Finite Group Databasebeta

← #27275628

Subgroups — #27275628

Order 13568.

834 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#20275139C212yesmaximal, complemented
2#20275008C211yesmaximal, characteristic
3#20274703C211yesmaximal, characteristic
4#20273287C212yesmaximal, complemented
5C53×#3501C211yesmaximal, characteristic
6C23×#88322C2211yescharacteristic
7C2×#13133141—22no—
8C2×#13133132—21no—
9C2×C22⋊#88322—21no—
10C2×C22⋊#88322C2211yescharacteristic
11#16810483C2212yes—
12#16810390—21no—
13#16810385—22no—
14#16810378—21no—
15#16810360—24no—
16C53×#554—24no—
17C53×#554—22no—
18D4×Q8×C53—22no—
19C53×#544C412yescomplemented
20C53×#543—21no—
21C53×#543C412yescomplemented
22C53×#533—21no—
23C2×C22⋊Q8×C53—21no—
24C2×C22⋊Q8×C53C2212yes—
25C53×#518—21no—
26C2×C4×D4×C53C2211yescharacteristic
27C22×#88322—21no—
28C22×#88322—22no—
29C22×#88322—21no—
30C22×#88322—42no—
31C2×C4⋊Dic53—22no—
32C2×C22⋊Dic53—22no—
33#13133141—44no—
34#13133132—42no—
35C22⋊#88322—22no—
36C22⋊#88322—42no—
37#13133124—22no—
38C22×Q8×C53—21no—
39C22×Q8×C53D412yes—
40C22×D4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C23×C4×C53C2311yescharacteristic
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53—21no—
45C4⋊Q8×C53—24no—
46#125×C53—22no—
47#125×C53—24no—
48C22⋊Q8×C53—22no—
49C22⋊Q8×C53—22no—
50C22⋊Q8×C53—24no—
51C22⋊Q8×C53—24no—
52C22⋊Q8×C53—24no—
53C22⋊Q8×C53—24no—
54C22⋊Q8×C53—22no—
55C22⋊Q8×C53—22no—
56C22⋊Q8×C53—22no—
57C22⋊Q8×C53—22no—
58C22⋊Q8×C53C2×C414yes—
59C22⋊Q8×C53D414yes—
60C4×Q8×C53—22no—
61C4×Q8×C53—22no—
62C4×D4×C53—24no—
63C4×D4×C53—24no—
64C4×D4×C53—22no—
65C4×D4×C53—22no—
66#117×C53—22no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53C2×C411yescharacteristic
71C2×C4⋊C4×C53D412yes—
72C2×C22⋊C4×C53—22no—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53C2×C411yescharacteristic
75C2×C22⋊C4×C53D411yescharacteristic
76C2×C42×C53D411yescharacteristic
77C24×C53C22⋊C411yescharacteristic
78C24×C53C2×D411yescharacteristic
79C2×Q8×C53—22no—
80C2×Q8×C53—22no—
81C2×Q8×C53—22no—
82C2×Q8×C53—42no—
83C2×Q8×C53—42no—
84C2×Q8×C53—81no—
85C2×Q8×C53—81no—
86C2×Q8×C53C22⋊C414yes—
87C2×D4×C53—22no—
88C2×D4×C53—22no—
89C2×D4×C53SD1612yes—
90C2×D4×C53Q1612yes—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53C22⋊C411yescharacteristic
102C22×C4×C53C22⋊C411yescharacteristic
103C22×C4×C53C22⋊C411yescharacteristic
104C22×C4×C53C22×C411yescharacteristic
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D412yes—
108C22×Dic53—42no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—24no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—24no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53C22⋊C414yes—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—24no—
122C22⋊C4×C53—24no—
123C22⋊C4×C53—22no—
124C22⋊C4×C53—22no—
125C42×C53—24no—
126C42×C53—21no—
127C42×C53—21no—
128C22⋊Dic53—42no—
129C22⋊Dic53—42no—
130C4⋊Dic53—42no—
131C4⋊Dic53—42no—
132C2×#88322—22no—
133C2×#88322—24no—
134C2×#88322—41no—
135C2×#88322—42no—
136C2×#88322—41no—
137C2×#88322—41no—
138C2×#88322—41no—
139C2×#88322—41no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53#10312yescomplemented
147C23×C53C2×C22⋊C411yescharacteristic
148C23×C53#12012yes—
149C23×C53C2×SD1611yescharacteristic
150C23×C53C2×Q1611yescharacteristic
151C23×C53#13712yes—
152Q8×C53—44no—
153Q8×C53—42no—
154Q8×C53—42no—
155Q8×C53—82no—
156Q8×C53—82no—
157D4×C53—24no—
158D4×C53—24no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—22no—
163C2×C4×C53—21no—
164C2×C4×C53—22no—
165C2×C4×C53—21no—
166C2×C4×C53—24no—
167C2×C4×C53—24no—
168C2×C4×C53—21no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—81no—
176C2×C4×C53—82no—
177C2×C4×C53—82no—
178C2×C4×C53—82no—
179C2×C4×C53—82no—
180C2×C4×C53#10312yescomplemented
181C2×C4×C53C2×C22⋊C412yes—
182C2×C4×C53#12011yescharacteristic
183C2×C4×C53#12011yescharacteristic
184C2×Dic53—82no—
185C2×Dic53—82no—
186C2×Dic53—82no—
187#88322—42no—
188#88322—81no—
189#8351—531nomaximal, Sylow
190C22×C53—22no—
191C22×C53—21no—
192C22×C53—22no—
193C22×C53—21no—
194C22×C53—22no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—81no—
202C22×C53#37911yescharacteristic
203C22×C53C2×#10311yescharacteristic
204C22×C53#41911yescharacteristic
205C22×C53#44812yes—
206C22×C53#45112yes—
207C4×C53—22no—
208C4×C53—21no—
209C4×C53—22no—
210C4×C53—41no—
211C4×C53—44no—
212C4×C53—41no—
