Finite Group Databasebeta

← #27275088

Subgroups — #27275088

Order 13568.

522 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#20274491C211yesmaximal, characteristic, complemented
2C8×C22⋊Dic53C211yesmaximal, characteristic, complemented
3#20273206C211yesmaximal, characteristic
4#20273206C211yesmaximal, characteristic
5#20272602C211yesmaximal, characteristic, complemented
6#20272600C211yesmaximal, characteristic, complemented
7C53×#2929C211yesmaximal, characteristic
8C2×C22⋊#88322C2211yescharacteristic
9C2×C22⋊#88322C2211yescharacteristic
10C2×#13133127—21no—
11C2×C8×Dic53—21no—
12C4×C22⋊Dic53C2211yescharacteristic
13C4×C22⋊Dic53C2211yescharacteristic
14#16810377—21no—
15#16810377—21no—
16#16810360C412yescomplemented
17#16810360C412yescomplemented
18#16810357C412yescomplemented
19#16810357C412yescomplemented
20C2×#131×C53C411yescharacteristic, complemented
21C23×C8×C53C2211yescharacteristic
22C53×#525C2211yescharacteristic
23C53×#435—21no—
24C53×#435—21no—
25C8×D4×C53—21no—
26C8×D4×C53—21no—
27C53×#424C411yescharacteristic, complemented
28C53×#408C411yescharacteristic, complemented
29C53×#407C2211yescharacteristic
30C53×#405C411yescharacteristic, complemented
31C22×#88322—21no—
32C22×#88322—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C22⋊#88322—21no—
38C22⋊#88322—21no—
39C22⋊#88322C2×C412yes—
40C22⋊#88322C2×C412yes—
41#13133127—42no—
42C8×Dic53—42no—
43#13133124C2×C412yes—
44#13133124C2×C412yes—
45C2×#41×C53C2×C411yescharacteristic
46C23×C4×C53C2311yescharacteristic
47#131×C53—42no—
48C2×M16×C53—21no—
49C2×M16×C53C2×C411yescharacteristic
50C2×M16×C53C2×C411yescharacteristic
51C22×C8×C53—21no—
52C22×C8×C53—42no—
53C22×C8×C53C2×C411yescharacteristic
54C22×C8×C53C2×C411yescharacteristic
55C22×C8×C53D411yescharacteristic
56C22×C8×C53D411yescharacteristic
57C22×C8×C53D411yescharacteristic
58C22×C8×C53D411yescharacteristic
59#123×C53—21no—
60C22⋊Q8×C53C2×C411yescharacteristic
61C22⋊Q8×C53C2×C411yescharacteristic
62#121×C53C2×C411yescharacteristic
63#121×C53C2×C411yescharacteristic
64#120×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67#117×C53C2×C411yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C22⋊C8×C53—21no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74#97×C53—21no—
75C4×C8×C53—21no—
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—41no—
86C22×C4×C53—41no—
87C22×C4×C53C22×C411yescharacteristic
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×Dic53—22no—
94C22×Dic53—22no—
95M16×C53—42no—
96M16×C53—41no—
97M16×C53—41no—
98C2×C8×C53—22no—
99C2×C8×C53—22no—
100C2×C8×C53—22no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—42no—
107C2×C8×C53C22⋊C411yescharacteristic
108C2×C8×C53C22⋊C411yescharacteristic
109C2×C8×C53C22⋊C411yescharacteristic
110C2×C8×C53C22⋊C411yescharacteristic
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53C4211yescharacteristic
114C4⋊C4×C53C4211yescharacteristic
115C22⋊C4×C53—21no—
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C42×C53—21no—
121C22⋊Dic53—24no—
122C22⋊Dic53—42no—
123C4×Dic53—41no—
124C4×Dic53—41no—
125C4×Dic53—41no—
126C4×Dic53—41no—
127C2×#88322—22no—
128C2×#88322—22no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
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×C53C2×C4211yescharacteristic
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×Dic53—24no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C8×C53—21no—
170C8×C53—21no—
171C8×C53—21no—
172C8×C53—41no—
173C8×C53—41no—
174C8×C53#9912yes—
175#88322—42no—
176#88322—42no—
177#8240—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#44312yes—
191C22×C53#44412yes—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—41no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53C2×#9911yescharacteristic
199C4×C53#41011yescharacteristic
200Dic53—44no—
201Dic53—82no—
202#2929D5311yescharacteristic
203#1752—531no—
204C8×C22⋊C4—531no—
205#1501—531no—
206#1501—531no—
207#1298—531no—
208#1297—531no—
209C2×C53—22no—
210C2×C53—22no—
211C2×C53—41no—
212C2×C53—81no—
213C2×C53C4×#9911yescharacteristic
214C2×C53#176411yescharacteristic
215C2×C53#176511yescharacteristic
216C2×#131Dic5311yescharacteristic, complemented
217C23×C8C2×D5311yescharacteristic
