Finite Group Databasebeta

← #27276840

Subgroups — #27276840

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274695C211yesmaximal, characteristic
2C2×#16810360C212yesmaximal
3#20272942C211yesmaximal, characteristic
4#20272593C212yesmaximal
5C4×C22⋊Q8×C53C211yesmaximal, characteristic
6C2×C4⋊#88322—21no—
7C2×C22⋊#88322C2212yes—
8C2×#13133124—21no—
9C2×#13133124—21no—
10C2×#13133124C2212yes—
11#16810378—21no—
12#16810360—24no—
13C2×C22⋊Q8×C53C2211yescharacteristic
14C2×C4×Q8×C53C411yescharacteristic, complemented
15C22×C42×C53C2211yescharacteristic
16C53×#391C411yescharacteristic, complemented
17C53×#389C411yescharacteristic, complemented
18C53×#387—21no—
19C53×#385—21no—
20C53×#380C2211yescharacteristic
21C4×C4⋊C4×C53—21no—
22C4×C4⋊C4×C53C411yescharacteristic, complemented
23C4×C22⋊C4×C53—21no—
24C23×Dic53—22no—
25C22×#88322—22no—
26C4⋊#88322—42no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—24no—
30#13133124—22no—
31#13133124—22no—
32#13133124—42no—
33C22×Q8×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37C22⋊Q8×C53—21no—
38C22⋊Q8×C53—21no—
39C22⋊Q8×C53D412yes—
40C22⋊Q8×C53Q812yes—
41C4×Q8×C53—42no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C42×C53—21no—
51C2×C42×C53—21no—
52C2×C42×C53—42no—
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53C2×C411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53D411yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62C24×C53C2×D411yescharacteristic
63C2×Q8×C53—21no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53C4⋊C412yes—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×Q811yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×Dic53—24no—
90C22×Dic53—42no—
91C22×Dic53—42no—
92C22×Dic53—42no—
93C22×Dic53—42no—
94C22×Dic53—42no—
95C22×Dic53—42no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—42no—
103C4⋊C4×C53C4⋊C412yes—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—42no—
107C42×C53—22no—
108C42×C53—22no—
109C42×C53—41no—
110C42×C53—41no—
111C42×C53—42no—
112C42×C53—42no—
113C2×#88322—42no—
114C2×#88322—42no—
115C2×#88322—42no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53C2×C22⋊C411yescharacteristic
121C23×C53#12111yescharacteristic
122C23×C53C22⋊Q811yescharacteristic
123C23×C53#13512yes—
124C23×C53#13712yes—
125Q8×C53—42no—
126Q8×C53—42no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—44no—
148C2×C4×C53—44no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53#9912yes—
160C2×C4×C53#10412yes—
161C2×C4×C53C2×C4⋊C411yescharacteristic
162C2×C4×C53#11711yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53C22⋊Q811yescharacteristic
165C2×Dic53—44no—
166C2×Dic53—44no—
167C2×Dic53—44no—
168C2×Dic53—82no—
169C2×Dic53—82no—
170C2×Dic53—82no—
171C2×Dic53—82no—
172#88322—82no—
173#8576—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—22no—
179C22×C53—22no—
180C22×C53—22no—
181C22×C53—22no—
182C22×C53#38011yescharacteristic
183C22×C53C2×#9911yescharacteristic
184C22×C53#41011yescharacteristic
185C22×C53#41611yescharacteristic
186C22×C53#41911yescharacteristic
187C22×C53C2×#10411yescharacteristic
188C22×C53#42111yescharacteristic
189C22×C53#46412yes—
190C22×C53#47912yes—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—22no—
196C4×C53—22no—
197C4×C53—42no—
198C4×C53—42no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—81no—
202C4×C53—81no—
203Dic53—84no—
204C4×C22⋊Q8D5311yescharacteristic
205#1797—531no—
206C2×#328—532no—
207#1434—531no—
208#1294—532no—
209C2×C53—22no—
210C2×C53—22no—
211C2×C53#150111yescharacteristic
212C2×C53#150611yescharacteristic
213C2×C53#151111yescharacteristic
214C2×C53#152111yescharacteristic
215C2×C53#179911yescharacteristic
216C2×C53#180511yescharacteristic
217C2×C53#180611yescharacteristic
218C2×C22⋊Q8C2×D5311yescharacteristic
219C2×C4×Q8Dic5311yescharacteristic, complemented
220C22×C42C2×D5311yescharacteristic
221C2×C4⋊C8—1061no—
222C2×C22⋊C8—532no—
