Finite Group Databasebeta

← #28314346

Subgroups — #28314346

Order 13568.

944 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291405C212yesmaximal, complemented
2#20289993C212yesmaximal, complemented
3#20289987C214yesmaximal, complemented
4#20289970C211yesmaximal, characteristic
5#20289966C212yesmaximal, complemented
6#20289950C211yesmaximal, characteristic
7#20289946C212yesmaximal, complemented
8C2×#16811308C211yesmaximal, characteristic
9#16811648—24no—
10#16811643C2214yes—
11#16811639C2214yes—
12#16811630—22no—
13#16811594C2212yes—
14D53⋊(C22⋊C8)C2212yes—
15#16811588—22no—
16#16811469C2212yes—
17#16811469C2212yes—
18#16811468C2214yes—
19#16811468C2214yes—
20#16811466C2211yescharacteristic
21#16811466C2212yes—
22#16811466C2211yescharacteristic
23#16811308—22no—
24#16811308—22no—
25#16811308—24no—
26C2×#13133212C412yes—
27C2×#13133208C411yescharacteristic
28C2×#13133207C2212yes—
29C2×#13133200C412yes—
30C2×#13133197C2211yescharacteristic
31C2×#13133196C411yescharacteristic
32C2×#13133182C411yescharacteristic
33C2×C4×C22⋊D53C2211yescharacteristic
34C2×#13133176C2212yes—
35C2×#13133175C2211yescharacteristic
36C22×C4⋊C4×C53C411yescharacteristic
37C2×#11719936—22no—
38C2×D53⋊C8—22no—
39#13133284—24no—
40C4⋊#88324—24no—
41#13133274—22no—
42#13133274—22no—
43C22⋊#88324C2312yes—
44C22⋊#88324C2312yes—
45C22⋊#88324C2314yes—
46#13133272—24no—
47C4×#88324—24no—
48C22×Dic106C2×C411yescharacteristic
49C22×C22⋊D53C2×C411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51#13133212—24no—
52#13133212—24no—
53#13133208—22no—
54#13133208—22no—
55#13133207—24no—
56#13133207—24no—
57#13133200—24no—
58#13133200—24no—
59#13133197—22no—
60#13133197—22no—
61#13133196—22no—
62#13133196—22no—
63#13133182—22no—
64#13133182—21no—
65#13133182—21no—
66C4×C22⋊D53—22no—
67C4×C22⋊D53—22no—
68#13133176—24no—
69#13133176—24no—
70#13133175D412yes—
71#13133175D412yes—
72#13133175D414yes—
73C2×C4⋊Dic53C2×C412yes—
74C2×C4⋊Dic53C2×C411yescharacteristic
75C2×C4⋊Dic53C2312yes—
76C2×#11719919C2×C412yes—
77C2×#11719919C2×C412yes—
78C2×#11719919C2×C412yes—
79C2×#11719919C2311yescharacteristic
80C2×C22⋊Dic53C2×C412yes—
81C2×C22⋊Dic53C2311yescharacteristic
82C2×#11719921C2×C411yescharacteristic
83C2×#11719921C2×C412yes—
84C2×#11719921C2×C412yes—
85C2×C4×Dic53C2×C412yes—
86C2×C4×Dic53C2×C411yescharacteristic
87C2×C4×Dic53C2312yes—
88C23×C4×C53C2×C412yes—
89C23×C4×C53C2×C411yescharacteristic
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53C2×C411yescharacteristic
94C2×C4⋊C4×C53C2×C412yes—
95C2×C4⋊C4×C53C2×C411yescharacteristic
96C24×C53C22×C411yescharacteristic
97C23×D53C22×C411yescharacteristic
98C22×C4×C53—24no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53C22×C412yes—
106C22×C4×C53C22×C412yes—
107C22×C4×C53C22×C411yescharacteristic
108C22×C4×C53C22×C411yescharacteristic
109C2×C22⋊D53—22no—
110C2×C22⋊D53—22no—
111C2×C22⋊D53—41no—
112C22×Dic53C22×C411yescharacteristic
113C22×Dic53C22×C411yescharacteristic
114C22×Dic53C22×C412yes—
115C22×Dic53C22×C412yes—
116C22×Dic53C2411yescharacteristic
117C2×C4×D53—42no—
118C2×C4×D53C2×D412yes—
119C2×C4×D53C2×D412yes—
120C2×Dic106—21no—
121C2×Dic106—21no—
122C2×Dic106—42no—
123C2×#88324—22no—
124C2×#88324—24no—
125C2×#88324—22no—
126#11719936—44no—
127D53⋊C8—44no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53C22⋊C412yes—
