Finite Group Databasebeta

← #27354210

Subgroups — #27354210

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284716C211yesmaximal, characteristic
2#20283793C211yesmaximal, characteristic, complemented
3#20283738C211yesmaximal, characteristic, complemented
4#20273578C212yesmaximal, complemented
5#20273543C212yesmaximal, complemented
6#20273538C212yesmaximal, complemented
7#20273509C212yesmaximal, complemented
8#20273157C211yesmaximal, characteristic, complemented
9#20273131C211yesmaximal, characteristic, complemented
10#20273010C211yesmaximal, characteristic, complemented
11C53×#1572C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14#16810937C412yes—
15#16810904C412yes—
16#16810819C412yes—
17#16810759C412yes—
18C4×D424C2212yes—
19C4×D424C2212yes—
20#16810700C2212yescomplemented
21#16810696—22no—
22C8×D212—22no—
23#16810688C2212yes—
24#16810688C2212yes—
25#16810688C2212yescomplemented
26#16810676C2211yescharacteristic
27#16810588—22no—
28C22⋊C8×D53—22no—
29C4×#11719917C2211yescharacteristic
30#16810350C2212yes—
31#16810350C2212yes—
32#16810350C2212yescomplemented
33#16810350C2212yescomplemented
34#16810345C2211yescharacteristic, complemented
35#16810345C2211yescharacteristic, complemented
36#16810344C2211yescharacteristic, complemented
37C8×#88322C2211yescharacteristic, complemented
38C53×#432C2211yescharacteristic
39C53×#431C2211yescharacteristic
40C53×#431C2211yescharacteristic
41C53×#335C2212yescomplemented
42C53×#334C2212yescomplemented
43C2×#11719945—21no—
44C22×D212C2×C411yescharacteristic
45C22×C4×D53—21no—
46C2×D424C2×C412yes—
47C2×D424C2×C412yes—
48C2×#11719917—21no—
49C2×#11719912—22no—
50C2×C8×D53—22no—
51C2×C8×D53—22no—
52C2×C8×D53—22no—
53#13133217C2×C411yescharacteristic
54#13133214C2×C411yescharacteristic
55#13133205—21no—
56#13133191—21no—
57#13133191C2×C412yes—
58C4×D212—21no—
59C4×D212—21no—
60C4×D212—21no—
61C4×D212C2×C412yes—
62C4×D212C2312yes—
63C4×D212C2312yes—
64C2×C4⋊Dic53C2×C411yescharacteristic
65C2×#11719921—21no—
66#13133151C2×C412yes—
67#13133151C2×C412yes—
68C4⋊#88322D412yes—
69#13133139C2×C412yes—
70#13133139C2×C412yes—
71#13133139C2×C412yes—
72#13133139C2×C412yes—
73C22⋊#88322D412yes—
74#13133129—22no—
75#13133129—22no—
76#13133129—22no—
77#13133129—22no—
78#13133128—21no—
79C4×#88322C2311yescharacteristic
80C4×#88322C2311yescharacteristic
81C4×#88322C2311yescharacteristic
82C4×#88322C2311yescharacteristic
83C2×C42×C53C2311yescharacteristic
84C4⋊C8×C53C2×C412yes—
85C4⋊C8×C53C2312yes—
86C4⋊C8×C53C2312yes—
87C22⋊C8×C53—21no—
88C22⋊C8×C53—21no—
89C22⋊C8×C53C2×C412yes—
90#97×C53—21no—
91C4×C8×C53—21no—
92C4×C8×C53C2311yescharacteristic
93C4×C8×C53C2311yescharacteristic
94C23×D53—21no—
95C22×C4×C53C22×C411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C2×C22⋊D53—22no—
99C22×Dic53—21no—
100#11719945—82no—
101C2×D212—21no—
102C2×D212—24no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212C22×C412yes—
106C2×D212C22×C412yes—
107C2×C4×D53—22no—
108C2×C4×D53—22no—
109C2×C4×D53—22no—
110C2×C4×D53—82no—
111C2×Dic106—21no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53C22×C412yes—
115C2×C8×C53C22×C412yes—
116C42×C53C22×C411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119C42×C53C2411yescharacteristic