223#391Dic5311yescharacteristic, complemented
224#389Dic5311yescharacteristic, complemented
225#387—21no—
226#385—21no—
227#380C2×D5311yescharacteristic
228C4×C4⋊C4—21no—
229C4×C4⋊C4Dic5311yescharacteristic, complemented
230C4×C22⋊C4—21no—
231C2×#95—532no—
232C2×#95—1061no—
233C2×#95—1061no—
234#336—1061no—
235#328—1064no—
236C53#857611yescharacteristic, complemented, Sylow
237C22×Q8C2×Dic5311yescharacteristic
238C23×C4—1062no—
239C23×C4C22⋊D5311yescharacteristic
240C23×C4C22⋊D5311yescharacteristic
241C23×C4C22×D5311yescharacteristic
242C22×C8—1062no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8Dic10612yes—
246C22⋊Q8D21212yes—
247C4×Q8—42no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×Dic5311yescharacteristic
252C2×C4⋊C4C2×Dic5311yescharacteristic
253C2×C4⋊C4C2×Dic5311yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—42no—
259C2×C42C2×Dic5311yescharacteristic
260C2×C42C2×Dic5311yescharacteristic
261C2×C42C22⋊D5311yescharacteristic
262C2×C42C22⋊D5311yescharacteristic
263C4⋊C8—2122no—
264C22⋊C8—1062no—
265C22⋊C8—1062no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—1064no—
272#95—1062no—
273#95—1062no—
274#95—2122no—
275C24C2×C22⋊D5311yescharacteristic
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8C4⋊Dic5312yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—1064no—
293C22×C4—2122no—
294C22×C4—2122no—
295C22×C4—2122no—
296C22×C4—2122no—
297C22×C4—2122no—
298C22×C4—2122no—
299C22×C4C22⋊Dic5311yescharacteristic
300C22×C4C22⋊Dic5311yescharacteristic
301C22×C4C22⋊Dic5311yescharacteristic
302C22×C4C22⋊Dic5311yescharacteristic
303C22×C4C2×Dic10611yescharacteristic
304C22×C4C2×D21211yescharacteristic
305C22×C4#1171994511yescharacteristic
306C22×C4#1171994511yescharacteristic
307C22×C4C22×Dic5311yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309C2×C8—2122no—
310C2×C8—2122no—
311C2×C8—2122no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4C4⋊Dic5312yes—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—42no—
323C42—22no—
324C42—22no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23C2×C22⋊Dic5311yescharacteristic
334C23#1313319111yescharacteristic
335C23#1313319611yescharacteristic
336C23#1313323712yes—
337C23#1313324812yes—
338Q8—42no—
339Q8—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—4242no—
373C2×C4—4242no—
374C2×C4—4242no—
375C2×C4—4242no—
376C2×C4—2124no—
377C2×C4—2124no—
378C2×C4—2124no—
379C2×C4#1313313212yes—
380C2×C4#1313314412yes—
381C2×C4C2×C4⋊Dic5311yescharacteristic
382C2×C4#1313317611yescharacteristic
383C2×C4#1313319111yescharacteristic
384C2×C4#1313319611yescharacteristic
385C8—4242no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22#1681048311yescharacteristic
395C22C2×#1313313211yescharacteristic
396C22#1681059711yescharacteristic
397C22#1681061911yescharacteristic
398C22#1681063211yescharacteristic
399C22C2×#1313314411yescharacteristic
400C22#1681064011yescharacteristic
401C22#1681084512yes—
402C22#1681093012yes—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—22no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—41no—
412C4—41no—
413C4—81no—
414C4—81no—
415C4—4244no—
416C2—22no—
417C2—22no—
418C2#2027320611yescharacteristic
419C2#2027322711yescharacteristic
420C2#2027324811yescharacteristic
421C2#2027329211yescharacteristic
422C2#2027470311yescharacteristic
423C2#2027473011yescharacteristic
424C2#2027473511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC4×C22⋊Q8D53
hypocenterC53#8576
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
14. C2×C4×Q8×C53C4semidirect
16. C53×#391C4semidirect
17. C53×#389C4semidirect
22. C4×C4⋊C4×C53C4semidirect
219. C2×C4×Q8Dic53semidirect
223. #391Dic53semidirect
224. #389Dic53semidirect
229. C4×C4⋊C4Dic53semidirect
236. C53#8576semidirect

Maximal subgroups

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

subgroupindex
#202746952
C2×#168103602
#202729422
#202725932
C4×C22⋊Q8×C532
#857653