213C4×C53—44no—
214C4×C53B(4,𝔽2)11yescharacteristic
215C4×C53#45811yescharacteristic
216Dic53—162no—
217#3501D5311yescharacteristic
218#1892—532no—
219#1869—531no—
220#1799—531no—
221#1520—532no—
222C2×C53—22no—
223C2×C53—21no—
224C2×C53—21no—
225C2×C53—22no—
226C2×C53—42no—
227C2×C53#178711yescharacteristic
228C2×C53#179011yescharacteristic
229C2×C53#179411yescharacteristic
230C23×C8—531no—
231#554—24no—
232#554—22no—
233D4×Q8—22no—
234#544Dic5312yescomplemented
235#543—21no—
236#543Dic5312yescomplemented
237#533—21no—
238C2×C22⋊Q8—21no—
239C2×C22⋊Q8C2×D5312yes—
240#518—21no—
241C2×C4×D4C2×D5311yescharacteristic
242C2×#103—1062no—
243C2×#99—1061no—
244C2×C22⋊C8—531no—
245C2×C22⋊C8—1061no—
246#380—532no—
247#342—1061no—
248#340—1062no—
249#336—1061no—
250#328—1064no—
251C53#835111yescharacteristic, complemented, Sylow
252C22×Q8—21no—
253C22×Q8C22⋊D5312yes—
254C22×D4C22⋊D5311yescharacteristic
255C23×C4C22⋊D5311yescharacteristic
256C23×C4C22×D5311yescharacteristic
257C22×C8—1061no—
258C22×C8—1062no—
259C22×C8—1061no—
260C22×C8—2122no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—24no—
264#125—22no—
265#125—24no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8C2×Dic5314yes—
277C22⋊Q8C22⋊D5314yes—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284#117—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—1062no—
289C2×C4⋊C4C2×Dic5311yescharacteristic
290C2×C4⋊C4C22⋊D5312yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—1062no—
294C2×C22⋊C4C2×Dic5311yescharacteristic
295C2×C22⋊C4C22⋊D5311yescharacteristic
296C2×C42C22⋊D5311yescharacteristic
297#103—2124no—
298#99—2122no—
299C22⋊C8—1062no—
300C22⋊C8—2122no—
301#95—1062no—
302C24C22⋊Dic5311yescharacteristic
303C24C2×C22⋊D5311yescharacteristic
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—81no—
310C2×Q8—81no—
311C2×Q8C22⋊Dic5314yes—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4#1171992412yes—
315C2×D4#1171992512yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—2122no—
327C22×C4C22⋊Dic5311yescharacteristic
328C22×C4C22⋊Dic5311yescharacteristic
329C22×C4C22⋊Dic5311yescharacteristic
330C22×C4C22×Dic5311yescharacteristic
331C22×C4C2×C22⋊D5312yes—
332C22×C4C2×C22⋊D5311yescharacteristic
333C22×C4C2×C22⋊D5312yes—
334C2×C8—1062no—
335C2×C8—1064no—
336C2×C8—2121no—
337C2×C8—2121no—
338C2×C8—2121no—
339C2×C8—2122no—
340C2×C8—2121no—
341C2×C8—2121no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—2122no—
353C4⋊C4—2122no—
354C4⋊C4C22⋊Dic5314yes—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—2122no—
361C22⋊C4—2122no—
362C42—24no—
363C42—21no—
364C42—21no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23#1313314112yescomplemented
372C23C2×C22⋊Dic5311yescharacteristic
373C23#1313318612yes—
374C23C2×#1171992411yescharacteristic
375C23C2×#1171992511yescharacteristic
376C23#1313324812yes—
377Q8—44no—
378Q8—42no—
379Q8—42no—
380Q8—82no—
381Q8—82no—
382D4—24no—
383D4—24no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—4242no—
406C2×C4—4242no—
407C2×C4—4242no—
408C2×C4#1313314112yescomplemented
409C2×C4C2×C22⋊Dic5312yes—
410C2×C4#1313318611yescharacteristic
411C2×C4#1313318611yescharacteristic
412C8—4241no—
413C8—2122no—
414C22—21no—
415C22—22no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—81no—
426C22#1681047911yescharacteristic
427C22C2×#1313314111yescharacteristic
428C22#1681063211yescharacteristic
429C22#1681076212yes—
430C22#1681077912yes—
431C4—22no—
432C4—21no—
433C4—22no—
434C4—41no—
435C4—44no—
436C4—41no—
437C4—44no—
438C4—8482no—
439C4#1681081011yescharacteristic
440C4#1681081311yescharacteristic
441C2—22no—
442C2—21no—
443C2—21no—
444C2—22no—
445C2—42no—
446C2#2027465011yescharacteristic
447C2#2027466311yescharacteristic
448C2#2027467911yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3501C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#3501D53
hypocenterC53#8351
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #20275139C2semidirect
4. #20273287C2semidirect
19. C53×#544C4semidirect
21. C53×#543C4semidirect
146. C23×C53#103semidirect
180. C2×C4×C53#103semidirect
234. #544Dic53semidirect
236. #543Dic53semidirect
251. C53#8351semidirect
371. C23#13133141semidirect
408. C2×C4#13133141semidirect

Maximal subgroups

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

subgroupindex
#202751392
#202750082
#202747032
#202732872
C53×#35012
#835153