Finite Group Databasebeta

← #27367474

Subgroups — #27367474

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#20286035C212yesmaximal, complemented
2#20283690C211yesmaximal, characteristic, complemented
3#20274252C212yesmaximal, complemented
4#20274245C212yesmaximal, complemented
5#20274116C211yesmaximal, characteristic, complemented
6#20274111C211yesmaximal, characteristic, complemented
7#20274072C211yesmaximal, characteristic, complemented
8#20274067C211yesmaximal, characteristic, complemented
9C2×#16810376C211yesmaximal, characteristic, complemented
10#20273410C212yesmaximal, complemented
11C53×#3169C211yesmaximal, characteristic, complemented
12#16811157C2211yescharacteristic
13#16810923—22no—
14#16810915C2212yes—
15#16810910C2212yes—
16#16810857—22no—
17#16810850—22no—
18#16810838—22no—
19#16810820C2212yes—
20#16810815C2212yes—
21#16810753C2211yescharacteristic
22C8×Dic106C2211yescharacteristic
23#16810702C2211yescharacteristic
24#16810694—21no—
25C8×D212C2211yescharacteristic
26C8×C22⋊D53—21no—
27#16810682—21no—
28#16810680C2211yescharacteristic
29C2×C4⋊#88322C2212yes—
30#16810577—21no—
31#16810376C2212yes—
32#16810376C2211yescharacteristic
33#16810376C2211yescharacteristic
34#16810372C2212yes—
35#16810372C2212yes—
36#16810367C2212yes—
37#16810367C2212yes—
38C53×#532C2212yes—
39C53×#500C2211yescharacteristic
40C53×#493C2211yescharacteristic
41C53×#486C2212yes—
42C53×#466—21no—
43C53×#466—21no—
44C4×D8×C53—21no—
45C4×D8×C53—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×#11719923—22no—
48C2×#11719922—22no—
49C2×#11719912—21no—
50C22×#88322—22no—
51C2×C8×D53—21no—
52#13133211—21no—
53#13133209D411yescharacteristic
54#13133205D411yescharacteristic
55#13133200—21no—
56#13133196D411yescharacteristic
57#13133191D411yescharacteristic
58C4×Dic106C2311yescharacteristic
59C4×D212C2311yescharacteristic
60C4×C22⋊D53—21no—
61#13133174—21no—
62#13133150—22no—
63#13133148—22no—
64C4⋊#88322—22no—
65C4⋊#88322C2311yescharacteristic
66C4⋊#88322C2312yes—
67C4⋊#88322C2312yes—
68C4⋊#88322C2311yescharacteristic
69#13133146—21no—
70#13133146—21no—
71#13133140—22no—
72#13133138—22no—
73#13133138—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76#13133129—21no—
77#13133129—21no—
78#13133128—21no—
79#13133127—21no—
80C8×Dic53—21no—
81C2×#41×C53—22no—
82C2×D8×C53—21no—
83C2×D8×C53—21no—
84C22×C8×C53D411yescharacteristic
85C22×C8×C53D411yescharacteristic
86C4⋊Q8×C53C2312yes—
87#127×C53C2312yes—
88#124×C53—22no—
89#121×C53—22no—
90#121×C53—22no—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93C2×C42×C53C2311yescharacteristic
94#107×C53—21no—
95#107×C53—21no—
96#102×C53—22no—
97#102×C53—22no—
98C4×C8×C53D411yescharacteristic
99C4×C8×C53D411yescharacteristic
100C4×C8×C53C2311yescharacteristic
101C4×C8×C53C2311yescharacteristic
102#41×C53—84no—
103C2×Q8×C53—22no—
104C2×D4×C53—22no—
105C2×D4×C53—22no—
106C2×D4×C53—22no—
107C2×D4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×C22⋊D53—21no—
113C2×D212C2×D411yescharacteristic
114C2×C4×D53—21no—
115C2×Dic106C2×D411yescharacteristic
116D8×C53—42no—
117D8×C53—42no—
118C2×C8×C53—22no—
119C2×C8×C53—41no—
120C2×C8×C53C2×D411yescharacteristic
121C2×C8×C53C2×D411yescharacteristic
122C2×C8×C53C2×D411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C22⋊C4×C53—42no—
