Finite Group Databasebeta

← #27369383

Subgroups — #27369383

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#20285633C211yesmaximal, characteristic
2#20284187C211yesmaximal, characteristic, complemented
3#20283796C211yesmaximal, characteristic, complemented
4#20274157C214yesmaximal, complemented
5#20274073C214yesmaximal, complemented
6#20273489C211yesmaximal, characteristic, complemented
7#20273383C211yesmaximal, characteristic, complemented
8#20273365C211yesmaximal, characteristic, complemented
9C2×C53×#460C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11#16811173C2211yescharacteristic
12#16810999C2212yes—
13#16810969C2212yes—
14#16810845—22no—
15#16810833C2212yescomplemented
16#16810823C2212yescomplemented
17#16810815—22no—
18#16810757—22no—
19Q8×#88322—22no—
20Q8×#88322C2214yescomplemented
21#16810676—22no—
22#16810676C2211yescharacteristic
23#16810676C2211yescharacteristic
24#16810647C2211yescharacteristic
25#16810622—21no—
26#16810375C2214yescomplemented
27#16810370C2212yescomplemented
28#16810367C2214yes—
29#16810366C2212yescomplemented
30C22×SD16×C53—21no—
31C2×#127×C53C2211yescharacteristic
32C2×C4×Q8×C53C2211yescharacteristic
33C53×#460—22no—
34C53×#460C2214yes—
35C2×C4⋊C8×C53C2211yescharacteristic
36C2×#102×C53—21no—
37C2×#11719945—21no—
38#13133248—44no—
39C2×#11719925—22no—
40C2×#11719924—22no—
41C2×#11719922—24no—
42C2×#11719917—21no—
43#13133217C2311yescharacteristic
44#13133214C2311yescharacteristic
45#13133205—21no—
46#13133191—21no—
47#13133191—21no—
48C4×D212—21no—
49C4×D212—21no—
50C4⋊#88322—21no—
51C4⋊#88322—24no—
52C4⋊#88322D412yes—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55#13133140—22no—
56#13133138—24no—
57#13133137—22no—
58C22⋊#88322—21no—
59C22⋊#88322D412yes—
60C4×#88322—21no—
61C4×#88322—24no—
62C4×#88322C2312yes—
63C22×Q8×C53D411yescharacteristic
64C22×D4×C53—21no—
65C22×D4×C53—21no—
66C22×D4×C53—21no—
67C2×SD16×C53—24no—
68C2×SD16×C53—44no—
69C22×C8×C53—21no—
70#127×C53—21no—
71#127×C53—21no—
72#127×C53—21no—
73#127×C53C2312yes—
74C4×Q8×C53—21no—
75C4×Q8×C53—21no—
76C4×Q8×C53D412yes—
77C4×Q8×C53C2312yes—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53D411yescharacteristic
80C2×C42×C53—21no—
81C2×C42×C53C2311yescharacteristic
82C4⋊C8×C53—21no—
83C4⋊C8×C53C2312yes—
84#102×C53—24no—
85C24×C53—41no—
86C2×Q8×C53—21no—
87C2×Q8×C53—22no—
88C2×Q8×C53D814yescomplemented
89C2×Q8×C53C2×D412yes—
90C2×D4×C53—22no—
91C2×D4×C53—22no—
92C2×D4×C53—22no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—42no—
97C2×D4×C53—42no—
98C2×D4×C53—82no—
99C2×D4×C53—82no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C2×C22⋊D53—41no—
106#11719945—42no—
107#11719945—42no—
108C2×D212—21no—
109C2×D212—21no—
110C2×D212—21no—
111C2×C4×D53—41no—
112C2×Dic106—21no—
113SD16×C53—48no—
114C2×C8×C53—22no—
115C2×C8×C53—42no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53C2×D412yes—
120C42×C53—22no—
121C42×C53—22no—
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125C42×C53C2411yescharacteristic
126#11719925—44no—
127#11719924—44no—
128#11719922—84no—
129#11719921—41no—
130#11719921—41no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133#11719917—42no—
134C2×#88322—22no—
135C2×#88322—22no—
136C2×#88322C2×D414yes—
137C23×C53—42no—
138C23×C53—82no—
139C23×C53—82no—
140C23×C53—82no—
141C23×C53#12111yescharacteristic
142C22×D53—41no—
143Q8×C53—22no—
144Q8×C53—24no—
145Q8×C53C2×D814yes—
146D4×C53—42no—
147D4×C53—42no—
148D4×C53—44no—
149D4×C53—82no—
150D4×C53—82no—
151D4×C53—82no—
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—81no—
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53C2×D812yes—
163C2×C4×C53#13512yes—
164C2×C4×C53C22×D411yescharacteristic
165C2×C4×C53C22×D411yescharacteristic
166C2×C4×C53C2×#4111yescharacteristic
167C22⋊D53—82no—
168C2×Dic53—41no—
169D212—41no—
170D212—41no—
171D212—42no—
172D212—81no—
173C4×D53—82no—
174Dic106—41no—
175Dic106—41no—
176C8×C53—42no—
177#88322—24no—
178#88322—42no—
179#88322—42no—
180#19398—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—82no—
184C22×C53—82no—
185C22×C53—82no—
186C22×C53—164no—
187C22×C53#46612yes—
188C22×C53C2×#12111yescharacteristic
189C2×D53—81no—
190C2×D53—81no—
191C2×D53—81no—
192C4×C53—21no—
193C4×C53—24no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—42no—
