Finite Group Databasebeta

← #28299819

Subgroups — #28299819

Order 13568.

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

normal subgroups only

This group has 488 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#20290442C211yesmaximal, characteristic
2C4×#13133274C211yesmaximal, characteristic
3#20289536C212yesmaximal, complemented
4#20289534C212yesmaximal, complemented
5#20283053C211yesmaximal, characteristic
6C2×#13133274—21no—
7C2×#13133274—21no—
8C2×#13133272—21no—
9C2×C4×#88324—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133197C411yescharacteristic
13C2×C42×D53C2211yescharacteristic
14#16810529C411yescharacteristic
15#16810528C2211yescharacteristic
16#16810519C2211yescharacteristic
17#16810510—21no—
18#16810509—21no—
19#16810494—21no—
20#16810485—21no—
21#16810480C411yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133274—24no—
26#13133274—24no—
27#13133272—42no—
28C4×#88324—42no—
29C22×Dic106C2×C411yescharacteristic
30C22×C4×D53C2×C411yescharacteristic
31C22×C4×D53D411yescharacteristic
32C22×C4×D53D411yescharacteristic
33#13133197—42no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53C2311yescharacteristic
42C42×D53—42no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53#95×C53—21no—
54C23×D53C22⋊C411yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×Dic106—22no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C2×#88324—24no—
78C2×#88324—24no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—42no—
86C42×C53—41no—
87#11719921—42no—
88#11719921—42no—
89C4⋊Dic53—22no—
90C4⋊Dic53—42no—
91#11719919—42no—
92C4×Dic53—22no—
93C4×Dic53—22no—
94C4×Dic53—41no—
95C23×C53C2×C22⋊C411yescharacteristic
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—21no—
99C22×D53—21no—
100C22×D53—21no—
101C22×D53—21no—
102C22×D53#10412yes—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53#10412yes—
117C2×C4×C53#13112yes—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53#9912yes—
131C2×Dic53#13512yes—
132C2×Dic53#13712yes—
133C4×D53—44no—
134C4×D53—44no—
135C4×D53—44no—
136Dic106—82no—
137#88324—44no—
138#88324—44no—
139#7675—531nomaximal, Sylow
140C22×C53#40811yescharacteristic
141C22×C53C2×#9911yescharacteristic
142C22×C53#41311yescharacteristic
143C22×C53#41611yescharacteristic
144C22×C53#41911yescharacteristic
145C22×C53C2×#10411yescharacteristic
146C22×C53C2×#10411yescharacteristic
147C2×D53—22no—
148C2×D53—22no—
149C2×D53—22no—
150C2×D53—41no—
151C2×D53—41no—
152C2×D53—41no—
153C2×D53—41no—
154C4×C53—22no—
155C4×C53—22no—
156C4×C53—41no—
157C4×C53—81no—
158C4×C53—81no—
159Dic53—22no—
160Dic53—22no—
161Dic53—22no—
162Dic53—81no—
163Dic53—81no—
164#2855—531no—
165#1749—531no—
166C4×C22⋊C8—531no—
167#1272—532no—
168#1271—532no—
169C2×C53#146811yescharacteristic
170C2×C53#148511yescharacteristic
171C2×C53#149411yescharacteristic
172C2×C53#150111yescharacteristic
173C2×C53#150811yescharacteristic
174C2×C53#152411yescharacteristic
175C2×C53#155411yescharacteristic
176D53—42no—
177C2×C22⋊Q8—531no—
178C22×C42—531no—
179C2×C22⋊C8—1061no—
180C2×C22⋊C8—1061no—
181C2×#97—1061no—
182C2×C4×C8—1061no—
183#391—531no—
184#391—531no—
185#389—531no—
186#389#615711yescharacteristic
187#387—1061no—
188#387—1061no—
189#385—1061no—
190#381—1061no—
191#380—531no—
192#336—532no—
193#336—532no—
194C53#767511yescharacteristic, complemented, Sylow
195C22×Q8—531no—
196C23×C4—531no—
197C23×C4—531no—
198C23×C4—531no—
199C22×C8—1062no—
200C22×C8—1062no—
201C22⋊Q8—2122no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4C2×#615711yescharacteristic
207C2×C4⋊C4C2×#615711yescharacteristic
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1061no—
210C2×C42—531no—
211C2×C42—531no—
212C2×C42—531no—
213C2×C42—2122no—
214C2×C42C2×#615711yescharacteristic
215C22⋊C8—1064no—
216C22⋊C8—1064no—
217#97—2122no—
218C4×C8—2122no—
219#95—21no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226C24—531no—
227C2×Q8—1062no—
228C2×Q8—2121no—
229C2×Q8—2121no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—1062no—
236C22×C4—1062no—
237C22×C4—1062no—
238C22×C4—1062no—
239C22×C4—1062no—
240C22×C4—1062no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—2121no—
244C22×C4—2121no—
245C22×C4—2121no—
246C22×C4C22⋊#615711yescharacteristic
247C22×C4C22⋊#615711yescharacteristic
248C22×C4C22×#615711yescharacteristic
249C2×C8—1064no—
250C2×C8—1064no—
251C2×C8—2122no—
252C2×C8—2122no—
253C2×C8—2122no—
254C2×C8—2122no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—1062no—
259C4⋊C4—2122no—
260C4⋊C4—2122no—
261C22⋊C4—2122no—
262C22⋊C4—2122no—
263C42—41no—
264C42—1062no—
265C42—1062no—
266C42—2121no—
267C23—532no—
268C23—1061no—
269C23—1061no—
270C23—1061no—
271C23—1061no—
272C23—1061no—
273C23—1061no—
274C23C2×C22⋊#615711yescharacteristic
275Q8—4242no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—532no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—2124no—
299C2×C4—2124no—
300C2×C4—2124no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4#1313328112yes—
308C2×C4#1313330512yes—
309C8—2124no—
310C8—2124no—
311C22—1062no—
312C22—1062no—
313C22—1062no—
314C22—2121no—
315C22—2121no—
316C22—2121no—
317C22—2121no—
318C22#1681159811yescharacteristic
319C22D53⋊#9911yescharacteristic
320C22#1681160911yescharacteristic
321C22#1681161311yescharacteristic
322C22#1681161611yescharacteristic
323C22C2×#1313328111yescharacteristic
324C22D53⋊#10411yescharacteristic
325C4—22no—
326C4—22no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—1062no—
331C4—1062no—
332C4—1062no—
333C4—4241no—
334C4—4241no—
335C2—2122no—
336C2#2028998311yescharacteristic
337C2#2029003911yescharacteristic
338C2#2029006911yescharacteristic
339C2#2029009111yescharacteristic
340C2#2029011311yescharacteristic
341C2#2029016511yescharacteristic
342C2#2029024011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7675
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #20289536C2semidirect
4. #20289534C2semidirect
194. C53#7675semidirect

Maximal subgroups

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

subgroupindex
#202904422
C4×#131332742
#202895362
#202895342
#202830532
#767553