Finite Group Databasebeta

← #27281844

Subgroups — #27281844

Order 13568.

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

normal subgroups only

This group has 726 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#20278369C211yesmaximal, characteristic
2#20277301C211yesmaximal, characteristic, complemented
3#20273245C212yesmaximal
4#20272569C212yesmaximal, complemented
5C53×#1784C211yesmaximal, characteristic, complemented
6C2×#13133206C412yescomplemented
7C2×#13133191C2211yescharacteristic
8C2×#13133176—21no—
9#16810534—22no—
10#16810530—24no—
11#16810530C412yescomplemented
12#16810502—21no—
13#16810502—21no—
14#16810486—21no—
15#16810486C2212yes—
16C4×C22⋊Dic53C2211yescharacteristic
17#16810358—24no—
18C23×Q8×C53C2211yescharacteristic
19C2×#101×C53—22no—
20C53×#407C2212yes—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—21no—
23#13133206—44no—
24#13133191—22no—
25#13133191—22no—
26#13133176—44no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—22no—
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×#11719921—21no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53C2×C412yes—
36#13133124—24no—
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40#13133124—21no—
41#13133124D414yes—
42C22×Q8×C53—22no—
43C22×Q8×C53—22no—
44C22×Q8×C53—44no—
45C22×Q8×C53C2×C412yes—
46C22×Q8×C53D412yes—
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C23×C4×C53C2311yescharacteristic
50C2×M16×C53—22no—
51#101×C53—44no—
52C22⋊C8×C53—22no—
53C22⋊C8×C53—22no—
54C24×C53C2×D411yescharacteristic
55C23×D53—21no—
56C2×Q8×C53—24no—
57C2×Q8×C53—24no—
58C2×Q8×C53—24no—
59C2×Q8×C53—24no—
60C2×Q8×C53—42no—
61C2×Q8×C53—42no—
62C2×Q8×C53—48no—
63C2×Q8×C53—44no—
64C2×Q8×C53—44no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—24no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53C22⋊C412yes—
73C22×C4×C53C22⋊C412yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D412yes—
77C22×C4×C53#4112yes—
78C2×C22⋊D53—44no—
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C22×Dic53—21no—
82C22×Dic53—22no—
83C22×Dic53—24no—
84C2×D212—22no—
85C2×D212—42no—
86M16×C53—44no—
87C2×C8×C53—42no—
88C22⋊Dic53—42no—
89C22⋊Dic53—44no—
90#11719921—44no—
91#11719921—44no—
92#11719921—42no—
93C4⋊Dic53—22no—
94C4⋊Dic53—44no—
95C4×Dic53—44no—
96C4×Dic53—42no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—41no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53#12011yescharacteristic
102C23×C53#12311yescharacteristic
103C22×D53—42no—
104C22×D53—42no—
105C22×D53—41no—
106C22×D53—42no—
107Q8×C53—48no—
108Q8×C53—48no—
109C2×C4×C53—24no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53#9914yes—
124C2×C4×C53C4×D412yes—
125C2×C4×C53#12112yes—
126C22⋊D53—82no—
127C22⋊D53—82no—
128C2×Dic53—44no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—44no—
132C2×Dic53—42no—
133C2×Dic53—41no—
134C2×Dic53—42no—
135C2×Dic53—44no—
136D212—82no—
137C8×C53—82no—
138#9466—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—44no—
147C22×C53#38611yescharacteristic
148C22×C53C2×#9912yes—
149C22×C53#45412yes—
150C2×D53—84no—
151C2×D53—81no—
152C2×D53—81no—
153C2×D53—81no—
154C4×C53—22no—
155C4×C53—22no—
156C4×C53—24no—
157C4×C53—42no—
158Dic53—84no—
159Dic53—82no—
160Dic53—81no—
161#2436—531no—
162#2310—531no—
163#1784D5311yescharacteristic, complemented
164#1510—532no—
165#1284—532no—
166C2×C53—22no—
167C2×C53—22no—
168C2×C53—41no—
169C2×C53#188811yescharacteristic
170C2×C53#189811yescharacteristic
171C2×C53#191211yescharacteristic
172D53—161no—
173C23×Q8C2×D5311yescharacteristic
174C2×#124—532no—
175C2×#121—531no—
176C2×#117—1061no—
177C2×#101—22no—
178#407C2×D5312yes—
179#393—1062no—
180#391—532no—
181#391—1064no—
182#386—1061no—
183#386—1061no—
184#381—532no—
185#381—1061no—
186C4×C22⋊C4—531no—
187#327—1064no—
188C53#946611yescharacteristic, complemented, Sylow
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—44no—
192C22×Q8C4×D5312yes—
193C22×Q8C22⋊D5312yes—
194C22×D4—531no—
195C22×D4—1061no—
196C23×C4D21211yescharacteristic
197C23×C4D21211yescharacteristic
198C23×C4C22×D5311yescharacteristic
199C2×M16—22no—
200#124—2124no—
201#121—1062no—
202#121—1062no—
203#117—2124no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—1062no—
206C2×C22⋊C4—1062no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1062no—
210C2×C22⋊C4—1062no—
211C2×C42—532no—
212C2×C42—1062no—
213#101—44no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216#95—534no—
217#95—1064no—
218#95—1062no—
219#95—1061no—
220#95—1062no—
221#95—1062no—
222C24—1061no—
223C24C2×D21211yescharacteristic
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—48no—
233C2×D4—1062no—
234C2×D4—2122no—
235C2×D4—2124no—
236C2×D4—2121no—
237C2×D4—2121no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1061no—
246C22×C4—1062no—
247C22×C4—1064no—
248C22×C4#1171992112yes—
249C22×C4#1171992112yes—
250C22×C4C2×C4×D5311yescharacteristic
251C22×C4D4×D5312yes—
252C22×C4#1171994712yes—
253C22×C4C2×C22⋊D5311yescharacteristic
254M16—44no—
255C2×C8—42no—
256C4⋊C4—1062no—
257C4⋊C4—2124no—
258C22⋊C4—2124no—
259C22⋊C4—2124no—
260C22⋊C4—2122no—
261C22⋊C4—2124no—
262C22⋊C4—2122no—
263C42—2122no—
264C42—2124no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—2122no—
269C23—2122no—
270C23—2121no—
271C23—2122no—
272C23C2×#1171992111yescharacteristic
273C23#1313318711yescharacteristic
274C23#1313320111yescharacteristic
275Q8—48no—
276Q8—48no—
277D4—4242no—
278D4—4242no—
279D4—4242no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—2124no—
295C2×C4—2122no—
296C2×C4—2124no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4—2121no—
300C2×C4—2124no—
301C2×C4—2122no—
302C2×C4#1313313114yes—
303C2×C4#1313317712yes—
304C2×C4#1313318912yes—
305C8—82no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—4244no—
315C22—4241no—
316C22—4241no—
317C22—4241no—
318C22#1681050011yescharacteristic
319C22C2×#1313313112yes—
320C22#1681079812yes—
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—4244no—
326C4—4242no—
327C4—4241no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—8481no—
332C2#2027511611yescharacteristic
333C2#2027516611yescharacteristic
334C2#2027525011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1784C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1784D53
hypocenterC53#9466
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. #20277301C2semidirect
4. #20272569C2semidirect
5. C53×#1784C2semidirect
6. C2×#13133206C4semidirect
11. #16810530C4semidirect
163. #1784D53semidirect
188. C53#9466semidirect

Maximal subgroups

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

subgroupindex
#202783692
#202773012
#202732452
#202725692
C53×#17842
#946653