Finite Group Databasebeta

← #27366776

Subgroups — #27366776

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285040C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic
3C2×#16810754C211yesmaximal, characteristic, complemented
4#20274299C212yesmaximal, complemented
5#20274285C212yesmaximal, complemented
6#20274271C212yesmaximal, complemented
7#20274257C212yesmaximal, complemented
8#20273461C211yesmaximal, characteristic, complemented
9#20273372C211yesmaximal, characteristic, complemented
10#20273362C211yesmaximal, characteristic
11C53×#3104C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16810980—21no—
14#16810980—21no—
15#16810975C2211yescharacteristic
16#16810975C2211yescharacteristic
17#16810971C2211yescharacteristic
18#16810971C2211yescharacteristic
19#16810941—21no—
20#16810908C2212yes—
21#16810823C2212yes—
22#16810762—21no—
23#16810754—21no—
24#16810754—21no—
25#16810754C2212yes—
26#16810754C2212yes—
27C4×#11719924—22no—
28#16810676C2211yescharacteristic
29C2×C4⋊#88322—21no—
30C2×C4⋊#88322C2211yescharacteristic
31#16810617—21no—
32#16810617—21no—
33C2×#13133128—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yes—
36#16810373C2212yes—
37#16810372C2212yes—
38#16810369C2212yes—
39#16810368C2212yes—
40#16810366C2212yes—
41C53×#533C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#484—21no—
44C53×#477C2212yes—
45C53×#462C2212yes—
46C53×#451—21no—
47C4×Q16×C53—22no—
48C53×#431C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719924—22no—
51C22×#88322—21no—
52C22×#88322D411yescharacteristic
53C22×#88322D411yescharacteristic
54#13133217C2311yescharacteristic
55#13133214C2311yescharacteristic
56#13133205—21no—
57#13133191—21no—
58#13133191—21no—
59C4×D212—21no—
60C4×D212—21no—
61#13133148—22no—
62C4⋊#88322—21no—
63C4⋊#88322—22no—
64C4⋊#88322—22no—
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133141—22no—
68#13133140—21no—
69#13133140—21no—
70#13133140—21no—
71#13133140—21no—
72#13133137—22no—
73#13133137—21no—
74#13133137—21no—
75#13133137—21no—
76#13133137—21no—
77C22⋊#88322—21no—
78#13133128—22no—
79#13133128—22no—
80C4×#88322—21no—
81C4×#88322D411yescharacteristic
82C4×#88322D411yescharacteristic
83C4×#88322C2311yescharacteristic
84C4×#88322C2311yescharacteristic
85C22×Q8×C53D411yescharacteristic
86C2×Q16×C53—22no—
87C4⋊Q8×C53C2311yescharacteristic
88C4⋊Q8×C53C2311yescharacteristic
89#125×C53—21no—
90C22⋊Q8×C53—21no—
91C22⋊Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×Q8×C53D412yes—
97C4×Q8×C53C2312yes—
98#117×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C42×C53—21no—
102C2×C42×C53C2311yescharacteristic
103#107×C53—22no—
104C4⋊C8×C53C2312yes—
105#103×C53—22no—
106#103×C53—22no—
107C22⋊C8×C53—21no—
108C4×C8×C53—21no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—42no—
113C2×Q8×C53C2×D412yes—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C2×C22⋊D53—41no—
120#11719945—42no—
121#11719945—42no—
122C2×D212—21no—
123C2×D212—21no—
124C2×D212—21no—
125C2×C4×D53—41no—
126C2×Dic106—21no—
127Q16×C53—44no—
128C2×C8×C53—22no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53C2×D412yes—
139C22⋊C4×C53—41no—
140C42×C53—22no—
141C42×C53—22no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147#11719924—44no—
148#11719921—41no—
149#11719921—41no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C2×#88322—22no—
153C2×#88322—22no—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322—42no—
157C2×#88322C2×D411yescharacteristic
158C2×#88322C2×D411yescharacteristic
159C2×#88322C2×D411yescharacteristic
160C2×#88322C2×D411yescharacteristic
161C23×C53#12111yescharacteristic
162C22×D53—41no—
163Q8×C53—22no—
164Q8×C53—22no—
165Q8×C53—42no—
166Q8×C53—42no—
167Q8×C53—81no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—22no—
171C2×C4×C53—41no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—41no—
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
184C22⋊D53—82no—
185C2×Dic53—41no—
186D212—41no—
187D212—41no—
188D212—42no—
189D212—81no—
190C4×D53—82no—
191Dic106—41no—
192Dic106—41no—
193C8×C53—42no—
194#88322—21no—
195#88322—21no—
196#88322—21no—
197#88322—21no—
198#88322—42no—
199#88322—42no—
200#19155—531nomaximal, Sylow
201C22×C53—21no—
202C22×C53—21no—
203C22×C53#46412yes—
204C22×C53C2×#12111yescharacteristic
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—81no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—42no—
212C4×C53—42no—
213C4×C53—81no—
214C4×C53—81no—
215C4×C53#46412yes—
216C4×C53C2×#13511yescharacteristic
217C4×C53C2×#13711yescharacteristic
218C4×C53#57512yes—
219Dic53—81no—
220#3131—531no—
221#3104D5311yescharacteristic, complemented
222#3103—531no—
223C2×#446—531no—
224#1704—532no—
225#1702—532no—
