Finite Group Databasebeta

← #27294563

Subgroups — #27294563

Order 13568.

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

normal subgroups only

This group has 444 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#20274448C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272736C212yesmaximal
4#20272715C212yesmaximal
5C53×#2599C211yesmaximal, characteristic
6C2×#13133176—21no—
7#16810680—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×C4×Q8×C53C411yescharacteristic, complemented
13C2×#117×C53C2211yescharacteristic
14C53×#398C411yescharacteristic, complemented
15C53×#397C2211yescharacteristic
16C53×#395C411yescharacteristic, complemented
17C53×#393C2211yescharacteristic
18C53×#388—21no—
19C53×#387—21no—
20C53×#387—21no—
21C53×#385—21no—
22C53×#383C411yescharacteristic, complemented
23#13133176—24no—
24#13133176—42no—
25C2×C4⋊Dic53—41no—
26C2×C22⋊Dic53—22no—
27C2×C4×Dic53—41no—
28C4⋊#88322—41no—
29C22⋊#88322—22no—
30#13133128—41no—
31C22×Q8×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35#117×C53—41no—
36#117×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53C2311yescharacteristic
47C2×C42×C53—21no—
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×Q8×C53—42no—
60C2×Q8×C53—41no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—42no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—42no—
94C22⋊Dic53—24no—
95C22⋊Dic53—42no—
96C4⋊Dic53—44no—
97C4×Dic53—44no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53AGL(1,ℤ/8)11yescharacteristic
105C23×C53#13711yescharacteristic
106Q8×C53—42no—
107Q8×C53—42no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
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—41no—
136C2×C4×C53#9911yescharacteristic
137C2×C4×C53#9911yescharacteristic
138C2×C4×C53#10412yes—
139C2×Dic53—44no—
140C2×Dic53—82no—
141C2×Dic53—82no—
142#88322—82no—
143#11756—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—81no—
153C22×C53C2×#9911yescharacteristic
154C22×C53#41711yescharacteristic
155C22×C53C2×#10411yescharacteristic
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—42no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—81no—
165C4×C53—81no—
166C4×C53—81no—
167Dic53—84no—
168#2599D5311yescharacteristic
169#1740—531no—
170#1464—531no—
171#1351—532no—
172#1344—532no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—41no—
176C2×C53—41no—
177C2×C53#150411yescharacteristic
178C2×C53#212011yescharacteristic
179C2×C53#212911yescharacteristic
180C2×C4×Q8Dic5311yescharacteristic, complemented
181C2×#117—1061no—
182C2×#117C2×D5311yescharacteristic
183#432—1061no—
184#398Dic5311yescharacteristic, complemented
185#397C2×D5311yescharacteristic
186#395Dic5311yescharacteristic, complemented
187#393C2×D5311yescharacteristic
188#388—21no—
189#387—21no—
190#387—21no—
191#385—21no—
192#383Dic5311yescharacteristic, complemented
193#342—532no—
194#342—1061no—
195#324—1061no—
196#323—532no—
197C53#1175611yescharacteristic, complemented, Sylow
198C22×Q8C2×Dic5311yescharacteristic
199C23×C4C22⋊D5311yescharacteristic
200C4×Q8—41no—
201C4×Q8—41no—
202#117—41no—
203#117—41no—
204#117—1064no—
205#117—2122no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—2121no—
210C2×C4⋊C4C2×Dic5311yescharacteristic
211C2×C4⋊C4C2×Dic5311yescharacteristic
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—1062no—
216C2×C22⋊C4C22⋊D5311yescharacteristic
217C2×C22⋊C4C22×D5311yescharacteristic
218C2×C42—21no—
219C2×C42—2121no—
220C2×C42C2×Dic5311yescharacteristic
221C2×C42C22⋊D5311yescharacteristic
222C4⋊C8—2121no—
223C22⋊C8—1062no—
224#97—2121no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95C2×Dic5311yescharacteristic
232#95C2×Dic5311yescharacteristic
233C24C2×C22⋊D5311yescharacteristic
234C2×Q8—42no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—2122no—
246C22×C4C22⋊Dic5311yescharacteristic
247C22×C4C22⋊Dic5311yescharacteristic
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22⋊Dic5311yescharacteristic
250C22×C4C22×Dic5311yescharacteristic
251C22×C4C2×C22⋊D5311yescharacteristic
252C2×C8—2122no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—2124no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—1064no—
267C22⋊C4—2122no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—42no—
273C42—2124no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊Dic5311yescharacteristic
279C23#1313324211yescharacteristic
280C23#1313324711yescharacteristic
281Q8—42no—
282Q8—42no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
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—4242no—
312C2×C4—4242no—
313C2×C4—2124no—
314C2×C4#1313313211yescharacteristic
315C2×C4#1313313211yescharacteristic
316C2×C4#1313314412yes—
317C8—4242no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#1313313211yescharacteristic
328C22#1681062411yescharacteristic
329C22C2×#1313314411yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—42no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—4244no—
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2#2027321911yescharacteristic
347C2#2027635611yescharacteristic
348C2#2027639011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2599C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2599D53
hypocenterC53#11756
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
12. C2×C4×Q8×C53C4semidirect
14. C53×#398C4semidirect
16. C53×#395C4semidirect
22. C53×#383C4semidirect
180. C2×C4×Q8Dic53semidirect
184. #398Dic53semidirect
186. #395Dic53semidirect
192. #383Dic53semidirect
197. C53#11756semidirect

Maximal subgroups

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

subgroupindex
#202744482
#202730492
#202727362
#202727152
C53×#25992
#1175653