Finite Group Databasebeta

← #27294268

Subgroups — #27294268

Order 13568.

552 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 115 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#20274448C211yesmaximal, characteristic, complemented
2#20273029C211yesmaximal, characteristic, complemented
3#20272715C212yesmaximal, complemented
4#20272712C212yesmaximal
5C53×#2392C211yesmaximal, characteristic
6C2×#13133176—21no—
7#16810676—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×C4×D4×C53—21no—
13C2×C4×D4×C53C411yescharacteristic, complemented
14C2×#117×C53C2211yescharacteristic
15C53×#398C411yescharacteristic, complemented
16C53×#397C2211yescharacteristic
17C53×#396—21no—
18C53×#394—21no—
19C53×#394C411yescharacteristic, complemented
20C53×#392C2211yescharacteristic
21C4×C4⋊C4×C53C411yescharacteristic, complemented
22C4×C22⋊C4×C53—21no—
23#13133176—24no—
24#13133176—42no—
25C2×C4⋊Dic53—41no—
26C2×C22⋊Dic53—22no—
27C2×C4×Dic53—41no—
28C4⋊#88322—41no—
29C22⋊#88322—22no—
30C4×#88322—41no—
31C22×D4×C53—21no—
32C22×D4×C53C2×C411yescharacteristic
33C23×C4×C53—21no—
34C23×C4×C53D411yescharacteristic
35C4×D4×C53—42no—
36C4×D4×C53—42no—
37C4×D4×C53—41no—
38C4×D4×C53—41no—
39#117×C53—41no—
40#117×C53—41no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
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×C53C2×C411yescharacteristic
55C2×C42×C53D411yescharacteristic
56#95×C53—21no—
57#95×C53—21no—
58#95×C53C2×C411yescharacteristic
59#95×C53C2×C411yescharacteristic
60C24×C53—21no—
61C24×C53C2×D411yescharacteristic
62C2×D4×C53—42no—
63C2×D4×C53—41no—
64C2×D4×C53—42no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53—42no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53D812yes—
83C22×C4×C53SD1612yes—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×Dic53—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—42no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—42no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—42no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—42no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—42no—
100C42×C53—21no—
101C42×C53—21no—
102C42×C53—21no—
103C42×C53—21no—
104C42×C53—42no—
105C22⋊Dic53—24no—
106C22⋊Dic53—42no—
107C4⋊Dic53—44no—
108C4×Dic53—44no—
109C2×#88322—42no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—22no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53C2×D811yescharacteristic
123C23×C53C2×SD1611yescharacteristic
124D4×C53—42no—
125D4×C53—44no—
126D4×C53—44no—
127D4×C53—42no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—82no—
152C2×C4×C53—82no—
153C2×C4×C53#9911yescharacteristic
154C2×C4×C53#9911yescharacteristic
155C2×C4×C53#10212yes—
156C2×C4×C53#10212yes—
157C2×C4×C53#10412yes—
158C2×Dic53—44no—
159C2×Dic53—82no—
160C2×Dic53—82no—
161#88322—82no—
162#11691—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—42no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53C2×#9911yescharacteristic
180C22×C53C2×#10211yescharacteristic
181C22×C53C2×#10411yescharacteristic
182C4×C53—22no—
183C4×C53—42no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53#32712yes—
191C4×C53#32712yes—
192Dic53—84no—
193#2392D5311yescharacteristic
194#1740—531no—
195#1458—531no—
196#1344—532no—
197#1343—532no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—41no—
201C2×C53—41no—
202C2×C53—82no—
203C2×C53C2×#32711yescharacteristic
204C2×C53#212011yescharacteristic
205C2×C53#212611yescharacteristic
206C2×C4×D4—21no—
207C2×C4×D4Dic5311yescharacteristic, complemented
208C2×#117—1061no—
