Finite Group Databasebeta

← #27294385

Subgroups — #27294385

Order 13568.

448 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 115 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#20274448C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272739C214yesmaximal
4C53×#2628C211yesmaximal, characteristic
5C2×#13133176—21no—
6#16810676—21no—
7#16810390—21no—
8#16810390C2212yes—
9#16810349—21no—
10#16810347C2212yes—
11C2×C4⋊Q8×C53C411yescharacteristic, complemented
12C2×#117×C53C2211yescharacteristic
13C53×#400C412yescomplemented
14C53×#399C2212yes—
15C53×#395—21no—
16C53×#393—21no—
17C53×#389—21no—
18C53×#382C411yescharacteristic, complemented
19C53×#380—21no—
20#13133176—24no—
21#13133176—42no—
22C2×C4⋊Dic53—41no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—41no—
25C4⋊#88322—41no—
26C22⋊#88322—22no—
27C4×#88322—41no—
28C22×Q8×C53—21no—
29C23×C4×C53D411yescharacteristic
30C4⋊Q8×C53—41no—
31C4⋊Q8×C53—41no—
32#117×C53—41no—
33#117×C53—41no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53C2×C412yes—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53C2311yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C2×C42×C53D411yescharacteristic
45#95×C53—22no—
46#95×C53—21no—
47#95×C53—22no—
48#95×C53C2×C412yes—
49C24×C53C2×D411yescharacteristic
50C2×Q8×C53—42no—
51C2×Q8×C53—82no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53SD1612yes—
62C22×C4×C53Q1612yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—82no—
73C22⋊C4×C53—41no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—82no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80C22⋊Dic53—24no—
81C22⋊Dic53—42no—
82C4⋊Dic53—44no—
83C4×Dic53—44no—
84C2×#88322—42no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53C2×SD1611yescharacteristic
91C23×C53C2×Q1611yescharacteristic
92Q8×C53—84no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53#9911yescharacteristic
114C2×C4×C53#9911yescharacteristic
115C2×C4×C53#10312yes—
116C2×C4×C53#10312yes—
117C2×C4×C53#10412yes—
118C2×Dic53—44no—
119C2×Dic53—82no—
120C2×Dic53—82no—
121#88322—82no—
122#11718—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—81no—
132C22×C53C2×#9911yescharacteristic
133C22×C53C2×#10311yescharacteristic
134C22×C53C2×#10411yescharacteristic
135C4×C53—22no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—82no—
141C4×C53—82no—
142C4×C53#32812yes—
143C4×C53#32812yes—
144Dic53—84no—
145#2628D5311yescharacteristic
146#1740—531no—
147#1458—531no—
148#1352—534no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—41no—
152C2×C53—41no—
153C2×C53C2×#32811yescharacteristic
154C2×C53#212311yescharacteristic
155C2×C53#212811yescharacteristic
156C2×C4⋊Q8Dic5311yescharacteristic, complemented
157C2×#117—1061no—
158C2×#117C2×D5311yescharacteristic
159#431—1061no—
160#400Dic5312yescomplemented
161#399C2×D5312yes—
162#395—21no—
163#393—21no—
164#389—21no—
165#382Dic5311yescharacteristic, complemented
166#380—21no—
167#342—532no—
168#342—1061no—
169#324—1061no—
170#323—532no—
171C53#1171811yescharacteristic, complemented, Sylow
172C22×Q8—21no—
173C23×C4C22⋊D5311yescharacteristic
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—41no—
177#117—41no—
178#117—1064no—
179#117—2122no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—2121no—
184C2×C4⋊C4C2×Dic5312yes—
185C2×C4⋊C4C2×Dic5311yescharacteristic
186C2×C4⋊C4C22⋊D5311yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1062no—
189C2×C22⋊C4C22⋊D5311yescharacteristic
190C2×C22⋊C4C22×D5311yescharacteristic
191C2×C42—2121no—
192C2×C42C2×Dic5311yescharacteristic
193C2×C42C22⋊D5311yescharacteristic
194C4⋊C8—2121no—
195C22⋊C8—1062no—
196C4×C8—2121no—
197#95—22no—
198#95—21no—
199#95—22no—
200#95C2×Dic5312yes—
201C24C2×C22⋊D5311yescharacteristic
202C2×Q8—42no—
203C2×Q8—82no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—2122no—
210C22×C4#1171992412yes—
211C22×C4#1171992512yes—
212C22×C4C22⋊Dic5311yescharacteristic
213C22×C4C22⋊Dic5311yescharacteristic
214C22×C4C22⋊Dic5311yescharacteristic
215C22×C4C22⋊Dic5311yescharacteristic
216C22×C4C22×Dic5311yescharacteristic
217C22×C4C2×C22⋊D5311yescharacteristic
218C2×C8—2122no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C4⋊C4—2124no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C22⋊C4—1064no—
231C22⋊C4—2122no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—2124no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×C22⋊Dic5311yescharacteristic
242C23C2×#1171992411yescharacteristic
243C23C2×#1171992511yescharacteristic
244Q8—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—4242no—
266C2×C4—4242no—
267C2×C4—2124no—
268C2×C4#1313313211yescharacteristic
269C2×C4#1313313211yescharacteristic
270C2×C4#1313314112yes—
271C2×C4#1313314112yes—
272C2×C4#1313314412yes—
273C8—4242no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#1313313211yescharacteristic
284C22C2×#1313314111yescharacteristic
285C22C2×#1313314411yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4—4244no—
294C4#1681036012yes—
295C4#1681036012yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#1681036011yescharacteristic
301C2#2027636911yescharacteristic
302C2#2027638711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2628C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2628D53
hypocenterC53#11718
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C53C4semidirect
13. C53×#400C4semidirect
18. C53×#382C4semidirect
156. C2×C4⋊Q8Dic53semidirect
160. #400Dic53semidirect
165. #382Dic53semidirect
171. C53#11718semidirect

Maximal subgroups

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

subgroupindex
#202744482
#202730292
#202727392
C53×#26282
#1171853