Finite Group Databasebeta

← #27367443

Subgroups — #27367443

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285687C212yesmaximal, complemented
2#20285294C211yesmaximal, characteristic, complemented
3#20274283C212yesmaximal, complemented
4#20274276C212yesmaximal, complemented
5#20274109C211yesmaximal, characteristic, complemented
6#20274104C211yesmaximal, characteristic, complemented
7#20274075C211yesmaximal, characteristic, complemented
8#20274070C211yesmaximal, characteristic, 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#16810376C2211yescharacteristic
30#16810376C2211yescharacteristic
31#16810374C2212yes—
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—21no—
159Q8×C53—21no—
160D4×C53—21no—
161D4×C53—21no—
162D4×C53—42no—
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—21no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53C22×D411yescharacteristic
177C2×C4×C53C22×D411yescharacteristic
178C2×C4×C53C2×#4111yescharacteristic
179C22⋊D53—84no—
180C2×Dic53—42no—
181D212—42no—
182D212—42no—
183D212—42no—
184D212—42no—
185D212—82no—
186C4×D53—84no—
187Dic106—42no—
188Dic106—42no—
189C8×C53—21no—
190C8×C53—21no—
191C8×C53—21no—
192C8×C53—41no—
193C8×C53—41no—
194C8×C53#13512yes—
195#88322—42no—
196#88322—42no—
197#19219—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53C2×#12111yescharacteristic
205C2×D53—82no—
206C2×D53—82no—
207C2×D53—82no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—41no—
215C4×C53C2×#13511yescharacteristic
216C4×C53#57511yescharacteristic
217C4×C53#57611yescharacteristic
218C4×C53#57611yescharacteristic
219Dic53—82no—
220#3212—532no—
221#3169—531no—
222#2966D5311yescharacteristic, complemented
223#1702—532no—
224#1701—532no—
225#1674—531no—
226#1673—531no—
227#1668—531no—
228#1667—531no—
229#1564—531no—
230#1553—532no—
231C2×C53—41no—
232C2×C53—81no—
233C2×C53#303511yescharacteristic
234C2×C53#305811yescharacteristic
235C2×C53#306711yescharacteristic
236D53—162no—
237#532—532no—
238#529C2×D5311yescharacteristic
239#500—531no—
240#495—1062no—
241#493—531no—
242#488—1062no—
243#486—532no—
244#466—1061no—
245#466—1061no—
246#462—532no—
247#461—532no—
248#460—532no—
249#459—532no—
250#452—1062no—
251#449—1062no—
252#446C2×D5311yescharacteristic
253C8×Q8C2×D5311yescharacteristic
254C4×D8—1061no—
255C4×D8—1061no—
256#436C2×D5311yescharacteristic
257#435—21no—
258C8×D4—21no—
259C8×D4C2×D5311yescharacteristic
260#433—21no—
261#432C2×D5311yescharacteristic
262#431—532no—
263#405—21no—
264C2×C4×C8C2×D5311yescharacteristic
265#335—531no—
266#335—531no—
267#334—532no—
268#332—532no—
269#332—532no—
270#331—532no—
271#331—532no—
272C53#1921911yescharacteristic, complemented, Sylow
273C2×#41—1062no—
274C2×Q16—1062no—
275C2×SD16—1062no—
276C2×SD16—1062no—
277C2×D8—1062no—
278C2×D8—1061no—
279C2×D8—1061no—
280C2×M16—21no—
281C22×C8—21no—
282C22×C8C22⋊D5311yescharacteristic
283C22×C8C22⋊D5311yescharacteristic
284C4⋊Q8—532no—
285#127—532no—
286#126—21no—
287#125D21211yescharacteristic
288#124—1062no—
289#124D21211yescharacteristic
290#123—21no—
291C22⋊Q8D21211yescharacteristic
292#121—1062no—
293#121—1062no—
294#121D21211yescharacteristic
295C4×Q8C22×D5311yescharacteristic
296C4×D4—21no—
297C4×D4—1062no—
298C4×D4—1062no—
299C4×D4C22×D5311yescharacteristic
300#117—21no—
301C2×C42C22×D5311yescharacteristic
