Finite Group Databasebeta

← #27367370

Subgroups — #27367370

Order 13568.

646 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#20285653C211yesmaximal, characteristic
2C2×#16810971C211yesmaximal, characteristic
3#20283650C211yesmaximal, characteristic, complemented
4#20274257C214yesmaximal, complemented
5#20274133C214yesmaximal, complemented
6#20273461C211yesmaximal, characteristic, complemented
7#20273396C211yesmaximal, characteristic, complemented
8#20273382C211yesmaximal, characteristic
9C53×#3230C211yesmaximal, characteristic, complemented
10C2×#13133214C2211yescharacteristic
11#16810997—21no—
12#16810971—21no—
13#16810971—21no—
14#16810971—21no—
15#16810971C2212yes—
16#16810971C2212yes—
17#16810823C2214yes—
18#16810773—22no—
19#16810754—21no—
20#16810754C2212yescomplemented
21Q8×#88322—21no—
22Q8×#88322C2212yescomplemented
23#16810676C2211yescharacteristic
24#16810650—21no—
25#16810647C2211yescharacteristic
26C2×#13133137—21no—
27C2×#13133137—21no—
28#16810579—21no—
29C2×C4×#88322C2211yescharacteristic
30#16810375C2212yescomplemented
31#16810373C2212yescomplemented
32#16810372C2214yescomplemented
33#16810366C2214yes—
34C53×#533C2211yescharacteristic
35C2×C4⋊Q8×C53C2211yescharacteristic
36C53×#500—21no—
37C53×#487—21no—
38C53×#484—22no—
39C53×#477C2214yescomplemented
40C53×#431C2211yescharacteristic
41C22×D212—21no—
42C22×D212—21no—
43C22×D212—21no—
44C2×#11719924—24no—
45C2×#11719917—21no—
46C22×#88322D411yescharacteristic
47C22×#88322D411yescharacteristic
48#13133214—21no—
49#13133214—21no—
50#13133214—21no—
51#13133214C2312yes—
52C4⋊#88322—21no—
53C4⋊#88322—22no—
54C4⋊#88322—22no—
55C4⋊#88322C2312yes—
56C4⋊#88322C2312yes—
57#13133137—24no—
58#13133137—22no—
59#13133137—22no—
60#13133137—22no—
61#13133137—22no—
62C22⋊#88322—21no—
63#13133128—22no—
64C4×#88322—21no—
65C4×#88322—22no—
66C4×#88322D411yescharacteristic
67C4×#88322D411yescharacteristic
68C4×#88322C2311yescharacteristic
69C4×#88322C2311yescharacteristic
70C22×Q8×C53—21no—
71C4⋊Q8×C53—21no—
72C4⋊Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53D411yescharacteristic
75C4⋊Q8×C53D411yescharacteristic
76C4⋊Q8×C53C2312yes—
77#125×C53—21no—
78C22⋊Q8×C53—21no—
79C22⋊Q8×C53D412yes—
80C4×Q8×C53—21no—
81C4×Q8×C53C2312yes—
82#117×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C42×C53C2311yescharacteristic
86#107×C53—24no—
87C4⋊C8×C53C2312yes—
88#103×C53—24no—
89C22⋊C8×C53—21no—
90C4×C8×C53—21no—
91C23×D53—41no—
92C2×Q8×C53—21no—
93C2×Q8×C53—22no—
94C2×Q8×C53—41no—
95C2×Q8×C53—82no—
96C2×Q8×C53C2×D412yes—
97C22×C4×C53—41no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53#4111yescharacteristic
101C2×D212—22no—
102C2×D212—22no—
103C2×D212—22no—
104C2×D212—42no—
105C2×D212—42no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C2×D212—82no—
110C2×D212—82no—
111C2×C8×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53C2×D412yes—
122C22⋊C4×C53—22no—
123C42×C53—22no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C42×C53C2411yescharacteristic
128#11719924—84no—
129#11719917—42no—
130C2×#88322—22no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—22no—
134C2×#88322C2×D411yescharacteristic
135C2×#88322C2×D411yescharacteristic
136C2×#88322C2×D411yescharacteristic
137C2×#88322C2×D411yescharacteristic
138C23×C53#12111yescharacteristic
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—82no—
142C22×D53—82no—
143Q8×C53—24no—
144Q8×C53—42no—
145Q8×C53—82no—
146Q8×C53—82no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—42no—
152C2×C4×C53—82no—
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#13512yes—
157C2×C4×C53#13712yes—
158C2×C4×C53C22×D411yescharacteristic
159C2×C4×C53C22×D411yescharacteristic
160C2×C4×C53C2×#4111yescharacteristic
161D212—42no—
162D212—42no—
163D212—44no—
164D212—82no—
165D212—82no—
166D212—82no—
167D212—82no—
168C8×C53—42no—
169#88322—21no—
170#88322—21no—
171#88322—21no—
172#88322—21no—
173#88322—42no—
174#88322—42no—
175#19212—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53#46412yes—
179C22×C53C2×#12111yescharacteristic
180C2×D53—82no—
181C2×D53—82no—
182C2×D53—82no—
183C2×D53—164no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—42no—
187C4×C53—42no—
188C4×C53—82no—
189C4×C53#46412yes—
190C4×C53#47212yes—
191C4×C53C2×#13511yescharacteristic
192C4×C53C2×#13711yescharacteristic
193C4×C53#57712yes—
194#3230D5311yescharacteristic, complemented
195#3208—531no—
196C2×#486—531no—
197#2960—531no—
198#1698—534no—
