Finite Group Databasebeta

← #27279002

Subgroups — #27279002

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283885C211yesmaximal, characteristic
2#20275697C211yesmaximal, characteristic, complemented
3#20273201C211yesmaximal, characteristic
4#20273188C211yesmaximal, characteristic, complemented
5#20272609C211yesmaximal, characteristic, complemented
6#20272605C211yesmaximal, characteristic, complemented
7C53×#3172C211yesmaximal, characteristic
8#16810754—21no—
9#16810701—21no—
10#16810695—21no—
11#16810695—21no—
12#16810684—21no—
13#16810680C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810650—21no—
16#16810647C2211yescharacteristic
17#16810469C2211yescharacteristic
18#16810382—21no—
19#16810382—21no—
20#16810360—21no—
21#16810357—21no—
22Q8⋊#88322—21no—
23Q8⋊#88322—21no—
24C53×#533C2211yescharacteristic
25C53×#529C2211yescharacteristic
26C53×#501C411yescharacteristic, complemented
27C53×#497C411yescharacteristic, complemented
28C53×#491C411yescharacteristic, complemented
29C53×#489C411yescharacteristic, complemented
30C53×#470—21no—
31C53×#468—21no—
32C4×Q16×C53—21no—
33C4×SD16×C53—21no—
34C4×M16×C53C2211yescharacteristic
35C2×#11719917—21no—
36C2×#11719917D411yescharacteristic
37C2×#11719917D411yescharacteristic
38C22×#88322—21no—
39C2×C4×Dic53—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D411yescharacteristic
46C4⋊#88322D411yescharacteristic
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51#13133128—21no—
52#13133128—21no—
53C4×#88322—21no—
54C4×#88322—21no—
55#13133124—21no—
56#13133124—21no—
57C2×Q16×C53—21no—
58C2×SD16×C53—21no—
59C2×M16×C53—21no—
60C4⋊Q8×C53C2×C411yescharacteristic
61C4⋊Q8×C53C2×C411yescharacteristic
62#126×C53—21no—
63#125×C53—21no—
64#125×C53C2×C411yescharacteristic
65#124×C53C2×C411yescharacteristic
66#123×C53—21no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53D411yescharacteristic
70#121×C53D411yescharacteristic
71C4×Q8×C53—21no—
72C4×Q8×C53—21no—
73C4×Q8×C53D411yescharacteristic
74C4×D4×C53—21no—
75C4×D4×C53D411yescharacteristic
76#117×C53—21no—
77#117×C53—21no—
78C2×C42×C53C2311yescharacteristic
79#107×C53—21no—
80#106×C53—21no—
81#103×C53—21no—
82#103×C53—21no—
83#103×C53—21no—
84#102×C53—21no—
85#97×C53C2×C411yescharacteristic
86#97×C53C2×C411yescharacteristic
87C4×C8×C53—21no—
88C2×Q8×C53—21no—
89C2×Q8×C53—21no—
90C2×Q8×C53C22⋊C411yescharacteristic
91C2×D4×C53—21no—
92C2×D4×C53C22⋊C411yescharacteristic
93C22×C4×C53—21no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C22×Dic53—41no—
98Q16×C53—42no—
99SD16×C53—42no—
100M16×C53—42no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53C22⋊C411yescharacteristic
114C4⋊C4×C53C22⋊C411yescharacteristic
115C22⋊C4×C53—21no—
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C22⋊C4×C53—41no—
121C42×C53—21no—
122C42×C53—41no—
123C42×C53C22×C411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C4×Dic53—22no—
128C4×Dic53—41no—
129#11719917—41no—
130#11719917—41no—
131#11719917—81no—
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—21no—
135C2×#88322—21no—
136C2×#88322—41no—
137C2×#88322—81no—
138C23×C53—21no—
139C23×C53#12111yescharacteristic
140Q8×C53—41no—
141Q8×C53—42no—
142Q8×C53—41no—
143Q8×C53—41no—
144D4×C53—41no—
145D4×C53—81no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53C2×C22⋊C411yescharacteristic
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12311yescharacteristic
163C2×Dic53—42no—
164C2×Dic53—81no—
165C2×Dic53—81no—
166C8×C53—22no—
167C8×C53—41no—
168#88322—41no—
169#88322—41no—
170#88322—41no—
171#88322—41no—
172#88322—81no—
173#8998—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53#38611yescharacteristic
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—81no—
190C4×C53—81no—
191C4×C53#45511yescharacteristic
192C4×C53#45611yescharacteristic
193Dic53—82no—
194#3172D5311yescharacteristic
195#2995—531no—
196#1995—531no—
197#1500—531no—
198#1497—531no—
199#1300—531no—
200#1299—531no—
201C2×C53—41no—
202C2×C53—81no—
203C2×C53#188111yescharacteristic
204C2×C53#188211yescharacteristic
205C2×C53#188411yescharacteristic
206#533C2×D5311yescharacteristic