133C4⋊C4×C53C22⋊C411yescharacteristic
134C4⋊C4×C53C22⋊C411yescharacteristic
135C22⋊Dic53—24no—
136C22⋊Dic53—22no—
137#11719921—22no—
138#11719921—24no—
139#11719921C22⋊C414yes—
140#11719921C22⋊C414yes—
141C4⋊Dic53—24no—
142C4⋊Dic53—24no—
143C4⋊Dic53C22⋊C412yes—
144C4⋊Dic53C22⋊C412yes—
145#11719919—22no—
146#11719919—22no—
147#11719919—24no—
148#11719919—24no—
149#11719919—22no—
150C4×Dic53—22no—
151C4×Dic53—22no—
152C4×Dic53—21no—
153C4×Dic53—21no—
154C4×Dic53C2×D414yes—
155C4×Dic53C2×D414yes—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—41no—
159C23×C53—41no—
160C23×C53#13112yes—
161C23×C53C23×C411yescharacteristic
162C22×D53—21no—
163C22×D53—21no—
164C22×D53—41no—
165C22×D53—41no—
166C22×D53C2×C22⋊C412yes—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—21no—
171C2×C4×C53—22no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—44no—
175C2×C4×C53—44no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53C2×C22⋊C411yescharacteristic
180C2×C4×C53C2×C22⋊C412yes—
181C2×C4×C53C2×C22⋊C412yes—
182C2×C4×C53C2×C22⋊C411yescharacteristic
183C22⋊D53—44no—
184C2×Dic53—22no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—41no—
190C2×Dic53—42no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53C2×C22⋊C414yes—
194C2×Dic53C22×D412yes—
195C4×D53—24no—
196C4×D53—42no—
197Dic106—42no—
198Dic106—42no—
199#88324—44no—
200#88324—42no—
201#88324—42no—
202#17413—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—22no—
206C22×C53—42no—
207C22×C53—42no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53C22×C22⋊C411yescharacteristic
211C22×C53C2×#13111yescharacteristic
212C22×C53#56811yescharacteristic
213C2×D53—22no—
214C2×D53—22no—
215C2×D53—22no—
216C2×D53—81no—
217C2×D53—81no—
218C4×C53—22no—
219C4×C53—22no—
220C4×C53—21no—
221C4×C53—42no—
222C4×C53—41no—
223C4×C53#41012yes—
224Dic53—22no—
225Dic53—24no—
226Dic53—41no—
227Dic53—42no—
228Dic53—41no—
229C2×#552—531no—
230#2924—532no—
231#1472—532no—
232#1470—534no—
233#1465—531no—
234#1464—532no—
235#1459—531no—
236#1458—532no—
237C2×C53—21no—
238C2×C53—21no—
239C2×C53—42no—
240C2×C53#288011yescharacteristic
241C2×C53C2×#41011yescharacteristic
242C2×C53#288911yescharacteristic
243D53—42no—
244#552—1062no—
245#552—1062no—
246#552—1064no—
247C2×#126—532no—
248C2×#125—532no—
249C2×#125—531no—
250C2×#123—532no—
251C2×C22⋊Q8—531no—
252C2×C22⋊Q8—531no—
253C2×C4×Q8—531no—
254C2×C4×D4—531no—
255C2×#117—532no—
256C2×#117—531no—
257C22×C4⋊C4#615711yescharacteristic
258#433—1064no—
259#432—534no—
260#431—534no—
261#424—1062no—
262#407—532no—
263C2×C22⋊C8—532no—
264#405—1062no—
265#324—534no—
266#324—532no—
267#324—534no—
268#324—532no—
269#323—531no—
270#323—532no—
271#323—531no—
272C53#1741311yescharacteristic, complemented, Sylow
273C22×Q8—531no—
274C22×D4—531no—
275C23×C4—531no—
276C23×C4C2×#615712yes—
277C23×C4C2×#615711yescharacteristic
278C2×M16—1062no—
279C22×C8—1062no—
280#126—1064no—
281#126—1064no—
282#125—1064no—
283#125—1064no—
284#125—1062no—
285#125—1062no—
286#123—1064no—
287#123—1064no—
288C22⋊Q8—1062no—
289C22⋊Q8—1062no—
290C22⋊Q8—1062no—
291C22⋊Q8—1062no—
292C4×Q8—1062no—
293C4×Q8—1061no—
294C4×Q8—1061no—
295C4×D4—1062no—
296C4×D4—1062no—
297#117—532no—
298#117—532no—
299#117—534no—
300#117—1064no—
301#117—1064no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—531no—
306C2×C4⋊C4—532no—
