Finite Group Databasebeta

← #27275086

Subgroups — #27275086

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283410C211yesmaximal, characteristic
2#20274494C211yesmaximal, characteristic, complemented
3#20274493C211yesmaximal, characteristic, complemented
4#20273206C211yesmaximal, characteristic
5#20272603C211yesmaximal, characteristic, complemented
6#20272601C211yesmaximal, characteristic, complemented
7C53×#1501C211yesmaximal, characteristic
8C2×#13133224C411yescharacteristic, complemented
9C23×#88322C2211yescharacteristic
10#16810695—21no—
11#16810695—21no—
12D4×#88322—21no—
13D4×#88322—21no—
14#16810650C411yescharacteristic, complemented
15#16810590C411yescharacteristic, complemented
16#16810587C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18#16810579C411yescharacteristic, complemented
19C2×#13133128—21no—
20C2×#13133127—21no—
21C4×C22⋊Dic53C2211yescharacteristic
22#16810378—21no—
23#16810377—21no—
24#16810360C412yescomplemented
25#16810357C412yescomplemented
26C53×#525C2211yescharacteristic
27C2×C22⋊C8×C53C2211yescharacteristic
28C4×C22⋊C4×C53C2211yescharacteristic
29C53×#328C412yescomplemented
30C53×#327C412yescomplemented
31#13133224—42no—
32C2×#11719917—21no—
33C2×#11719917C2×C411yescharacteristic
34C2×#11719917C2×C411yescharacteristic
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322—42no—
38C22×#88322C2×C411yescharacteristic
39C22×#88322C2×C411yescharacteristic
40C22×#88322D411yescharacteristic
41C22×#88322D411yescharacteristic
42C22×#88322D411yescharacteristic
43C22×#88322D411yescharacteristic
44C2×C22⋊Dic53—21no—
45C2×C4×Dic53—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322C2×C412yes—
54#13133128—21no—
55#13133128—42no—
56#13133127—42no—
57C4×#88322—21no—
58#13133124C2×C412yes—
59C2×#41×C53C2×C411yescharacteristic
60C23×C4×C53C2311yescharacteristic
61C22×C8×C53—21no—
62#123×C53—21no—
63C22⋊Q8×C53C2×C411yescharacteristic
64C22⋊Q8×C53C2×C411yescharacteristic
65#121×C53C2×C411yescharacteristic
66#121×C53C2×C411yescharacteristic
67#120×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70#117×C53C2×C411yescharacteristic
71C2×C22⋊C4×C53—21no—
72C2×C42×C53—21no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53C2×C412yes—
75#95×C53C2×C412yes—
76C24×C53#4111yescharacteristic
77#41×C53—41no—
78#41×C53—41no—
79C2×Q8×C53C4211yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53C4211yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×Dic53—22no—
95C2×C8×C53—22no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53C4211yescharacteristic
101C4⋊C4×C53C4211yescharacteristic
102C22⋊C4×C53—21no—
103C22⋊C4×C53—21no—
104C22⋊C4×C53—21no—
105C22⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—42no—
108C42×C53—21no—
109C42×C53—41no—
110C42×C53—41no—
111C22⋊Dic53—42no—
112C4×Dic53—41no—
113C4×Dic53—41no—
114#11719917—42no—
115#11719917—41no—
116#11719917—41no—
117C2×#88322—22no—
118C2×#88322—22no—
119C2×#88322—22no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—22no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—42no—
129C2×#88322C22⋊C411yescharacteristic
130C2×#88322C22⋊C411yescharacteristic
131C2×#88322C22⋊C411yescharacteristic
132C2×#88322C22⋊C411yescharacteristic
133C23×C53—22no—
134C23×C53—22no—
135C23×C53—21no—
136C23×C53—41no—
137C23×C53#11711yescharacteristic
138C23×C53C4×D411yescharacteristic
139C23×C53C4×D411yescharacteristic
140Q8×C53—41no—
141D4×C53—41no—
142D4×C53—42no—
143D4×C53—42no—
144D4×C53—42no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53C2×C4211yescharacteristic
163C2×C4×C53C2×C22⋊C411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C8×C53—42no—
170#88322—21no—
171#88322—21no—
172#88322—21no—
173#88322—42no—
174#88322—41no—
175#88322—41no—
176#88322#9912yes—
177#8239—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—44no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53C4×C22⋊C411yescharacteristic
190C22×C53#44412yes—
191C22×C53#44412yes—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—41no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—82no—
199C4×C53C2×#9911yescharacteristic
200C4×C53#41011yescharacteristic
201Dic53—82no—
202#2929—531no—
203#1752—531no—
204#1752—531no—
205#1501—531no—
206#1501D5311yescharacteristic
207#1298—531no—
208#1297—531no—
209C2×C53—22no—
210C2×C53—22no—
211C2×C53—41no—
212C2×C53—81no—
213C2×C53C4×#9911yescharacteristic
