Finite Group Databasebeta

← #27294451

Subgroups — #27294451

Order 13568.

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

normal subgroups only

This group has 442 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
1C4×#13133132C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272739C212yesmaximal, complemented
4#20272715C212yesmaximal, complemented
5C53×#2633C211yesmaximal, characteristic
6C2×#13133132—21no—
7C2×C22⋊#88322—21no—
8C4×C22⋊Dic53—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×#125×C53C411yescharacteristic, complemented
13C2×C4×D4×C53C2211yescharacteristic
14C53×#399C411yescharacteristic, complemented
15C53×#399C2211yescharacteristic
16C53×#398—21no—
17C53×#397C411yescharacteristic, complemented
18C53×#397C2211yescharacteristic
19C53×#394—21no—
20C53×#385—21no—
21C53×#385—21no—
22C53×#383C411yescharacteristic, complemented
23C22×#88322—41no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—41no—
26#13133132—24no—
27C22⋊#88322—22no—
28C22⋊#88322—41no—
29#13133124—41no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53D411yescharacteristic
32C23×C4×C53D411yescharacteristic
33#125×C53—41no—
34#125×C53—41no—
35C4×D4×C53—41no—
36C4×D4×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53C2311yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—41no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—81no—
71C22×C4×C53—81no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—81no—
85C4⋊C4×C53—81no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—81no—
91C22⋊C4×C53—81no—
92C42×C53—21no—
93C42×C53—21no—
94C22⋊Dic53—24no—
95C22⋊Dic53—42no—
96C4×Dic53—82no—
97C2×#88322—42no—
98C2×#88322—82no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#10412yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C23×C53#13512yes—
110D4×C53—44no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53#9911yescharacteristic
138C2×C4×C53#9911yescharacteristic
139C2×Dic53—44no—
140C2×Dic53—82no—
141C2×Dic53—82no—
142#88322—82no—
143#11732—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53C2×#9911yescharacteristic
157C22×C53#41611yescharacteristic
158C22×C53C2×#10411yescharacteristic
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—81no—
166C4×C53—81no—
167C4×C53—81no—
168C4×C53—81no—
169Dic53—84no—
170#2633D5311yescharacteristic
171C4×#99—531no—
172#1459—531no—
173#1352—532no—
174#1344—532no—
175C2×C53—21no—
176C2×C53—21no—
177C2×C53—42no—
178C2×C53—42no—
179C2×C53#150111yescharacteristic
180C2×C53#212311yescharacteristic
181C2×C53#212711yescharacteristic
182C2×#125Dic5311yescharacteristic, complemented
183C2×C4×D4C2×D5311yescharacteristic
184C2×#99—1061no—
185C2×C22⋊C8—1061no—
186#399Dic5311yescharacteristic, complemented
187#399C2×D5311yescharacteristic
188#398—21no—
189#397Dic5311yescharacteristic, complemented
190#397C2×D5311yescharacteristic
191#394—21no—
192#385—21no—
193#385—21no—
194#383Dic5311yescharacteristic, complemented
195C4×C22⋊C4—1061no—
196#342—532no—
197#324—1061no—
198#323—532no—
199C53#1173211yescharacteristic, complemented, Sylow
200C22×D4C22⋊D5311yescharacteristic
201C23×C4C22⋊D5311yescharacteristic
202C23×C4C22⋊D5311yescharacteristic
203C22×C8—2121no—
204#125—41no—
205#125—41no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×Dic5311yescharacteristic
213C2×C4⋊C4C2×Dic5311yescharacteristic
214C2×C4⋊C4C22⋊D5311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—1062no—
220C2×C22⋊C4C2×Dic5311yescharacteristic
221C2×C22⋊C4C22×D5311yescharacteristic
222C2×C42—2121no—
223C2×C42C2×Dic5311yescharacteristic
224#99—1064no—
225C22⋊C8—1062no—
226C22⋊C8—2121no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—2121no—
232#95C2×Dic5311yescharacteristic
233#95C2×Dic5311yescharacteristic
234C24C22⋊Dic5311yescharacteristic
235C24C2×C22⋊D5311yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—2122no—
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22⋊Dic5311yescharacteristic
252C22×C4C22⋊Dic5311yescharacteristic
253C22×C4C22×Dic5311yescharacteristic
254C22×C4C2×C22⋊D5311yescharacteristic
255C2×C8—4242no—
256C2×C8—2122no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—1064no—
272C22⋊C4—2122no—
273C42—21no—
274C42—21no—
275C42—4242no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#1313314412yes—
285C23C2×C22⋊Dic5311yescharacteristic
286C23#1313323712yes—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—4242no—
315C2×C4—4242no—
316C2×C4—2124no—
317C2×C4#1313313211yescharacteristic
318C2×C4#1313313211yescharacteristic
319C8—4242no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22C2×#1313313211yescharacteristic
333C22#1681061911yescharacteristic
334C22C2×#1313314411yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—4244no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—42no—
350C2#2027320611yescharacteristic
351C2#2027636911yescharacteristic
352C2#2027638411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2633C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2633D53
hypocenterC53#11732
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2633

Complements

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

NH ≅ G/Ntype
3. #20272739C2semidirect
4. #20272715C2semidirect
12. C2×#125×C53C4semidirect
14. C53×#399C4semidirect
17. C53×#397C4semidirect
22. C53×#383C4semidirect
182. C2×#125Dic53semidirect
186. #399Dic53semidirect
189. #397Dic53semidirect
194. #383Dic53semidirect
199. C53#11732semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730332
#202727392
#202727152
C53×#26332
#1173253