197C4×C53#46612yes—
198C4×C53C22×D811yescharacteristic
199C4×C53C2×#13511yescharacteristic
200C4×C53#57712yes—
201Dic53—81no—
202#3205—531no—
203#3037—531no—
204C2×#460D5311yescharacteristic, complemented
205#2982—531no—
206#1682—534no—
207#1668—534no—
208#1571—531no—
209#1545—531no—
210#1540—531no—
211C2×C53—22no—
212C2×C53—162no—
213C2×C53#304311yescharacteristic
214C2×C53C2×#46611yescharacteristic
215C2×C53#306811yescharacteristic
216D53—161no—
217C2×#137—1061no—
218C22×SD16—21no—
219#532—531no—
220C2×#127C2×D5311yescharacteristic
221C2×C4×Q8C2×D5311yescharacteristic
222#493—532no—
223#486—532no—
224#464—1062no—
225#462—532no—
226#460—22no—
227#460—532no—
228#460C2×D5314yes—
229#459—1062no—
230#447—1062no—
231C8×Q8—534no—
232C8×Q8—1062no—
233#431—531no—
234#431—531no—
235#431—1062no—
236#423—531no—
237C2×C4⋊C8C2×D5311yescharacteristic
238#417—1061no—
239C2×#102—21no—
240#335—534no—
241#332—532no—
242#331—534no—
243#331—532no—
244C53#1939811yescharacteristic, complemented, Sylow
245C2×#41—1061no—
246C22×Q8C22⋊D5311yescharacteristic
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250#137—2124no—
251C2×Q16—1062no—
252C2×SD16—24no—
253C2×SD16—44no—
254C2×SD16—1062no—
255C2×D8—1064no—
256C2×M16—1061no—
257C22×C8—21no—
258C4⋊Q8—531no—
259#127—21no—
260#127—21no—
261#127—21no—
262#127—531no—
263#127C22×D5312yes—
264#124—1061no—
265#121—1061no—
266#121—1061no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C22⋊D5312yes—
270C4×Q8C22×D5312yes—
271C4×D4—1061no—
272C4×D4—1061no—
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—1062no—
285#102—24no—
286#102—1064no—
287#102—1062no—
288C22⋊C8—532no—
289C22⋊C8—1061no—
290C4×C8—532no—
291C4×C8—1064no—
292C4×C8—1061no—
293C24—41no—
294#41—2122no—
295#41—2122no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8—1061no—
299C2×Q8#1171992214yescomplemented
300C2×Q8C2×C22⋊D5312yes—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4—1061no—
312C2×D4—1061no—
313C2×D4—1061no—
314C2×D4—2121no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2121no—
318C22×C4D4×D5311yescharacteristic
319C22×C4#1171994711yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321Q16—2124no—
322SD16—48no—
323SD16—2124no—
324D8—4244no—
325M16—2122no—
326C2×C8—22no—
327C2×C8—42no—
328C2×C8—534no—
329C2×C8—1062no—
330C2×C8—1062no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C4⋊C4C2×C22⋊D5312yes—
337C22⋊C4—2121no—
338C22⋊C4—2121no—
339C42—22no—
340C42—22no—
341C42D4×D5311yescharacteristic
342C42#1171994711yescharacteristic
343C42C2×C22⋊D5311yescharacteristic
344C42C23×D5311yescharacteristic
345C23—42no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23—2121no—
350C23#1313318911yescharacteristic
351Q8—24no—
352Q8—22no—
353Q8—2121no—
354Q8—2121no—
355Q8C2×#1171992214yes—
356D4—42no—
357D4—42no—
358D4—44no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—82no—
363D4—4241no—
364D4—4242no—
365D4—2121no—
366D4—2121no—
367D4—2122no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—4242no—
375C2×C4—2121no—
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—82no—
391C22—82no—
392C22—82no—
393C22—164no—
394C22—4241no—
395C22—4241no—
396C22—4241no—
397C22#1681085712yes—
398C22C2×#1313318911yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—24no—
403C4—42no—
404C4—4241no—
405C4#1681085712yes—
406C4C22×#1171992211yescharacteristic
407C4C2×#1313323711yescharacteristic
408C4#1681142912yes—
409C2—22no—
410C2—162no—
411C2—8481no—
412C2#2028423311yescharacteristic
413C2C2×#1681085711yescharacteristic
414C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#460C2
socleC22×C53C2×#121
prime radical#20285633C2
hypercenterC2×#460D53
hypocenterC53#19398
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #20284187C2semidirect
3. #20283796C2semidirect
4. #20274157C2semidirect
5. #20274073C2semidirect
6. #20273489C2semidirect
7. #20273383C2semidirect
8. #20273365C2semidirect
9. C2×C53×#460C2semidirect
15. #16810833C22semidirect
16. #16810823C22semidirect
20. Q8×#88322C22semidirect
26. #16810375C22semidirect
27. #16810370C22semidirect
29. #16810366C22semidirect
88. C2×Q8×C53D8semidirect
204. C2×#460D53semidirect
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
#202856332
#202841872
#202837962
#202741572
#202740732
#202734892
#202733832
#202733652
C2×C53×#4602
#1939853