Finite Group Databasebeta

← #28299931

Subgroups — #28299931

Order 13568.

500 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 92 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
2#20290441C211yesmaximal, characteristic
3#20289536C212yesmaximal, complemented
4#20289535C212yesmaximal, complemented
5#20283063C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133215C411yescharacteristic
13C2×#13133196C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810530C2211yescharacteristic
16#16810528C2211yescharacteristic
17#16810518—21no—
18#16810518—21no—
19#16810492C411yescharacteristic
20#16810481—21no—
21#16810481—21no—
22C2×C22⋊Q8×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324—42no—
27#13133272—42no—
28C22×Dic106C2×C411yescharacteristic
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133215—42no—
32#13133196—42no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×C22⋊Dic53—21no—
40C2×C4×Dic53—42no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×Q8×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C22⋊Q8×C53—42no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C2×Q8×C53—22no—
57C2×Q8×C53—41no—
58C2×Q8×C53—41no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—41no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77C22×Dic53C2×D411yescharacteristic
78C2×Dic106—22no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×#88324—24no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—41no—
89C22⋊C4×C53—42no—
90C22⋊Dic53—42no—
91C4⋊Dic53—41no—
92#11719919—42no—
93#11719919—42no—
94#11719919—42no—
95#11719919—42no—
96#11719919—42no—
97C4×Dic53—22no—
98C4×Dic53—22no—
99C4×Dic53—41no—
100C4×Dic53—41no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53#10012yes—
108C23×C53C2×C22⋊C411yescharacteristic
109Q8×C53—82no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—44no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#13112yes—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—44no—
132C2×Dic53—44no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53#9912yes—
140C2×Dic53#10412yes—
141C2×Dic53#13712yes—
142C2×Dic53#13712yes—
143Dic106—82no—
144#88324—44no—
145#88324—82no—
146#7707—531nomaximal, Sylow
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53#40811yescharacteristic
155C22×C53C2×#9911yescharacteristic
156C22×C53C2×#10011yescharacteristic
157C22×C53#41911yescharacteristic
158C22×C53#41911yescharacteristic
159C22×C53C2×#10411yescharacteristic
160C22×C53#42111yescharacteristic
161C4×C53—22no—
162C4×C53—41no—
163C4×C53—81no—
164C4×C53—81no—
165Dic53—22no—
166Dic53—22no—
167Dic53—22no—
168Dic53—22no—
169Dic53—81no—
170Dic53—81no—
171#2857—531no—
172#1817—531no—
173#1749—531no—
174#1272—532no—
175#1272—532no—
176C2×C53—42no—
177C2×C53#147111yescharacteristic
178C2×C53#148511yescharacteristic
179C2×C53#149711yescharacteristic
180C2×C53#150811yescharacteristic
181C2×C53#151111yescharacteristic
182C2×C53#153011yescharacteristic
183C2×C53#155811yescharacteristic
184C2×C4⋊Q8—531no—
185C2×C22⋊Q8—531no—
186C2×C22⋊Q8#615711yescharacteristic
187C22×C42—531no—
188C2×C22⋊C8—1061no—
189C2×C22⋊C8—1061no—
190C2×#97—1061no—
191#391—531no—
192#391—531no—
193#389—1061no—
194#389—1061no—
195#384—531no—
196#380—1061no—
197#380—1061no—
198#336—532no—
199#336—532no—
200#336—1061no—
201C53#770711yescharacteristic, complemented, Sylow
202C22×Q8—531no—
203C22×Q8C2×#615711yescharacteristic
204C23×C4—531no—
205C23×C4—531no—
206C23×C4C2×#615711yescharacteristic
207C22×C8—1062no—
208C22×C8—1062no—
209C4⋊Q8—2122no—
210C22⋊Q8—42no—
211C22⋊Q8—2122no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
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—2122no—
227C22⋊C8—1064no—
228C22⋊C8—2122no—
229#97—2122no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234C24C22⋊#615711yescharacteristic
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—1062no—
239C2×Q8—2121no—
240C2×Q8—2121no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
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×C4C22⋊#615711yescharacteristic
259C22×C4C22×#615711yescharacteristic
260C2×C8—1064no—
261C2×C8—2122no—
262C2×C8—2122no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2122no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—2121no—
269C4⋊C4—2122no—
270C4⋊C4—2122no—
271C4⋊C4—2122no—
272C4⋊C4—2122no—
273C4⋊C4—2122no—
274C22⋊C4—42no—
275C22⋊C4—2122no—
276C42—1062no—
277C42—1062no—
278C42—2121no—
279C42—2121no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23#1313327712yes—
287C23C2×C22⋊#615711yescharacteristic
288Q8—82no—
289Q8—4242no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—532no—
303C2×C4—532no—
304C2×C4—532no—
305C2×C4—532no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—2124no—
315C2×C4—2124no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4#1313330412yes—
323C8—4242no—
324C8—2124no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22#1681159611yescharacteristic
333C22D53⋊#9911yescharacteristic
334C22C2×#1313327711yescharacteristic
335C22#1681161611yescharacteristic
336C22#1681161611yescharacteristic
337C22D53⋊#10411yescharacteristic
338C22#1681162011yescharacteristic
339C4—22no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—1062no—
344C4—1062no—
345C4—1062no—
346C4—1062no—
347C4—4241no—
348C4—4241no—
349C2—42no—
350C2#2028999111yescharacteristic
351C2#2029003711yescharacteristic
352C2#2029007711yescharacteristic
353C2#2029011311yescharacteristic
354C2#2029012411yescharacteristic
355C2#2029018411yescharacteristic
356C2#2029024811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#7707
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289536C2semidirect
4. #20289535C2semidirect
201. C53#7707semidirect

Maximal subgroups

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

subgroupindex
#202906052
#202904412
#202895362
#202895352
#202830632
#770753