Finite Group Databasebeta

← #28299941

Subgroups — #28299941

Order 13568.

570 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290569C211yesmaximal, characteristic, complemented
2#20290378C211yesmaximal, characteristic, complemented
3#20289536C212yesmaximal, complemented
4#20289531C212yesmaximal, complemented
5#20283014C211yesmaximal, characteristic
6C22×#11719940—21no—
7C2×C4⋊#88324—21no—
8C2×C22⋊#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133204C411yescharacteristic
13C2×#13133196C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810528C2211yescharacteristic
16#16810526C2211yescharacteristic
17#16810517—21no—
18#16810517—21no—
19#16810503—21no—
20#16810503—21no—
21#16810488C411yescharacteristic
22C2×#121×C53C411yescharacteristic
23C2×#11719940—24no—
24C2×#11719940—42no—
25C2×#11719940—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28C4⋊#88324—42no—
29C22⋊#88324—42no—
30C22×Dic106C2×C411yescharacteristic
31C23×Dic53D411yescharacteristic
32C23×Dic53D411yescharacteristic
33#13133204—42no—
34#13133196—42no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C4×Dic53—42no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×D4×C53—21no—
52C22×D4×C53C2×C411yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54#121×C53—42no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53—21no—
57C24×C53—21no—
58C24×C53C22⋊C411yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—42no—
63C2×D4×C53—42no—
64C2×D4×C53—42no—
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—41no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C2×D411yescharacteristic
84C22×Dic53C2×D411yescharacteristic
85C2×Dic106—22no—
86C2×Dic106—41no—
87C2×Dic106—41no—
88#11719940—48no—
89C2×#88324—24no—
90C2×#88324—42no—
91C2×#88324—42no—
92C2×#88324—42no—
93C2×#88324—42no—
94C2×#88324—42no—
95C4⋊C4×C53—41no—
96C22⋊C4×C53—42no—
97C22⋊Dic53—42no—
98C22⋊Dic53—42no—
99C22⋊Dic53—42no—
100C22⋊Dic53—41no—
101C22⋊Dic53—41no—
102C22⋊Dic53—41no—
103C22⋊Dic53—41no—
104C4⋊Dic53—41no—
105#11719919—42no—
106C4×Dic53—22no—
107C4×Dic53—22no—
108C4×Dic53—41no—
109C4×Dic53—41no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53#10412yes—
124C23×C53C2×C22⋊C411yescharacteristic
125D4×C53—82no—
126D4×C53—84no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—44no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53#10012yes—
136C2×C4×C53#13112yes—
137C2×Dic53—24no—
138C2×Dic53—24no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53#9912yes—
157C2×Dic53AGL(1,ℤ/8)11yescharacteristic
158C2×Dic53AGL(1,ℤ/8)11yescharacteristic
159C2×Dic53#13711yescharacteristic
160C2×Dic53#13711yescharacteristic
161Dic106—82no—
162#88324—44no—
163#88324—82no—
164#7710—531nomaximal, Sylow
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53#40811yescharacteristic
180C22×C53C2×#9911yescharacteristic
181C22×C53C2×#10011yescharacteristic
182C22×C53#41711yescharacteristic
183C22×C53#41711yescharacteristic
184C22×C53C2×#10411yescharacteristic
185C22×C53#42111yescharacteristic
186C4×C53—22no—
187C4×C53—22no—
188C4×C53—81no—
189Dic53—21no—
190Dic53—22no—
191Dic53—21no—
192Dic53—21no—
193Dic53—21no—
194Dic53—41no—
195Dic53—81no—
196Dic53—81no—
197#2850—531no—
198#1797—531no—
199#1728—531no—
200#1272—532no—
201#1270—532no—
202C2×C53—42no—
203C2×C53—161no—
204C2×C53#147111yescharacteristic
205C2×C53#148311yescharacteristic
206C2×C53#149511yescharacteristic
207C2×C53#150411yescharacteristic
208C2×C53#151311yescharacteristic
209C2×C53#152711yescharacteristic
210C2×C53#156011yescharacteristic
211C22×M16—1061no—
212C2×#124—531no—
213C2×C22⋊Q8—531no—
214C2×#121#615711yescharacteristic
215C22×C42—531no—
216C2×C4⋊C8—1061no—
217C2×C22⋊C8—1061no—
218#391—531no—
219#390—531no—
220#388—1061no—
221#388—1061no—
222#386—1061no—
223#386—1061no—
224#382—531no—
225#336—532no—
226#336—532no—
227#336—1061no—
228C53#771011yescharacteristic, complemented, Sylow
229C22×Q8—531no—
230C22×D4—21no—
231C22×D4C2×#615711yescharacteristic
232C23×C4—531no—
233C23×C4—531no—
234C23×C4C2×#615711yescharacteristic
235C2×M16—1064no—
236C2×M16—2122no—
237C2×M16—2122no—
238C22×C8—1062no—
239C22×C8—1062no—
240#124—2122no—
241C22⋊Q8—2122no—
242#121—42no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4C2×#615711yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—1061no—
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C22⋊C4—1061no—
252C2×C42—531no—
253C2×C42—531no—
254C2×C42—531no—
255C2×C42—531no—
256C2×C42—2122no—
257C4⋊C8—2122no—
258C22⋊C8—2122no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262#95—1061no—
263C24—21no—
264C24C22⋊#615711yescharacteristic
265C2×Q8—1062no—
266C2×Q8—2121no—
267C2×Q8—2121no—
268C2×D4—22no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—1062no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1062no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4C22⋊#615711yescharacteristic
293C22×C4C22×#615711yescharacteristic
294M16—2128no—
295C2×C8—1064no—
296C2×C8—2122no—
297C2×C8—2122no—
298C2×C8—2122no—
299C2×C8—2122no—
300C2×C8—2122no—
301C4⋊C4—41no—
302C4⋊C4—2121no—
303C4⋊C4—2122no—
304C22⋊C4—42no—
305C22⋊C4—2122no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C22⋊C4—2121no—
310C22⋊C4—2122no—
311C22⋊C4—2122no—
312C42—1062no—
313C42—1062no—
314C42—2121no—
315C42—2121no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#1313328112yes—
330C23C2×C22⋊#615711yescharacteristic
331Q8—4242no—
332D4—82no—
333D4—84no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—44no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—532no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—531no—
347C2×C4—1064no—
348C2×C4—1064no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4#1313327712yes—
367C2×C4#1313330412yes—
368C8—4242no—
369C8—2124no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#1681159611yescharacteristic
385C22D53⋊#9911yescharacteristic
386C22C2×#1313327711yescharacteristic
387C22#1681161411yescharacteristic
388C22#1681161411yescharacteristic
389C22C2×#1313328111yescharacteristic
390C22#1681162111yescharacteristic
391C4—22no—
392C4—22no—
393C4—81no—
394C4—1061no—
395C4—1062no—
396C4—1061no—
397C4—1061no—
398C4—1061no—
399C4—4241no—
400C4—4241no—
401C4—2121no—
402C2—42no—
403C2—161no—
404C2#2028999111yescharacteristic
405C2#2029003011yescharacteristic
406C2#2029007011yescharacteristic
407C2#2029010011yescharacteristic
408C2#2029013111yescharacteristic
409C2#2029017411yescharacteristic
410C2#2029025211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#121×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#121#6157
hypocenterC53#7710
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20290569C2semidirect
2. #20290378C2semidirect
3. #20289536C2semidirect
4. #20289531C2semidirect
228. C53#7710semidirect

Maximal subgroups

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

subgroupindex
#202905692
#202903782
#202895362
#202895312
#202830142
#771053