Finite Group Databasebeta

← #27369388

Subgroups — #27369388

Order 13568.

740 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#20285630C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic, complemented
3#20283796C211yesmaximal, characteristic, complemented
4#20274161C214yesmaximal, complemented
5#20274073C214yesmaximal, complemented
6#20273491C211yesmaximal, characteristic, complemented
7#20273384C211yesmaximal, characteristic, complemented
8#20273364C211yesmaximal, characteristic, complemented
9C53×#3037C211yesmaximal, characteristic, complemented
10C22×#11719924—21no—
11C2×#13133214C2211yescharacteristic
12#16810999C2212yes—
13#16810971C2212yes—
14#16810823—22no—
15#16810823C2214yes—
16#16810818—22no—
17#16810756—22no—
18Q8×#88322—22no—
19Q8×#88322C2214yescomplemented
20#16810676—22no—
21#16810676C2211yescharacteristic
22#16810676C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133137—21no—
25#16810375C2214yescomplemented
26#16810369C2212yescomplemented
27#16810367C2212yescomplemented
28#16810366C2214yes—
29C2×#137×C53—21no—
30C53×#532C2211yescharacteristic
31C2×C4×Q8×C53C2211yescharacteristic
32C53×#464—22no—
33C53×#462C2212yescomplemented
34C53×#460C2212yescomplemented
35C53×#423C2211yescharacteristic
36C53×#417—21no—
37C22×D212—21no—
38C22×D212—21no—
39C22×D212—21no—
40C2×#11719924—24no—
41C2×#11719924—44no—
42C2×#11719922—24no—
43C22×#88322—21no—
44#13133214—21no—
45#13133214—21no—
46#13133214—21no—
47#13133214C2312yes—
48C4⋊#88322—21no—
49C4⋊#88322—24no—
50C4⋊#88322D412yes—
51C4⋊#88322C2312yes—
52C4⋊#88322C2312yes—
53#13133137—24no—
54#13133137—24no—
55C22⋊#88322—21no—
56C22⋊#88322D412yes—
57C4×#88322—21no—
58C4×#88322—24no—
59C4×#88322C2312yes—
60C2×#41×C53—21no—
61C22×Q8×C53D411yescharacteristic
62#137×C53—44no—
63C2×Q16×C53—22no—
64C2×SD16×C53—22no—
65C2×M16×C53—21no—
66C4⋊Q8×C53C2311yescharacteristic
67#127×C53C2311yescharacteristic
68#124×C53—21no—
69#121×C53—21no—
70#121×C53—21no—
71C4×Q8×C53—21no—
72C4×Q8×C53—21no—
73C4×Q8×C53D412yes—
74C4×Q8×C53C2312yes—
75C4×D4×C53—21no—
76C4×D4×C53—21no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53D411yescharacteristic
79C2×C42×C53—21no—
80C2×C42×C53C2311yescharacteristic
81C4⋊C8×C53—21no—
82C4⋊C8×C53C2312yes—
83#103×C53—22no—
84#102×C53—22no—
85C23×D53—41no—
86#41×C53—42no—
87#41×C53—42no—
88C2×Q8×C53—21no—
89C2×Q8×C53—22no—
90C2×Q8×C53—21no—
91C2×Q8×C53D814yescomplemented
92C2×Q8×C53C2×D412yes—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—41no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—41no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C2×D212—22no—
104C2×D212—22no—
105C2×D212—22no—
106C2×D212—42no—
107C2×D212—42no—
108C2×D212—41no—
109C2×D212—41no—
110C2×D212—41no—
111C2×D212—82no—
112C2×D212—82no—
113Q16×C53—44no—
114SD16×C53—44no—
115M16×C53—42no—
116C2×C8×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53C2×D412yes—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C42×C53—22no—
126C42×C53—22no—
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53#4111yescharacteristic
130C42×C53C2411yescharacteristic
131#11719924—48no—
132#11719922—84no—
133C2×#88322—22no—
134C2×#88322—22no—
135C2×#88322—42no—
136C2×#88322C2×D414yes—
137C23×C53—41no—
138C23×C53#12111yescharacteristic
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—82no—
142C22×D53—82no—
143Q8×C53—22no—
144Q8×C53—24no—
145Q8×C53—41no—
146Q8×C53—41no—
147Q8×C53C2×D814yes—
148D4×C53—41no—
149D4×C53—41no—
150D4×C53—42no—
151D4×C53—81no—
152D4×C53—82no—
153C2×C4×C53—22no—
154C2×C4×C53—24no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—81no—
160C2×C4×C53—82no—
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53C2×D812yes—
165C2×C4×C53#13512yes—
166C2×C4×C53C22×D411yescharacteristic
167C2×C4×C53C22×D411yescharacteristic
168C2×C4×C53C2×#4111yescharacteristic
169D212—42no—
170D212—42no—
171D212—44no—
172D212—82no—
173D212—82no—
174D212—82no—
175D212—82no—
176C8×C53—42no—
177#88322—24no—
178#88322—42no—
179#88322—42no—
180#19398—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53#46612yes—
187C22×C53C2×#12111yescharacteristic
188C2×D53—82no—
189C2×D53—82no—
190C2×D53—82no—
191C2×D53—164no—
192C4×C53—21no—
193C4×C53—24no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—42no—
197C4×C53—81no—
198C4×C53#46612yes—
199C4×C53C22×D811yescharacteristic
200C4×C53C2×#13511yescharacteristic
201C4×C53#57712yes—
202#3205—531no—
203#3037D5311yescharacteristic, complemented
204C2×#460—531no—
205#2982—531no—
206#1682—534no—
207#1668—534no—
208#1571—531no—
209#1545—531no—
210#1540—531no—
211C2×C53—22no—
212C2×C53—161no—
213C2×C53#304311yescharacteristic
