Finite Group Databasebeta

← #28302915

Subgroups — #28302915

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290571C211yesmaximal, characteristic
2#20290569C211yesmaximal, characteristic
3C2×#16811469C212yesmaximal
4#20289553C212yesmaximal
5C2×#16810504C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4⋊#88324—21no—
8C2×C22⋊#88324C2212yes—
9C2×C22⋊#88324C2212yes—
10#16811489—21no—
11#16811489—21no—
12#16811469—24no—
13C22×#11719919C2211yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810504—21no—
16#16810504—21no—
17#16810504—21no—
18#16810504—21no—
19C2×#13133124C411yescharacteristic
20C2×#13133124C411yescharacteristic
21C2×#13133124C411yescharacteristic
22C2×#13133124C2211yescharacteristic
23C22×C4⋊C4×C53C411yescharacteristic
24C22×#88324—22no—
25C22×#88324—22no—
26#13133284—42no—
27C4⋊#88324—42no—
28C22⋊#88324—22no—
29C22⋊#88324—22no—
30C22⋊#88324—22no—
31C22⋊#88324—22no—
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2×C411yescharacteristic
35C23×Dic53C2311yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919D412yes—
41C2×#11719919Q812yes—
42C2×C4×Dic53—42no—
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55C23×C4×C53C2×C411yescharacteristic
56C23×C4×C53C2×C411yescharacteristic
57C23×C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—42no—
61C24×C53C22×C411yescharacteristic
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C4⋊C412yes—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—22no—
79C22×Dic53—22no—
80C22×Dic53—22no—
81C22×Dic53—22no—
82C22×Dic53—21no—
83C22×Dic53—22no—
84C22×Dic53—41no—
85C22×Dic53—41no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53C22⋊C411yescharacteristic
90C22×Dic53C22⋊C411yescharacteristic
91C22×Dic53C4⋊C412yes—
92C22×Dic53C2×D411yescharacteristic
93C22×Dic53C2×D411yescharacteristic
94C22×Dic53C2×D411yescharacteristic
95C22×Dic53C2×Q811yescharacteristic
96C22×Dic53#4111yescharacteristic
97C22×Dic53#4111yescharacteristic
98C2×#88324—42no—
99C2×#88324—42no—
100C2×#88324—42no—
101C2×#88324—42no—
102C2×#88324—42no—
103C2×#88324—42no—
104C4⋊C4×C53—42no—
105C4⋊C4×C53—42no—
106#11719919—22no—
107#11719919—22no—
108#11719919—22no—
109#11719919—22no—
110C4×Dic53—22no—
111C4×Dic53—22no—
112C4×Dic53—41no—
113C4×Dic53—41no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53C2×C22⋊C411yescharacteristic
119C23×C53C2×C4⋊C411yescharacteristic
120C23×C53#11711yescharacteristic
121C23×C53#13112yes—
122C23×C53#13112yes—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—44no—
130C2×C4×C53—42no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—82no—
138C2×C4×C53#9912yes—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—22no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—24no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—44no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—82no—
162C2×Dic53#10112yes—
163C2×Dic53#12111yescharacteristic
164C2×Dic53#12111yescharacteristic
165C2×Dic53C22⋊Q811yescharacteristic
166C2×Dic53C22⋊Q811yescharacteristic
167#88324—82no—
168#88324—82no—
169#8562—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—22no—
178C22×C53#38011yescharacteristic
179C22×C53#40811yescharacteristic
180C22×C53#40811yescharacteristic
181C22×C53C2×#9911yescharacteristic
182C22×C53#41011yescharacteristic
183C22×C53C2×#10111yescharacteristic
184C22×C53#41311yescharacteristic
185C22×C53#42412yes—
186C22×C53#43312yes—
187C4×C53—22no—
188C4×C53—42no—
189C4×C53—41no—
190C4×C53—81no—
191Dic53—21no—
192Dic53—21no—
193Dic53—21no—
194Dic53—21no—
195Dic53—22no—
196Dic53—42no—
197Dic53—41no—
198Dic53—81no—
199C2×#387—531no—
200#1797—531no—
201#1797—531no—
202C2×#324—532no—
