Finite Group Databasebeta

← #27539154

Subgroups — #27539154

Order 13568.

612 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#20285285C211yesmaximal, characteristic, complemented
2#20276912C211yesmaximal, characteristic, complemented
3#20276895C211yesmaximal, characteristic, complemented
4#20276872C212yesmaximal, complemented
5#20276770C212yesmaximal, complemented
6#20276644C211yesmaximal, characteristic, complemented
7#20276627C211yesmaximal, characteristic
8#20276589C212yesmaximal, complemented
9#20276579C212yesmaximal, complemented
10#20276518C211yesmaximal, characteristic, complemented
11C2×C4×D8×C53C211yesmaximal, characteristic, complemented
12#16811157—21no—
13C2×#13133161C2212yes—
14C2×#13133158C2212yes—
15#16811025C2211yescharacteristic, complemented
16#16811007C2211yescharacteristic, complemented
17#16810978—21no—
18#16810864C2211yescharacteristic
19#16810859C2211yescharacteristic
20#16810852—21no—
21C4×Dic212C2211yescharacteristic, complemented
22C4×#11719913—21no—
23C4×D424C2211yescharacteristic, complemented
24#16810450C2212yescomplemented
25#16810448C2212yescomplemented
26C4⋊#11719908C2212yescomplemented
27#16810426C2212yescomplemented
28#16810424C2212yes—
29#16810424C2212yes—
30#16810423C2212yescomplemented
31C22⋊#11719908C2211yescharacteristic
32C22⋊#11719908C2211yescharacteristic
33#16810399C2212yescomplemented
34C22×D8×C53C2211yescharacteristic
35C2×C4×D4×C53—21no—
36C4×D8×C53—21no—
37C4×D8×C53—21no—
38C4×D8×C53C2212yes—
39C4×D8×C53C2212yes—
40C2×#107×C53C2211yescharacteristic
41C2×#102×C53—21no—
42C2×C4×C8×C53C2211yescharacteristic
43C2×Dic212C2311yescharacteristic
44C2×#11719913—21no—
45C2×D424C2311yescharacteristic
46#13133211—41no—
47#13133209—21no—
48#13133205—21no—
49#13133200—41no—
50#13133196—21no—
51#13133191—21no—
52C4×Dic106—21no—
53C4×D212—21no—
54C4×C22⋊D53—41no—
55#13133174—41no—
56#13133161—24no—
57#13133158—24no—
58C2×#11719908C2312yes—
59C2×#11719908C2312yes—
60#13133151C2311yescharacteristic
61#13133151C2311yescharacteristic
62#13133149—21no—
63#13133142—21no—
64#13133142—21no—
65#13133139—21no—
66#13133139—21no—
67C22×D4×C53—21no—
68C23×C4×C53—41no—
69C2×D8×C53—24no—
70C2×D8×C53—21no—
71C2×D8×C53C2312yes—
72C22×C8×C53D411yescharacteristic
73C22×C8×C53C2311yescharacteristic
74C4×D4×C53—22no—
75C4×D4×C53—22no—
76C4×D4×C53—41no—
77C4×D4×C53—41no—
78C2×C4⋊C4×C53—21no—
79C2×C22⋊C4×C53—41no—
80C2×C42×C53D411yescharacteristic
81#107×C53—21no—
82#107×C53C2312yes—
83#102×C53—22no—
84#102×C53—22no—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C24×C53—41no—
90C2×D4×C53—24no—
91C2×D4×C53—22no—
92C2×D4×C53—42no—
93C2×D4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—42no—
96C22×C4×C53—42no—
97C22×C4×C53—84no—
98C22×C4×C53D811yescharacteristic
99C22×C4×C53D811yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C2×C22⋊D53—41no—
102C2×D212—21no—
103C2×C4×D53—41no—
104C2×Dic106—21no—
105D8×C53—24no—
106D8×C53—22no—
107D8×C53#4114yes—
108C2×C8×C53—22no—
109C2×C8×C53—41no—
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2411yescharacteristic
114C4⋊C4×C53—22no—
115C4⋊C4×C53—41no—
116C22⋊C4×C53—42no—
117C22⋊C4×C53—42no—
118C42×C53D811yescharacteristic
119C42×C53D811yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C22⋊Dic53—41no—
