Finite Group Databasebeta

← #27294697

Subgroups — #27294697

Order 13568.

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

normal subgroups only

This group has 460 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#20273053C211yesmaximal, characteristic
3#20272739C212yesmaximal, complemented
4#20272736C212yesmaximal, complemented
5C53×#2580C211yesmaximal, characteristic
6C2×#13133132—21no—
7#16810587—21no—
8C4×C22⋊Dic53—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×C4×Q8×C53C411yescharacteristic, complemented
13C2×C4×D4×C53C2211yescharacteristic
14C53×#399C411yescharacteristic, complemented
15C53×#399C2211yescharacteristic
16C53×#393C411yescharacteristic, complemented
17C53×#393C2211yescharacteristic
18C53×#390—21no—
19C53×#389—21no—
20C53×#385—21no—
21C53×#385—21no—
22C53×#382C411yescharacteristic, complemented
23C2×#11719917—41no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—41no—
26#13133132—24no—
27C22⋊#88322—22no—
28C22⋊#88322—41no—
29#13133124—41no—
30C22×Q8×C53C2×C411yescharacteristic
31C22×D4×C53D411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C4×Q8×C53—41no—
35C4×Q8×C53—41no—
36C4×D4×C53—41no—
37C4×D4×C53—41no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
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×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
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—42no—
60C2×Q8×C53—41no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—41no—
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—21no—
71C22×C4×C53—21no—
72C22×C4×C53—81no—
73C22×C4×C53—81no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—81no—
86C4⋊C4×C53—81no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—81no—
92C22⋊C4×C53—81no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—42no—
96C22⋊Dic53—24no—
97C22⋊Dic53—42no—
98C4×Dic53—82no—
99#11719917—82no—
100C2×#88322—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53#13712yes—
112Q8×C53—42no—
113Q8×C53—42no—
114D4×C53—44no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
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#9911yescharacteristic
141C2×C4×C53#9911yescharacteristic
142C2×C4×C53#10412yes—
143C2×Dic53—44no—
144C2×Dic53—82no—
145C2×Dic53—82no—
146#88322—82no—
147#11784—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—22no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53C2×#9911yescharacteristic
161C22×C53#41911yescharacteristic
162C22×C53C2×#10411yescharacteristic
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—22no—
166C4×C53—42no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—81no—
171C4×C53—81no—
172C4×C53—81no—
173Dic53—84no—
174#2580D5311yescharacteristic
175C4×#99—531no—
176#1465—531no—
177#1352—532no—
178#1351—532no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—42no—
182C2×C53—81no—
183C2×C53#150811yescharacteristic
184C2×C53#212011yescharacteristic
185C2×C53#212711yescharacteristic
186C2×C4×Q8Dic5311yescharacteristic, complemented
187C2×C4×D4C2×D5311yescharacteristic
188C2×#99—1061no—
189#407—1061no—
190#399Dic5311yescharacteristic, complemented
191#399C2×D5311yescharacteristic
192#393Dic5311yescharacteristic, complemented
193#393C2×D5311yescharacteristic
194#390—21no—
195#389—21no—
196#385—21no—
197#385—21no—
198#382Dic5311yescharacteristic, complemented
199C4×C22⋊C4—1061no—
200#342—532no—
201#324—1061no—
202#323—532no—
203C53#1178411yescharacteristic, complemented, Sylow
204C22×Q8C2×Dic5311yescharacteristic
205C22×D4C22⋊D5311yescharacteristic
206C23×C4C22⋊D5311yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C2×M16—2121no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×D4—41no—
212C4×D4—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×Dic5311yescharacteristic
218C2×C4⋊C4C22⋊D5311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—1062no—
224C2×C22⋊C4C2×Dic5311yescharacteristic
225C2×C22⋊C4C22×D5311yescharacteristic
226C2×C42—21no—
227C2×C42—2121no—
228C2×C42C2×Dic5311yescharacteristic
229#99—1064no—
230C22⋊C8—1062no—
231C22⋊C8—2121no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—2121no—
236#95C2×Dic5311yescharacteristic
237#95C2×Dic5311yescharacteristic
238C24C22⋊Dic5311yescharacteristic
239C24C2×C22⋊D5311yescharacteristic
240C2×Q8—42no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—2122no—
256C22×C4C22⋊Dic5311yescharacteristic
257C22×C4C22⋊Dic5311yescharacteristic
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22×Dic5311yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261M16—4242no—
262C2×C8—2122no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C22⋊C4—1064no—
277C22⋊C4—2122no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—4242no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊Dic5311yescharacteristic
292C23#1313324812yes—
293Q8—42no—
294Q8—42no—
295D4—44no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
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—81no—
320C2×C4—81no—
321C2×C4—4242no—
322C2×C4—4242no—
323C2×C4—2124no—
324C2×C4#1313313211yescharacteristic
325C2×C4#1313313211yescharacteristic
326C2×C4#1313314412yes—
327C8—4242no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22C2×#1313313211yescharacteristic
341C22#1681063211yescharacteristic
342C22C2×#1313314411yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—4244no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—81no—
358C2#2027323511yescharacteristic
359C2#2027635611yescharacteristic
360C2#2027638411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2580C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2580D53
hypocenterC53#11784
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #20272739C2semidirect
4. #20272736C2semidirect
12. C2×C4×Q8×C53C4semidirect
14. C53×#399C4semidirect
16. C53×#393C4semidirect
22. C53×#382C4semidirect
186. C2×C4×Q8Dic53semidirect
190. #399Dic53semidirect
192. #393Dic53semidirect
198. #382Dic53semidirect
203. C53#11784semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730532
#202727392
#202727362
C53×#25802
#1178453