Finite Group Databasebeta

← #27294853

Subgroups — #27294853

Order 13568.

626 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274703C211yesmaximal, characteristic, complemented
2#20273056C211yesmaximal, characteristic, complemented
3#20272736C212yesmaximal, complemented
4#20272712C212yesmaximal, complemented
5C53×#2388C211yesmaximal, characteristic
6#16810680—21no—
7C2×#13133132—21no—
8#16810483—21no—
9#16810390C2212yes—
10#16810347—21no—
11#16810347C2212yes—
12C22×#41×C53C411yescharacteristic, complemented
13C2×C4×D4×C53—21no—
14C2×C4×D4×C53C2211yescharacteristic
15C53×#395C411yescharacteristic, complemented
16C53×#394—21no—
17C53×#394C411yescharacteristic, complemented
18C53×#393C2211yescharacteristic
19C53×#392—21no—
20C53×#392C2211yescharacteristic
21C4×C22⋊C4×C53—21no—
22C4×C22⋊C4×C53C411yescharacteristic, complemented
23C2×C4⋊Dic53—41no—
24C2×C22⋊Dic53—22no—
25C4⋊#88322—41no—
26#13133132—42no—
27C22⋊#88322—22no—
28#13133128—41no—
29#13133124—41no—
30C2×#41×C53—44no—
31C2×#41×C53—41no—
32C2×#41×C53—41no—
33C22×Q8×C53C2×C411yescharacteristic
34C22×D4×C53—21no—
35C22×D4×C53—21no—
36C22×D4×C53C2×C411yescharacteristic
37C22×D4×C53D411yescharacteristic
38C23×C4×C53—21no—
39C23×C4×C53C2×C411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C4×D4×C53—41no—
42C4×D4×C53—41no—
43C4×D4×C53—44no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53D411yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53C2×C411yescharacteristic
58#95×C53C2×C411yescharacteristic
59C24×C53—21no—
60C24×C53C22⋊C411yescharacteristic
61C24×C53C2×D411yescharacteristic
62#41×C53—44no—
63#41×C53—44no—
64#41×C53—44no—
65#41×C53—44no—
66C2×Q8×C53—42no—
67C2×Q8×C53—41no—
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C2×D4×C53—42no—
73C2×D4×C53—42no—
74C2×D4×C53—41no—
75C2×D4×C53—42no—
76C2×D4×C53—42no—
77C2×D4×C53—42no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—42no—
81C22×C4×C53—24no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—42no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53C22⋊C411yescharacteristic
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53C22⋊C411yescharacteristic
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×Dic53—42no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—42no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—42no—
109C42×C53—21no—
110C42×C53—21no—
111C42×C53—42no—
112C22⋊Dic53—42no—
113C22⋊Dic53—42no—
114C4⋊Dic53—82no—
115C2×#88322—42no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—22no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53C2×C22⋊C411yescharacteristic
133C23×C53AGL(1,ℤ/8)11yescharacteristic
134C23×C53#13711yescharacteristic
135Q8×C53—42no—
136Q8×C53—42no—
137D4×C53—42no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—44no—
141D4×C53—44no—
142D4×C53—48no—
143D4×C53—42no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—44no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—44no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—81no—
169C2×C4×C53#9911yescharacteristic
170C2×C4×C53#9911yescharacteristic
171C2×Dic53—82no—
172C2×Dic53—82no—
173C2×Dic53—82no—
174#88322—82no—
175#11820—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—42no—
183C22×C53—42no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—42no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53C2×#9911yescharacteristic
198C22×C53#41711yescharacteristic
199C22×C53#42111yescharacteristic
200C4×C53—21no—
201C4×C53—42no—
202C4×C53—42no—
203C4×C53—41no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53—81no—
208C4×C53AΣL(1,𝔽16)12yes—
209Dic53—162no—
210#2388D5311yescharacteristic
211#1799—531no—
212#1466—531no—
213#1351—532no—
214#1343—532no—
215C2×C53—21no—
216C2×C53—21no—