226#1700—532no—
227#1698—532no—
228#1564—531no—
229#1542—531no—
230#1540—531no—
231C2×C53—22no—
232C2×C53C2×#46411yescharacteristic
233C2×C53#305411yescharacteristic
234C2×C53#306711yescharacteristic
235D53—161no—
236#533C2×D5311yescharacteristic
237#532—531no—
238C2×C4×Q8C2×D5311yescharacteristic
239#488—1061no—
240#488—1061no—
241#487—531no—
242#487—531no—
243#486—531no—
244#486—531no—
245#484—21no—
246#481—1061no—
247#477C2×D5312yes—
248#475—532no—
249#462C2×D5312yes—
250#460—532no—
251#451—21no—
252#448—1061no—
253#446—532no—
254#446—532no—
255#446—1061no—
256#446—1061no—
257C4×Q16—22no—
258C4×SD16—1062no—
259#431—531no—
260#431C2×D5311yescharacteristic
261C2×C4⋊C8—531no—
262C2×C4⋊C8—1061no—
263#416—1061no—
264#416—1061no—
265C2×#97—1061no—
266C2×C4×C8—531no—
267#335—532no—
268#334—532no—
269#333—532no—
270#332—532no—
271#332—532no—
272#331—532no—
273C53#1915511yescharacteristic, complemented, Sylow
274C2×#41—1061no—
275C22×Q8C22⋊D5311yescharacteristic
276C2×Q16—22no—
277C2×SD16—1062no—
278C22×C8—531no—
279C22×C8—531no—
280C22×C8—1061no—
281C4⋊Q8—531no—
282C4⋊Q8C22×D5311yescharacteristic
283C4⋊Q8C22×D5311yescharacteristic
284#127—531no—
285#125—21no—
286#124—1061no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289#121—1061no—
290#121—1061no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8C22⋊D5312yes—
296C4×Q8C22×D5312yes—
297C4×D4—1061no—
298C4×D4—1061no—
299#117—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C22⋊D5311yescharacteristic
302C2×C42—21no—
303C2×C42C22×D5311yescharacteristic
304#107—22no—
305#106—1062no—
306C4⋊C8—532no—
307C4⋊C8—532no—
308C4⋊C8—1062no—
309C4⋊C8—1062no—
310C4⋊C8—1061no—
311C4⋊C8C22×D5312yes—
312#103—22no—
313#103—22no—
314#103—1061no—
315#103—1061no—
316#103—1061no—
317#103—1061no—
318#103—1062no—
319#102—1061no—
320#102—1061no—
321#102—1061no—
322#102—1061no—
323#102—1062no—
324C22⋊C8—21no—
325C22⋊C8—1061no—
326#97—1062no—
327#97—1062no—
328C4×C8—21no—
329C4×C8—531no—
330C4×C8—531no—
331C4×C8—531no—
332C4×C8—531no—
333C4×C8—1061no—
334#41—2122no—
335#41—2122no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—42no—
340C2×Q8—1061no—
341C2×Q8C2×C22⋊D5312yes—
342C2×D4—1061no—
343C2×D4—1061no—
344C2×D4—1061no—
345C2×D4—2121no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—2121no—
349C22×C4D4×D5311yescharacteristic
350C22×C4#1171994711yescharacteristic
351C22×C4C2×C22⋊D5311yescharacteristic
352Q16—44no—
353SD16—2124no—
354C2×C8—22no—
355C2×C8—531no—
356C2×C8—531no—
357C2×C8—531no—
358C2×C8—531no—
359C2×C8—1062no—
360C2×C8—1062no—
361C2×C8—1062no—
362C2×C8—1062no—
363C2×C8—2122no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
375C4⋊C4C2×C22⋊D5312yes—
376C22⋊C4—41no—
377C22⋊C4—2121no—
378C22⋊C4—2121no—
379C42—22no—
380C42—22no—
381C42—41no—
382C42D4×D5311yescharacteristic
383C42#1171994711yescharacteristic
384C42C2×C22⋊D5311yescharacteristic
385C42C23×D5311yescharacteristic
386C23—2121no—
387C23#1313318911yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—42no—
391Q8—42no—
392Q8—81no—
393Q8—2121no—
394Q8—2121no—
395D4—4241no—
396D4—4242no—
397D4—2121no—
398D4—2121no—
399D4—2122no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—4242no—
409C2×C4—2121no—
410C2×C4#1313318911yescharacteristic
411C2×C4#1313318911yescharacteristic
412C2×C4#1313318911yescharacteristic
413C2×C4#1313324012yes—
414C2×C4#1313324912yes—
415C2×C4C2×D4×D5311yescharacteristic
416C2×C4C22×C22⋊D5311yescharacteristic
417C2×C4C2×#1171994711yescharacteristic
418C8—42no—
419C8—1061no—
420C8—1061no—
421C8—1061no—
422C8—1061no—
423C8—2122no—
424C8—2122no—
425C22—21no—
426C22—21no—
427C22—4241no—
428C22—4241no—
429C22—4241no—
430C22#1681084712yes—
431C22C2×#1313318911yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—42no—
436C4—42no—
437C4—81no—
438C4—81no—
439C4—4241no—
440C4#1681084712yes—
441C4C2×#1313324011yescharacteristic
442C4C2×#1313324911yescharacteristic
443C4#1681142112yes—
444C2—22no—
445C2—8481no—
446C2C2×#1681084711yescharacteristic
447C2#2028432111yescharacteristic
448C2#2028443211yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3104C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3104D53
hypocenterC53#19155
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. C2×#16810754C2semidirect
4. #20274299C2semidirect
5. #20274285C2semidirect
6. #20274271C2semidirect
7. #20274257C2semidirect
8. #20273461C2semidirect
9. #20273372C2semidirect
11. C53×#3104C2semidirect
221. #3104D53semidirect
273. C53#19155semidirect

Maximal subgroups

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

subgroupindex
#202850402
#202847462
C2×#168107542
#202742992
#202742852
#202742712
#202742572
#202734612
#202733722
#202733622
C53×#31042
#1915553