127C22⋊C4×C53—42no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53C2411yescharacteristic
132C22⋊Dic53—21no—
133#11719923—82no—
134#11719922—82no—
135#11719921—21no—
136#11719921—21no—
137#11719921—21no—
138#11719921—21no—
139C4⋊Dic53C2×D411yescharacteristic
140C4⋊Dic53C2×D411yescharacteristic
141#11719919—21no—
142#11719919—21no—
143#11719919—21no—
144#11719919—21no—
145C4×Dic53—21no—
146C2×#88322—21no—
147C2×#88322—21no—
148C2×#88322—22no—
149C2×#88322—22no—
150C2×#88322—82no—
151#11719912—42no—
152C8×D53—42no—
153C23×C53—42no—
154C23×C53#12111yescharacteristic
155C22×D53—21no—
156Q8×C53—82no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—42no—
160D4×C53—42no—
161D4×C53—82no—
162D4×C53—84no—
163C2×C4×C53—22no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—42no—
169C2×C4×C53—84no—
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—42no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180D212—21no—
181D212—21no—
182C4×D53—41no—
183C4×D53—41no—
184Dic106—21no—
185Dic106—21no—
186C8×C53—21no—
187C8×C53—21no—
188C8×C53—21no—
189C8×C53#13512yes—
190#88322—41no—
191#88322—41no—
192#88322—42no—
193#88322—42no—
194#19222—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—82no—
198C22×C53—82no—
199C22×C53—82no—
200C22×C53#47212yes—
201C22×C53C2×#12111yescharacteristic
202C2×D53—41no—
203C2×D53—41no—
204C2×D53—41no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—82no—
210C4×C53C2×#13511yescharacteristic
211C4×C53#57511yescharacteristic
212C4×C53#57611yescharacteristic
213C4×C53#57811yescharacteristic
214Dic53—41no—
215Dic53—41no—
216Dic53—41no—
217#3247—532no—
218#3169D5311yescharacteristic, complemented
219#2966—531no—
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—162no—
230C2×C53#303511yescharacteristic
231C2×C53#306111yescharacteristic
232C2×C53C2×#47211yescharacteristic
233D53—81no—
234#532C2×D5312yes—
235#529—531no—
236#500C2×D5311yescharacteristic
237#493C2×D5311yescharacteristic
238#486C2×D5312yes—
239#478—1062no—
240#477—532no—
241#476—532no—
242#466—21no—
243#466—21no—
244#466—1062no—
245#465—1062no—
246#463—1062no—
247#460—532no—
248#459—532no—
249#446—531no—
250C8×Q8—531no—
251C4×D8—21no—
252C4×D8—21no—
253#436—531no—
254#435—1061no—
255C8×D4—531no—
256C8×D4—1061no—
257#433—1061no—
258#432—531no—
259C2×C4⋊C8—532no—
260#405—1061no—
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—22no—
271C2×SD16—1062no—
272C2×D8—21no—
273C2×D8—21no—
274C2×D8—1062no—
275C2×M16—1061no—
276C22×C8—1062no—
277C22×C8—1061no—
278C22×C8C22⋊D5311yescharacteristic
279C22×C8C22⋊D5311yescharacteristic
280C4⋊Q8C22×D5312yes—
281#127C22×D5312yes—
282#126—1061no—
283#125—531no—
284#124—22no—
285#124—531no—
286#123—1061no—
287C22⋊Q8—531no—
288#121—22no—
289#121—22no—
290#121—531no—
291C4×Q8—531no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—531no—
295C4×D4—1061no—
296#117—1061no—
297C2×C42C22×D5311yescharacteristic
298#107—21no—
299#107—21no—
300#107—1062no—
301#106—1062no—
302C4⋊C8—531no—
303C4⋊C8—531no—
304C4⋊C8—532no—
305C4⋊C8—532no—
306C4⋊C8—1061no—
307C4⋊C8—1061no—
308C4⋊C8—1062no—
309#103—1062no—
310#102—22no—
311#102—22no—
312#102—1062no—
313#102—1062no—
314#102—1062no—
315C22⋊C8—1061no—
316C22⋊C8—1061no—
