Finite Group Databasebeta

← #27367704

Subgroups — #27367704

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286057C211yesmaximal, characteristic, complemented
2#20285089C211yesmaximal, characteristic
3#20283664C211yesmaximal, characteristic, complemented
4#20274281C212yesmaximal, complemented
5#20274253C212yesmaximal, complemented
6#20274157C212yesmaximal, complemented
7#20274129C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9#20273370C211yesmaximal, characteristic, complemented
10C2×#16810367C211yesmaximal, characteristic
11C53×#3208C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16810987—22no—
14#16810969C2212yes—
15#16810969C2212yes—
16#16810918C2212yescomplemented
17#16810915—21no—
18#16810862—22no—
19#16810833C2212yescomplemented
20#16810830—21no—
21#16810754—22no—
22#16810754C2212yescomplemented
23Q8×#88322C2212yescomplemented
24#16810650—21no—
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27#16810628—21no—
28#16810628—21no—
29#16810579—21no—
30C4×#11719917C2211yescharacteristic
31#16810375C2212yescomplemented
32#16810373C2212yescomplemented
33#16810371C2211yescharacteristic, complemented
34#16810371C2211yescharacteristic, complemented
35#16810370C2211yescharacteristic, complemented
36#16810370C2211yescharacteristic, complemented
37#16810367C2212yes—
38#16810367C2212yes—
39C53×#533C2211yescharacteristic
40C2×#127×C53C2211yescharacteristic
41C53×#492—21no—
42C53×#486—21no—
43C53×#460C2212yes—
44C53×#460C2212yes—
45C53×#450—22no—
46C53×#431C2211yescharacteristic
47C2×#11719925—22no—
48C2×#11719917D411yescharacteristic
49C2×#11719917D411yescharacteristic
50C22×#88322—21no—
51C22×#88322—21no—
52#13133217C2311yescharacteristic
53#13133217C2311yescharacteristic
54#13133209—21no—
55#13133196—21no—
56#13133196—21no—
57C4×Dic106—21no—
58C4×Dic106—21no—
59#13133176—21no—
60#13133150—22no—
61C4⋊#88322—21no—
62C4⋊#88322—21no—
63C4⋊#88322—22no—
64C4⋊#88322—22no—
65C4⋊#88322C2311yescharacteristic
66C4⋊#88322C2312yes—
67C4⋊#88322C2311yescharacteristic
68#13133141—22no—
69#13133140—22no—
70#13133138—22no—
71#13133138—22no—
72#13133138—22no—
73#13133138—22no—
74#13133128—22no—
75#13133128D412yes—
76C4×#88322—22no—
77C4×#88322C2312yes—
78C22×D4×C53—21no—
79C22×D4×C53—21no—
80C22×D4×C53—21no—
81C2×SD16×C53—22no—
82C2×SD16×C53—22no—
83C4⋊Q8×C53D411yescharacteristic
84C4⋊Q8×C53D411yescharacteristic
85#127×C53—21no—
86#127×C53—21no—
87#127×C53—21no—
88#127×C53C2312yes—
89#125×C53—21no—
90C22⋊Q8×C53—21no—
91C22⋊Q8×C53D412yes—
92C4×Q8×C53—21no—
93C4×Q8×C53C2312yes—
94#117×C53—21no—
95C2×C42×C53C2311yescharacteristic
96C4⋊C8×C53C2311yescharacteristic
97C4⋊C8×C53C2311yescharacteristic
98#102×C53—22no—
99#102×C53—22no—
100C22⋊C8×C53—21no—
101C4×C8×C53—21no—
102C24×C53—41no—
103C2×Q8×C53—21no—
104C2×Q8×C53C2×D412yes—
105C2×D4×C53—22no—
106C2×D4×C53—22no—
107C2×D4×C53—22no—
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—42no—
112C2×D4×C53—42no—
113C2×D4×C53—81no—
114C2×D4×C53—81no—
115C2×D4×C53—82no—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C2×Dic106—21no—
120C2×Dic106—21no—
121SD16×C53—82no—
122SD16×C53—82no—
123C2×C8×C53—21no—
124C2×C8×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53C2×D412yes—
131C22⋊C4×C53—22no—
132C42×C53—22no—