207#529C2×D5311yescharacteristic
208#501Dic5311yescharacteristic, complemented
209#497Dic5311yescharacteristic, complemented
210#491Dic5311yescharacteristic, complemented
211#489Dic5311yescharacteristic, complemented
212#470—21no—
213#468—21no—
214#446—1061no—
215C4×Q16—21no—
216C4×SD16—21no—
217#436—1061no—
218#435—1061no—
219#435—1061no—
220#433—1061no—
221#432—531no—
222#431—531no—
223#424—1061no—
224#423—531no—
225C4×M16C2×D5311yescharacteristic
226#376—531no—
227#339—1061no—
228#339—1061no—
229#328—1061no—
230#327—1061no—
231Q8⋊C8—1061no—
232Q8⋊C8—1061no—
233C53#899811yescharacteristic, complemented, Sylow
234C2×Q16—21no—
235C2×SD16—21no—
236C2×M16—21no—
237C2×M16—531no—
238C2×M16—531no—
239C2×M16—1061no—
240C22×C8—1061no—
241C4⋊Q8C2×Dic5311yescharacteristic
242C4⋊Q8C2×Dic5311yescharacteristic
243#126—21no—
244#125—21no—
245#125C2×Dic5311yescharacteristic
246#124C2×Dic5311yescharacteristic
247#123—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8C22⋊D5311yescharacteristic
251#121C22⋊D5311yescharacteristic
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C22⋊D5311yescharacteristic
255C4×D4—21no—
256C4×D4C22⋊D5311yescharacteristic
257#117—21no—
258#117—21no—
259C2×C42—1061no—
260C2×C42C22×D5311yescharacteristic
261#107—21no—
262#106—21no—
263C4⋊C8—531no—
264C4⋊C8—531no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270#103—21no—
271#103—21no—
272#103—21no—
273#102—21no—
274C22⋊C8—1061no—
275C22⋊C8—1061no—
276C22⋊C8—1061no—
277C22⋊C8—1061no—
278#97—1061no—
279#97—1061no—
280#97C2×Dic5311yescharacteristic
281#97C2×Dic5311yescharacteristic
282C4×C8—21no—
283C4×C8—1061no—
284C4×C8—1061no—
285#95—1061no—
286#95—1061no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8C22⋊Dic5311yescharacteristic
290C2×D4—21no—
291C2×D4C22⋊Dic5311yescharacteristic
292C22×C4—21no—
293C22×C4—2121no—
294C22×C4D4×D5311yescharacteristic
295C22×C4#1171994711yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297Q16—42no—
298SD16—42no—
299M16—42no—
300M16—4241no—
301M16—2121no—
302M16—2121no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—1061no—
306C2×C8—1061no—
307C2×C8—1061no—
308C2×C8—1061no—
309C2×C8—4241no—
310C2×C8—2121no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4C22⋊Dic5311yescharacteristic
322C4⋊C4C22⋊Dic5311yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—41no—
329C42—21no—
330C42—41no—
331C42—1062no—
332C42—2121no—
333C42D4×D5311yescharacteristic
334C42#1171994711yescharacteristic
335C42C22×Dic5311yescharacteristic
336C42C2×C22⋊D5311yescharacteristic
337C23—21no—
338C23#1313318911yescharacteristic
339Q8—41no—
340Q8—41no—
341Q8—41no—
342Q8—42no—
343D4—41no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4—2122no—
359C2×C4C2×C22⋊Dic5311yescharacteristic
360C2×C4D4×Dic5311yescharacteristic
361C2×C4D4×Dic5311yescharacteristic
362C2×C4#1313318511yescharacteristic
363C2×C4#1313318911yescharacteristic
364C2×C4#1313319811yescharacteristic
365C8—22no—
366C8—41no—
367C8—4241no—
368C8—2121no—
369C8—2121no—
370C8—2121no—
371C8—2121no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22#1681050311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C4—4242no—
390C4#1681079911yescharacteristic
391C4#1681080411yescharacteristic
392C2—41no—
393C2—81no—
394C2#2027507911yescharacteristic
395C2#2027508611yescharacteristic
396C2#2027509911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3172C2
socleC22×C53#386
prime radical#121×C53D4
cosocleC2×C42×C53C23
hypercenter#3172D53
hypocenterC53#8998
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3172

Complements

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

NH ≅ G/Ntype
2. #20275697C2semidirect
4. #20273188C2semidirect
5. #20272609C2semidirect
6. #20272605C2semidirect
26. C53×#501C4semidirect
27. C53×#497C4semidirect
28. C53×#491C4semidirect
29. C53×#489C4semidirect
208. #501Dic53semidirect
209. #497Dic53semidirect
210. #491Dic53semidirect
211. #489Dic53semidirect
233. C53#8998semidirect

Maximal subgroups

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

subgroupindex
#202838852
#202756972
#202732012
#202731882
#202726092
#202726052
C53×#31722
#899853