302#107—1061no—
303#107—1061no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—532no—
307C4⋊C8—532no—
308C4⋊C8C22×D5311yescharacteristic
309C4⋊C8C22×D5311yescharacteristic
310#103—1062no—
311#103—1062no—
312#102—1062no—
313#102—1062no—
314#102—1062no—
315#102—1062no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—1062no—
320#97—21no—
321#97—21no—
322C4×C8—21no—
323C4×C8—1062no—
324C4×C8C22⋊D5311yescharacteristic
325C4×C8C22⋊D5311yescharacteristic
326C4×C8C22×D5311yescharacteristic
327C4×C8C22×D5311yescharacteristic
328#41—4242no—
329#41—4242no—
330C2×Q8—1062no—
331C2×Q8C2×D21211yescharacteristic
332C2×D4—21no—
333C2×D4—1062no—
334C2×D4—1062no—
335C2×D4—1062no—
336C2×D4—2122no—
337C2×D4C2×D21211yescharacteristic
338C22×C4—21no—
339C22×C4—2122no—
340C22×C4C2×D21211yescharacteristic
341C22×C4#1171994511yescharacteristic
342C22×C4C2×C22⋊D5311yescharacteristic
343Q16—4242no—
344SD16—4242no—
345SD16—4242no—
346D8—4242no—
347D8—2122no—
348D8—2122no—
349M16—42no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—1062no—
356C2×C8—1062no—
357C2×C8C2×C22⋊D5311yescharacteristic
358C2×C8C2×C22⋊D5311yescharacteristic
359C2×C8C2×C22⋊D5311yescharacteristic
360C2×C8C2×C22⋊D5311yescharacteristic
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—1062no—
366C4⋊C4—1062no—
367C4⋊C4C2×D21211yescharacteristic
368C4⋊C4C2×D21211yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—2122no—
375C22⋊C4—2122no—
376C42—21no—
377C42C2×D21211yescharacteristic
378C42#1171994511yescharacteristic
379C42C2×C22⋊D5311yescharacteristic
380C42C23×D5311yescharacteristic
381C23—21no—
382C23—2122no—
383C23#1313319111yescharacteristic
384Q8—21no—
385Q8—21no—
386Q8—2122no—
387Q8—2122no—
388D4—21no—
389D4—21no—
390D4—42no—
391D4—4242no—
392D4—4244no—
393D4—2122no—
394D4—2122no—
395D4—2122no—
396D4—2122no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—4244no—
408C2×C4—2122no—
409C2×C4#1313319111yescharacteristic
410C2×C4#1313319111yescharacteristic
411C2×C4#1313319111yescharacteristic
412C2×C4C22×C22⋊D5311yescharacteristic
413C2×C4C22×D21211yescharacteristic
414C2×C4C2×#1171994511yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8—41no—
419C8—41no—
420C8—2122no—
421C8—2122no—
422C8#1313323712yes—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—4242no—
430C22—4242no—
431C22—4242no—
432C22C2×#1313319111yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—4242no—
441C4C2×#1313323711yescharacteristic
442C4#1681142111yescharacteristic
443C4#1681142811yescharacteristic
444C4#1681142811yescharacteristic
445C2—41no—
446C2—81no—
447C2—8482no—
448C2#2028417011yescharacteristic
449C2#2028436311yescharacteristic
450C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2966C2
socleC22×C53C2×#121
hypercenter#2966D53
hypocenterC53#19219
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. #20274283C2semidirect
4. #20274276C2semidirect
5. #20274109C2semidirect
6. #20274104C2semidirect
7. #20274075C2semidirect
8. #20274070C2semidirect
9. #20273463C2semidirect
10. #20273421C2semidirect
11. C53×#2966C2semidirect
222. #2966D53semidirect
272. C53#19219semidirect

Maximal subgroups

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

subgroupindex
#202856872
#202852942
#202742832
#202742762
#202741092
#202741042
#202740752
#202740702
#202734632
#202734212
C53×#29662
#1921953