Finite Group Databasebeta

← #27280569

Subgroups — #27280569

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283771C211yesmaximal, characteristic
2#20273219C211yesmaximal, characteristic
3C2×#16810360C211yesmaximal, characteristic, complemented
4#20273079C211yesmaximal, characteristic, complemented
5#20272603C211yesmaximal, characteristic, complemented
6#20272557C211yesmaximal, characteristic, complemented
7C53×#3190C211yesmaximal, characteristic
8#16810695—21no—
9#16810695—21no—
10D4×#88322—21no—
11D4×#88322—21no—
12#16810684—21no—
13#16810590—21no—
14#16810587C2211yescharacteristic
15C2×C22⋊#88322C2211yescharacteristic
16C2×C22⋊#88322C2211yescharacteristic
17C2×#13133124C2211yescharacteristic
18#16810360—21no—
19#16810360—21no—
20#16810360—21no—
21#16810357—21no—
22#16810349—21no—
23#16810349—21no—
24C53×#525C2211yescharacteristic
25C2×C22⋊Q8×C53C2211yescharacteristic
26C53×#485—21no—
27C53×#483C411yescharacteristic, complemented
28C53×#480C411yescharacteristic, complemented
29C53×#467—21no—
30C53×#465—21no—
31C53×#452—21no—
32C53×#451C411yescharacteristic, complemented
33C53×#448C411yescharacteristic, complemented
34C2×C22⋊C8×C53C2211yescharacteristic
35C23×Dic53—21no—
36C2×#11719917—21no—
37C2×#11719917—21no—
38C22×#88322—21no—
39C22×#88322—21no—
40C22×#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322D411yescharacteristic
50C22⋊#88322D411yescharacteristic
51C22⋊#88322D411yescharacteristic
52C22⋊#88322D411yescharacteristic
53#13133128—21no—
54C4×#88322—21no—
55#13133124—21no—
56#13133124—21no—
57C2×#41×C53D411yescharacteristic
58C22×Q8×C53C2×C411yescharacteristic
59C23×C4×C53C2311yescharacteristic
60C2×Q16×C53—21no—
61C2×SD16×C53—21no—
62C22×C8×C53—21no—
63#123×C53—21no—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53C2×C411yescharacteristic
69C22⋊Q8×C53D411yescharacteristic
70#121×C53C2×C411yescharacteristic
71#121×C53D411yescharacteristic
72#120×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75#117×C53D411yescharacteristic
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53C2×C411yescharacteristic
78C2×C22⋊C4×C53—21no—
79#107×C53—21no—
80#106×C53—21no—
81#103×C53—21no—
82#103×C53—21no—
83#103×C53—21no—
84#102×C53—21no—
85C22⋊C8×C53—21no—
86C22⋊C8×C53C2×C411yescharacteristic
87C22⋊C8×C53C2×C411yescharacteristic
88C24×C53#4111yescharacteristic
89#41×C53—81no—
90C2×Q8×C53—21no—
91C2×Q8×C53—21no—
92C2×Q8×C53—81no—
93C2×Q8×C53C22⋊C411yescharacteristic
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53C22⋊C411yescharacteristic
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—81no—
102C22×C4×C53C22×C411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×Dic53—22no—
109C22×Dic53—41no—
110C22×Dic53—41no—
111C22×Dic53—81no—
112C22×Dic53—81no—
113Q16×C53—81no—
114SD16×C53—81no—
115C2×C8×C53—21no—
116C2×C8×C53—21no—
117C2×C8×C53—81no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53C22⋊C411yescharacteristic
125C4⋊C4×C53C22⋊C411yescharacteristic
126C22⋊C4×C53—21no—
127C22⋊C4×C53—21no—
128C22⋊C4×C53—21no—
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C22⋊C4×C53—41no—
132C22⋊C4×C53—41no—
133C42×C53—21no—
134#11719917—81no—
135#11719917—81no—
136C2×#88322—21no—
137C2×#88322—21no—
138C2×#88322—21no—
139C2×#88322—21no—
140C2×#88322—41no—
141C2×#88322—81no—
142C2×#88322—81no—
143C2×#88322—81no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—41no—
150C23×C53C4×D411yescharacteristic
151C23×C53#12111yescharacteristic
152C23×C53#12311yescharacteristic
153Q8×C53—41no—
154Q8×C53—41no—
155Q8×C53—41no—
156Q8×C53—81no—
157D4×C53—41no—
158D4×C53—41no—
159D4×C53—41no—
160D4×C53—41no—
161D4×C53—41no—
162D4×C53—81no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—81no—
179C2×C4×C53C2×C22⋊C411yescharacteristic
180C2×C4×C53C4×D411yescharacteristic
181C2×C4×C53#12011yescharacteristic
182C2×C4×C53#12111yescharacteristic
183C2×Dic53—42no—
184C2×Dic53—81no—
185C2×Dic53—82no—
186C2×Dic53—81no—
187C2×Dic53—82no—
188C8×C53—41no—
189C8×C53—41no—
190#88322—41no—
191#88322—41no—
192#88322—41no—
193#88322—41no—
194#88322—81no—
195#9263—531nomaximal, Sylow
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53#38611yescharacteristic
211C22×C53#44311yescharacteristic
212C22×C53#44411yescharacteristic
213C22×C53#47111yescharacteristic
214C22×C53#47311yescharacteristic
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—41no—
220C4×C53—41no—
221C4×C53—41no—
222C4×C53—81no—
223C4×C53—81no—
224Dic53—82no—
225#3190D5311yescharacteristic
226#2977—531no—
227#1504—531no—
228C2×#328—531no—
229#1472—531no—
230#1298—531no—
231#1280—531no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53—41no—
237C2×C53—81no—
238C2×C53#187611yescharacteristic
239C2×C53#190811yescharacteristic
240C2×C53#191311yescharacteristic
