Finite Group Databasebeta

← #27287120

Subgroups — #27287120

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279954C211yesmaximal, characteristic, complemented
2#20275642C211yesmaximal, characteristic
3#20275641C211yesmaximal, characteristic
4#20275521C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×C4⋊#88322C2211yescharacteristic
10C2×#13133139—22no—
11C2×#13133139—21no—
12C2×#13133139—21no—
13C2×#13133137—21no—
14C2×#13133137—21no—
15#16810565—22no—
16#16810521—21no—
17#16810519—21no—
18#16810493C2211yescharacteristic
19#16810489—21no—
20#16810488—21no—
21C4×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊C8×C53C2211yescharacteristic
23C2×C4×C8×C53C2211yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C22×D212—21no—
26C22×D212—21no—
27C22×D212—21no—
28C22×#88322—21no—
29#13133214—21no—
30#13133214—21no—
31#13133214—21no—
32#13133214Q812yes—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53D412yes—
38C2×#11719919—22no—
39C2×C4×Dic53—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42#13133139—42no—
43#13133139—42no—
44#13133139—44no—
45#13133137—42no—
46#13133137—42no—
47#13133124—22no—
48#13133124—22no—
49#13133124—22no—
50#13133124—21no—
51C22×C8×C53—21no—
52C22×C8×C53—21no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C42×C53—21no—
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4×C8×C53—21no—
60C4×C8×C53—21no—
61#95×C53—21no—
62C23×D53—41no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—22no—
71C2×D212—22no—
72C2×D212—22no—
73C2×D212—22no—
74C2×D212—42no—
75C2×D212—42no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×C8×C53—22no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C42×C53—41no—
98C42×C53—41no—
99C42×C53C2×Q811yescharacteristic
100C42×C53#4111yescharacteristic
101C42×C53#4111yescharacteristic
102C42×C53#4111yescharacteristic
103C4⋊Dic53—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—41no—
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110C4⋊Dic53—41no—
111C4⋊Dic53—42no—
112#11719919—82no—
113C4×Dic53—41no—
114C4×Dic53—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C23×C53#12011yescharacteristic
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122C22×D53—81no—
123C22×D53—81no—
124C22×D53—81no—
125C22×D53—81no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53C22⋊Q811yescharacteristic
135C2×C4×C53C22⋊Q811yescharacteristic
136C2×C4×C53C22⋊Q811yescharacteristic
137C2×C4×C53#12411yescharacteristic
138C2×C4×C53#12411yescharacteristic
139C2×C4×C53#12411yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×C4×C53#13511yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—41no—
155C2×Dic53—42no—
156C2×Dic53—41no—
157D212—44no—
158D212—42no—
159D212—42no—
160D212—42no—
161D212—42no—
162D212—82no—
163D212—81no—
164D212—81no—
165D212—81no—
166D212—81no—
167C8×C53—42no—
168C8×C53—81no—
169#88322—81no—
170#10351—531nomaximal, Sylow
171C22×C53#39311yescharacteristic
172C22×C53#44911yescharacteristic
173C22×C53#44911yescharacteristic
174C22×C53#44911yescharacteristic
175C22×C53#44911yescharacteristic
176C22×C53#45311yescharacteristic
177C22×C53#45411yescharacteristic
178C2×D53—82no—
179C2×D53—82no—
180C2×D53—82no—
181C2×D53—161no—
182C2×D53—161no—
183C2×D53—161no—
184C2×D53—161no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—81no—
188C4×C53—81no—
189C4×C53#47811yescharacteristic
190C4×C53#47811yescharacteristic
191C4×C53#47811yescharacteristic
192C4×C53#47811yescharacteristic
193C4×C53#48811yescharacteristic
194C4×C53#48811yescharacteristic
195C4×C53#49011yescharacteristic
196C4×C53#49011yescharacteristic
197Dic53—82no—
198Dic53—81no—
199Dic53—81no—
200#2571—531no—
201#1982—531no—
202#1982—531no—
203#1957—531no—
204#1296—531no—
205#1279—531no—
206#1279D5311yescharacteristic, complemented
207C2×C53#162011yescharacteristic
208C2×C53#162411yescharacteristic
209C2×C53#162811yescharacteristic
210C2×C53#162811yescharacteristic
211C2×C53#202511yescharacteristic
212C2×C53#202511yescharacteristic
213C2×C53#202711yescharacteristic
214D53—162no—
215C2×#127—531no—
216C2×C4⋊C8—531no—
217C2×C4⋊C8C2×D5311yescharacteristic
218C2×#102—1062no—
219C2×#102—1061no—
220C2×#102—1061no—
221C2×#102—1061no—
222C2×#102—1061no—
223C2×C4×C8C2×D5311yescharacteristic
