Finite Group Databasebeta

← #27367517

Subgroups — #27367517

Order 13568.

630 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#20285687C212yesmaximal, complemented
2#20285294C211yesmaximal, characteristic, complemented
3#20274233C211yesmaximal, characteristic, complemented
4#20274228C211yesmaximal, characteristic, complemented
5#20274199C211yesmaximal, characteristic, complemented
6#20274194C211yesmaximal, characteristic, complemented
7#20274159C212yesmaximal, complemented
8#20274152C212yesmaximal, complemented
9#20273463C211yesmaximal, characteristic, complemented
10#20273421C212yesmaximal, complemented
11C53×#2966C211yesmaximal, characteristic, complemented
12#16811173C2212yes—
13#16811028C2211yescharacteristic
14#16811005—22no—
15#16811000C2211yescharacteristic
16#16810980—22no—
17#16810970C2212yes—
18#16810859—21no—
19#16810859—21no—
20#16810833C2212yes—
21#16810828C2212yes—
22#16810823C2212yes—
23#16810818C2212yes—
24#16810783—22no—
25#16810766—22no—
26C4×D424—21no—
27C4×D424—21no—
28#16810676C2212yes—
29#16810376C2212yes—
30#16810374C2211yescharacteristic
31#16810374C2211yescharacteristic
32#16810368C2212yes—
33#16810368C2212yes—
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×#11719925—22no—
49C2×#11719923—22no—
50C2×#11719924—22no—
51C2×D424—21no—
52C2×D424—21no—
53C2×#11719922—22no—
54#13133217C2312yes—
55#13133214C2312yes—
56#13133205—22no—
57#13133191—22no—
58#13133191—22no—
59C4×D212—22no—
60C4×D212—22no—
61#13133151—21no—
62#13133151—21no—
63C4⋊#88322C2312yes—
64C4⋊#88322C2312yes—
65#13133140—22no—
66#13133140—22no—
67#13133139—22no—
68#13133139—22no—
69#13133137—22no—
70#13133137—22no—
71C22⋊#88322—22no—
72C4×#88322—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—82no—
111#11719945—82no—
112C2×D212—22no—
113C2×D212—22no—
114C2×D212—22no—
115C2×C4×D53—42no—
116C2×Dic106—22no—
117M16×C53—42no—
118C2×C8×C53—22no—
119C2×C8×C53—21no—
120C2×C8×C53—21no—
121C2×C8×C53—41no—
122C2×C8×C53—42no—
123C2×C8×C53C2×D411yescharacteristic
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53C2×D411yescharacteristic
132C4⋊C4×C53C2×D411yescharacteristic
133C22⋊C4×C53—21no—
134C22⋊C4×C53—21no—
135C22⋊C4×C53—21no—
136C22⋊C4×C53—21no—
137C22⋊C4×C53—21no—
138C42×C53—21no—
139C42×C53C2×D411yescharacteristic
140C42×C53C2×D411yescharacteristic
141C42×C53#4111yescharacteristic
142C42×C53C2411yescharacteristic
143#11719925—82no—
144#11719924—82no—
145#11719923—82no—
146#11719922—82no—
147#11719921—42no—
148#11719921—42no—
149C4⋊Dic53—22no—
150C4⋊Dic53—22no—
151C2×#88322—22no—
152C2×#88322—22no—
153D424—42no—
154D424—42no—
155C23×C53—21no—
156C23×C53#12111yescharacteristic
157C22×D53—42no—
158Q8×C53—22no—
159D4×C53—22no—
160D4×C53—42no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—22no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C2×#4111yescharacteristic
176C22⋊D53—84no—
177C2×Dic53—42no—
178D212—42no—
179D212—42no—
180D212—42no—
181D212—42no—
182D212—82no—
183C4×D53—84no—
184Dic106—42no—
185Dic106—42no—
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#19225—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
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#57612yes—
215Dic53—82no—
216#3212—532no—
217#3169—531no—
218#2966D5311yescharacteristic, complemented
219#1694—531no—
220#1693—531no—
221#1688—531no—
222#1687—531no—
223#1682—532no—
224#1681—532no—
225#1564—531no—
226#1553—532no—
227C2×C53—41no—
228C2×C53—81no—
229C2×C53#303511yescharacteristic
230C2×C53#305811yescharacteristic
231C2×C53#306711yescharacteristic
232D53—162no—
233#532—532no—
234#529C2×D5311yescharacteristic
235#500—531no—
236#495—1062no—
237#493—531no—
238#488—1062no—
239#486—532no—
240#466—1061no—
241#466—1061no—
242#462—532no—
243#461—532no—
244#460—532no—
