Finite Group Databasebeta

← #27294266

Subgroups — #27294266

Order 13568.

552 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277951C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic, complemented
3#20272714C212yesmaximal, complemented
4#20272711C212yesmaximal
5C53×#1740C211yesmaximal, characteristic, complemented
6C2×C4×D212C411yescharacteristic, complemented
7C2×C4×C22⋊D53—21no—
8#16810676—21no—
9#16810553C411yescharacteristic, complemented
10#16810548C2211yescharacteristic
11#16810545—21no—
12#16810540C411yescharacteristic, complemented
13#16810539—21no—
14#16810532C2211yescharacteristic
15C4×C4⋊Dic53C411yescharacteristic, complemented
16C4×#11719921—21no—
17#16810349—21no—
18#16810347C2212yes—
19C2×#117×C53—21no—
20C2×#117×C53C2211yescharacteristic
21C53×#342—21no—
22C53×#342C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×C4×D53—21no—
26C4×D212—41no—
27C4×D212—41no—
28C4×C22⋊D53—42no—
29C4×C22⋊D53—42no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×C4×Dic53—21no—
39C4⋊#88322—41no—
40C22⋊#88322—22no—
41C4×#88322—41no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47#117×C53—24no—
48#117×C53—41no—
49#117×C53—41no—
50#117×C53—42no—
51C2×C4⋊C4×C53—41no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53C2311yescharacteristic
57C2×C42×C53—41no—
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C24×C53C2×D411yescharacteristic
61C23×D53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53D812yes—
71C22×C4×C53SD1612yes—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C2×C22⋊D53—42no—
75C2×C22⋊D53—42no—
76C2×C22⋊D53—41no—
77C2×C22⋊D53—41no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C2×D212—41no—
82C2×D212—42no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—42no—
86C2×C4×D53—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—44no—
90C22⋊C4×C53—24no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—42no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—44no—
99C22⋊Dic53—41no—
100C22⋊Dic53—41no—
101#11719921—42no—
102#11719921—42no—
103#11719921—42no—
104#11719921—42no—
105C4⋊Dic53—42no—
106C4⋊Dic53—41no—
107#11719919—42no—
108C4×Dic53—42no—
109C2×#88322—42no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53C2×D811yescharacteristic
116C23×C53C2×SD1611yescharacteristic
117C22×D53—22no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—44no—
139C2×C4×C53—82no—
140C2×C4×C53—82no—
141C2×C4×C53#9911yescharacteristic
142C2×C4×C53#9911yescharacteristic
143C2×C4×C53#10212yes—
144C2×C4×C53#10212yes—
145C2×C4×C53#10412yes—
146C22⋊D53—44no—
147C22⋊D53—44no—
148C2×Dic53—22no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157D212—42no—
158D212—42no—
159C4×D53—82no—
160C4×D53—82no—
161#88322—82no—
162#11691—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—81no—
172C22×C53C2×#9911yescharacteristic
173C22×C53C2×#10211yescharacteristic
174C22×C53C2×#10411yescharacteristic
175C2×D53—42no—
176C2×D53—42no—
177C2×D53—42no—
178C2×D53—81no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—22no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—84no—
188C4×C53#32712yes—
189C4×C53#32712yes—
190Dic53—42no—
191Dic53—81no—
192Dic53—81no—
193#2392—531no—
194#1740D5311yescharacteristic, complemented
195#1458—531no—
196#1344—532no—
197#1343—532no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—41no—
201C2×C53—41no—
202C2×C53C2×#32711yescharacteristic
203C2×C53#212011yescharacteristic
204C2×C53#212611yescharacteristic
205D53—82no—
206C2×C4×D4—531no—
207C2×C4×D4—1061no—
208C2×#117—21no—
209C2×#117C2×D5311yescharacteristic
210#431—1061no—
211#398—531no—
212#397—531no—
213#396—1061no—