224#400—1062no—
225#389—1061no—
226#389—1061no—
227#384—531no—
228#382—1061no—
229#382—1061no—
230C4×C4⋊C4—531no—
231C4×C4⋊C4C2×D5311yescharacteristic
232C53#1035111yescharacteristic, complemented, Sylow
233C22×D4—1061no—
234C22×D4—1061no—
235C22×D4—1061no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—1061no—
239#127—532no—
240#127—1061no—
241#127—1061no—
242#127—1061no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—532no—
246C2×C4⋊C4—1062no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4D21211yescharacteristic
250C2×C4⋊C4D21211yescharacteristic
251C2×C42—21no—
252C2×C42—1061no—
253C2×C42C22×D5311yescharacteristic
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—1061no—
257C4⋊C8—1061no—
258#102—2122no—
259#102—2122no—
260#102—2122no—
261#102—2122no—
262#102—2124no—
263C4×C8—21no—
264C4×C8—21no—
265#95—21no—
266#95—1062no—
267#95—1062no—
268#95—1062no—
269#95—1061no—
270C24—2121no—
271C2×D4—1062no—
272C2×D4—1062no—
273C2×D4—1062no—
274C2×D4—2122no—
275C2×D4—2122no—
276C2×D4—2121no—
277C2×D4—2121no—
278C2×D4—2121no—
279C2×D4—2121no—
280C2×D4—2121no—
281C2×D4—2121no—
282C2×D4—2121no—
283C2×D4—2121no—
284C2×D4—2121no—
285C2×D4—2121no—
286C2×D4—2121no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1062no—
292C22×C4C2×D21211yescharacteristic
293C22×C4D4×D5311yescharacteristic
294C22×C4D4×D5311yescharacteristic
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—2121no—
302C2×C8—2121no—
303C2×C8—2121no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—4242no—
313C4⋊C4—2122no—
314C4⋊C4—2121no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C4⋊C4—2121no—
318C42—41no—
319C42—41no—
320C42—2121no—
321C42—2121no—
322C42#1171994511yescharacteristic
323C42#1171994511yescharacteristic
324C42#1171994711yescharacteristic
325C42Q8×D5311yescharacteristic
326C23—4241no—
327C23—4241no—
328C23—4241no—
329C23—4241no—
330C23—4241no—
331C23—4241no—
332C23—2122no—
333C23#1313318711yescharacteristic
334D4—4242no—
335D4—4241no—
336D4—4241no—
337D4—4241no—
338D4—4241no—
339D4—2124no—
340D4—2122no—
341D4—2122no—
342D4—2122no—
343D4—2122no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2122no—
356C2×C4—2121no—
357C2×C4—2122no—
358C2×C4—2121no—
359C2×C4—2122no—
360C2×C4—2121no—
361C2×C4#1313319511yescharacteristic
362C2×C4#1313319511yescharacteristic
363C2×C4#1313319711yescharacteristic
364C2×C4#1313320311yescharacteristic
365C2×C4#1313320311yescharacteristic
366C2×C4#1313320511yescharacteristic
367C2×C4#1313323811yescharacteristic
368C2×C4#1313323911yescharacteristic
369C2×C4#1313323911yescharacteristic
370C2×C4#1313324011yescharacteristic
371C2×C4#1313324111yescharacteristic
372C2×C4#1313324411yescharacteristic
373C2×C4#1313324511yescharacteristic
374C2×C4#1313324511yescharacteristic
375C8—42no—
376C8—81no—
377C8—4241no—
378C22—8481no—
379C22—8481no—
380C22—8481no—
381C22—8481no—
382C22—4242no—
383C22—4242no—
384C22—4242no—
385C22#1681053611yescharacteristic
386C22#1681076911yescharacteristic
387C22#1681077011yescharacteristic
388C22#1681077011yescharacteristic
389C22#1681077111yescharacteristic
390C22#1681079211yescharacteristic
391C22#1681079711yescharacteristic
392C4—21no—
393C4—21no—
394C4—81no—
395C4—81no—
396C4—4242no—
397C4—4241no—
398C4—4241no—
399C4#1681092211yescharacteristic
400C4#1681092411yescharacteristic
401C4#1681092411yescharacteristic
402C4#1681092611yescharacteristic
403C4#1681097811yescharacteristic
404C4#1681097911yescharacteristic
405C4#1681098611yescharacteristic
406C4#1681098811yescharacteristic
407C2—8482no—
408C2#2027377611yescharacteristic
409C2#2027380011yescharacteristic
410C2#2027382711yescharacteristic
411C2#2027382811yescharacteristic
412C2#2027586011yescharacteristic
413C2#2027586211yescharacteristic
414C2#2027587011yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#1279D53
hypocenterC53#10351
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. #20279954C2semidirect
5. #20272599C2semidirect
6. #20272555C2semidirect
7. C53×#1279C2semidirect
206. #1279D53semidirect
232. C53#10351semidirect

Maximal subgroups

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

subgroupindex
#202799542
#202756422
#202756412
#202755212
#202725992
#202725552
C53×#12792
#1035153