Finite Group Databasebeta

← #27367480

Subgroups — #27367480

Order 13568.

628 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286041C212yesmaximal, complemented
2#20285294C211yesmaximal, characteristic, complemented
3#20274255C212yesmaximal, complemented
4#20274248C212yesmaximal, complemented
5#20274119C211yesmaximal, characteristic, complemented
6#20274114C211yesmaximal, characteristic, complemented
7#20274075C211yesmaximal, characteristic, complemented
8#20274070C211yesmaximal, characteristic, complemented
9C2×#16810376C211yesmaximal, characteristic, complemented
10#20273411C212yesmaximal, complemented
11C53×#2966C211yesmaximal, characteristic, complemented
12#16811173C2212yes—
13#16811028C2211yescharacteristic
14#16811000C2211yescharacteristic
15#16810970C2212yes—
16#16810920—22no—
17#16810918C2212yes—
18#16810913C2212yes—
19#16810859—21no—
20#16810859—21no—
21#16810857—22no—
22#16810854—22no—
23#16810835—22no—
24#16810823C2212yes—
25#16810818C2212yes—
26C4×D424—21no—
27C4×D424—21no—
28C2×C4⋊#88322C2212yes—
29#16810376C2212yes—
30#16810376C2211yescharacteristic
31#16810376C2211yescharacteristic
32#16810371C2212yes—
33#16810371C2212yes—
34#16810366C2212yes—
35#16810366C2212yes—
36C53×#529C2211yescharacteristic
37C53×#446C2211yescharacteristic
38C8×Q8×C53C2211yescharacteristic
39C53×#436C2211yescharacteristic
40C53×#435—21no—
41C8×D4×C53—21no—
42C8×D4×C53C2211yescharacteristic
43C53×#433—21no—
44C53×#432C2211yescharacteristic
45C53×#405—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×#11719945—22no—
48C2×#11719924—22no—
49C2×D424—21no—
50C2×D424—21no—
51C2×#11719922—22no—
52C22×#88322—22no—
53#13133217C2312yes—
54#13133214C2312yes—
55#13133205—22no—
56#13133191—22no—
57#13133191—22no—
58C4×D212—22no—
59C4×D212—22no—
60#13133151—21no—
61#13133151—21no—
62#13133150—22no—
63#13133148—22no—
64C4⋊#88322—22no—
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133141—22no—
68#13133139—22no—
69#13133139—22no—
70#13133138—22no—
71#13133137—22no—
72#13133137—22no—
73C2×M16×C53—21no—
74C22×C8×C53—21no—
75C22×C8×C53D411yescharacteristic
76C22×C8×C53D411yescharacteristic
77#126×C53—21no—
78#125×C53D411yescharacteristic
79#124×C53D411yescharacteristic
80#123×C53—21no—
81C22⋊Q8×C53D411yescharacteristic
82#121×C53D411yescharacteristic
83C4×Q8×C53C2311yescharacteristic
84C4×D4×C53—21no—
85C4×D4×C53C2311yescharacteristic
86#117×C53—21no—
87C2×C42×C53C2311yescharacteristic
88C4⋊C8×C53—21no—
89C4⋊C8×C53—21no—
90C4⋊C8×C53C2311yescharacteristic
91C4⋊C8×C53C2311yescharacteristic
92C22⋊C8×C53—21no—
93C22⋊C8×C53—21no—
94C22⋊C8×C53—21no—
95#97×C53—21no—
96#97×C53—21no—
97C4×C8×C53—21no—
98C4×C8×C53D411yescharacteristic
99C4×C8×C53D411yescharacteristic
100C4×C8×C53C2311yescharacteristic
101C4×C8×C53C2311yescharacteristic
102C2×Q8×C53C2×D411yescharacteristic
103C2×D4×C53—21no—
104C2×D4×C53C2×D411yescharacteristic
105C22×C4×C53—21no—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C2×C22⋊D53—42no—
110#11719945—84no—
111C2×D212—22no—
112C2×D212—22no—
113C2×D212—22no—
114C2×C4×D53—42no—
115C2×Dic106—22no—
116M16×C53—42no—
117C2×C8×C53—22no—
118C2×C8×C53—21no—
119C2×C8×C53—21no—
120C2×C8×C53—41no—
121C2×C8×C53—42no—
122C2×C8×C53C2×D411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2×D411yescharacteristic