218#525C2×D5311yescharacteristic
219#435—21no—
220#435—21no—
221C8×D4—21no—
222C8×D4—21no—
223#424Dic5311yescharacteristic, complemented
224#408Dic5311yescharacteristic, complemented
225#407C2×D5311yescharacteristic
226C2×C22⋊C8—531no—
227C2×C22⋊C8—531no—
228#405Dic5311yescharacteristic, complemented
229C2×#97—1061no—
230C2×C4×C8—1061no—
231C4×C22⋊C4—531no—
232C4×C22⋊C4—531no—
233#336—1061no—
234#336—1061no—
235#328—532no—
236#328—532no—
237#327—532no—
238#327—532no—
239C53#824011yescharacteristic, complemented, Sylow
240C2×#41C2×Dic5311yescharacteristic
241C23×C4C22×D5311yescharacteristic
242#131—42no—
243C2×M16—21no—
244C2×M16C2×Dic5311yescharacteristic
245C2×M16C2×Dic5311yescharacteristic
246C22×C8—21no—
247C22×C8—42no—
248C22×C8—1061no—
249C22×C8—1061no—
250C22×C8C2×Dic5311yescharacteristic
251C22×C8C2×Dic5311yescharacteristic
252C22×C8C22⋊D5311yescharacteristic
253C22×C8C22⋊D5311yescharacteristic
254C22×C8C22⋊D5311yescharacteristic
255C22×C8C22⋊D5311yescharacteristic
256#123—21no—
257C22⋊Q8C4×D5311yescharacteristic
258C22⋊Q8C4×D5311yescharacteristic
259#121C4×D5311yescharacteristic
260#121C4×D5311yescharacteristic
261#120—21no—
262C4×D4—21no—
263C4×D4—21no—
264#117C2×Dic5311yescharacteristic
265C2×C22⋊C4—1061no—
266C2×C22⋊C4—1061no—
267C2×C42—1061no—
268C2×C42—1061no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—532no—
276C22⋊C8—532no—
277C22⋊C8—1061no—
278C22⋊C8—1061no—
279#97—21no—
280#97—2122no—
281C4×C8—21no—
282C4×C8—2122no—
283#95—532no—
284#95—532no—
285C24#1171994511yescharacteristic
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—41no—
295C22×C4—41no—
296C22×C4—1062no—
297C22×C4—1062no—
298C22×C4C2×C4×D5311yescharacteristic
299C22×C4C2×C4×D5311yescharacteristic
300C22×C4#1171994511yescharacteristic
301C22×C4C22×Dic5311yescharacteristic
302C22×C4C2×C22⋊D5311yescharacteristic
303C22×C4C2×C22⋊D5311yescharacteristic
304M16—42no—
305M16—41no—
306M16—41no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—1062no—
317C2×C8—1062no—
318C2×C8—2121no—
319C2×C8—2121no—
320C2×C8—2121no—
321C2×C8—2121no—
322C2×C8C22⋊Dic5311yescharacteristic
323C2×C8C22⋊Dic5311yescharacteristic
324C2×C8C22⋊Dic5311yescharacteristic
325C2×C8C22⋊Dic5311yescharacteristic
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—1064no—
336C22⋊C4—2122no—
337C42—21no—
338C42—2121no—
339C42—2121no—
340C42—2121no—
341C42—2121no—
342C23—22no—
343C23—22no—
344C23—21no—
345C23—41no—
346C23#1313317611yescharacteristic
347C23C4×C22⋊D5311yescharacteristic
348C23C4×C22⋊D5311yescharacteristic
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—1064no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4C2×C4×Dic5311yescharacteristic
375C2×C4C2×C22⋊Dic5311yescharacteristic
376C2×C4C4×C22⋊D5311yescharacteristic
377C2×C4C4×C22⋊D5311yescharacteristic
378C8—21no—
379C8—21no—
380C8—21no—
381C8—41no—
382C8—41no—
383C8—2122no—
384C8—2122no—
385C8#1313313212yes—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22C4×C22⋊Dic5311yescharacteristic
398C22#1681073812yes—
399C22#1681074312yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—4242no—
407C4—2124no—
408C4C2×#1313313211yescharacteristic
409C4#1681059711yescharacteristic
410C2—22no—
411C2—22no—
412C2—41no—
413C2—81no—
414C2C4×#1313313211yescharacteristic
415C2#2027455011yescharacteristic
416C2#2027455711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊Dic53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#2929C2
socleC22×C53C4×C22⋊C4
prime radicalC53×#525C22
cosocleC23×C4×C53C23
hypercenter#2929D53
hypocenterC53#8240
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #20274491C2semidirect
2. C8×C22⋊Dic53C2semidirect
5. #20272602C2semidirect
6. #20272600C2semidirect
16. #16810360C4semidirect
17. #16810360C4semidirect
18. #16810357C4semidirect
19. #16810357C4semidirect
20. C2×#131×C53C4semidirect
27. C53×#424C4semidirect
28. C53×#408C4semidirect
30. C53×#405C4semidirect
216. C2×#131Dic53semidirect
223. #424Dic53semidirect
224. #408Dic53semidirect
228. #405Dic53semidirect
239. C53#8240semidirect

Maximal subgroups

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

subgroupindex
#202744912
C8×C22⋊Dic532
#202732062
#202726022
#202726002
C53×#29292
#824053