209C2×#117C2×D5311yescharacteristic
210#431—1061no—
211#398Dic5311yescharacteristic, complemented
212#397C2×D5311yescharacteristic
213#396—21no—
214#394—21no—
215#394Dic5311yescharacteristic, complemented
216#392C2×D5311yescharacteristic
217C4×C4⋊C4Dic5311yescharacteristic, complemented
218C4×C22⋊C4—21no—
219#342—532no—
220#342—1061no—
221#324—1061no—
222#323—532no—
223C53#1169111yescharacteristic, complemented, Sylow
224C22×D4—21no—
225C22×D4C2×Dic5311yescharacteristic
226C23×C4—21no—
227C23×C4C22⋊D5311yescharacteristic
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—42no—
231C4×D4—42no—
232#117—41no—
233#117—41no—
234#117—1064no—
235#117—2122no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—2121no—
239C2×C4⋊C4C2×Dic5311yescharacteristic
240C2×C4⋊C4C2×Dic5311yescharacteristic
241C2×C4⋊C4C22⋊D5311yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—1062no—
248C2×C22⋊C4C22⋊D5311yescharacteristic
249C2×C22⋊C4C22×D5311yescharacteristic
250C2×C42—21no—
251C2×C42—2121no—
252C2×C42C2×Dic5311yescharacteristic
253C2×C42C22⋊D5311yescharacteristic
254C4⋊C8—2121no—
255C22⋊C8—1062no—
256C4×C8—2121no—
257#95—21no—
258#95—21no—
259#95C2×Dic5311yescharacteristic
260#95C2×Dic5311yescharacteristic
261C24—21no—
262C24C2×C22⋊D5311yescharacteristic
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—2122no—
280C22×C4#1171992212yes—
281C22×C4#1171992312yes—
282C22×C4C22⋊Dic5311yescharacteristic
283C22×C4C22⋊Dic5311yescharacteristic
284C22×C4C22⋊Dic5311yescharacteristic
285C22×C4C22⋊Dic5311yescharacteristic
286C22×C4C22×Dic5311yescharacteristic
287C22×C4C2×C22⋊D5311yescharacteristic
288C2×C8—2122no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—2124no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—1064no—
304C22⋊C4—2122no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—42no—
310C42—2124no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23C2×C22⋊Dic5311yescharacteristic
323C23C2×#1171992211yescharacteristic
324C23C2×#1171992311yescharacteristic
325D4—42no—
326D4—44no—
327D4—44no—
328D4—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—4242no—
355C2×C4—4242no—
356C2×C4—2124no—
357C2×C4#1313313211yescharacteristic
358C2×C4#1313313211yescharacteristic
359C2×C4#1313313812yes—
360C2×C4#1313313812yes—
361C2×C4#1313314412yes—
362C8—4242no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C2×#1313313211yescharacteristic
380C22C2×#1313313811yescharacteristic
381C22C2×#1313314411yescharacteristic
382C4—22no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—4244no—
391C4#1681035712yes—
392C4#1681035712yes—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—82no—
398C2C2×#1681035711yescharacteristic
399C2#2027635611yescharacteristic
400C2#2027638011yescharacteristic
subgroupG/H
centerC2×C4#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2392C2
socleC22×C53C2×#99
prime radicalC53×#392C22
cosocleC2×C22⋊C4×C53C23
hypercenter#2392D53
hypocenterC53#11691
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2392

Complements

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

NH ≅ G/Ntype
1. #20274448C2semidirect
2. #20273029C2semidirect
3. #20272715C2semidirect
13. C2×C4×D4×C53C4semidirect
15. C53×#398C4semidirect
19. C53×#394C4semidirect
21. C4×C4⋊C4×C53C4semidirect
207. C2×C4×D4Dic53semidirect
211. #398Dic53semidirect
215. #394Dic53semidirect
217. C4×C4⋊C4Dic53semidirect
223. C53#11691semidirect

Maximal subgroups

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

subgroupindex
#202744482
#202730292
#202727152
#202727122
C53×#23922
#1169153