Finite Group Databasebeta

← #28299495

Subgroups — #28299495

Order 13568.

522 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 112 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#20290605C211yesmaximal, characteristic
2C4×C22⋊#88324C211yesmaximal, characteristic
3#20289536C212yesmaximal, complemented
4#20289534C212yesmaximal, complemented
5#20277243C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133196C411yescharacteristic
13C2×#13133182C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810528C2211yescharacteristic
16#16810519C2211yescharacteristic
17#16810505—21no—
18#16810495—21no—
19C4⋊C4×Dic53C411yescharacteristic
20#16810474—21no—
21C22⋊C4×Dic53—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324—42no—
27C4×#88324—42no—
28C22×Dic106C2×C411yescharacteristic
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133196—22no—
32#13133196C814yescomplemented
33#13133182—42no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—42no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C23×C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53—21no—
54#95×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C2×C812yes—
62C22×C4×C53M1612yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—41no—
71C22×Dic53—41no—
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C22×Dic53C2×D411yescharacteristic
76C2×Dic106—21no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×Dic106C2×C812yes—
80C2×#88324—24no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—42no—
89C22⋊Dic53—24no—
90C4⋊Dic53—21no—
91C4⋊Dic53C2×C812yes—
92#11719919—24no—
93#11719919—42no—
94#11719919—42no—
95C4×Dic53—22no—
96C4×Dic53—22no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C4×Dic53—42no—
100C4×Dic53—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53C22⋊C812yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53C22⋊C812yes—
122C2×C4×C53C22×C811yescharacteristic
123C2×C4×C53C2×M1611yescharacteristic
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—44no—
135C2×Dic53—44no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53#9912yes—
141C2×Dic53#10412yes—
142C2×Dic53#13512yes—
143C2×Dic53#13712yes—
144Dic106—42no—
145Dic106—42no—
146#88324—44no—
147#88324—82no—
148#7583—531nomaximal, Sylow
149C22×C53—22no—
150C22×C53—22no—
151C22×C53—22no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53C2×C22⋊C811yescharacteristic
157C22×C53C2×#9911yescharacteristic
158C22×C53#41311yescharacteristic
159C22×C53#41611yescharacteristic
160C22×C53#41911yescharacteristic
161C22×C53C2×#10411yescharacteristic
162C22×C53#42111yescharacteristic
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—22no—
166C4×C53—81no—
167C4×C53—81no—
168Dic53—22no—
169Dic53—22no—
170Dic53—22no—
171Dic53—22no—
172Dic53—42no—
173Dic53—42no—
174C4×C22⋊Q8—531no—
175#1817—531no—
176C4×C22⋊C8—531no—
177#1272—532no—
178#1271—532no—
179C2×C53—42no—
180C2×C53#145911yescharacteristic
181C2×C53#147611yescharacteristic
182C2×C53#147911yescharacteristic
183C2×C53#150811yescharacteristic
184C2×C53#151611yescharacteristic
185C2×C53#152911yescharacteristic
186C2×C53#155411yescharacteristic
187C2×C22⋊Q8—531no—
188C2×C4×Q8—531no—
189C22×C42—531no—
190C2×C22⋊C8—1061no—
191C2×C22⋊C8—1061no—
192C2×C4×C8—1061no—
193#391—531no—
194#389—531no—
195#387—1061no—
196#385—1061no—
197#380#615711yescharacteristic
198C4×C4⋊C4—531no—
199C4×C4⋊C4—1061no—
200C4×C22⋊C4—1061no—
201#336—532no—
202#336—532no—
203#336—1061no—
204C53#758311yescharacteristic, complemented, Sylow
205C22×Q8—531no—
206C23×C4—531no—
207C23×C4—531no—
208C23×C4C2×#615711yescharacteristic
209C22×C8—1062no—
210C22×C8—1062no—
211C22⋊Q8—534no—
212C22⋊Q8—1062no—
213C4×Q8—2122no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4C2×#615711yescharacteristic
219C2×C4⋊C4C2×#615711yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—1061no—
222C2×C42—531no—
223C2×C42—531no—
224C2×C42—531no—
225C2×C42—531no—
226C2×C42—1061no—
227C2×C42—1061no—
228C2×C42—2122no—
229C22⋊C8—1064no—
230C22⋊C8—2122no—
231C4×C8—2122no—
232#95—21no—
233#95—1061no—
234#95—1061no—
235#95—1061no—
236#95—1061no—
237C24C22⋊#615711yescharacteristic
238C2×Q8—532no—
239C2×Q8—1061no—
240C2×Q8—2121no—
241C2×Q8—2121no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—1062no—
251C22×C4—1062no—
252C22×C4—1062no—
253C22×C4—1062no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—2121no—
257C22×C4—2121no—
258C22×C4D53⋊C812yes—
259C22×C4#1171993612yes—
260C22×C4C22⋊#615711yescharacteristic
261C22×C4C22×#615711yescharacteristic
262C2×C8—1064no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2122no—
266C2×C8—2122no—
267C2×C8—2122no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—532no—
271C4⋊C4—1064no—
272C4⋊C4—1061no—
273C4⋊C4—2122no—
274C4⋊C4—2122no—
275C22⋊C4—42no—
276C22⋊C4—1064no—
277C42—1062no—
278C42—1062no—
279C42—2121no—
280C42—2121no—
281C42—2122no—
282C42—2122no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1313327412yes—
290C23C2×C22⋊#615711yescharacteristic
291Q8—2122no—
292Q8—2122no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—532no—
306C2×C4—532no—
307C2×C4—532no—
308C2×C4—532no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1062no—
318C2×C4—1062no—
319C2×C4—2124no—
320C2×C4—2124no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4#1313327412yes—
326C2×C4C2×D53⋊C811yescharacteristic
327C2×C4C2×#1171993611yescharacteristic
328C8—4242no—
329C8—2124no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C2×#1313327411yescharacteristic
338C22D53⋊#9911yescharacteristic
339C22#1681161011yescharacteristic
340C22#1681161311yescharacteristic
341C22#1681161611yescharacteristic
342C22D53⋊#10411yescharacteristic
343C22#1681162111yescharacteristic
344C4—22no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—81no—
349C4—1062no—
350C4—1062no—
351C4—1062no—
352C4—1062no—
353C4—2122no—
354C4—2122no—
355C2—42no—
356C2#2028995311yescharacteristic
357C2#2029000811yescharacteristic
358C2#2029001911yescharacteristic
359C2#2029011311yescharacteristic
360C2#2029014011yescharacteristic
361C2#2029018111yescharacteristic
362C2#2029024111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#7583
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289536C2semidirect
4. #20289534C2semidirect
32. #13133196C8semidirect
204. C53#7583semidirect

Maximal subgroups

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

subgroupindex
#202906052
C4×C22⋊#883242
#202895362
#202895342
#202772432
#758353