199#1678—534no—
200#1564—531no—
201#1548—531no—
202#1545—531no—
203C2×C53—22no—
204C2×C53C2×#46411yescharacteristic
205C2×C53#305711yescharacteristic
206C2×C53C2×#47211yescharacteristic
207D53—162no—
208#533C2×D5311yescharacteristic
209C2×C4⋊Q8C2×D5311yescharacteristic
210C2×#127—531no—
211#500—21no—
212#492—1061no—
213#487—21no—
214#486—532no—
215#486—532no—
216#486—1061no—
217#486—1061no—
218#486—1061no—
219#484—22no—
220#477C2×D5314yescomplemented
221#460—534no—
222#450—1062no—
223#446—532no—
224#446—1061no—
225C8×Q8—532no—
226C8×Q8—1061no—
227#431—531no—
228#431C2×D5311yescharacteristic
229#424—1061no—
230#423—531no—
231C2×#102—1061no—
232C2×#102—1061no—
233#405—1061no—
234C2×C4×C8—531no—
235#335—532no—
236#334—532no—
237#333—534no—
238#331—534no—
239C53#1921211yescharacteristic, complemented, Sylow
240C22×Q8—21no—
241C22×D4—1061no—
242C22×D4—1061no—
243C22×D4—1061no—
244C2×SD16—1064no—
245C2×M16—1061no—
246C22×C8—531no—
247C22×C8—531no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22⋊D5311yescharacteristic
252C4⋊Q8C22⋊D5311yescharacteristic
253C4⋊Q8C22×D5312yes—
254#127—532no—
255#127—1061no—
256#127—1061no—
257#127—1061no—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8C22⋊D5312yes—
261C4×Q8—21no—
262C4×Q8C22×D5312yes—
263#117—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C42C22×D5311yescharacteristic
267#107—24no—
268C4⋊C8—532no—
269C4⋊C8—532no—
270C4⋊C8—1062no—
271C4⋊C8—1062no—
272C4⋊C8—1061no—
273C4⋊C8C22×D5312yes—
274#103—24no—
275#102—1062no—
276#102—1062no—
277#102—1062no—
278#102—1062no—
279#102—1064no—
280C22⋊C8—21no—
281C22⋊C8—1061no—
282#97—1062no—
283C4×C8—21no—
284C4×C8—531no—
285C4×C8—531no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—1062no—
289C4×C8—1061no—
290C24—2121no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8C2×C22⋊D5312yes—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—1062no—
299C2×D4—4242no—
300C2×D4—4242no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2122no—
305C2×D4—2122no—
306C22×C4—41no—
307C22×C4D4×D5311yescharacteristic
308C22×C4#1171994711yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310SD16—4244no—
311M16—2122no—
312C2×C8—22no—
313C2×C8—531no—
314C2×C8—531no—
315C2×C8—531no—
316C2×C8—531no—
317C2×C8—1062no—
318C2×C8—1062no—
319C2×C8—1062no—
320C2×C8—1062no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4C2×C22⋊D5312yes—
331C22⋊C4—22no—
332C42—22no—
333C42D4×D5311yescharacteristic
334C42#1171994711yescharacteristic
335C42C2×C22⋊D5311yescharacteristic
336C42C23×D5311yescharacteristic
337C23—4242no—
338C23—4242no—
339C23—4242no—
340C23—2122no—
341C23#1313318911yescharacteristic
342Q8—24no—
343Q8—42no—
344Q8—82no—
345Q8—82no—
346D4—4242no—
347D4—4242no—
348D4—4242no—
349D4—4242no—
350D4—2122no—
351D4—2122no—
352D4—2124no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4#1313318911yescharacteristic
360C2×C4#1313318911yescharacteristic
361C2×C4#1313318911yescharacteristic
362C2×C4#1313324012yes—
363C2×C4#1313324912yes—
364C2×C4C2×D4×D5311yescharacteristic
365C2×C4C22×C22⋊D5311yescharacteristic
366C2×C4C2×#1171994711yescharacteristic
367C8—42no—
368C8—1061no—
369C8—1061no—
370C8—1061no—
371C8—1061no—
372C8—2122no—
373C8—2122no—
374C22—21no—
375C22—21no—
376C22—8484no—
377C22—4242no—
378C22—4242no—
379C22—4242no—
380C22#1681084712yes—
381C22C2×#1313318911yescharacteristic
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4#1681084712yes—
388C4#1681089112yes—
389C4C2×#1313324011yescharacteristic
390C4C2×#1313324911yescharacteristic
391C4#1681143112yes—
392C2—22no—
393C2—8482no—
394C2C2×#1681084711yescharacteristic
395C2#2028435611yescharacteristic
396C2C2×#1681089111yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3230C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#3230D53
hypocenterC53#19212
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
3. #20283650C2semidirect
4. #20274257C2semidirect
5. #20274133C2semidirect
6. #20273461C2semidirect
7. #20273396C2semidirect
9. C53×#3230C2semidirect
20. #16810754C22semidirect
22. Q8×#88322C22semidirect
30. #16810375C22semidirect
31. #16810373C22semidirect
32. #16810372C22semidirect
39. C53×#477C22semidirect
194. #3230D53semidirect
220. #477C2×D53semidirect
239. C53#19212semidirect

Maximal subgroups

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

subgroupindex
#202856532
C2×#168109712
#202836502
#202742572
#202741332
#202734612
#202733962
#202733822
C53×#32302
#1921253