Finite Group Databasebeta

← #27295233

Subgroups — #27295233

Order 13568.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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#20274703C211yesmaximal, characteristic
2#20273036C211yesmaximal, characteristic
3#20272739C212yesmaximal, complemented
4#20272736C212yesmaximal, complemented
5C53×#2564C211yesmaximal, characteristic
6#16810676—21no—
7C2×#13133132—21no—
8#16810483—21no—
9#16810390C2212yes—
10#16810347—21no—
11#16810347C2212yes—
12C2×C22⋊Q8×C53C411yescharacteristic, complemented
13C2×C4×D4×C53C2211yescharacteristic
14C53×#400C411yescharacteristic, complemented
15C53×#399C2211yescharacteristic
16C53×#395C411yescharacteristic, complemented
17C53×#393C2211yescharacteristic
18C53×#387—21no—
19C53×#386—21no—
20C53×#385—21no—
21C53×#380—21no—
22C2×#95×C53C411yescharacteristic, complemented
23C2×C4⋊Dic53—41no—
24C2×C22⋊Dic53—22no—
25C4⋊#88322—41no—
26#13133132—42no—
27C22⋊#88322—22no—
28C4×#88322—41no—
29#13133124—41no—
30C22×Q8×C53C2×C411yescharacteristic
31C22×D4×C53D411yescharacteristic
32C23×C4×C53—21no—
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C22⋊Q8×C53—41no—
36C22⋊Q8×C53—41no—
37C4×D4×C53—41no—
38C4×D4×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53C2311yescharacteristic
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×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×Q8×C53—41no—
60C2×Q8×C53—81no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53SD1612yes—
64C2×D4×C53Q1612yes—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53—81no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×Dic53—42no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—81no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—81no—
95C42×C53—21no—
96C42×C53—21no—
97C22⋊Dic53—42no—
98C22⋊Dic53—42no—
99C4⋊Dic53—82no—
100C2×#88322—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53#10312yes—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53C2×SD1611yescharacteristic
112C23×C53C2×Q1611yescharacteristic
113Q8×C53—82no—
114D4×C53—42no—
115D4×C53—42no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
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—41no—
137C2×C4×C53—41no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53—84no—
142C2×C4×C53#9911yescharacteristic
143C2×C4×C53#9911yescharacteristic
144C2×C4×C53#10312yes—
145C2×Dic53—82no—
146C2×Dic53—82no—
147C2×Dic53—82no—
148#88322—82no—
149#11902—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—21no—
155C22×C53—42no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—42no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—81no—
164C22×C53C2×#9911yescharacteristic
165C22×C53C2×#10311yescharacteristic
166C22×C53#42111yescharacteristic
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—22no—
170C4×C53—42no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53—81no—
177Dic53—162no—
178#2564D5311yescharacteristic
179#1799—531no—
180#1460—531no—
181#1352—532no—
182#1351—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—41no—
186C2×C53—42no—
187C2×C53—41no—
188C2×C53#152011yescharacteristic
189C2×C53#212111yescharacteristic
190C2×C53#212611yescharacteristic
191C2×C22⋊Q8Dic5311yescharacteristic, complemented
192C2×C4×D4C2×D5311yescharacteristic
193#431—1061no—
194C2×#99—1061no—
195#400Dic5311yescharacteristic, complemented
196#399C2×D5311yescharacteristic
197#395Dic5311yescharacteristic, complemented
198#393C2×D5311yescharacteristic
199#387—21no—
200#386—21no—
201#385—21no—
202#380—21no—
203#380—1061no—
204C2×#95Dic5311yescharacteristic, complemented
205#342—532no—
206#323—532no—
207#323—1061no—
208C53#1190211yescharacteristic, complemented, Sylow
209C22×Q8C2×Dic5311yescharacteristic
210C22×D4C22⋊D5311yescharacteristic
211C23×C4—21no—
212C23×C4C2×Dic5311yescharacteristic
213C23×C4C22⋊D5311yescharacteristic
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C4×D4—41no—
217C4×D4—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—2121no—
222C2×C4⋊C4C2×Dic5311yescharacteristic
223C2×C4⋊C4C2×Dic5311yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4C22⋊D5311yescharacteristic
229C2×C22⋊C4C22×D5311yescharacteristic
230C2×C42C22⋊D5311yescharacteristic
231C4⋊C8—2121no—
232#99—2122no—
233C22⋊C8—1062no—
234C4×C8—2121no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—2121no—
241#95C2×Dic5311yescharacteristic
242#95C2×Dic5311yescharacteristic
243C24C22⋊Dic5311yescharacteristic
244C24C2×C22⋊D5311yescharacteristic
245C2×Q8—41no—
246C2×Q8—81no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4#1171992412yes—
250C2×D4#1171992512yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—2122no—
265C22×C4C22⋊Dic5311yescharacteristic
266C22×C4C22⋊Dic5311yescharacteristic
267C22×C4C22⋊Dic5311yescharacteristic
268C22×C4C22×Dic5311yescharacteristic
269C22×C4C2×C22⋊D5311yescharacteristic
270C2×C8—2122no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—81no—
277C4⋊C4—4242no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—81no—
283C22⋊C4—2122no—
284C22⋊C4—2122no—
285C42—21no—
286C42—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23#1313314112yes—
296C23C2×C22⋊Dic5311yescharacteristic
297C23C2×#1171992411yescharacteristic
298C23C2×#1171992511yescharacteristic
299Q8—82no—
300D4—42no—
301D4—42no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—84no—
328C2×C4—4242no—
329C2×C4—4242no—
330C2×C4—4242no—
331C2×C4#1313313211yescharacteristic
332C2×C4#1313313211yescharacteristic
333C2×C4#1313314112yes—
334C8—4242no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22C2×#1313313211yescharacteristic
350C22C2×#1313314111yescharacteristic
351C22#1681064011yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—8482no—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—42no—
367C2—41no—
368C2#2027328711yescharacteristic
369C2#2027636111yescharacteristic
370C2#2027638011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2564C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2564D53
hypocenterC53#11902
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #20272739C2semidirect
4. #20272736C2semidirect
12. C2×C22⋊Q8×C53C4semidirect
14. C53×#400C4semidirect
16. C53×#395C4semidirect
22. C2×#95×C53C4semidirect
191. C2×C22⋊Q8Dic53semidirect
195. #400Dic53semidirect
197. #395Dic53semidirect
204. C2×#95Dic53semidirect
208. C53#11902semidirect

Maximal subgroups

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

subgroupindex
#202747032
#202730362
#202727392
#202727362
C53×#25642
#1190253