241#525C2×D5311yescharacteristic
242C2×C22⋊Q8C2×D5311yescharacteristic
243#485—21no—
244#483Dic5311yescharacteristic, complemented
245#480Dic5311yescharacteristic, complemented
246#467—21no—
247#465—21no—
248#452—21no—
249#451Dic5311yescharacteristic, complemented
250#448Dic5311yescharacteristic, complemented
251#435—1061no—
252#435—1061no—
253C8×D4—1061no—
254C8×D4—1061no—
255#433—1061no—
256#408—1061no—
257#407—531no—
258C2×C22⋊C8—531no—
259C2×C22⋊C8—531no—
260C2×C22⋊C8C2×D5311yescharacteristic
261C2×#95—531no—
262#328—1061no—
263#328—1061no—
264#328—1061no—
265#327—1061no—
266#324—1061no—
267#324—1061no—
268C53#926311yescharacteristic, complemented, Sylow
269C2×#41C22⋊D5311yescharacteristic
270C22×Q8C2×Dic5311yescharacteristic
271C23×C4—1061no—
272C23×C4C22×D5311yescharacteristic
273C2×Q16—21no—
274C2×SD16—21no—
275C2×M16—1061no—
276C2×M16—1061no—
277C22×C8—21no—
278C22×C8—1061no—
279C22×C8—1061no—
280C22×C8—1061no—
281#123—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286C22⋊Q8C2×Dic5311yescharacteristic
287C22⋊Q8C22⋊D5311yescharacteristic
288#121C2×Dic5311yescharacteristic
289#121C22⋊D5311yescharacteristic
290#120—21no—
291C4×D4—21no—
292C4×D4—21no—
293#117C22⋊D5311yescharacteristic
294C2×C4⋊C4—21no—
295C2×C4⋊C4C2×Dic5311yescharacteristic
296C2×C22⋊C4—21no—
297#107—21no—
298#106—21no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301#103—21no—
302#103—21no—
303#103—21no—
304#102—21no—
305C22⋊C8—21no—
306C22⋊C8—531no—
307C22⋊C8—531no—
308C22⋊C8—531no—
309C22⋊C8—531no—
310C22⋊C8—1061no—
311C22⋊C8—1061no—
312C22⋊C8—1061no—
313C22⋊C8—1061no—
314C22⋊C8—1061no—
315C22⋊C8—1061no—
316C22⋊C8C2×Dic5311yescharacteristic
317C22⋊C8C2×Dic5311yescharacteristic
318#97—1061no—
319C4×C8—1061no—
320#95—1061no—
321#95—1061no—
322C24#1171994711yescharacteristic
323#41—81no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—81no—
327C2×Q8C22⋊Dic5311yescharacteristic
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4C22⋊Dic5311yescharacteristic
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—81no—
336C22×C4—1062no—
337C22×C4—4241no—
338C22×C4—4241no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4D4×D5311yescharacteristic
342C22×C4D4×D5311yescharacteristic
343C22×C4#1171994711yescharacteristic
344C22×C4C22×Dic5311yescharacteristic
345C22×C4C2×C22⋊D5311yescharacteristic
346C22×C4C2×C22⋊D5311yescharacteristic
347Q16—81no—
348SD16—81no—
349M16—4241no—
350M16—4241no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—81no—
354C2×C8—1061no—
355C2×C8—1061no—
356C2×C8—1061no—
357C2×C8—1061no—
358C2×C8—4241no—
359C2×C8—4241no—
360C2×C8—4241no—
361C2×C8—2121no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4C22⋊Dic5311yescharacteristic
369C4⋊C4C22⋊Dic5311yescharacteristic
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—21no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C42—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—41no—
384C23D4×Dic5311yescharacteristic
385C23#1313318911yescharacteristic
386C23#1313319811yescharacteristic
387Q8—41no—
388Q8—41no—
389Q8—41no—
390Q8—81no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—81no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—81no—
413C2×C4—4241no—
414C2×C4—4242no—
415C2×C4—4241no—
416C2×C4—4242no—
417C2×C4—2122no—
418C2×C4C2×C22⋊Dic5311yescharacteristic
419C2×C4D4×Dic5311yescharacteristic
420C2×C4#1313318511yescharacteristic
421C2×C4#1313318911yescharacteristic
422C8—41no—
423C8—41no—
424C8—4241no—
425C8—2121no—
426C8—2121no—
427C8—2121no—
428C8—2121no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22#1681050311yescharacteristic
444C22#1681074011yescharacteristic
445C22#1681074511yescharacteristic
446C22#1681088811yescharacteristic
447C22#1681089611yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—81no—
456C4—81no—
457C4—4242no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—81no—
464C2#2027504411yescharacteristic
465C2#2027523211yescharacteristic
466C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3190C2
socleC22×C53#386
prime radicalC53×#525C22
cosocleC23×C4×C53C23
hypercenter#3190D53
hypocenterC53#9263
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3190

Complements

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

NH ≅ G/Ntype
3. C2×#16810360C2semidirect
4. #20273079C2semidirect
5. #20272603C2semidirect
6. #20272557C2semidirect
27. C53×#483C4semidirect
28. C53×#480C4semidirect
32. C53×#451C4semidirect
33. C53×#448C4semidirect
244. #483Dic53semidirect
245. #480Dic53semidirect
249. #451Dic53semidirect
250. #448Dic53semidirect
268. C53#9263semidirect

Maximal subgroups

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

subgroupindex
#202837712
#202732192
C2×#168103602
#202730792
#202726032
#202725572
C53×#31902
#926353