214#394—531no—
215#394—1061no—
216#392—531no—
217C4×C4⋊C4—531no—
218C4×C22⋊C4—1061no—
219#342—21no—
220#342C2×D5312yes—
221#324—1061no—
222#323—532no—
223C53#1169111yescharacteristic, complemented, Sylow
224C22×D4—531no—
225C22×D4—1061no—
226C23×C4—1061no—
227C23×C4C22⋊D5311yescharacteristic
228C4×D4—2121no—
229C4×D4—2121no—
230C4×D4—2122no—
231C4×D4—2122no—
232#117—24no—
233#117—41no—
234#117—41no—
235#117—42no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—1061no—
239C2×C4⋊C4—1061no—
240C2×C4⋊C4C4×D5311yescharacteristic
241C2×C4⋊C4D21211yescharacteristic
242C2×C22⋊C4—22no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C22⋊C4—1061no—
247C2×C22⋊C4—1061no—
248C2×C22⋊C4C22⋊D5311yescharacteristic
249C2×C22⋊C4C22×D5311yescharacteristic
250C2×C42—41no—
251C2×C42—1061no—
252C2×C42C4×D5311yescharacteristic
253C2×C42D21211yescharacteristic
254C4⋊C8—2121no—
255C22⋊C8—1062no—
256C4×C8—2121no—
257#95—531no—
258#95—531no—
259#95—1061no—
260#95—1061no—
261C24—1061no—
262C24C2×C22⋊D5311yescharacteristic
263C2×D4—2121no—
264C2×D4—2122no—
265C2×D4—2122no—
266C2×D4—2122no—
267C2×D4—2121no—
268C2×D4—2121no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2122no—
279C22×C4—2122no—
280C22×C4#1171992111yescharacteristic
281C22×C4#1171992111yescharacteristic
282C22×C4#1171992111yescharacteristic
283C22×C4#1171992111yescharacteristic
284C22×C4#1171992212yes—
285C22×C4#1171992412yes—
286C22×C4C2×C4×D5311yescharacteristic
287C22×C4C2×D21211yescharacteristic
288C2×C8—2122no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—44no—
292C4⋊C4—2121no—
293C4⋊C4—2122no—
294C4⋊C4—2122no—
295C22⋊C4—24no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—2122no—
300C22⋊C4—2122no—
301C22⋊C4—2122no—
302C22⋊C4—2121no—
303C22⋊C4—2121no—
304C22⋊C4—2122no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—44no—
310C42—2122no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—1062no—
316C23—2121no—
317C23—2121no—
318C23—2121no—
319C23—2121no—
320C23—2121no—
321C23—2121no—
322C23C2×#1171992111yescharacteristic
323C23C2×#1171992211yescharacteristic
324C23C2×#1171992411yescharacteristic
325D4—2122no—
326D4—2122no—
327D4—2124no—
328D4—2124no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—1062no—
347C2×C4—4242no—
348C2×C4—4242no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4#1313313111yescharacteristic
358C2×C4#1313313111yescharacteristic
359C2×C4#1313313712yes—
360C2×C4#1313313712yes—
361C2×C4#1313314512yes—
362C8—4242no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—4241no—
373C22—4241no—
374C22—4241no—
375C22—4241no—
376C22—2122no—
377C22—2122no—
378C22—2122no—
379C22C2×#1313313111yescharacteristic
380C22C2×#1313313711yescharacteristic
381C22C2×#1313314511yescharacteristic
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—84no—
388C4—4241no—
389C4—4241no—
390C4—2122no—
391C4#1681035612yes—
392C4#1681035612yes—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—4242no—
398C2C2×#1681035611yescharacteristic
399C2#2027635411yescharacteristic
400C2#2027637811yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1740C2
socleC22×C53C2×#99
prime radical#16810532C22
cosocleC2×C22⋊C4×C53C23
hypercenter#1740D53
hypocenterC53#11691
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. #20273029C2semidirect
3. #20272714C2semidirect
5. C53×#1740C2semidirect
6. C2×C4×D212C4semidirect
9. #16810553C4semidirect
12. #16810540C4semidirect
15. C4×C4⋊Dic53C4semidirect
194. #1740D53semidirect
223. C53#11691semidirect

Maximal subgroups

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

subgroupindex
#202779512
#202730292
#202727142
#202727112
C53×#17402
#1169153