Finite Group Databasebeta

← #27299252

Subgroups — #27299252

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283419C211yesmaximal, characteristic
2#20276356C211yesmaximal, characteristic
3#20274518C211yesmaximal, characteristic, complemented
4#20274517C211yesmaximal, characteristic, complemented
5#20272837C211yesmaximal, characteristic, complemented
6#20272835C211yesmaximal, characteristic, complemented
7C53×#2120C211yesmaximal, characteristic
8C2×#13133224C2211yescharacteristic
9C22×#11719917C411yescharacteristic, complemented
10#16810701—21no—
11#16810701—21no—
12#16810695—21no—
13#16810695—21no—
14#16810650C411yescharacteristic, complemented
15#16810607C2211yescharacteristic
16#16810597C2211yescharacteristic
17#16810590C2211yescharacteristic
18#16810587C411yescharacteristic, complemented
19#16810581—21no—
20#16810579—21no—
21#16810579C411yescharacteristic, complemented
22#16810413C412yescomplemented
23#16810410C412yescomplemented
24#16810395—21no—
25#16810393—21no—
26C53×#525C2211yescharacteristic
27C53×#413C2211yescharacteristic
28C53×#410C2211yescharacteristic
29C53×#352C412yescomplemented
30C53×AΣL(1,𝔽16)C412yescomplemented
31#13133224—41no—
32#13133224—41no—
33C2×#11719917—21no—
34C2×#11719917—21no—
35C2×#11719917—21no—
36C2×#11719917—41no—
37C2×#11719917—41no—
38C2×#11719917C2×C411yescharacteristic
39C2×#11719917C2×C411yescharacteristic
40C2×#11719917D411yescharacteristic
41C2×#11719917D411yescharacteristic
42C22×#88322C2×C411yescharacteristic
43C22×#88322C2×C411yescharacteristic
44C22×#88322D411yescharacteristic
45C22×#88322D411yescharacteristic
46#13133176—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49#13133136—21no—
50#13133134C2×C412yes—
51#13133132—21no—
52#13133132C2×C412yes—
53C22⋊#88322—21no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56C22⋊#88322—21no—
57#13133128—21no—
58#13133128—41no—
59#13133127—41no—
60C4×#88322—21no—
61C4×#88322—41no—
62C8×Dic53—41no—
63C2×#41×C53C2311yescharacteristic
64C23×C4×C53C2×C411yescharacteristic
65C2×M16×C53—21no—
66#123×C53C2×C411yescharacteristic
67#123×C53C2×C411yescharacteristic
68C22⋊Q8×C53—21no—
69#121×C53—21no—
70#120×C53C2×C411yescharacteristic
71#120×C53C2×C411yescharacteristic
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74#117×C53—21no—
75#117×C53C2×C411yescharacteristic
76#101×C53—21no—
77#100×C53C2×C412yes—
78#99×C53—21no—
79#99×C53C2×C412yes—
80C24×C53C4211yescharacteristic
81#41×C53—41no—
82#41×C53—41no—
83C2×Q8×C53#4111yescharacteristic
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53C22×C411yescharacteristic
87C2×D4×C53C22×C411yescharacteristic
88C2×D4×C53#4111yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53C4211yescharacteristic
94C22×C4×C53C22×C411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97M16×C53—22no—
98M16×C53—41no—
99C2×C8×C53—41no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—41no—
104C22⋊C4×C53—21no—
105C22⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—21no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53C4211yescharacteristic
110C22⋊C4×C53C4211yescharacteristic
111C42×C53—21no—
112C42×C53—41no—
113C22⋊Dic53—22no—
114C4⋊Dic53—41no—
115C4×Dic53—41no—
116#11719917—22no—
117#11719917—41no—
118#11719917—41no—
119#11719917—41no—
120#11719917—42no—
121#11719917—41no—
122#11719917—41no—
123#11719917—41no—
124#11719917—41no—
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322—21no—
132C2×#88322—21no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322C22⋊C411yescharacteristic
136C2×#88322C22⋊C411yescharacteristic
137C2×#88322C22⋊C411yescharacteristic
138C2×#88322C22⋊C411yescharacteristic
139C23×C53—21no—
140C23×C53—41no—
141C23×C53—81no—
142C23×C53C2×C4211yescharacteristic
143C23×C53C4×D411yescharacteristic
144C23×C53C4×D411yescharacteristic
145Q8×C53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—41no—
149D4×C53—81no—
150D4×C53—81no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—42no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53C2×C22⋊C411yescharacteristic
166C2×C4×C53#11711yescharacteristic
167C2×C4×C53C4×D411yescharacteristic
168C2×C4×C53C4×D411yescharacteristic
169C2×Dic53—42no—
170C8×C53—42no—
171#88322—21no—
172#88322—21no—
173#88322—21no—
174#88322—42no—
175#88322—41no—
176#88322—41no—
177#88322#9911yescharacteristic
178#88322#9911yescharacteristic
179#12624—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—82no—
190C22×C53C4×C22⋊C411yescharacteristic
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—41no—
195C4×C53—41no—
196C4×C53—41no—
197C4×C53—82no—
198C4×C53C2×#9911yescharacteristic