214C2×C53#176511yescharacteristic
215C2×C53#176511yescharacteristic
216C2×#131—531no—
217C23×C8—531no—
218#525C2×D5311yescharacteristic
219#435—1061no—
220#435—1061no—
221C8×D4—1061no—
222C8×D4—1061no—
223#424—531no—
224#408—531no—
225#407—531no—
226C2×C22⋊C8—531no—
227C2×C22⋊C8C2×D5311yescharacteristic
228#405—531no—
229C2×#97—1061no—
230C2×#97—1061no—
231C4×C22⋊C4—531no—
232C4×C22⋊C4C2×D5311yescharacteristic
233#336—1061no—
234#336—1061no—
235#328—532no—
236#328Dic5312yescomplemented
237#327—532no—
238#327Dic5312yescomplemented
239C53#823911yescharacteristic, complemented, Sylow
240C2×#41C4×D5311yescharacteristic
241C23×C4C22×D5311yescharacteristic
242#131—2122no—
243C2×M16—531no—
244C2×M16—531no—
245C2×M16—1061no—
246C22×C8—21no—
247C22×C8—531no—
248C22×C8—531no—
249C22×C8—531no—
250C22×C8—531no—
251C22×C8—531no—
252C22×C8—531no—
253C22×C8—1061no—
254C22×C8—1061no—
255C22×C8—2122no—
256#123—21no—
257C22⋊Q8C4×D5311yescharacteristic
258C22⋊Q8C2×Dic5311yescharacteristic
259#121C4×D5311yescharacteristic
260#121C2×Dic5311yescharacteristic
261#120—21no—
262C4×D4—21no—
263C4×D4—21no—
264#117C4×D5311yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—1061no—
267C2×C42—21no—
268C2×C42—1061no—
269C4⋊C8—1061no—
270C4⋊C8—1061no—
271C22⋊C8—21no—
272C22⋊C8—532no—
273C22⋊C8—1061no—
274C22⋊C8—1061no—
275C22⋊C8—1061no—
276C22⋊C8—1061no—
277C22⋊C8—1061no—
278C22⋊C8C2×Dic5312yes—
279#97—1061no—
280#97—2122no—
281#97—2122no—
282C4×C8—1061no—
283#95—532no—
284#95C2×Dic5312yes—
285C24#1171994711yescharacteristic
286#41—41no—
287#41—41no—
288C2×Q8C4×Dic5311yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4C4×Dic5311yescharacteristic
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—1062no—
298C22×C4C2×C4×D5311yescharacteristic
299C22×C4C2×C4×D5311yescharacteristic
300C22×C4D4×D5311yescharacteristic
301C22×C4D4×D5311yescharacteristic
302C22×C4#1171994711yescharacteristic
303C22×C4C22×Dic5311yescharacteristic
304M16—2122no—
305M16—2121no—
306M16—2121no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—531no—
311C2×C8—531no—
312C2×C8—531no—
313C2×C8—531no—
314C2×C8—1062no—
315C2×C8—1062no—
316C2×C8—1062no—
317C2×C8—1061no—
318C2×C8—1061no—
319C2×C8—1062no—
320C2×C8—2122no—
321C2×C8—2121no—
322C2×C8—2121no—
323C2×C8—2121no—
324C2×C8—2121no—
325C2×C8—2121no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4C4×Dic5311yescharacteristic
329C4⋊C4C4×Dic5311yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—42no—
336C22⋊C4—2122no—
337C42—21no—
338C42—41no—
339C42—41no—
340C42—2121no—
341C42—2121no—
342C23—22no—
343C23—22no—
344C23—21no—
345C23—41no—
346C23#1313317311yescharacteristic
347C23#1313317711yescharacteristic
348C23D4×Dic5311yescharacteristic
349Q8—41no—
350D4—41no—
351D4—42no—
352D4—42no—
353D4—42no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4C2×C4×Dic5311yescharacteristic
375C2×C4C22⋊C4×D5311yescharacteristic
376C2×C4#1313317711yescharacteristic
377C2×C4D4×Dic5311yescharacteristic
378C8—42no—
379C8—532no—
380C8—1061no—
381C8—1061no—
382C8—1061no—
383C8—2122no—
384C8—2121no—
385C8—2121no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22C22⋊C4×Dic5311yescharacteristic
398C22#1681074212yes—
399C22#1681074512yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—82no—
407C4—4242no—
408C4#99×D5311yescharacteristic
409C4#1681059511yescharacteristic
410C2—22no—
411C2—22no—
412C2—41no—
413C2—81no—
414C2#99×Dic5311yescharacteristic
415C2#2027456011yescharacteristic
416C2#2027456211yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1501C2
socleC22×C53C4×C22⋊C4
prime radicalC53×#525C22
cosocleC23×C4×C53C23
hypercenter#1501D53
hypocenterC53#8239
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1501

Complements

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

NH ≅ G/Ntype
2. #20274494C2semidirect
3. #20274493C2semidirect
5. #20272603C2semidirect
6. #20272601C2semidirect
8. C2×#13133224C4semidirect
14. #16810650C4semidirect
15. #16810590C4semidirect
18. #16810579C4semidirect
24. #16810360C4semidirect
25. #16810357C4semidirect
29. C53×#328C4semidirect
30. C53×#327C4semidirect
236. #328Dic53semidirect
238. #327Dic53semidirect
239. C53#8239semidirect

Maximal subgroups

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

subgroupindex
#202834102
#202744942
#202744932
#202732062
#202726032
#202726012
C53×#15012
#823953