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53C2×D411yescharacteristic
131C4⋊C4×C53C2×D411yescharacteristic
132C22⋊C4×C53—21no—
133C22⋊C4×C53—21no—
134C22⋊C4×C53—21no—
135C22⋊C4×C53—21no—
136C22⋊C4×C53—21no—
137C42×C53—21no—
138C42×C53C2×D411yescharacteristic
139C42×C53C2×D411yescharacteristic
140C42×C53#4111yescharacteristic
141C42×C53C2411yescharacteristic
142#11719924—82no—
143#11719922—82no—
144#11719921—42no—
145#11719921—42no—
146C4⋊Dic53—22no—
147C4⋊Dic53—22no—
148C2×#88322—22no—
149C2×#88322—22no—
150C2×#88322—82no—
151D424—42no—
152D424—42no—
153C23×C53—21no—
154C23×C53#12111yescharacteristic
155C22×D53—42no—
156Q8×C53—21no—
157Q8×C53—21no—
158D4×C53—21no—
159D4×C53—21no—
160D4×C53—42no—
161C2×C4×C53—22no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C22×D411yescharacteristic
176C2×C4×C53C2×#4111yescharacteristic
177C22⋊D53—84no—
178C2×Dic53—42no—
179D212—42no—
180D212—42no—
181D212—42no—
182D212—42no—
183D212—82no—
184C4×D53—84no—
185Dic106—82no—
186C8×C53—21no—
187C8×C53—21no—
188C8×C53—21no—
189C8×C53—41no—
190C8×C53—41no—
191C8×C53#13512yes—
192#88322—42no—
193#88322—42no—
194#19222—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53#47212yes—
201C22×C53C2×#12111yescharacteristic
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—82no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53—41no—
212C4×C53C2×#13511yescharacteristic
213C4×C53#57511yescharacteristic
214C4×C53#57611yescharacteristic
215C4×C53#57811yescharacteristic
216Dic53—82no—
217#3247—532no—
218#3169—531no—
219#2966D5311yescharacteristic, complemented
220#1698—532no—
221#1697—532no—
222#1676—531no—
223#1675—531no—
224#1668—531no—
225#1667—531no—
226C2×#335—531no—
227#1551—532no—
228C2×C53—22no—
229C2×C53—81no—
230C2×C53#303511yescharacteristic
231C2×C53#306111yescharacteristic
232C2×C53C2×#47211yescharacteristic
233D53—162no—
234#532—532no—
235#529C2×D5311yescharacteristic
236#500—531no—
237#493—531no—
238#486—532no—
239#478—1062no—
240#477—532no—
241#476—532no—
242#466—1061no—
243#466—1061no—
244#466—1062no—
245#465—1062no—
246#463—1062no—
247#460—532no—
248#459—532no—
249#446C2×D5311yescharacteristic
250C8×Q8C2×D5311yescharacteristic
251C4×D8—1061no—
252C4×D8—1061no—
253#436C2×D5311yescharacteristic
254#435—21no—
255C8×D4—21no—
256C8×D4C2×D5311yescharacteristic
257#433—21no—
258#432C2×D5311yescharacteristic
259C2×C4⋊C8—532no—
260#405—21no—
261C2×C4×C8C2×D5311yescharacteristic
262#335—531no—
263#335—531no—
264#335—532no—
265#333—532no—
266#333—532no—
267#331—532no—
268#331—532no—
269C53#1922211yescharacteristic, complemented, Sylow
270C2×#41—1062no—
271C2×SD16—1062no—
272C2×D8—1062no—
273C2×D8—1061no—
274C2×D8—1061no—
275C2×M16—21no—
276C22×C8—21no—
277C22×C8—1062no—
278C22×C8C22⋊D5311yescharacteristic
279C22×C8C22⋊D5311yescharacteristic
280C4⋊Q8—532no—
281#127—532no—
282#126—21no—
283#125D21211yescharacteristic
284#124—1062no—
285#124D21211yescharacteristic
286#123—21no—
287C22⋊Q8D21211yescharacteristic
288#121—1062no—
289#121—1062no—
290#121D21211yescharacteristic
291C4×Q8C22×D5311yescharacteristic
292C4×D4—21no—
293C4×D4—1062no—
294C4×D4—1062no—
295C4×D4C22×D5311yescharacteristic
296#117—21no—
297C2×C42C22×D5311yescharacteristic
298#107—1061no—
299#107—1061no—
300#107—1062no—
301#106—1062no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—532no—