123#11719921—41no—
124#11719921—41no—
125#11719921—41no—
126#11719921—41no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129#11719919—41no—
130#11719919—41no—
131#11719919—41no—
132#11719919—41no—
133C4×Dic53—41no—
134Dic212—22no—
135D424—22no—
136#11719913—42no—
137#11719908—22no—
138#11719908—22no—
139C23×C53—44no—
140C23×C53—42no—
141C23×C53—82no—
142C23×C53—82no—
143C23×C53C2×D811yescharacteristic
144C22×D53—41no—
145D4×C53—24no—
146D4×C53—44no—
147D4×C53—42no—
148C2×C4×C53—22no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—84no—
154C2×C4×C53—84no—
155C2×C4×C53—82no—
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C2×D811yescharacteristic
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53C2×D811yescharacteristic
160C2×C4×C53C2×D811yescharacteristic
161C2×C4×C53C22×D411yescharacteristic
162C22⋊D53—82no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166D212—41no—
167D212—41no—
168C4×D53—82no—
169Dic106—41no—
170Dic106—41no—
171C8×C53—21no—
172C8×C53—21no—
173C8×C53—21no—
174C8×C53C2×#4112yes—
175#33674—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—44no—
179C22×C53—88no—
180C22×C53—82no—
181C22×C53—82no—
182C22×C53#50912yes—
183C22×C53C22×D811yescharacteristic
184C2×D53—81no—
185C2×D53—81no—
186C2×D53—81no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—82no—
193C4×C53#52511yescharacteristic
194C4×C53#57511yescharacteristic
195Dic53—81no—
196Dic53—81no—
197Dic53—81no—
198#3168—531no—
199C2×C4×D8D5311yescharacteristic, complemented
200#2235—531no—
201#2232—531no—
202#2228—532no—
203#2210—532no—
204#2187—531no—
205#2184—531no—
206#2177—532no—
207#2175—532no—
208#2163—531no—
209C2×C53—22no—
210C2×C53—84no—
211C2×C53#310011yescharacteristic
212C2×C53C2×#50911yescharacteristic
213C2×C53#341411yescharacteristic
214D53—161no—
215C22×D8C2×D5311yescharacteristic
216#529—1061no—
217C2×C4×D4—21no—
218C2×SD32—532no—
219C2×D16—532no—
220#499—531no—
221#495—531no—
222#488—1061no—
223#467—531no—
224#466—531no—
225#465—1061no—
226C4×Q16—531no—
227C4×SD16—1061no—
228C4×D8—21no—
229C4×D8—21no—
230C4×D8—531no—
231C4×D8C2×D5312yes—
232C4×D8C2×D5312yes—
233C2×#107C2×D5311yescharacteristic
234C2×#102—21no—
235C2×C4×C8C2×D5311yescharacteristic
236#367—532no—
237#366—532no—
238C4⋊C16—532no—
239#358—532no—
240#357—532no—
241#357—532no—
242#357—532no—
243C22⋊C16—531no—
244C22⋊C16—531no—
245#346—532no—
246C53#3367411yescharacteristic, complemented, Sylow
247C22×D4—21no—
248C23×C4—41no—
249C2×Q16—531no—
250C2×SD16—1061no—
251C2×D8—24no—
252C2×D8—21no—
253C2×D8—531no—
254C2×D8C22×D5312yes—
255C22×C8C22⋊D5311yescharacteristic
256C22×C8C22×D5311yescharacteristic
257#126—2121no—
258#125—1061no—
259#124—1061no—
260#123—2121no—
261C22⋊Q8—1061no—
262#121—1061no—
263C4×Q8—1061no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—1061no—
269C4×D4—2121no—
270#117—2121no—
271C2×C4⋊C4—21no—
272C2×C22⋊C4—41no—
273C2×C42C22⋊D5311yescharacteristic
274SD32—1064no—
275D16—1064no—
276C2×C16—532no—
277C2×C16—532no—
278#107—21no—
279#107—531no—
280#107—531no—
281#107C22×D5312yes—
282#106—1061no—
283#103—1061no—
284#103—1061no—
285#102—22no—
286#102—22no—
287#102—1061no—
288#102—1061no—
