Finite Group Databasebeta

← #27294841

Subgroups — #27294841

Order 13568.

386 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280528C211yesmaximal, characteristic
2#20274795C211yesmaximal, characteristic
3#20272746C212yesmaximal, complemented
4#20272743C212yesmaximal, complemented
5C53×#1463C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7#16810587—21no—
8#16810562C411yescharacteristic
9#16810562C2211yescharacteristic
10#16810556—21no—
11#16810556—21no—
12#16810543C411yescharacteristic
13#16810543C2211yescharacteristic
14#16810496—21no—
15#16810486—21no—
16#16810483C411yescharacteristic
17#16810392—21no—
18#16810392C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—22no—
25#13133196—41no—
26#13133196—41no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C22⋊#88322—41no—
35C22⋊#88322—41no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×C53—41no—
48C2×C4⋊C4×C53—41no—
49C2×C4⋊C4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53C2311yescharacteristic
54C22⋊C8×C53—22no—
55C22⋊C8×C53—41no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—41no—
65C22×C4×C53—81no—
66C22×C4×C53—81no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×Dic106—41no—
78C2×Dic106—81no—
79M16×C53—82no—
80C2×C8×C53—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C22⋊Dic53—42no—
86C22⋊Dic53—41no—
87C22⋊Dic53—41no—
88C4⋊Dic53—41no—
89C4⋊Dic53—81no—
90#11719919—42no—
91#11719919—81no—
92#11719919—81no—
93#11719917—42no—
94#11719917—42no—
95C2×#88322—42no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53#9911yescharacteristic
101C23×C53#9911yescharacteristic
102C23×C53C2×C22⋊C411yescharacteristic
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53—81no—
122C2×C4×C53#13712yes—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—42no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134Dic106—82no—
135C8×C53—82no—
136#88322—82no—
137#11817—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53C2×#9911yescharacteristic
149C22×C53#41911yescharacteristic
150C22×C53#42111yescharacteristic
151C4×C53—22no—
152C4×C53—41no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—41no—
157Dic53—81no—
158Dic53—81no—
159Dic53—81no—
160Dic53—81no—
161#2627—531no—
162#1821—531no—
163#1463D5311yescharacteristic
164#1354—532no—
165#1353—532no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—41no—
169C2×C53—41no—
170C2×C53#151111yescharacteristic
171C2×C53#212111yescharacteristic
172C2×C53#213311yescharacteristic
173C2×C22⋊Q8—531no—
174C22×C4⋊C4C2×D5311yescharacteristic
175#407—21no—
176#407—1061no—
177#400—531no—
178#400—531no—
179#398—1061no—
180#398—1061no—
181#395—531no—
182#395—531no—
183#385—1061no—
184#381—1061no—
185#380—531no—
186#343—532no—
187#343—1061no—
188#324—21no—
189#324C2×D5312yes—
190C53#1181711yescharacteristic, complemented, Sylow
191C22×Q8—531no—
192C23×C4C4×D5311yescharacteristic
193C23×C4D21211yescharacteristic
194C23×C4C22⋊D5311yescharacteristic
195C2×M16—41no—
196C2×M16—1062no—
197C22⋊Q8—2121no—
198C22⋊Q8—2121no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4C4×D5311yescharacteristic
207C2×C4⋊C4D21211yescharacteristic
208C2×C4⋊C4C22⋊D5311yescharacteristic
209C2×C4⋊C4C22×D5311yescharacteristic
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1061no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—2121no—
215C22⋊C8—2121no—
216#95—531no—
217#95—531no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224C24#1171992111yescharacteristic
225C2×Q8—4241no—
226C2×Q8—2121no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4C2×C4×D5311yescharacteristic
245C22×C4C2×D21211yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—82no—
248M16—2122no—
249M16—2122no—
250C2×C8—42no—
251C2×C8—2122no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—4241no—
257C4⋊C4—4241no—
258C4⋊C4—4241no—
259C4⋊C4—2121no—
260C4⋊C4—2122no—
261C22⋊C4—2122no—
262C22⋊C4—2121no—
263C22⋊C4—2121no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1313313111yescharacteristic
269C23#1313313111yescharacteristic
270C23C2×#1171992111yescharacteristic
271Q8—4242no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2122no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4#1313325012yes—
303C8—82no—
304C8—4242no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1313313111yescharacteristic
316C22#1681063311yescharacteristic
317C22#1681063911yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—4241no—
325C4—4241no—
326C4—4241no—
327C4—4241no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#2027324911yescharacteristic
333C2#2027635911yescharacteristic
334C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1463C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1463D53
hypocenterC53#11817
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
3. #20272746C2semidirect
4. #20272743C2semidirect
190. C53#11817semidirect

Maximal subgroups

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

subgroupindex
#202805282
#202747952
#202727462
#202727432
C53×#14632
#1181753