Finite Group Databasebeta

← #27281665

Subgroups — #27281665

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#20283711C211yesmaximal, characteristic
2#20273330C211yesmaximal, characteristic, complemented
3#20273279C211yesmaximal, characteristic, complemented
4#20273253C211yesmaximal, characteristic
5#20272614C211yesmaximal, characteristic, complemented
6#20272597C211yesmaximal, characteristic, complemented
7C53×#3181C211yesmaximal, characteristic
8#16810754—21no—
9#16810695—21no—
10#16810695—21no—
11D4×#88322—21no—
12#16810684—21no—
13#16810680C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810579—21no—
16C2×#13133128C2211yescharacteristic
17#16810489C2211yescharacteristic
18#16810365—21no—
19#16810365—21no—
20#16810360—21no—
21#16810360—21no—
22#16810360—21no—
23#16810357—21no—
24C53×#533C2211yescharacteristic
25C53×#529C2211yescharacteristic
26C53×#501C411yescharacteristic, complemented
27C53×#497C411yescharacteristic, complemented
28C53×#491C411yescharacteristic, complemented
29C53×#489C411yescharacteristic, complemented
30C53×#467—21no—
31C53×#465—21no—
32C53×#441—21no—
33C53×#440—21no—
34C2×#97×C53C2211yescharacteristic
35C2×#11719917—21no—
36C22×#88322—21no—
37C22×#88322—21no—
38C2×C4⋊Dic53—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C22⋊#88322D411yescharacteristic
47C22⋊#88322D411yescharacteristic
48#13133128—21no—
49#13133128—21no—
50#13133128—21no—
51#13133128D411yescharacteristic
52#13133128D411yescharacteristic
53C4×#88322—21no—
54C4×#88322—21no—
55#13133124—21no—
56#13133124—21no—
57C2×Q16×C53—21no—
58C2×SD16×C53—21no—
59C22×C8×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×C53—21no—
86#97×C53C2×C411yescharacteristic
87#97×C53C2×C411yescharacteristic
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—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53—42no—
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—81no—
128#11719917—42no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322—21no—
132C2×#88322—21no—
133C2×#88322—42no—
134C2×#88322—42no—
135C2×#88322—41no—
136C23×C53—21no—
137C23×C53#12111yescharacteristic
138Q8×C53—41no—
139Q8×C53—41no—
140Q8×C53—41no—
141Q8×C53—81no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—41no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53C2×C22⋊C411yescharacteristic
157C2×C4×C53C4×D411yescharacteristic
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12311yescharacteristic
162C2×Dic53—81no—
163C2×Dic53—81no—
164C2×Dic53—81no—
165C8×C53—22no—
166C8×C53—41no—
167#88322—22no—
168#88322—41no—
169#88322—41no—
170#88322—41no—
171#88322—81no—
172#9431—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53#38611yescharacteristic
179C22×C53#47111yescharacteristic
180C22×C53#47311yescharacteristic
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—41no—
190C4×C53—81no—
191C4×C53—81no—
192Dic53—161no—
193#3181D5311yescharacteristic
194#2968—531no—
195#1531—531no—
196#1518—531no—
197#1512—531no—
198#1302—531no—
199#1295—531no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—81no—
203C2×C53#188411yescharacteristic
204C2×C53#190611yescharacteristic
205C2×C53#190811yescharacteristic
206#533C2×D5311yescharacteristic
207#529C2×D5311yescharacteristic
208#501Dic5311yescharacteristic, complemented
209#497Dic5311yescharacteristic, complemented
210#491Dic5311yescharacteristic, complemented
211#489Dic5311yescharacteristic, complemented
212#467—21no—
213#465—21no—
214#446—1061no—
215#441—21no—
216#440—21no—
217#435—1061no—
218#435—1061no—
219C8×D4—1061no—
220#433—1061no—
221#432—531no—
222#431—531no—
223#405—1061no—
224C2×#97—531no—
225C2×#97C2×D5311yescharacteristic
226#382—531no—
227#330—1061no—
228#330—1061no—
229#328—1061no—
230#328—1061no—
231#328—1061no—
232#327—1061no—
233C53#943111yescharacteristic, complemented, Sylow
234C2×Q16—21no—
235C2×SD16—21no—
236C2×M16—1061no—
237C22×C8—21no—
238C22×C8—1061no—
239C22×C8—1061no—
240C4⋊Q8C2×Dic5311yescharacteristic
241C4⋊Q8C2×Dic5311yescharacteristic
242#126—21no—
243#125—21no—
244#125C2×Dic5311yescharacteristic
245#124C2×Dic5311yescharacteristic
246#123—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8C22⋊D5311yescharacteristic
250#121C22⋊D5311yescharacteristic
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D5311yescharacteristic
254C4×D4—21no—
255C4×D4C22⋊D5311yescharacteristic
256#117—21no—
257#117—21no—
258C2×C4⋊C4—1061no—
259C2×C42C22×D5311yescharacteristic
260#107—21no—
261#106—21no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266#103—21no—
267#103—21no—
268#103—21no—
269#102—21no—
270C22⋊C8—531no—
271C22⋊C8—531no—
272C22⋊C8—1061no—
273C22⋊C8—1061no—
274C22⋊C8—1061no—
275#97—21no—
276#97—531no—
277#97—531no—
278#97—1061no—
279#97—1061no—
280#97—1061no—
281#97C2×Dic5311yescharacteristic
282#97C2×Dic5311yescharacteristic
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—2122no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—42no—
303C2×C8—1061no—
304C2×C8—1061no—
305C2×C8—1061no—
306C2×C8—1061no—
307C2×C8—2122no—
308C2×C8—2122no—
309C2×C8—2121no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—4241no—
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—
331C42D4×D5311yescharacteristic
332C42#1171994711yescharacteristic
333C42C22×Dic5311yescharacteristic
334C42C2×C22⋊D5311yescharacteristic
335C23—21no—
336C23#1313318911yescharacteristic
337Q8—41no—
338Q8—41no—
339Q8—41no—
340Q8—81no—
341D4—41no—
342D4—41no—
343D4—41no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4C2×C22⋊Dic5311yescharacteristic
359C2×C4D4×Dic5311yescharacteristic
360C2×C4D4×Dic5311yescharacteristic
361C2×C4#1313318511yescharacteristic
362C2×C4#1313318911yescharacteristic
363C2×C4#1313319811yescharacteristic
364C8—22no—
365C8—41no—
366C8—1062no—
367C8—4241no—
368C8—2121no—
369C8—2121no—
370C8—2121no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22#1681050311yescharacteristic
377C22#1681088811yescharacteristic
378C22#1681089611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—8481no—
391C2—21no—
392C2—21no—
393C2—81no—
394C2#2027509911yescharacteristic
395C2#2027521811yescharacteristic
396C2#2027523211yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3181C2
socleC22×C53#386
prime radical#121×C53D4
cosocleC2×C42×C53C23
hypercenter#3181D53
hypocenterC53#9431
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3181

Complements

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

NH ≅ G/Ntype
2. #20273330C2semidirect
3. #20273279C2semidirect
5. #20272614C2semidirect
6. #20272597C2semidirect
26. C53×#501C4semidirect
27. C53×#497C4semidirect
28. C53×#491C4semidirect
29. C53×#489C4semidirect
208. #501Dic53semidirect
209. #497Dic53semidirect
210. #491Dic53semidirect
211. #489Dic53semidirect
233. C53#9431semidirect

Maximal subgroups

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

subgroupindex
#202837112
#202733302
#202732792
#202732532
#202726142
#202725972
C53×#31812
#943153