203#1280—532no—
204C2×C53—22no—
205C2×C53—22no—
206C2×C53#146811yescharacteristic
207C2×C53#147011yescharacteristic
208C2×C53#147211yescharacteristic
209C2×C53#147311yescharacteristic
210C2×C53#179811yescharacteristic
211C2×C53#179911yescharacteristic
212C2×C53#180111yescharacteristic
213C22×C4⋊C4—531no—
214C22×C4⋊C4#615711yescharacteristic
215C22×C42—531no—
216C2×C4⋊C8—1061no—
217C2×C4⋊C8—1061no—
218C2×C22⋊C8—532no—
219C2×C22⋊C8—532no—
220#387—1061no—
221#387—1061no—
222#387—1061no—
223#387—1061no—
224C2×#95—531no—
225C2×#95—531no—
226C2×#95—531no—
227C2×#95—531no—
228#336—1061no—
229#336—1061no—
230#324—1064no—
231C53#856211yescharacteristic, complemented, Sylow
232C23×C4—531no—
233C23×C4—531no—
234C23×C4—531no—
235C23×C4—531no—
236C23×C4C2×#615711yescharacteristic
237C23×C4C2×#615711yescharacteristic
238C23×C4C2×#615711yescharacteristic
239C22×C8—1062no—
240C22×C8—1062no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—532no—
245C2×C4⋊C4—532no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—1061no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—531no—
254C2×C42—2122no—
255C4⋊C8—2122no—
256C4⋊C8—2122no—
257C22⋊C8—1062no—
258C22⋊C8—1062no—
259C22⋊C8—1062no—
260C22⋊C8—1062no—
261#95—1061no—
262#95—1061no—
263#95—1061no—
264#95—1061no—
265#95—1061no—
266#95—1061no—
267#95—1061no—
268#95—1061no—
269C24C22×#615711yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—532no—
290C22×C4—1062no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1062no—
294C22×C4—1062no—
295C22×C4—1062no—
296C22×C4—1062no—
297C22×C4—1061no—
298C22×C4—2121no—
299C22×C4—2121no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4C4⋊#615712yes—
304C22×C4C22⋊#615711yescharacteristic
305C22×C4C22⋊#615711yescharacteristic
306C2×C8—2122no—
307C2×C8—2122no—
308C2×C8—2122no—
309C2×C8—2122no—
310C2×C8—2122no—
311C2×C8—2122no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—1062no—
315C4⋊C4—1062no—
316C4⋊C4—1062no—
317C4⋊C4—1062no—
318C42—1062no—
319C42—1062no—
320C42—2121no—
321C42—2121no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23C2×C22⋊#615711yescharacteristic
327C23C2×C4⋊#615711yescharacteristic
328C23#1313329611yescharacteristic
329C23#1313330412yes—
330C23#1313330512yes—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—82no—
346C2×C4—531no—
347C2×C4—531no—
348C2×C4—531no—
349C2×C4—531no—
350C2×C4—532no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1062no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1064no—
362C2×C4—4242no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2124no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2122no—
374C2×C4#1313327512yes—
375C8—4242no—
376C8—4242no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22#1681156511yescharacteristic
386C22#1681159611yescharacteristic
387C22#1681159811yescharacteristic
388C22C2×#1313327511yescharacteristic
389C22#1681160211yescharacteristic
390C22D53⋊#10111yescharacteristic
391C22#1681161011yescharacteristic
392C22#1681163012yes—
393C22#1681164812yes—
394C4—22no—
395C4—42no—
396C4—41no—
397C4—81no—
398C4—1061no—
399C4—1061no—
400C4—1061no—
401C4—1061no—
402C4—1062no—
403C4—4241no—
404C4—2122no—
405C4—2121no—
406C2—22no—
407C2—22no—
408C2#2028998011yescharacteristic
409C2#2028998711yescharacteristic
410C2#2028999511yescharacteristic
411C2#2028999811yescharacteristic
412C2#2029057411yescharacteristic
413C2#2029057511yescharacteristic
414C2#2029058211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8562
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
231. C53#8562semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202905692
C2×#168114692
#202895532
C2×#168105042
#856253