4⋊Dic53—21no—
114C4⋊Dic53—22no—
115C4⋊Dic53—21no—
116C4⋊Dic53—41no—
117C4⋊Dic53—42no—
118#11719919—42no—
119#11719919—42no—
120C4×Dic53—41no—
121Dic212—21no—
122Dic212—21no—
123D424—22no—
124D424—22no—
125D424—21no—
126D424—21no—
127D424—42no—
128D424—44no—
129#11719913—42no—
130#11719913—44no—
131C16×C53—21no—
132C16×C53—21no—
133C16×C53—22no—
134C23×C53C2×D811yescharacteristic
135C22×D53—41no—
136C22×D53—42no—
137C22×D53—82no—
138C22×D53—161no—
139C22×D53—161no—
140C2×C4×C53#12412yes—
141C2×C4×C53C2×D812yes—
142C2×C4×C53C2×D812yes—
143C2×C4×C53C2×D811yescharacteristic
144C2×C4×C53C2×SD1612yes—
145C2×C4×C53C22×D411yescharacteristic
146C22⋊D53—82no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—42no—
150C2×Dic53—41no—
151C2×Dic53—161no—
152D212—42no—
153D212—41no—
154D212—42no—
155D212—41no—
156D212—82no—
157D212—162no—
158D212—162no—
159C4×D53—82no—
160Dic106—41no—
161Dic106—41no—
162Dic106—162no—
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#48612yes—
182C4×C53C2×#12411yescharacteristic
183C4×C53C22×SD1611yescharacteristic
184Dic53—81no—
185Dic53—81no—
186Dic53—82no—
187Dic53—81no—
188#3164—531no—
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#2108D5311yescharacteristic, complemented
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—1061no—
209#535—1061no—
210C2×M32C2×D5311yescharacteristic
211C22×C16C2×D5311yescharacteristic
212#501—1062no—
213#500—532no—
214#499—532no—
215#496—1062no—
216#495—532no—
217#493—532no—
218#428—1061no—
219#427—531no—
220C2×#107—531no—
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#346C2×D5312yescomplemented
233C4×C16C2×D5312yescomplemented
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—1061no—
246C2×D8—1062no—
247C2×D8—1062no—
248C2×D8—1062no—
249C2×M16D21211yescharacteristic
250C22×C8C22×D5311yescharacteristic
251C4⋊Q8—1062no—
252C4⋊Q8—2121no—
253#127—1062no—
254#125—1062no—
255#125—2121no—
256#124—1062no—
257C22⋊Q8—2122no—
258#121—2121no—
259#121—2121no—
260#120—2122no—
261#117—1061no—
262#117D21211yescharacteristic
263C2×C4⋊C4—1061no—
264M32—22no—
265C2×C16—22no—
266C2×C16C22×D5311yescharacteristic
267C2×C16C22×D5312yes—
268C2×C16C22×D5311yescharacteristic
269#107—532no—
270#107—531no—
271#107—531no—
272#107—1061no—
273#107—1062no—
274#106—1061no—
275#106—1062no—
276#97D21212yes—
277C4×C8C22×D5312yes—
278C24—2121no—
279#41—2122no—
280#41—2122no—
281C2×Q8—1061no—
282C2×Q8—2122no—
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—2121no—
292C22×C4—2121no—
293C22×C4C2×D21211yescharacteristic
294Q16—1061no—
295Q16—1061no—
296SD16—2122no—
297SD16—2124no—
298D8—1062no—
299D8—1062no—
300D8—1061no—
301D8—1061no—
302D8—2122no—
303D8—2124no—
304M16#1171991314yes—
305C2×C8C2×D21211yescharacteristic
306C2×C8C2×D21212yes—
307C2×C8#1171994512yes—
308C2×C8#1171994512yes—
309C2×C8C23×D5311yescharacteristic
310C4⋊C4—1061no—
311C4⋊C4—1062no—
312C4⋊C4—1061no—
313C4⋊C4—2121no—
314C4⋊C4—2122no—
315C4⋊C4—2122no—
316C4⋊C4—2122no—
317C4⋊C4D42411yescharacteristic
318C4⋊C4D42411yescharacteristic
319C22⋊C4—2121no—
320C22⋊C4—2122no—
321C22⋊C4—2122no—
322C22⋊C4D42412yes—
323C42—2121no—
324C42C2×D21212yes—
325C16—21no—
326C16—21no—
327C16—22no—
328C23—8481no—
329C23—8481no—
330C23—4242no—
331C23—2122no—
332C23—2121no—
333C23C2×D42411yescharacteristic
334Q8—8482no—
335Q8—2121no—
336Q8—2121no—
337D4—8482no—
338D4—8482no—
339D4—4242no—
340D4—4242no—
341D4—2122no—
342D4—2121no—
343D4—2121no—
344D4—2122no—
345C2×C4—8481no—
346C2×C4—4242no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2122no—
351C2×C4#1313320512yes—
352C2×C4C2×D42412yes—
353C2×C4C2×D42412yes—
354C2×C4C2×D42411yescharacteristic
355C2×C4C2×#1171991312yes—
356C2×C4C22×D21211yescharacteristic
357C8#1313320512yes—
358C8C2×#1171991314yes—
359C8C2×#1171994511yescharacteristic
360C8C2×#1171994511yescharacteristic
361C22—21no—
362C22—21no—
363C22—8484no—
364C22—4241no—
365C22—4242no—
366C22—4241no—
367C22—4242no—
368C22—4241no—
369C22—4242no—
370C22#1681097012yes—
371C22C22×D42411yescharacteristic
372C4—22no—
373C4—22no—
374C4—4241no—
375C4—4242no—
376C4—4241no—
377C4—4241no—
378C4#1681097012yes—
379C4C2×#1313320511yescharacteristic
380C4C22×#1171991311yescharacteristic
381C2—22no—
382C2—8482no—
383C2—8481no—
384C2C2×#1681097011yescharacteristic
385C2#2028785611yescharacteristic
386C2#2028786311yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2108C2
socleC22×C53C22×D8
prime radical#20285201C2
hypercenter#2108D53
hypocenterC53#33702
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108

Complements

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

NH ≅ G/Ntype
1. #20285254C2semidirect
3. #20276957C2semidirect
4. #20276952C2semidirect
5. #20276947C2semidirect
6. #20276939C2semidirect
7. #20276432C2semidirect
8. #20276428C2semidirect
9. #20276423C2semidirect
10. C2×#16810425C2semidirect
11. C53×#2108C2semidirect
18. #16811028C22semidirect
19. #16811025C22semidirect
26. #16810428C22semidirect
27. #16810428C22semidirect
28. #16810428C22semidirect
33. #16810425C22semidirect
34. #16810425C22semidirect
35. #16810425C22semidirect
39. C53×#346C22semidirect
40. C4×C16×C53C22semidirect
198. #2108D53semidirect
232. #346C2×D53semidirect
233. C4×C16C2×D53semidirect
234. C53#33702semidirect

Maximal subgroups

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

subgroupindex
#202852542
#202852012
#202769572
#202769522
#202769472
#202769392
#202764322
#202764282
#202764232
C2×#168104252
C53×#21082
#3370253