307C2×C4⋊C4—532no—
308C2×C4⋊C4—531no—
309C2×C4⋊C4—532no—
310C2×C4⋊C4—532no—
311C2×C4⋊C4—532no—
312C2×C4⋊C4C2×#615711yescharacteristic
313C2×C4⋊C4C2×#615712yes—
314C2×C4⋊C4C2×#615711yescharacteristic
315C2×C22⋊C4—532no—
316C2×C22⋊C4—531no—
317C2×C22⋊C4—532no—
318C2×C22⋊C4—531no—
319C2×C22⋊C4—532no—
320C2×C42—532no—
321C2×C42—532no—
322C2×C42—531no—
323C4⋊C8—1064no—
324C4⋊C8—1064no—
325C22⋊C8—532no—
326C22⋊C8—532no—
327C22⋊C8—534no—
328C22⋊C8—1062no—
329C22⋊C8—1062no—
330#97—1064no—
331C4×C8—1064no—
332C24—531no—
333C24C22×#615711yescharacteristic
334C2×Q8—1061no—
335C2×Q8—1061no—
336C2×Q8—2122no—
337C2×D4—1062no—
338C2×D4—1062no—
339C2×D4—2121no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—532no—
348C22×C4—531no—
349C22×C4—532no—
350C22×C4—532no—
351C22×C4—531no—
352C22×C4—531no—
353C22×C4—532no—
354C22×C4—2122no—
355C22×C4C22×#615712yes—
356C22×C4C22×#615712yes—
357C22×C4C22×#615711yescharacteristic
358C22×C4C22×#615711yescharacteristic
359M16—2124no—
360C2×C8—1062no—
361C2×C8—1064no—
362C2×C8—1062no—
363C2×C8—2124no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—532no—
369C4⋊C4—532no—
370C4⋊C4—1064no—
371C4⋊C4—1064no—
372C4⋊C4—1062no—
373C4⋊C4—1064no—
374C4⋊C4—1064no—
375C4⋊C4—1062no—
376C4⋊C4—1062no—
377C4⋊C4C22⋊#615712yes—
378C4⋊C4C22⋊#615711yescharacteristic
379C4⋊C4C22⋊#615711yescharacteristic
380C22⋊C4—534no—
381C22⋊C4—534no—
382C22⋊C4—1062no—
383C22⋊C4—1064no—
384C22⋊C4—1062no—
385C22⋊C4—1064no—
386C42—534no—
387C42—534no—
388C42—1062no—
389C42—1062no—
390C42—1061no—
391C42—1061no—
392C23—21no—
393C23—21no—
394C23—41no—
395C23—41no—
396C23—532no—
397C23—1061no—
398C23—1061no—
399C23—2121no—
400C23—2121no—
401C23#1313330412yes—
402C23C23×#615711yescharacteristic
403Q8—2122no—
404Q8—2122no—
405D4—2124no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—534no—
419C2×C4—532no—
420C2×C4—1064no—
421C2×C4—1061no—
422C2×C4—1062no—
423C2×C4—1061no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—2122no—
427C2×C4—2122no—
428C2×C4—2121no—
429C2×C4—2121no—
430C2×C4—2122no—
431C2×C4C2×C22⋊#615712yes—
432C2×C4C2×C22⋊#615712yes—
433C2×C4C2×C22⋊#615711yescharacteristic
434C2×C4C2×C22⋊#615711yescharacteristic
435C8—2122no—
436C8—2122no—
437C8—2124no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—42no—
442C22—42no—
443C22—41no—
444C22—41no—
445C22—1062no—
446C22—1062no—
447C22—1062no—
448C22—4241no—
449C22—4241no—
450C22C22×C22⋊#615711yescharacteristic
451C22C2×#1313330411yescharacteristic
452C22#1681170111yescharacteristic
453C4—22no—
454C4—22no—
455C4—21no—
456C4—42no—
457C4—41no—
458C4—1064no—
459C4—1062no—
460C4—2121no—
461C4—2121no—
462C4—2122no—
463C4#1681160212yes—
464C2—21no—
465C2—21no—
466C2—42no—
467C2—2122no—
468C2#2029130311yescharacteristic
469C2C2×#1681160211yescharacteristic
470C2#2029132811yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC22×C22⋊D53C2×C4
cosocleC22×Dic53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53#17413
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20291405C2semidirect
2. #20289993C2semidirect
3. #20289987C2semidirect
5. #20289966C2semidirect
7. #20289946C2semidirect
272. C53#17413semidirect

Maximal subgroups

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

subgroupindex
#202914052
#202899932
#202899872
#202899702
#202899662
#202899502
#202899462
C2×#168113082
#1741353