245#459—532no—
246#452—1062no—
247#449—1062no—
248#446C2×D5311yescharacteristic
249C8×Q8C2×D5311yescharacteristic
250C4×D8—1061no—
251C4×D8—1061no—
252#436C2×D5311yescharacteristic
253#435—21no—
254C8×D4—21no—
255C8×D4C2×D5311yescharacteristic
256#433—21no—
257#432C2×D5311yescharacteristic
258#431—532no—
259#405—21no—
260C2×C4×C8C2×D5311yescharacteristic
261#335—532no—
262#334—531no—
263#334—531no—
264#332—532no—
265#332—532no—
266#331—532no—
267#331—532no—
268C53#1922511yescharacteristic, complemented, Sylow
269C2×#41—1062no—
270C2×Q16—1062no—
271C2×SD16—1062no—
272C2×SD16—1062no—
273C2×D8—1062no—
274C2×D8—1061no—
275C2×D8—1061no—
276C2×M16—21no—
277C22×C8—21no—
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—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—532no—
303C4⋊C8—532no—
304C4⋊C8C22×D5311yescharacteristic
305C4⋊C8C22×D5311yescharacteristic
306#103—1062no—
307#103—1062no—
308#102—1062no—
309#102—1062no—
310#102—1062no—
311#102—1062no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C22⋊C8—1062no—
316#97—21no—
317#97—21no—
318C4×C8—21no—
319C4×C8—1062no—
320C4×C8C22⋊D5311yescharacteristic
321C4×C8C22⋊D5311yescharacteristic
322C4×C8C22×D5311yescharacteristic
323C4×C8C22×D5311yescharacteristic
324#41—4242no—
325#41—4242no—
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
339Q16—4242no—
340SD16—4242no—
341SD16—4242no—
342D8—4242no—
343D8—2122no—
344D8—2122no—
345M16—42no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—1062no—
352C2×C8—1062no—
353C2×C8C2×C22⋊D5311yescharacteristic
354C2×C8C2×C22⋊D5311yescharacteristic
355C2×C8C2×C22⋊D5311yescharacteristic
356C2×C8C2×C22⋊D5311yescharacteristic
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—1062no—
362C4⋊C4—1062no—
363C4⋊C4C2×D21211yescharacteristic
364C4⋊C4C2×D21211yescharacteristic
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—2122no—
371C22⋊C4—2122no—
372C42—21no—
373C42C2×D21211yescharacteristic
374C42#1171994511yescharacteristic
375C42C2×C22⋊D5311yescharacteristic
376C42C23×D5311yescharacteristic
377C23—21no—
378C23—2122no—
379C23#1313319111yescharacteristic
380Q8—22no—
381Q8—2122no—
382Q8—2122no—
383D4—22no—
384D4—42no—
385D4—4242no—
386D4—4244no—
387D4—2122no—
388D4—2122no—
389D4—2122no—
390D4—2122no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—4244no—
401C2×C4—2122no—
402C2×C4#1313319111yescharacteristic
403C2×C4#1313319111yescharacteristic
404C2×C4#1313319111yescharacteristic
405C2×C4C22×C22⋊D5311yescharacteristic
406C2×C4C22×D21211yescharacteristic
407C2×C4C2×#1171994511yescharacteristic
408C8—21no—
409C8—21no—
410C8—21no—
411C8—41no—
412C8—41no—
413C8—2122no—
414C8—2122no—
415C8#1313323712yes—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—4242no—
423C22—4242no—
424C22—4242no—
425C22C2×#1313319111yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—4242no—
434C4C2×#1313323711yescharacteristic
435C4#1681142111yescharacteristic
436C4#1681142812yes—
437C2—41no—
438C2—81no—
439C2—8482no—
440C2#2028417011yescharacteristic
441C2#2028436311yescharacteristic
442C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2966C2
socleC22×C53C2×#121
hypercenter#2966D53
hypocenterC53#19225
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. #20285687C2semidirect
2. #20285294C2semidirect
3. #20274233C2semidirect
4. #20274228C2semidirect
5. #20274199C2semidirect
6. #20274194C2semidirect
7. #20274159C2semidirect
8. #20274152C2semidirect
9. #20273463C2semidirect
10. #20273421C2semidirect
11. C53×#2966C2semidirect
218. #2966D53semidirect
268. C53#19225semidirect

Maximal subgroups

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

subgroupindex
#202856872
#202852942
#202742332
#202742282
#202741992
#202741942
#202741592
#202741522
#202734632
#202734212
C53×#29662
#1922553