120#11719921—22no—
121#11719921—22no—
122#11719921—22no—
123#11719921—22no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53C22×C412yes—
127C4⋊Dic53C22×C412yes—
128#11719917—42no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322C2×D412yes—
132C2×#88322C2×D412yes—
133D424—24no—
134D424—24no—
135#11719912—82no—
136C8×D53—82no—
137C8×D53—82no—
138C8×D53—82no—
139C23×C53C4×D411yescharacteristic
140C22×D53—22no—
141C22×D53—24no—
142C22×D53—22no—
143C22×D53—81no—
144C22×D53—81no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53C23×C411yescharacteristic
152C2×C4×C53C22×D411yescharacteristic
153C2×C4×C53C2×#4111yescharacteristic
154C22⋊D53—82no—
155C2×Dic53—22no—
156C2×Dic53—22no—
157C2×Dic53—81no—
158D212—24no—
159D212—22no—
160D212—22no—
161D212—22no—
162D212—81no—
163D212—81no—
164D212#13114yes—
165C4×D53—42no—
166C4×D53—42no—
167C4×D53—42no—
168C4×D53—42no—
169C4×D53—82no—
170Dic106—81no—
171C8×C53—22no—
172C8×C53—22no—
173C8×C53—42no—
174C8×C53—42no—
175#88322—22no—
176#88322—22no—
177#88322—22no—
178#88322—41no—
179#17922—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53C2×C4×D411yescharacteristic
184C2×D53—24no—
185C2×D53—42no—
186C2×D53—42no—
187C2×D53—42no—
188C2×D53—42no—
189C2×D53—42no—
190C2×D53—84no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53C2×#13112yes—
196C4×C53#57612yes—
197Dic53—42no—
198Dic53—42no—
199#3100—531no—
200#2981—531no—
201#2972—531no—
202#1586—532no—
203#1581—532no—
204#1580—532no—
205C8×D8—532no—
206#1572D5311yescharacteristic, complemented
207#1490—531no—
208#1484—531no—
209#1453—531no—
210C2×C53—41no—
211C2×C53#292911yescharacteristic
212C2×C53#294811yescharacteristic
213C2×C53#295611yescharacteristic
214D53—44no—
215D53—82no—
216#532—531no—
217C2×C4×D4—531no—
218#480—532no—
219#474—532no—
220#459—532no—
221#447—532no—
222C4×D8—532no—
223C4×D8—532no—
224#436—532no—
225#435—1062no—
226C8×D4—532no—
227C8×D4—532no—
228C8×D4—532no—
229C8×D4—1062no—
230#432C2×D5311yescharacteristic
231#431—531no—
232#431C2×D5311yescharacteristic
233#431C2×D5311yescharacteristic
234#408—1062no—
235C2×C22⋊C8—1062no—
236C4×M16—531no—
237#335C2×D5312yescomplemented
238#334C2×D5312yescomplemented
239D4⋊C8—532no—
240D4⋊C8—532no—
241D4⋊C8—532no—
242D4⋊C8—532no—
243#322—531no—
244#322—531no—
245#322—531no—
246C82—531no—
247C53#1792211yescharacteristic, complemented, Sylow
248C2×#41—1061no—
249C22×D4—531no—
250C23×C4—1061no—
251C2×D8—532no—
252C2×D8—532no—
253C2×M16—1062no—
254C2×M16—1061no—
255C22×C8—1062no—
256C22×C8—1062no—
257C22×C8—1062no—
258C4⋊Q8—531no—
259#127—531no—
260#124—1061no—
261#121—532no—
262#121—1061no—
263C4×D4—532no—
264C4×D4—532no—
265C4×D4—532no—
266C4×D4—1061no—
267C4×D4—1061no—
268C4×D4—1061no—
269C2×C4⋊C4—531no—
270C2×C22⋊C4—1061no—
271C2×C42C22×D5311yescharacteristic
272#107—532no—
273#107—532no—
274C4⋊C8—532no—
275C4⋊C8C4×D5312yes—
276C4⋊C8C22×D5312yes—
277C4⋊C8C22×D5312yes—
278#102—532no—
279#102—532no—
280#102—532no—
281#102—532no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—532no—
285C22⋊C8—1062no—
286C22⋊C8—1062no—
287C22⋊C8—1062no—
288C22⋊C8—1062no—
289C22⋊C8C4×D5312yes—
290#97—21no—
291#97—1061no—
292C4×C8—21no—
293C4×C8—531no—
294C4×C8—531no—
295C4×C8—531no—