317C22⋊C8—1061no—
318#97—1061no—
319#97—1061no—
320C4×C8—1061no—
321C4×C8C22⋊D5311yescharacteristic
322C4×C8C22⋊D5311yescharacteristic
323C4×C8C22×D5311yescharacteristic
324C4×C8C22×D5311yescharacteristic
325#41—84no—
326C2×Q8—22no—
327C2×Q8—531no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—42no—
332C2×D4—531no—
333C2×D4—1061no—
334C22×C4—42no—
335C22×C4—1061no—
336C22×C4D4×D5311yescharacteristic
337C22×C4#1171994711yescharacteristic
338C22×C4C2×C22⋊D5311yescharacteristic
339SD16—4242no—
340D8—42no—
341D8—42no—
342D8—4242no—
343M16—2122no—
344C2×C8—22no—
345C2×C8—41no—
346C2×C8—1062no—
347C2×C8—1062no—
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—4242no—
351C2×C8—2122no—
352C2×C8C2×C22⋊D5311yescharacteristic
353C2×C8C2×C22⋊D5311yescharacteristic
354C2×C8C2×C22⋊D5311yescharacteristic
355C2×C8C2×C22⋊D5311yescharacteristic
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—531no—
359C4⋊C4—531no—
360C4⋊C4—1061no—
361C4⋊C4—1061no—
362C4⋊C4—1061no—
363C4⋊C4—1061no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—1061no—
367C22⋊C4—1061no—
368C22⋊C4—1061no—
369C22⋊C4—1061no—
370C22⋊C4—1061no—
371C42—1061no—
372C42D4×D5311yescharacteristic
373C42#1171994711yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C42C23×D5311yescharacteristic
376C23—42no—
377C23—1061no—
378C23#1313318811yescharacteristic
379Q8—82no—
380Q8—1061no—
381Q8—1061no—
382D4—42no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—82no—
387D4—84no—
388D4—1061no—
389D4—1061no—
390D4—2122no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—84no—
398C2×C4—1061no—
399C2×C4—1061no—
400C2×C4—1061no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4#1313318811yescharacteristic
404C2×C4#1313318811yescharacteristic
405C2×C4#1313318811yescharacteristic
406C2×C4C2×D4×D5311yescharacteristic
407C2×C4C22×C22⋊D5311yescharacteristic
408C2×C4C2×#1171994711yescharacteristic
409C8—21no—
410C8—21no—
411C8—21no—
412C8—2121no—
413C8—2121no—
414C8—2122no—
415C8—2122no—
416C8#1313323712yes—
417C22—21no—
418C22—21no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—2121no—
423C22—2121no—
424C22—2121no—
425C22#1681089012yes—
426C22C2×#1313318811yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—82no—
432C4—2121no—
433C4—2121no—
434C4—2121no—
435C4C2×#1313323711yescharacteristic
436C4#1681142111yescharacteristic
437C4#1681142511yescharacteristic
438C4#1681143711yescharacteristic
439C2—22no—
440C2—162no—
441C2—4241no—
442C2#2028416411yescharacteristic
443C2#2028438111yescharacteristic
444C2C2×#1681089011yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3169C2
socleC22×C53C2×#121
hypercenter#3169D53
hypocenterC53#19222
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #20286035C2semidirect
2. #20283690C2semidirect
3. #20274252C2semidirect
4. #20274245C2semidirect
5. #20274116C2semidirect
6. #20274111C2semidirect
7. #20274072C2semidirect
8. #20274067C2semidirect
9. C2×#16810376C2semidirect
10. #20273410C2semidirect
11. C53×#3169C2semidirect
218. #3169D53semidirect
269. C53#19222semidirect

Maximal subgroups

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

subgroupindex
#202860352
#202836902
#202742522
#202742452
#202741162
#202741112
#202740722
#202740672
C2×#168103762
#202734102
C53×#31692
#1922253