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53#4111yescharacteristic
136C42×C53C2411yescharacteristic
137C22⋊Dic53—41no—
138#11719925—82no—
139C4⋊Dic53—21no—
140C4⋊Dic53—21no—
141C4⋊Dic53—21no—
142C4⋊Dic53—21no—
143#11719919—41no—
144#11719919—41no—
145C4×Dic53—41no—
146#11719917—22no—
147#11719917—22no—
148C2×#88322—21no—
149C2×#88322—21no—
150C2×#88322—22no—
151C2×#88322—42no—
152C2×#88322—81no—
153C2×#88322C2×D412yes—
154C2×#88322C2×D412yes—
155C23×C53—42no—
156C23×C53—82no—
157C23×C53—82no—
158C23×C53—82no—
159C23×C53#12111yescharacteristic
160Q8×C53—22no—
161Q8×C53—22no—
162Q8×C53—81no—
163D4×C53—42no—
164D4×C53—42no—
165D4×C53—42no—
166D4×C53—42no—
167D4×C53—82no—
168D4×C53—82no—
169D4×C53—82no—
170D4×C53—82no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—41no—
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#13512yes—
180C2×C4×C53#13712yes—
181C2×C4×C53C22×D411yescharacteristic
182C2×C4×C53C22×D411yescharacteristic
183C2×C4×C53C2×#4111yescharacteristic
184C2×Dic53—41no—
185C2×Dic53—41no—
186Dic106—41no—
187Dic106—41no—
188Dic106—81no—
189C8×C53—41no—
190C8×C53—41no—
191#88322—22no—
192#88322—22no—
193#88322—41no—
194#88322—41no—
195#88322—42no—
196#19242—531nomaximal, Sylow
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—82no—
200C22×C53—82no—
201C22×C53—82no—
202C22×C53—164no—
203C22×C53#46412yes—
204C22×C53C2×#12111yescharacteristic
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—42no—
209C4×C53—42no—
210C4×C53#46412yes—
211C4×C53C2×#13511yescharacteristic
212C4×C53C2×#13711yescharacteristic
213C4×C53#57611yescharacteristic
214C4×C53#57711yescharacteristic
215Dic53—81no—
216Dic53—81no—
217#3249—531no—
218#3208D5311yescharacteristic
219#3138—531no—
220#2962—531no—
221#1702—532no—
222#1698—532no—
223#1682—532no—
224#1678—532no—
225#1565—531no—
226#1542—531no—
227C2×#331—531no—
228C2×C53—22no—
229C2×C53—162no—
230C2×C53C2×#46411yescharacteristic
231C2×C53#306211yescharacteristic
232C2×C53#306811yescharacteristic
233#533—531no—
234#533C2×D5311yescharacteristic
235C2×#127C2×D5311yescharacteristic
236#492—21no—
237#490—1062no—
238#486—21no—
239#486—532no—
240#486—532no—
241#477—532no—
242#477—1061no—
243#467—1062no—
244#462—532no—
245#462—1061no—
246#460C2×D5312yes—
247#460C2×D5312yes—
248#450—22no—
249#446—532no—
250#446—1062no—
251C8×Q8—532no—
252#431C2×D5311yescharacteristic
253#424—1061no—
254C2×C4⋊C8—531no—
255C2×C4⋊C8—531no—
256#418—1061no—
257#418—1061no—
258#405—1061no—
259C4×M16—531no—
260#335—532no—
261#334—532no—
262#333—531no—
263#333—531no—
264#332—531no—
265#332—531no—
266#331—532no—
267#331—532no—
268C53#1924211yescharacteristic, complemented, Sylow
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—21no—
272C2×Q16—1062no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×M16—531no—
276C2×M16—531no—
277C22×C8—1061no—
278C22×C8—1061no—
279C4⋊Q8—531no—
280C4⋊Q8—531no—
281C4⋊Q8C22⋊D5311yescharacteristic
282C4⋊Q8C22⋊D5311yescharacteristic
283#127—21no—
284#127—21no—
285#127—21no—
286#127C22×D5312yes—
287#125—21no—
288#125—1061no—
289C22⋊Q8—21no—
290C22⋊Q8—1061no—
291C22⋊Q8—1061no—
292C22⋊Q8C22⋊D5312yes—
293C4×Q8—21no—
294C4×Q8—1061no—
295C4×Q8—1061no—
296C4×Q8C22×D5312yes—
297#117—21no—
298#117—1061no—
299C2×C42C22×D5311yescharacteristic
