Finite Group Databasebeta

← #27280145

Subgroups — #27280145

Order 13568.

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

normal subgroups only

This group has 612 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#20280520C211yesmaximal, characteristic
2#20273286C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272582C212yesmaximal, complemented
5C53×#2312C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7C2×C22⋊#88322C2212yes—
8#16810559C412yescomplemented
9#16810546—22no—
10#16810535C412yescomplemented
11#16810496—21no—
12#16810483C2211yescharacteristic
13#16810479—21no—
14#16810359—24no—
15#16810347—22no—
16C22×C22⋊C4×C53C2211yescharacteristic
17C53×#388—24no—
18C53×#380C2212yes—
19C53×#379—21no—
20C4×C22⋊C4×C53—21no—
21C22×Dic106C2×C411yescharacteristic
22C22×#88322—22no—
23#13133196—22no—
24#13133196—22no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C22⋊#88322—22no—
34C22⋊#88322—22no—
35#13133124—22no—
36#13133124C2×C412yes—
37C25×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53C2×C412yes—
47C2×C22⋊C4×C53D412yes—
48C2×C42×C53—22no—
49#95×C53—22no—
50#95×C53D414yes—
51C24×C53—21no—
52C24×C53—21no—
53C24×C53—22no—
54C24×C53—22no—
55C24×C53—81no—
56C24×C53—81no—
57C24×C53C22⋊C412yes—
58C24×C53C2×D411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—24no—
62C22×C4×C53—81no—
63C22×C4×C53—81no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53#4112yes—
69C22×Dic53—21no—
70C22×Dic53—22no—
71C22×Dic53—21no—
72C2×Dic106—22no—
73C2×Dic106—81no—
74C4⋊C4×C53—82no—
75C22⋊C4×C53—24no—
76C22⋊C4×C53—24no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—82no—
80C22⋊C4×C53—82no—
81C42×C53—82no—
82C22⋊Dic53—41no—
83C22⋊Dic53—41no—
84C22⋊Dic53—42no—
85C22⋊Dic53—82no—
86C22⋊Dic53—82no—
87C4⋊Dic53—22no—
88C4⋊Dic53—81no—
89#11719919—42no—
90#11719919—82no—
91C2×#88322—42no—
92C2×#88322—82no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53—42no—
104C23×C53—81no—
105C23×C53—81no—
106C23×C53—84no—
107C23×C53—84no—
108C23×C53C2×C22⋊C411yescharacteristic
109C23×C53#12011yescharacteristic
110C23×C53#12311yescharacteristic
111C23×C53#13512yes—
112C23×C53#13712yes—
113C2×C4×C53—22no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—44no—
121C2×C4×C53—44no—
122C2×C4×C53—44no—
123C2×C4×C53—81no—
124C2×C4×C53C4×D412yes—
125C2×C4×C53#12112yes—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—42no—
133Dic106—82no—
134#88322—82no—
135#9201—531nomaximal, Sylow
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53—44no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—82no—
149C22×C53—84no—
150C22×C53—84no—
151C22×C53#38611yescharacteristic
152C22×C53#41611yescharacteristic
153C22×C53#41911yescharacteristic
154C22×C53#45212yes—
155C22×C53#48512yes—
156C4×C53—22no—
157C4×C53—22no—
158C4×C53—42no—
159C4×C53—84no—
160Dic53—82no—
161Dic53—81no—
162Dic53—81no—
163#2626—531no—
164#2312D5311yescharacteristic
165#1520—532no—
166C2×#323—531no—
167#1289—532no—
168C2×C53—21no—
169C2×C53—22no—
170C2×C53—21no—
171C2×C53—22no—
172C2×C53—82no—
173C2×C53#187311yescharacteristic
174C2×C53#191111yescharacteristic
175C2×C53#191311yescharacteristic
176C2×C22⋊Q8—531no—
177C22×C22⋊C4C2×D5311yescharacteristic
178C2×C22⋊C8—532no—
179#399—532no—
180#396—1062no—
181#393—532no—
182#388—24no—
183#385—1061no—
184#380—531no—
185#380C2×D5312yes—
186#379—21no—
187#379—1061no—
188C4×C22⋊C4—21no—
189#328—1064no—
190#323—1062no—
191C53#920111yescharacteristic, complemented, Sylow
192C25C22⋊D5311yescharacteristic
193C22×Q8—531no—
194C23×C4C22⋊D5311yescharacteristic
195C23×C4C22×D5311yescharacteristic
196C22×C8—1062no—
197C22⋊Q8—1062no—
198C22⋊Q8—1062no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1062no—
203C2×C4⋊C4—1061no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—1062no—
210C2×C22⋊C4—1062no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4C4×D5312yes—
214C2×C22⋊C4D21212yes—
215C2×C42—22no—
216C22⋊C8—1062no—
217C22⋊C8—1062no—
218#95—22no—
219#95—532no—
220#95—1062no—
221#95C22⋊D5314yes—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24#1171992112yes—
229C24C2×C22⋊D5311yescharacteristic
230C2×Q8—1062no—
231C2×Q8—4241no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—1062no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4#1171992112yes—
241C22×C4C2×C4×D5311yescharacteristic
242C22×C4C2×D21211yescharacteristic
243C22×C4#1171994512yes—
244C22×C4C2×C22⋊D5312yes—
245C2×C8—4242no—
246C2×C8—2122no—
247C4⋊C4—82no—
248C4⋊C4—1062no—
249C4⋊C4—4242no—
250C4⋊C4—4241no—
251C4⋊C4—2122no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C22⋊C4—4242no—
259C22⋊C4—4242no—
260C22⋊C4—2121no—
261C22⋊C4—2121no—
262C22⋊C4—2122no—
263C42—82no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×#1171992111yescharacteristic
280C23#1313318611yescharacteristic
281C23#1313320011yescharacteristic
282C23#1313323712yes—
283C23#1313324712yes—
284Q8—4242no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—81no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4—2122no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2122no—
303C2×C4C4×C22⋊D5312yes—
304C2×C4#1313319112yes—
305C8—4242no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#1681050211yescharacteristic
322C22#1681061711yescharacteristic
323C22#1681063111yescharacteristic
324C22#1681078512yes—
325C22#1681096412yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—84no—
330C4—4242no—
331C4—4241no—
332C4—4241no—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#2027502711yescharacteristic
339C2#2027524711yescharacteristic
340C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2312C2
socleC22×C53#386
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#2312D53
hypocenterC53#9201
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #20273286C2semidirect
4. #20272582C2semidirect
8. #16810559C4semidirect
10. #16810535C4semidirect
191. C53#9201semidirect

Maximal subgroups

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

subgroupindex
#202805202
#202732862
C2×#168103472
#202725822
C53×#23122
#920153