296C4×C8—531no—
297C4×C8C22×D5311yescharacteristic
298C4×C8C22×D5311yescharacteristic
299C24—1061no—
300#41—4242no—
301C2×Q8—1061no—
302C2×D4—532no—
303C2×D4—532no—
304C2×D4—1064no—
305C2×D4—1062no—
306C2×D4—1061no—
307C2×D4—1061no—
308C2×D4—1061no—
309C22×C4—1061no—
310C22×C4—1062no—
311C22×C4—1062no—
312C22×C4—1062no—
313C22×C4—4242no—
314C22×C4C2×C4×D5311yescharacteristic
315C22×C4D4×D5311yescharacteristic
316C22×C4#1171994911yescharacteristic
317D8—1064no—
318D8—1064no—
319M16—4242no—
320M16—2122no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—532no—
324C2×C8—532no—
325C2×C8—1061no—
326C2×C8—1061no—
327C2×C8—4242no—
328C2×C8—4242no—
329C2×C8—4242no—
330C2×C8C2×C4×D5312yes—
331C2×C8C2×C4×D5312yes—
332C4⋊C4—532no—
333C4⋊C4—532no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C22⋊C4—1062no—
337C22⋊C4—1062no—
338C22⋊C4—1062no—
339C22⋊C4—1062no—
340C42C2×C4×D5311yescharacteristic
341C42D4×D5311yescharacteristic
342C42#1171994911yescharacteristic
343C42C23×D5311yescharacteristic
344C23—1064no—
345C23—1062no—
346C23—1062no—
347C23—4241no—
348C23—4241no—
349C23#1313317911yescharacteristic
350Q8—4241no—
351D4—534no—
352D4—1064no—
353D4—1062no—
354D4—1062no—
355D4—1062no—
356D4—4241no—
357D4—4241no—
358D4—4242no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—1062no—
363C2×C4—1062no—
364C2×C4—4242no—
365C2×C4—4241no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4#1313317911yescharacteristic
371C2×C4#1313317911yescharacteristic
372C2×C4#1313317911yescharacteristic
373C2×C4C22×C4×D5311yescharacteristic
374C2×C4C2×D4×D5311yescharacteristic
375C2×C4C2×#1171994911yescharacteristic
376C8—22no—
377C8—22no—
378C8—42no—
379C8—42no—
380C8—1062no—
381C8—1062no—
382C8—1062no—
383C8—2121no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—1064no—
388C22—4244no—
389C22—2122no—
390C22—2122no—
391C22—2122no—
392C22—2122no—
393C22—2122no—
394C22C2×#1313317911yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—2122no—
400C4—2122no—
401C4#131×D5312yes—
402C4#1681142512yes—
403C2—41no—
404C2—4242no—
405C2—2124no—
406C2#2028340811yescharacteristic
407C2#2028356611yescharacteristic
408C2#2028362411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1572C2
socleC22×C53C2×C4×D4
prime radical#20284716C2
cosocleC42×C53C24
hypercenter#1572D53
hypocenterC53#17922
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
2. #20283793C2semidirect
3. #20283738C2semidirect
4. #20273578C2semidirect
5. #20273543C2semidirect
6. #20273538C2semidirect
7. #20273509C2semidirect
8. #20273157C2semidirect
9. #20273131C2semidirect
10. #20273010C2semidirect
11. C53×#1572C2semidirect
20. #16810700C22semidirect
25. #16810688C22semidirect
32. #16810350C22semidirect
33. #16810350C22semidirect
34. #16810345C22semidirect
35. #16810345C22semidirect
36. #16810344C22semidirect
37. C8×#88322C22semidirect
41. C53×#335C22semidirect
42. C53×#334C22semidirect
206. #1572D53semidirect
237. #335C2×D53semidirect
238. #334C2×D53semidirect
247. C53#17922semidirect

Maximal subgroups

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

subgroupindex
#202847162
#202837932
#202837382
#202735782
#202735432
#202735382
#202735092
#202731572
#202731312
#202730102
C53×#15722
#1792253