214C2×C53C2×#46611yescharacteristic
215C2×C53#306811yescharacteristic
216D53—162no—
217C2×#137—21no—
218C22×SD16—1061no—
219#532C2×D5311yescharacteristic
220C2×#127—531no—
221C2×C4×Q8C2×D5311yescharacteristic
222#493—532no—
223#486—532no—
224#464—22no—
225#462C2×D5312yescomplemented
226#460—534no—
227#460—1062no—
228#460C2×D5312yescomplemented
229#459—1062no—
230#447—1062no—
231C8×Q8—534no—
232C8×Q8—1062no—
233#431—531no—
234#431—531no—
235#431—1062no—
236#423C2×D5311yescharacteristic
237C2×C4⋊C8—531no—
238#417—21no—
239C2×#102—1061no—
240#335—534no—
241#332—532no—
242#331—532no—
243#331—534no—
244C53#1939811yescharacteristic, complemented, Sylow
245C2×#41—21no—
246C22×Q8C22⋊D5311yescharacteristic
247C22×D4—1061no—
248C22×D4—1061no—
249C22×D4—1061no—
250#137—44no—
251C2×Q16—22no—
252C2×SD16—22no—
253C2×SD16—1064no—
254C2×SD16—2124no—
255C2×D8—1064no—
256C2×M16—21no—
257C22×C8—1061no—
258C4⋊Q8C22×D5311yescharacteristic
259#127—532no—
260#127—1061no—
261#127—1061no—
262#127—1061no—
263#127C22×D5311yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C22⋊D5312yes—
270C4×Q8C22×D5312yes—
271C4×D4—21no—
272C4×D4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C22⋊D5311yescharacteristic
275C2×C42—21no—
276C2×C42C22×D5311yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—532no—
279C4⋊C8—532no—
280C4⋊C8—532no—
281C4⋊C8—1064no—
282C4⋊C8—1061no—
283C4⋊C8C22×D5312yes—
284#103—22no—
285#102—22no—
286#102—1064no—
287#102—1064no—
288C22⋊C8—532no—
289C22⋊C8—1061no—
290C4×C8—532no—
291C4×C8—1064no—
292C4×C8—1061no—
293C24—2121no—
294#41—42no—
295#41—42no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—21no—
299C2×Q8#1171992214yescomplemented
300C2×Q8C2×C22⋊D5312yes—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—1062no—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—4242no—
309C2×D4—4242no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2122no—
314C2×D4—2122no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4D4×D5311yescharacteristic
319C22×C4#1171994711yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321Q16—44no—
322SD16—44no—
323SD16—2128no—
324D8—4244no—
325M16—42no—
326C2×C8—22no—
327C2×C8—534no—
328C2×C8—1062no—
329C2×C8—1062no—
330C2×C8—2122no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4C2×C22⋊D5312yes—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C42—22no—
340C42—22no—
341C42D4×D5311yescharacteristic
342C42#1171994711yescharacteristic
343C42C2×C22⋊D5311yescharacteristic
344C42C23×D5311yescharacteristic
345C23—41no—
346C23—4242no—
347C23—4242no—
348C23—4242no—
349C23—2122no—
350C23#1313318911yescharacteristic
351Q8—24no—
352Q8—22no—
353Q8—41no—
354Q8—41no—
355Q8C2×#1171992214yes—
356D4—41no—
357D4—41no—
358D4—42no—
359D4—81no—
360D4—82no—
361D4—4242no—
362D4—4242no—
363D4—4242no—
364D4—4242no—
365D4—2122no—
366D4—2122no—
367D4—2124no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4#1313318911yescharacteristic
377C2×C4#1313318911yescharacteristic
378C2×C4#1313318911yescharacteristic
379C2×C4C2×#1171992212yes—
380C2×C4#1313323712yes—
381C2×C4C2×D4×D5311yescharacteristic
382C2×C4C22×C22⋊D5311yescharacteristic
383C2×C4C2×#1171994711yescharacteristic
384C8—42no—
385C8—1064no—
386C8—2122no—
387C8—2122no—
388C22—21no—
389C22—21no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—8484no—
394C22—4242no—
395C22—4242no—
396C22—4242no—
397C22#1681085712yes—
398C22C2×#1313318911yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—24no—
403C4—42no—
404C4—81no—
405C4#1681085712yes—
406C4C22×#1171992211yescharacteristic
407C4C2×#1313323711yescharacteristic
408C4#1681143212yes—
409C2—22no—
410C2—161no—
411C2—8482no—
412C2#2028423811yescharacteristic
413C2C2×#1681085711yescharacteristic
414C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3037C2
socleC22×C53C2×#121
prime radical#20285630C2
hypercenter#3037D53
hypocenterC53#19398
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
2. C2×#16810823C2semidirect
3. #20283796C2semidirect
4. #20274161C2semidirect
5. #20274073C2semidirect
6. #20273491C2semidirect
7. #20273384C2semidirect
8. #20273364C2semidirect
9. C53×#3037C2semidirect
19. Q8×#88322C22semidirect
25. #16810375C22semidirect
26. #16810369C22semidirect
27. #16810367C22semidirect
33. C53×#462C22semidirect
34. C53×#460C22semidirect
91. C2×Q8×C53D8semidirect
203. #3037D53semidirect
225. #462C2×D53semidirect
228. #460C2×D53semidirect
244. C53#19398semidirect
299. C2×Q8#11719922semidirect

Maximal subgroups

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

subgroupindex
#202856302
C2×#168108232
#202837962
#202741612
#202740732
#202734912
#202733842
#202733642
C53×#30372
#1939853