305C4⋊C8—532no—
306C4⋊C8—1062no—
307C4⋊C8C22×D5311yescharacteristic
308C4⋊C8C22×D5311yescharacteristic
309#103—1062no—
310#102—1062no—
311#102—1062no—
312#102—1062no—
313#102—1062no—
314#102—1062no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318#97—21no—
319#97—21no—
320C4×C8—21no—
321C4×C8C22⋊D5311yescharacteristic
322C4×C8C22⋊D5311yescharacteristic
323C4×C8C22×D5311yescharacteristic
324C4×C8C22×D5311yescharacteristic
325#41—4244no—
326C2×Q8—1062no—
327C2×Q8C2×D21211yescharacteristic
328C2×D4—21no—
329C2×D4—1062no—
330C2×D4—1062no—
331C2×D4—1062no—
332C2×D4—2122no—
333C2×D4C2×D21211yescharacteristic
334C22×C4—21no—
335C22×C4—2122no—
336C22×C4C2×D21211yescharacteristic
337C22×C4#1171994511yescharacteristic
338C22×C4C2×C22⋊D5311yescharacteristic
339SD16—4242no—
340D8—4242no—
341D8—2122no—
342D8—2122no—
343M16—42no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—1062no—
350C2×C8—1062no—
351C2×C8—4242no—
352C2×C8C2×C22⋊D5311yescharacteristic
353C2×C8C2×C22⋊D5311yescharacteristic
354C2×C8C2×C22⋊D5311yescharacteristic
355C2×C8C2×C22⋊D5311yescharacteristic
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—1062no—
361C4⋊C4—1062no—
362C4⋊C4C2×D21211yescharacteristic
363C4⋊C4C2×D21211yescharacteristic
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—2122no—
370C22⋊C4—2122no—
371C42—21no—
372C42C2×D21211yescharacteristic
373C42#1171994511yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C42C23×D5311yescharacteristic
376C23—21no—
377C23—2122no—
378C23#1313319111yescharacteristic
379Q8—21no—
380Q8—21no—
381Q8—4242no—
382D4—21no—
383D4—21no—
384D4—42no—
385D4—4242no—
386D4—4244no—
387D4—2122no—
388D4—2122no—
389D4—2122no—
390D4—2122no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—4244no—
402C2×C4—2122no—
403C2×C4#1313319111yescharacteristic
404C2×C4#1313319111yescharacteristic
405C2×C4#1313319111yescharacteristic
406C2×C4C22×C22⋊D5311yescharacteristic
407C2×C4C22×D21211yescharacteristic
408C2×C4C2×#1171994511yescharacteristic
409C8—21no—
410C8—21no—
411C8—21no—
412C8—41no—
413C8—41no—
414C8—2122no—
415C8—2122no—
416C8#1313323712yes—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—4242no—
423C22—4242no—
424C22—4242no—
425C22#1681089312yes—
426C22C2×#1313319111yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—4242no—
435C4C2×#1313323711yescharacteristic
436C4#1681142111yescharacteristic
437C4#1681142811yescharacteristic
438C4#1681144011yescharacteristic
439C2—22no—
440C2—81no—
441C2—8482no—
442C2#2028417011yescharacteristic
443C2#2028438511yescharacteristic
444C2C2×#1681089311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2966C2
socleC22×C53C2×#121
hypercenter#2966D53
hypocenterC53#19222
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
1. #20286041C2semidirect
2. #20285294C2semidirect
3. #20274255C2semidirect
4. #20274248C2semidirect
5. #20274119C2semidirect
6. #20274114C2semidirect
7. #20274075C2semidirect
8. #20274070C2semidirect
9. C2×#16810376C2semidirect
10. #20273411C2semidirect
11. C53×#2966C2semidirect
219. #2966D53semidirect
269. C53#19222semidirect

Maximal subgroups

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

subgroupindex
#202860412
#202852942
#202742552
#202742482
#202741192
#202741142
#202740752
#202740702
C2×#168103762
#202734112
C53×#29662
#1922253