Finite Group Databasebeta

← #28308431

Subgroups — #28308431

Order 13568.

426 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289975C211yesmaximal, characteristic
2#20289951C211yesmaximal, characteristic
3#20289660C212yesmaximal, complemented
4#20289659C212yesmaximal, complemented
5#20279029C211yesmaximal, characteristic
6#16811643—21no—
7C2×C22⋊#88324—21no—
8#16811468—21no—
9#16811466—21no—
10#16811466C2212yes—
11#16811466C2212yes—
12C2×D4×Dic53C2211yescharacteristic
13C22×#11719919C411yescharacteristic
14#16810565—21no—
15#16810559—21no—
16#16810520—21no—
17#16810503C411yescharacteristic
18#16810498C2211yescharacteristic
19#16810496C2211yescharacteristic
20#16810481—21no—
21#16810481C411yescharacteristic
22C53×#386C411yescharacteristic
23C22×#88324—41no—
24#13133284—41no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27C22⋊#88324—41no—
28#13133272—41no—
29C23×Dic53C2×C411yescharacteristic
30C23×Dic53C2×C411yescharacteristic
31C23×Dic53D411yescharacteristic
32D4×Dic53—41no—
33D4×Dic53—41no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—41no—
41C2×#11719919—41no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53C2×C411yescharacteristic
44C2×C22⋊Dic53D411yescharacteristic
45C2×C22⋊Dic53C2311yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×D4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53C2×C411yescharacteristic
55#95×C53—21no—
56C24×C53C22⋊C411yescharacteristic
57C24×C53C22×C411yescharacteristic
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—41no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—41no—
64C22×C4×C53—81no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—42no—
73C22×Dic53—42no—
74C22×Dic53—41no—
75C22×Dic53—81no—
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C2×D411yescharacteristic
79C22×Dic53C2×D411yescharacteristic
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—82no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—81no—
85C22⋊Dic53—22no—
86C22⋊Dic53—41no—
87C22⋊Dic53—41no—
88C22⋊Dic53—81no—
89C4⋊Dic53—41no—
90C4⋊Dic53—81no—
91#11719919—42no—
92#11719919—42no—
93#11719919—41no—
94#11719919—41no—
95#11719919—81no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53#10112yes—
107C23×C53C2×C22⋊C411yescharacteristic
108C23×C53#13112yes—
109D4×C53—44no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—82no—
120C2×Dic53—22no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—81no—
137C2×Dic53—82no—
138C2×Dic53#9911yescharacteristic
139C2×Dic53#9911yescharacteristic
140#88324—82no—
141#88324—82no—
142#11576—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—81no—
155C22×C53#40811yescharacteristic
156C22×C53C2×#9911yescharacteristic
157C22×C53C2×#10111yescharacteristic
158C4×C53—41no—
159C4×C53—81no—
160C4×C53—81no—
161Dic53—21no—
162Dic53—41no—
163Dic53—41no—
164Dic53—41no—
165Dic53—81no—
166Dic53—81no—
167Dic53#35212yes—
168#2487—531no—
169#1466—531no—
170#1459—531no—
171#1326—532no—
172#1326—532no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—42no—
176C2×C53—42no—
177C2×C53#147211yescharacteristic
178C2×C53C2×#35211yescharacteristic
179C2×C53#212211yescharacteristic
180C2×C4×D4—531no—
181C22×C4⋊C4—531no—
182#432—1061no—
183C2×C22⋊C8—1061no—
184#400—1061no—
185#399—1061no—
186#389—1061no—
187#386—531no—
188#386#615711yescharacteristic
189#385—531no—
190#385—531no—
191#380—531no—
192#380—1061no—
193#324—1061no—
194#323—532no—
195#323—532no—
196#323—1061no—
197C53#1157611yescharacteristic, complemented, Sylow
198C22×D4C2×#615711yescharacteristic
199C23×C4—531no—
200C23×C4—531no—
201C23×C4—531no—
202C23×C4C2×#615711yescharacteristic
203C22×C8—2121no—
204C4×D4—2121no—
205C4×D4—2121no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—2121no—
213C2×C4⋊C4—2121no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—531no—
216C2×C22⋊C4—531no—
217C2×C22⋊C4—531no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4C2×#615711yescharacteristic
220C2×C42—531no—
221C4⋊C8—2121no—
222C22⋊C8—1062no—
223C22⋊C8—1062no—
224C22⋊C8—2121no—
225#97—2121no—
226#95—21no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231C24C22⋊#615711yescharacteristic
232C24C22×#615711yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—81no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—4241no—
251C22×C4—2122no—
252C22×C4—2122no—
253C22×C4—2121no—
254C22×C4C22⋊#615711yescharacteristic
255C2×C8—4242no—
256C2×C8—2122no—
257C2×C8—2122no—
258C4⋊C4—4241no—
259C4⋊C4—4241no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2122no—
264C4⋊C4—2122no—
265C22⋊C4—22no—
266C22⋊C4—81no—
267C22⋊C4—1062no—
268C22⋊C4—4241no—
269C22⋊C4—2121no—
270C22⋊C4—2121no—
271C42—1061no—
272C42—1061no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#1313327812yes—
282C23C2×C22⋊#615711yescharacteristic
283C23#1313330412yes—
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—531no—
296C2×C4—531no—
297C2×C4—1062no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—4241no—
302C2×C4—4242no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2122no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C8—4242no—
316C8—4242no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—22no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22#1681159611yescharacteristic
330C22D53⋊#9911yescharacteristic
331C22C2×#1313327811yescharacteristic
332C4—41no—
333C4—81no—
334C4—81no—
335C4—532no—
336C4—1061no—
337C4—4241no—
338C4—4241no—
339C4—2121no—
340C4—2121no—
341C4—2121no—
342C2—21no—
343C2—21no—
344C2—42no—
345C2—42no—
346C2#2028999411yescharacteristic
347C2D53⋊#35211yescharacteristic
348C2#2029097511yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC53×#386C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C22⋊Dic53C23
hypercenter#386#6157
hypocenterC53#11576
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #20289660C2semidirect
4. #20289659C2semidirect
197. C53#11576semidirect

Maximal subgroups

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

subgroupindex
#202899752
#202899512
#202896602
#202896592
#202790292
#1157653