Finite Group Databasebeta

← #27367388

Subgroups — #27367388

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 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#20274285C214yesmaximal, complemented
5#20274161C214yesmaximal, complemented
6#20273461C211yesmaximal, characteristic, complemented
7#20273400C211yesmaximal, characteristic, complemented
8#20273382C211yesmaximal, characteristic
9C53×#3209C211yesmaximal, 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#16810369C2214yescomplemented
33#16810366C2214yes—
34C53×#533C2211yescharacteristic
35C2×C4⋊Q8×C53C2211yescharacteristic
36C53×#494—21no—
37C53×#487—21no—
38C53×#462C2214yescomplemented
39C53×#451—22no—
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—
71C2×Q16×C53—24no—
72C4⋊Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53D411yescharacteristic
76C4⋊Q8×C53D411yescharacteristic
77C4⋊Q8×C53C2312yes—
78#125×C53—21no—
79C22⋊Q8×C53—21no—
80C22⋊Q8×C53D412yes—
81C4×Q8×C53—21no—
82C4×Q8×C53C2312yes—
83#117×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C42×C53C2311yescharacteristic
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—
111Q16×C53—84no—
112C2×C8×C53—22no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53C2×D412yes—
123C22⋊C4×C53—22no—
124C42×C53—22no—
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128C42×C53C2411yescharacteristic
129#11719924—84no—
130#11719917—42no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—22no—
134C2×#88322—22no—
135C2×#88322C2×D411yescharacteristic
136C2×#88322C2×D411yescharacteristic
137C2×#88322C2×D411yescharacteristic
138C2×#88322C2×D411yescharacteristic
139C23×C53#12111yescharacteristic
140C22×D53—42no—
141C22×D53—82no—
142C22×D53—82no—
143C22×D53—82no—
144Q8×C53—24no—
145Q8×C53—42no—
146Q8×C53—44no—
147Q8×C53—82no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—42no—
153C2×C4×C53—82no—
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#13512yes—
158C2×C4×C53#13712yes—
159C2×C4×C53C22×D411yescharacteristic
160C2×C4×C53C22×D411yescharacteristic
161C2×C4×C53C2×#4111yescharacteristic
162D212—42no—
163D212—42no—
164D212—44no—
165D212—82no—
166D212—82no—
167D212—82no—
168D212—82no—
169C8×C53—42no—
170#88322—21no—
171#88322—21no—
172#88322—21no—
173#88322—21no—
174#88322—42no—
175#88322—42no—
176#19214—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53#46412yes—
180C22×C53C2×#12111yescharacteristic
181C2×D53—82no—
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—164no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—42no—
188C4×C53—42no—
189C4×C53—82no—
190C4×C53#46412yes—
191C4×C53#47112yes—
192C4×C53C2×#13511yescharacteristic
193C4×C53C2×#13711yescharacteristic
194C4×C53#57612yes—
195#3209D5311yescharacteristic, complemented
196#3208—531no—
197C2×#486—531no—
198#2960—531no—
199#1702—534no—
200#1682—534no—
201#1564—531no—
202#1549—531no—
203#1545—531no—
204C2×C53—22no—
205C2×C53C2×#46411yescharacteristic
206C2×C53#305811yescharacteristic
207C2×C53C2×#47111yescharacteristic
208D53—162no—
209#533C2×D5311yescharacteristic
210C2×C4⋊Q8C2×D5311yescharacteristic
211C2×#127—531no—
212#494—21no—
213#492—1061no—
214#487—21no—
215#486—532no—
216#486—532no—
217#486—1061no—
218#486—1061no—
219#486—1061no—
220#462C2×D5314yescomplemented
221#460—534no—
222#451—22no—
223#450—1062no—
224#446—532no—
225#446—1061no—
226C8×Q8—532no—
227C8×Q8—1061no—
228#431—531no—
229#431C2×D5311yescharacteristic
230#424—1061no—
231#423—531no—
232C2×#102—1061no—
233C2×#102—1061no—
234#405—1061no—
235C2×C4×C8—531no—
236#335—532no—
237#334—532no—
238#332—534no—
239#331—534no—
240C53#1921411yescharacteristic, complemented, Sylow
