Finite Group Databasebeta

← #27295220

Subgroups — #27295220

Order 13568.

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

normal subgroups only

This group has 452 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#20274703C211yesmaximal, characteristic
2#20273036C211yesmaximal, characteristic
3#20272739C214yesmaximal, complemented
4C53×#2487C211yesmaximal, characteristic
5#16810676—21no—
6C2×#13133132—21no—
7#16810483—21no—
8#16810390C2212yes—
9#16810347—21no—
10#16810347C2212yes—
11C2×C4×D4×C53C2211yescharacteristic
12C22×C4⋊C4×C53C411yescharacteristic, complemented
13C53×#400C412yescomplemented
14C53×#399C2212yes—
15C53×#389—21no—
16C53×#386—21no—
17C53×#385—21no—
18C53×#380—21no—
19C53×#380C411yescharacteristic, complemented
20C2×C4⋊Dic53—41no—
21C2×C22⋊Dic53—22no—
22C4⋊#88322—41no—
23#13133132—42no—
24C22⋊#88322—22no—
25C4×#88322—41no—
26#13133124—41no—
27C22×D4×C53D411yescharacteristic
28C23×C4×C53—21no—
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C4×D4×C53—41no—
32C4×D4×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—41no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C2×C42×C53D411yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C24×C53C2×D411yescharacteristic
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53SD1612yes—
53C2×D4×C53Q1612yes—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—82no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—42no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—82no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—82no—
80C42×C53—21no—
81C42×C53—21no—
82C22⋊Dic53—42no—
83C22⋊Dic53—42no—
84C4⋊Dic53—82no—
85C2×#88322—42no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53#10312yes—
95C23×C53C2×C22⋊C411yescharacteristic
96C23×C53C2×SD1611yescharacteristic
97C23×C53C2×Q1611yescharacteristic
98D4×C53—42no—
99D4×C53—42no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—81no—
119C2×C4×C53—84no—
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#10312yes—
123C2×Dic53—82no—
124C2×Dic53—82no—
125C2×Dic53—82no—
126#88322—82no—
127#11899—531nomaximal, Sylow
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—22no—
132C22×C53—21no—
133C22×C53—42no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—81no—
142C22×C53C2×#9911yescharacteristic
143C22×C53C2×#10311yescharacteristic
144C22×C53#42111yescharacteristic
145C4×C53—21no—
146C4×C53—22no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—82no—
151C4×C53—82no—
152C4×C53#35212yes—
153Dic53—162no—
154#2487D5311yescharacteristic
155#1799—531no—
156#1460—531no—
157#1352—534no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—41no—
161C2×C53—42no—
162C2×C53—41no—
163C2×C53#152011yescharacteristic
164C2×C53C2×#35211yescharacteristic
165C2×C53#212811yescharacteristic
166C2×C4×D4C2×D5311yescharacteristic
167C22×C4⋊C4Dic5311yescharacteristic, complemented
168#431—1061no—
169C2×#99—1061no—
170#400Dic5312yescomplemented
171#399C2×D5312yes—
172#389—21no—
173#386—21no—
174#385—21no—
175#380—21no—
176#380—1061no—
177#380Dic5311yescharacteristic, complemented
178#342—532no—
179#323—532no—
180#323—1061no—
181C53#1189911yescharacteristic, complemented, Sylow
182C22×D4C22⋊D5311yescharacteristic
183C23×C4—21no—
184C23×C4C2×Dic5311yescharacteristic
185C23×C4C22⋊D5311yescharacteristic
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—2121no—
194C2×C4⋊C4C2×Dic5312yes—
195C2×C4⋊C4C2×Dic5311yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1062no—
199C2×C22⋊C4C22⋊D5311yescharacteristic
200C2×C22⋊C4C22×D5311yescharacteristic
201C2×C42C22⋊D5311yescharacteristic
202C4⋊C8—2121no—
203#99—2122no—
204C22⋊C8—1062no—
205C4×C8—2121no—
206#95—22no—
207#95—22no—
208#95—2121no—
209#95C2×Dic5312yes—
210C24C22⋊Dic5311yescharacteristic
211C24C2×C22⋊D5311yescharacteristic
212C2×D4—41no—
213C2×D4—41no—
214C2×D4#1171992412yes—
215C2×D4#1171992512yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—2122no—
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4C22⋊Dic5311yescharacteristic
230C22×C4C22×Dic5311yescharacteristic
231C22×C4C2×C22⋊D5311yescharacteristic
232C2×C8—2122no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—82no—
239C4⋊C4—4242no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C42—21no—
247C42—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#1313314112yes—
257C23C2×C22⋊Dic5311yescharacteristic
258C23C2×#1171992411yescharacteristic
259C23C2×#1171992511yescharacteristic
260D4—42no—
261D4—42no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—84no—
282C2×C4—4242no—
283C2×C4—4242no—
284C2×C4—4242no—
285C2×C4#1313313211yescharacteristic
286C2×C4#1313313211yescharacteristic
287C2×C4#1313314112yes—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1313313211yescharacteristic
304C22C2×#1313314111yescharacteristic
305C22#1681064011yescharacteristic
306C4—21no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4—8482no—
314C4#1681041312yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2#2027328711yescharacteristic
321C2C2×#1681041311yescharacteristic
322C2#2027638711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2487C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2487D53
hypocenterC53#11899
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #20272739C2semidirect
12. C22×C4⋊C4×C53C4semidirect
13. C53×#400C4semidirect
19. C53×#380C4semidirect
167. C22×C4⋊C4Dic53semidirect
170. #400Dic53semidirect
177. #380Dic53semidirect
181. C53#11899semidirect

Maximal subgroups

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

subgroupindex
#202747032
#202730362
#202727392
C53×#24872
#1189953