Finite Group Databasebeta

← #27271414

Subgroups — #27271414

Order 13568.

374 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 71 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#20273045C211yesmaximal, characteristic
2#20272983C211yesmaximal, characteristic
3C53×#1463C211yesmaximal, characteristic
4#16810587—21no—
5C2×#13133124—21no—
6C2×#13133124—21no—
7C2×#13133124C411yescharacteristic
8C2×#13133124C411yescharacteristic
9#16810349—21no—
10#16810349C412yescomplemented
11C22×C4⋊C4×C53C2211yescharacteristic
12C53×#407—21no—
13C53×#324—21no—
14C53×#324C412yescomplemented
15C23×Dic53—21no—
16C23×Dic53—21no—
17C2×#11719917—41no—
18C22⋊#88322—22no—
19C22⋊#88322—41no—
20#13133124—42no—
21#13133124—42no—
22#13133124—41no—
23#13133124—41no—
24C23×C4×C53C2×C411yescharacteristic
25C23×C4×C53D411yescharacteristic
26C23×C4×C53D411yescharacteristic
27C2×M16×C53—41no—
28C2×C4⋊C4×C53—41no—
29C2×C4⋊C4×C53—41no—
30C2×C4⋊C4×C53C2×C411yescharacteristic
31C2×C4⋊C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53D411yescharacteristic
33C2×C4⋊C4×C53Q811yescharacteristic
34C22⋊C8×C53—22no—
35C22⋊C8×C53—41no—
36C24×C53C22⋊C411yescharacteristic
37C22×C4×C53—21no—
38C22×C4×C53—21no—
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—41no—
44C22×C4×C53—41no—
45C22×C4×C53—41no—
46C22×C4×C53—41no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53C4211yescharacteristic
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C4⋊C411yescharacteristic
53C22×C4×C53C4⋊C411yescharacteristic
54C22×C4×C53C4⋊C411yescharacteristic
55C22×Dic53—22no—
56C22×Dic53—22no—
57C22×Dic53—41no—
58C22×Dic53—41no—
59C22×Dic53—41no—
60C22×Dic53—41no—
61C22×Dic53—41no—
62C22×Dic53—41no—
63C22×Dic53—41no—
64C22×Dic53—41no—
65C22×Dic53—41no—
66C22×Dic53—41no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69M16×C53—81no—
70M16×C53—81no—
71C2×C8×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76#11719917—81no—
77#11719917—81no—
78C2×#88322—42no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53#9511yescharacteristic
86C23×C53#9911yescharacteristic
87C23×C53#9911yescharacteristic
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—81no—
109C2×C4×C53—81no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—81no—
117C2×Dic53—81no—
118C2×Dic53—81no—
119C2×Dic53—81no—
120C2×Dic53—81no—
121C2×Dic53—81no—
122C2×Dic53—81no—
123C2×Dic53—81no—
124C8×C53—82no—
125#88322—82no—
126#7326—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53#34211yescharacteristic
142C22×C53#34411yescharacteristic
143C22×C53#34411yescharacteristic
144C22×C53#35311yescharacteristic
145C22×C53#35411yescharacteristic
146C22×C53#35511yescharacteristic
147C22×C53#35611yescharacteristic
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—41no—
154Dic53—82no—
155Dic53—82no—
156#1463—531no—
157#1463D5311yescharacteristic
158#1445—531no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53#130611yescharacteristic
166C2×C53#139311yescharacteristic
167C2×C53#139411yescharacteristic
168C22×C4⋊C4C2×D5311yescharacteristic
169#407—21no—
170#407—1061no—
171C2×#95—531no—
172C2×#95—531no—
173C2×#95—1061no—
174C2×#95—1061no—
175#324—21no—
176#324—532no—
177#324—1061no—
178#324Dic5312yescomplemented
179C53#732611yescharacteristic, complemented, Sylow
180C23×C4—1061no—
181C23×C4—1061no—
182C23×C4C4×D5311yescharacteristic
183C23×C4D21211yescharacteristic
184C23×C4C22⋊D5311yescharacteristic
185C2×M16—41no—
186C2×M16—2121no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4Dic10611yescharacteristic
190C2×C4⋊C4C4×D5311yescharacteristic
191C2×C4⋊C4C2×Dic5311yescharacteristic
192C2×C4⋊C4C22⋊D5311yescharacteristic
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—1062no—
196C22⋊C8—2121no—
197#95—2121no—
198#95—2121no—
199#95—2122no—
200#95—2122no—
201C24#1171992111yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4—2121no—
217C22×C4—2121no—
218C22×C4—2121no—
219C22×C4—2121no—
220C22×C4—2121no—
221C22×C4—2121no—
222C22×C4—2121no—
223C22×C4—2121no—
224C22×C4—2121no—
225C22×C4—2121no—
226C22×C4—2121no—
227C22×C4—2121no—
228C22×C4C4×Dic5311yescharacteristic
229C22×C4#1171991911yescharacteristic
230C22×C4#1171991911yescharacteristic
231C22×C4C4⋊Dic5311yescharacteristic
232C22×C4#1171992111yescharacteristic
233C22×C4C22⋊Dic5311yescharacteristic
234M16—81no—
235M16—81no—
236M16—4241no—
237M16—4241no—
238C2×C8—42no—
239C2×C8—2122no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1313312411yescharacteristic
251C23#1313313111yescharacteristic
252C23#1313313111yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—4241no—
276C2×C4—4241no—
277C2×C4—4241no—
278C2×C4—4241no—
279C2×C4—4241no—
280C2×C4—4241no—
281C2×C4—4241no—
282C2×C4—4241no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4—2122no—
286C2×C4—2122no—
287C2×C4—2122no—
288C2×C4—2122no—
289C8—82no—
290C8—4242no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1681038911yescharacteristic
306C22#1681039411yescharacteristic
307C22#1681039511yescharacteristic
308C22#1681041511yescharacteristic
309C22#1681041711yescharacteristic
310C22#1681041911yescharacteristic
311C22#1681042111yescharacteristic
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—4242no—
319C4—4242no—
320C2—21no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—21no—
325C2—21no—
326C2#2027262411yescharacteristic
327C2#2027283911yescharacteristic
328C2#2027284111yescharacteristic
subgroupG/H
centerC22#16810389
commutatorC22×C4×C53C42
FrattiniC22×C4⋊C4C2×D53
FittingC53×#1463C2
socleC22×C53#342
prime radicalC24×C53C22⋊C4
cosocleC22×C4⋊C4×C53C22
hypercenter#1463D53
hypocenterC53#7326
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
10. #16810349C4semidirect
14. C53×#324C4semidirect
178. #324Dic53semidirect
179. C53#7326semidirect

Maximal subgroups

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

subgroupindex
#202730452
#202729832
C53×#14632
#732653