199C4×C53#41011yescharacteristic
200Dic53—82no—
201#2930—531no—
202#2120—531no—
203#2120D5311yescharacteristic
204#1757—531no—
205#1757—531no—
206#1392—531no—
207#1391—531no—
208C2×C53—21no—
209C2×C53—41no—
210C2×C53—41no—
211C2×C53—41no—
212C2×C53—81no—
213C2×C53C4×#9911yescharacteristic
214C2×#131—531no—
215C22×M16—531no—
216#525C2×D5311yescharacteristic
217#436—1061no—
218#436—1061no—
219#435—1061no—
220#435—1061no—
221#424—531no—
222#413—531no—
223#413C2×D5311yescharacteristic
224#410—531no—
225#410C2×D5311yescharacteristic
226#408—531no—
227#407—531no—
228#405—531no—
229#405—1061no—
230#405—1061no—
231#352—532no—
232#352Dic5312yescomplemented
233AΣL(1,𝔽16)—532no—
234AΣL(1,𝔽16)Dic5312yescomplemented
235#344—1061no—
236#344—1061no—
237C53#1262411yescharacteristic, complemented, Sylow
238C2×#41C22×D5311yescharacteristic
239C23×C4C4×D5311yescharacteristic
240#131—2121no—
241#131—2121no—
242C2×M16—21no—
243C2×M16—531no—
244C2×M16—531no—
245C2×M16—531no—
246C2×M16—531no—
247C2×M16—1061no—
248C2×M16—1061no—
249C2×M16—1061no—
250C2×M16—2121no—
251C2×M16—2121no—
252C22×C8—531no—
253C22×C8—531no—
254C22×C8—531no—
255C22×C8—531no—
256#123C4×D5311yescharacteristic
257#123C2×Dic5311yescharacteristic
258C22⋊Q8—21no—
259#121—21no—
260#120C4×D5311yescharacteristic
261#120C2×Dic5311yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264#117—21no—
265#117—1061no—
266#117C4×D5311yescharacteristic
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269#101—21no—
270#101—1061no—
271#100—532no—
272#100C2×Dic5312yes—
273#99—21no—
274#99—532no—
275#99—1061no—
276#99C2×Dic5312yes—
277C22⋊C8—1061no—
278C22⋊C8—1061no—
279C22⋊C8—1061no—
280C22⋊C8—1061no—
281#97—1061no—
282#97—2121no—
283#97—2121no—
284C4×C8—1061no—
285C4×C8—2121no—
286C4×C8—2121no—
287C24C4×Dic5311yescharacteristic
288#41—41no—
289#41—41no—
290C2×Q8#1171994711yescharacteristic
291C2×D4—21no—
292C2×D4—21no—
293C2×D4C2×C4×D5311yescharacteristic
294C2×D4#1171994711yescharacteristic
295C2×D4C22×Dic5311yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C4×Dic5311yescharacteristic
301C22×C4C2×C4×D5311yescharacteristic
302C22×C4D4×D5311yescharacteristic
303C22×C4D4×D5311yescharacteristic
304M16—22no—
305M16—41no—
306M16—1062no—
307M16—2121no—
308M16—2121no—
309M16—2121no—
310M16—2121no—
311M16—2121no—
312M16—2122no—
313M16—2121no—
314M16—2121no—
315C2×C8—41no—
316C2×C8—531no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—531no—
320C2×C8—1061no—
321C2×C8—1061no—
322C2×C8—1061no—
323C2×C8—1061no—
324C2×C8—1061no—
325C2×C8—1061no—
326C2×C8—1061no—
327C2×C8—1061no—
328C2×C8—2121no—
329C2×C8—2121no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—2121no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—22no—
340C22⋊C4—1062no—
341C22⋊C4C4×Dic5311yescharacteristic
342C22⋊C4C4×Dic5311yescharacteristic
343C42—21no—
344C42—41no—
345C42—2121no—
346C23—21no—
347C23—41no—
348C23—81no—
349C23C2×C4×Dic5311yescharacteristic
350C23#1313317711yescharacteristic
351C23D4×Dic5311yescharacteristic
352Q8—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—2122no—
373C2×C4C22⋊C4×D5311yescharacteristic
374C2×C4#1313317311yescharacteristic
375C2×C4#1313317711yescharacteristic
376C2×C4D4×Dic5311yescharacteristic
377C8—42no—
378C8—531no—
379C8—531no—
380C8—1061no—
381C8—1061no—
382C8—1061no—
383C8—2122no—
384C8—2121no—
385C8—2121no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—82no—
396C22C22⋊C4×Dic5311yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—82no—
404C4—4242no—
405C4#99×D5311yescharacteristic
406C4#1681059511yescharacteristic
407C2—21no—
408C2—41no—
409C2—41no—
410C2—41no—
411C2—81no—
412C2#99×Dic5311yescharacteristic
subgroupG/H
centerC4#99×D53
commutatorC23×C53C2×C42
FrattiniC2×#41C22×D53
FittingC53×#2120C2
socleC2×C53C4×#99
prime radicalC53×#525C22
cosocleC2×#41×C53C23
hypercenter#2120D53
hypocenterC53#12624
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2120

Complements

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

NH ≅ G/Ntype
3. #20274518C2semidirect
4. #20274517C2semidirect
5. #20272837C2semidirect
6. #20272835C2semidirect
9. C22×#11719917C4semidirect
14. #16810650C4semidirect
18. #16810587C4semidirect
21. #16810579C4semidirect
22. #16810413C4semidirect
23. #16810410C4semidirect
29. C53×#352C4semidirect
30. C53×AΣL(1,𝔽16)C4semidirect
232. #352Dic53semidirect
234. AΣL(1,𝔽16)Dic53semidirect
237. C53#12624semidirect

Maximal subgroups

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

subgroupindex
#202834192
#202763562
#202745182
#202745172
#202728372
#202728352
C53×#21202
#1262453