Finite Group Databasebeta

← #27294581

Subgroups — #27294581

Order 13568.

436 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 83 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#20280413C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272738C212yesmaximal
4#20272714C212yesmaximal
5C53×#1740C211yesmaximal, characteristic
6C2×#13133209C411yescharacteristic, complemented
7#16810680—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810553C411yescharacteristic, complemented
11#16810548C2211yescharacteristic
12#16810518—21no—
13#16810517—21no—
14#16810507—21no—
15#16810496—21no—
16#16810469C411yescharacteristic, complemented
17#16810349—21no—
18#16810347C2212yes—
19C2×#117×C53—21no—
20C2×#117×C53C2211yescharacteristic
21C53×#342—21no—
22C53×#342C2212yes—
23#13133209—41no—
24#13133209—41no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—21no—
34C4⋊#88322—41no—
35C22⋊#88322—22no—
36#13133128—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45#117×C53—24no—
46#117×C53—41no—
47#117×C53—41no—
48#117×C53—42no—
49C2×C4⋊C4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53—22no—
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53C2311yescharacteristic
55C2×C42×C53—41no—
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—44no—
78C22⋊C4×C53—24no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—44no—
87C22⋊Dic53—42no—
88C22⋊Dic53—41no—
89C22⋊Dic53—41no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92#11719919—42no—
93#11719919—42no—
94#11719919—41no—
95#11719919—41no—
96#11719919—42no—
97C4×Dic53—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53AGL(1,ℤ/8)11yescharacteristic
105C23×C53#13711yescharacteristic
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—44no—
123C2×C4×C53—82no—
124C2×C4×C53—82no—
125C2×C4×C53#9911yescharacteristic
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#10412yes—
128C2×Dic53—22no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140#88322—82no—
141#11760—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—81no—
151C22×C53C2×#9911yescharacteristic
152C22×C53#41711yescharacteristic
153C22×C53C2×#10411yescharacteristic
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—84no—
162Dic53—42no—
163Dic53—81no—
164Dic53—81no—
165Dic53—81no—
166#2611—531no—
167#1740D5311yescharacteristic
168#1464—531no—
169#1352—532no—
170#1344—532no—
171C2×C53—21no—
172C2×C53—21no—
173C2×C53—41no—
174C2×C53—41no—
175C2×C53#150411yescharacteristic
176C2×C53#212311yescharacteristic
177C2×C53#212611yescharacteristic
178C2×#125—531no—
179C2×#117—21no—
180C2×#117C2×D5311yescharacteristic
181#432—1061no—
182#400—531no—
183#399—531no—
184#398—531no—
185#397—531no—
186#389—1061no—
187#388—1061no—
188#387—1061no—
189#385—1061no—
190#376—531no—
191#342—21no—
192#342C2×D5312yes—
193#324—1061no—
194#323—532no—
195C53#1176011yescharacteristic, complemented, Sylow
196C23×C4C22⋊D5311yescharacteristic
197#125—2121no—
198#125—2121no—
199#117—24no—
200#117—41no—
201#117—41no—
202#117—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4C4×D5311yescharacteristic
211C2×C4⋊C4D21211yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1061no—
215C2×C22⋊C4C22⋊D5311yescharacteristic
216C2×C22⋊C4C22×D5311yescharacteristic
217C2×C42—41no—
218C2×C42—1061no—
219C2×C42C4×D5311yescharacteristic
220C2×C42D21211yescharacteristic
221C4⋊C8—2121no—
222C22⋊C8—1062no—
223#97—2121no—
224#95—531no—
225#95—531no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231C24C2×C22⋊D5311yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4#1171992111yescharacteristic
246C22×C4C2×C4×D5311yescharacteristic
247C22×C4C2×D21211yescharacteristic
248C2×C8—2122no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—44no—
252C4⋊C4—2121no—
253C4⋊C4—2121no—
254C4⋊C4—2122no—
255C4⋊C4—2122no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2122no—
259C22⋊C4—24no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—2122no—
264C22⋊C4—2121no—
265C22⋊C4—2121no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—44no—
271C42—2122no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×#1171992111yescharacteristic
277C23#1313324311yescharacteristic
278C23#1313324811yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—1062no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4#1313313111yescharacteristic
311C2×C4#1313313111yescharacteristic
312C2×C4#1313314512yes—
313C8—4242no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#1313313111yescharacteristic
324C22#1681062211yescharacteristic
325C22C2×#1313314511yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—84no—
334C4—4241no—
335C4—4241no—
336C4—4241no—
337C4—2122no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#2027321711yescharacteristic
343C2#2027636811yescharacteristic
344C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1740C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1740D53
hypocenterC53#11760
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
6. C2×#13133209C4semidirect
8. #16810562C4semidirect
10. #16810553C4semidirect
16. #16810469C4semidirect
195. C53#11760semidirect

Maximal subgroups

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

subgroupindex
#202804132
#202730492
#202727382
#202727142
C53×#17402
#1176053