300#107—1062no—
301C4⋊C8—531no—
302C4⋊C8—531no—
303C4⋊C8—532no—
304C4⋊C8—1062no—
305C4⋊C8—1062no—
306C4⋊C8—1061no—
307C4⋊C8—1061no—
308C4⋊C8C22×D5311yescharacteristic
309C4⋊C8C22×D5311yescharacteristic
310#103—1062no—
311#103—1062no—
312#102—22no—
313#102—22no—
314#102—1062no—
315#102—1062no—
316#102—1062no—
317#102—1062no—
318C22⋊C8—21no—
319#97—532no—
320#97—1062no—
321C4×C8—21no—
322C4×C8—532no—
323C4×C8—1062no—
324C24—41no—
325C2×Q8—21no—
326C2×Q8—1061no—
327C2×Q8—1061no—
328C2×Q8C2×C22⋊D5312yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4D4×D5311yescharacteristic
341C22×C4#1171994711yescharacteristic
342C22×C4C2×C22⋊D5311yescharacteristic
343Q16—4242no—
344SD16—82no—
345SD16—82no—
346M16—1062no—
347M16—1062no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—532no—
351C2×C8—532no—
352C2×C8—1061no—
353C2×C8—1061no—
354C2×C8—1062no—
355C2×C8—4241no—
356C2×C8—2122no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—1061no—
363C4⋊C4—1061no—
364C4⋊C4—1061no—
365C4⋊C4—1061no—
366C4⋊C4—2121no—
367C4⋊C4—2121no—
368C4⋊C4C2×C22⋊D5312yes—
369C22⋊C4—22no—
370C22⋊C4—2121no—
371C42—22no—
372C42—2121no—
373C42D4×D5311yescharacteristic
374C42#1171994711yescharacteristic
375C42C2×C22⋊D5311yescharacteristic
376C42C23×D5311yescharacteristic
377C23—42no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23#1313318911yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—81no—
385Q8—4241no—
386Q8—2121no—
387Q8—2121no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4#1313318911yescharacteristic
404C2×C4#1313318911yescharacteristic
405C2×C4#1313318911yescharacteristic
406C2×C4#1313323812yes—
407C2×C4#1313324612yes—
408C2×C4C2×D4×D5311yescharacteristic
409C2×C4C22×C22⋊D5311yescharacteristic
410C2×C4C2×#1171994711yescharacteristic
411C8—41no—
412C8—41no—
413C8—1062no—
414C8—1062no—
415C8—2121no—
416C8—2121no—
417C8—2122no—
418C22—21no—
419C22—21no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—164no—
424C22#1681084312yes—
425C22C2×#1313318911yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—4241no—
432C4—4241no—
433C4#1681084312yes—
434C4C2×#1313323811yescharacteristic
435C4C2×#1313324611yescharacteristic
436C4#1681142611yescharacteristic
437C4#1681143411yescharacteristic
438C2—22no—
439C2—162no—
440C2C2×#1681084311yescharacteristic
441C2#2028439111yescharacteristic
442C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3208C2
socleC22×C53C2×#121
prime radicalC2×#127×C53C22
hypercenter#3208D53
hypocenterC53#19242
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #20286057C2semidirect
3. #20283664C2semidirect
4. #20274281C2semidirect
5. #20274253C2semidirect
6. #20274157C2semidirect
7. #20274129C2semidirect
8. #20273468C2semidirect
9. #20273370C2semidirect
16. #16810918C22semidirect
19. #16810833C22semidirect
22. #16810754C22semidirect
23. Q8×#88322C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810371C22semidirect
34. #16810371C22semidirect
35. #16810370C22semidirect
36. #16810370C22semidirect
268. C53#19242semidirect

Maximal subgroups

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

subgroupindex
#202860572
#202850892
#202836642
#202742812
#202742532
#202741572
#202741292
#202734682
#202733702
C2×#168103672
C53×#32082
#1924253