289C4×C8C22⋊D5311yescharacteristic
290C4×C8C22⋊D5311yescharacteristic
291C4×C8C22×D5311yescharacteristic
292C4×C8C22×D5311yescharacteristic
293C24—41no—
294C2×Q8—1061no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—1061no—
300C2×D4—2121no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—84no—
305C22×C4—2121no—
306C22×C4#1171992211yescharacteristic
307C22×C4#1171992211yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309Q16—1062no—
310SD16—2122no—
311D8—24no—
312D8—22no—
313D8—1062no—
314D8#1171994514yes—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8C2×C22⋊D5311yescharacteristic
318C2×C8C2×C22⋊D5311yescharacteristic
319C2×C8C2×C22⋊D5311yescharacteristic
320C2×C8C23×D5311yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4—2121no—
326C4⋊C4—2121no—
327C4⋊C4—2121no—
328C4⋊C4—2121no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—2121no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C22⋊C4—2121no—
335C22⋊C4—2121no—
336C42—2121no—
337C42#1171992211yescharacteristic
338C42#1171992211yescharacteristic
339C42C2×C22⋊D5311yescharacteristic
340C42C2×C22⋊D5311yescharacteristic
341C16—1062no—
342C16—1062no—
343C23—44no—
344C23—42no—
345C23—82no—
346C23—82no—
347C23—2121no—
348C23C2×#1171992211yescharacteristic
349Q8—2121no—
350Q8—2121no—
351D4—24no—
352D4—44no—
353D4—42no—
354D4—4242no—
355D4—2121no—
356D4—2121no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—4242no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4C2×#1171992211yescharacteristic
370C2×C4C2×#1171992211yescharacteristic
371C2×C4C2×#1171992211yescharacteristic
372C2×C4C2×#1171992211yescharacteristic
373C2×C4C2×#1171992211yescharacteristic
374C2×C4C22×C22⋊D5311yescharacteristic
375C8—21no—
376C8—21no—
377C8—21no—
378C8C2×#1171994512yes—
379C22—21no—
380C22—21no—
381C22—44no—
382C22—88no—
383C22—82no—
384C22—82no—
385C22—4241no—
386C22—4241no—
387C22—4241no—
388C22#1681105912yes—
389C22C22×#1171992211yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—82no—
396C4—4241no—
397C4—4241no—
398C4—4241no—
399C4#1681113411yescharacteristic
400C4#1681142111yescharacteristic
401C2—22no—
402C2—84no—
403C2—8481no—
404C2#2028471511yescharacteristic
405C2C2×#1681105911yescharacteristic
406C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C4×D8×C53C2
socleC22×C53C22×D8
prime radical#20276627C2
hypercenterC2×C4×D8D53
hypocenterC53#33674
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. #20285285C2semidirect
2. #20276912C2semidirect
3. #20276895C2semidirect
4. #20276872C2semidirect
5. #20276770C2semidirect
6. #20276644C2semidirect
8. #20276589C2semidirect
9. #20276579C2semidirect
10. #20276518C2semidirect
11. C2×C4×D8×C53C2semidirect
15. #16811025C22semidirect
16. #16811007C22semidirect
21. C4×Dic212C22semidirect
23. C4×D424C22semidirect
24. #16810450C22semidirect
25. #16810448C22semidirect
26. C4⋊#11719908C22semidirect
27. #16810426C22semidirect
30. #16810423C22semidirect
33. #16810399C22semidirect
199. C2×C4×D8D53semidirect
246. C53#33674semidirect

Maximal subgroups

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

subgroupindex
#202852852
#202769122
#202768952
#202768722
#202767702
#202766442
#202766272
#202765892
#202765792
#202765182
C2×C4×D8×C532
#3367453