217C2×C53—41no—
218C2×C53—41no—
219C2×C53—81no—
220C2×C53—82no—
221C2×C53#151311yescharacteristic
222C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
223C2×C53#212611yescharacteristic
224C22×#41Dic5311yescharacteristic, complemented
225C2×C4×D4—21no—
226C2×C4×D4C2×D5311yescharacteristic
227#432—1061no—
228C2×#99—1061no—
229#395Dic5311yescharacteristic, complemented
230#394—21no—
231#394Dic5311yescharacteristic, complemented
232#393C2×D5311yescharacteristic
233#392—21no—
234#392C2×D5311yescharacteristic
235#380—1061no—
236C4×C22⋊C4—21no—
237C4×C22⋊C4Dic5311yescharacteristic, complemented
238#342—532no—
239#323—532no—
240#323—1061no—
241C53#1182011yescharacteristic, complemented, Sylow
242C2×#41—44no—
243C2×#41—41no—
244C2×#41—41no—
245C22×Q8C2×Dic5311yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2×Dic5311yescharacteristic
249C22×D4C22⋊D5311yescharacteristic
250C23×C4—21no—
251C23×C4C2×Dic5311yescharacteristic
252C23×C4C22⋊D5311yescharacteristic
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—44no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—2121no—
258C2×C4⋊C4C2×Dic5311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—1062no—
265C2×C22⋊C4C22⋊D5311yescharacteristic
266C2×C22⋊C4C22×D5311yescharacteristic
267C2×C42—21no—
268C2×C42C22⋊D5311yescharacteristic
269C4⋊C8—2121no—
270#99—2122no—
271C22⋊C8—1062no—
272#97—2121no—
273#95—21no—
274#95—21no—
275#95—2121no—
276#95C2×Dic5311yescharacteristic
277#95C2×Dic5311yescharacteristic
278C24—21no—
279C24C22⋊Dic5311yescharacteristic
280C24C2×C22⋊D5311yescharacteristic
281#41—44no—
282#41—44no—
283#41—44no—
284#41—44no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—2122no—
314C22×C4C22⋊Dic5311yescharacteristic
315C22×C4C22⋊Dic5311yescharacteristic
316C22×C4C22⋊Dic5311yescharacteristic
317C22×C4C22×Dic5311yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319C2×C8—2122no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—4242no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—2122no—
331C22⋊C4—2122no—
332C42—21no—
333C42—21no—
334C42—42no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23C2×C22⋊Dic5311yescharacteristic
352C23#1313324211yescharacteristic
353C23#1313324711yescharacteristic
354Q8—42no—
355Q8—42no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—44no—
360D4—44no—
361D4—48no—
362D4—42no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—4242no—
389C2×C4—4242no—
390C2×C4—4242no—
391C2×C4#1313313211yescharacteristic
392C2×C4#1313313211yescharacteristic
393C8—4242no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—42no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22C2×#1313313211yescharacteristic
416C22#1681062411yescharacteristic
417C22#1681064011yescharacteristic
418C4—21no—
419C4—42no—
420C4—41no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—8482no—
427C4#1681041012yes—
428C2—21no—
429C2—21no—
430C2—41no—
431C2—41no—
432C2—82no—
433C2—81no—
434C2#2027325811yescharacteristic
435C2C2×#1681041011yescharacteristic
436C2#2027638011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
Fitting, prime radicalC53×#2388C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#2388D53
hypocenterC53#11820
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. #20274703C2semidirect
2. #20273056C2semidirect
3. #20272736C2semidirect
4. #20272712C2semidirect
12. C22×#41×C53C4semidirect
15. C53×#395C4semidirect
17. C53×#394C4semidirect
22. C4×C22⋊C4×C53C4semidirect
224. C22×#41Dic53semidirect
229. #395Dic53semidirect
231. #394Dic53semidirect
237. C4×C22⋊C4Dic53semidirect
241. C53#11820semidirect

Maximal subgroups

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

subgroupindex
#202747032
#202730562
#202727362
#202727122
C53×#23882
#1182053