241C22×Q8—21no—
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4—1061no—
245C2×Q16—24no—
246C2×SD16—1064no—
247C2×M16—1061no—
248C22×C8—531no—
249C22×C8—531no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8C22⋊D5311yescharacteristic
254C4⋊Q8C22⋊D5311yescharacteristic
255C4⋊Q8C22×D5312yes—
256#127—532no—
257#127—1061no—
258#127—1061no—
259#127—1061no—
260#125—21no—
261C22⋊Q8—21no—
262C22⋊Q8C22⋊D5312yes—
263C4×Q8—21no—
264C4×Q8C22×D5312yes—
265#117—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C42C22×D5311yescharacteristic
269C4⋊C8—532no—
270C4⋊C8—532no—
271C4⋊C8—1062no—
272C4⋊C8—1062no—
273C4⋊C8—1061no—
274C4⋊C8C22×D5312yes—
275#103—24no—
276#102—1062no—
277#102—1062no—
278#102—1062no—
279#102—1062no—
280#102—1064no—
281C22⋊C8—21no—
282C22⋊C8—1061no—
283#97—1062no—
284C4×C8—21no—
285C4×C8—531no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—531no—
289C4×C8—1062no—
290C4×C8—1061no—
291C24—2121no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8C2×C22⋊D5312yes—
297C2×D4—1062no—
298C2×D4—1062no—
299C2×D4—1062no—
300C2×D4—4242no—
301C2×D4—4242no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2122no—
306C2×D4—2122no—
307C22×C4—41no—
308C22×C4D4×D5311yescharacteristic
309C22×C4#1171994711yescharacteristic
310C22×C4C2×C22⋊D5311yescharacteristic
311Q16—84no—
312SD16—4244no—
313M16—2122no—
314C2×C8—22no—
315C2×C8—531no—
316C2×C8—531no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—1062no—
320C2×C8—1062no—
321C2×C8—1062no—
322C2×C8—1062no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4C2×C22⋊D5312yes—
333C22⋊C4—22no—
334C42—22no—
335C42D4×D5311yescharacteristic
336C42#1171994711yescharacteristic
337C42C2×C22⋊D5311yescharacteristic
338C42C23×D5311yescharacteristic
339C23—4242no—
340C23—4242no—
341C23—4242no—
342C23—2122no—
343C23#1313318911yescharacteristic
344Q8—24no—
345Q8—42no—
346Q8—44no—
347Q8—82no—
348D4—4242no—
349D4—4242no—
350D4—4242no—
351D4—4242no—
352D4—2122no—
353D4—2122no—
354D4—2124no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4#1313318911yescharacteristic
362C2×C4#1313318911yescharacteristic
363C2×C4#1313318911yescharacteristic
364C2×C4#1313324012yes—
365C2×C4#1313324912yes—
366C2×C4C2×D4×D5311yescharacteristic
367C2×C4C22×C22⋊D5311yescharacteristic
368C2×C4C2×#1171994711yescharacteristic
369C8—42no—
370C8—1061no—
371C8—1061no—
372C8—1061no—
373C8—1061no—
374C8—2122no—
375C8—2122no—
376C22—21no—
377C22—21no—
378C22—8484no—
379C22—4242no—
380C22—4242no—
381C22—4242no—
382C22#1681084712yes—
383C22C2×#1313318911yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—42no—
388C4—82no—
389C4#1681084712yes—
390C4#1681088812yes—
391C4C2×#1313324011yescharacteristic
392C4C2×#1313324911yescharacteristic
393C4#1681142612yes—
394C2—22no—
395C2—8482no—
396C2C2×#1681084711yescharacteristic
397C2#2028436011yescharacteristic
398C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3209C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#3209D53
hypocenterC53#19214
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
3. #20283650C2semidirect
4. #20274285C2semidirect
5. #20274161C2semidirect
6. #20273461C2semidirect
7. #20273400C2semidirect
9. C53×#3209C2semidirect
20. #16810754C22semidirect
22. Q8×#88322C22semidirect
30. #16810375C22semidirect
31. #16810373C22semidirect
32. #16810369C22semidirect
38. C53×#462C22semidirect
195. #3209D53semidirect
220. #462C2×D53semidirect
240. C53#19214semidirect

Maximal subgroups

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

subgroupindex
#202856532
C2×#168109712
#202836502
#202742852
#202741612
#202734612
#202734002
#202733822
C53×#32092
#1921453