Finite Group Databasebeta

← #27365362

Subgroups — #27365362

Order 13568.

850 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 171 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#20285921C212yesmaximal, complemented
2#20285101C211yesmaximal, characteristic
3#20274049C218yesmaximal, complemented
4#20273339C211yesmaximal, characteristic
5#20273317C212yesmaximal, complemented
6C2×C53×#563C211yesmaximal, characteristic
7#16811180C2211yescharacteristic
8#16811033C2214yescomplemented
9#16810993C2214yescomplemented
10#16810982C2214yes—
11#16810977—22no—
12#16810958—22no—
13#16810953—22no—
14#16810927—24no—
15#16810680C2212yes—
16#16810632—21no—
17#16810628—21no—
18C4×#11719917C2211yescharacteristic
19#16810365C2218yescomplemented
20#16810363C2214yes—
21C53×#563—24no—
22C53×#563—22no—
23C53×#563C2214yes—
24C2×C4⋊Q8×C53—21no—
25C2×#126×C53—22no—
26C2×#125×C53C2212yes—
27C2×#124×C53C2211yescharacteristic
28C2×#123×C53—21no—
29C2×C22⋊Q8×C53—22no—
30C2×#11719917—21no—
31#13133217C2312yes—
32#13133209—21no—
33#13133196—22no—
34C4×Dic106—22no—
35#13133176—21no—
36#13133150—24no—
37#13133148—24no—
38C4⋊#88322—22no—
39#13133141—24no—
40#13133140—24no—
41#13133140—24no—
42#13133138—24no—
43C22⋊#88322—22no—
44#13133128C2314yes—
45#13133128C2312yes—
46C4×#88322—21no—
47C22×Q8×C53—21no—
48C22×Q8×C53D411yescharacteristic
49C22×D4×C53D411yescharacteristic
50C23×C4×C53—21no—
51C4⋊Q8×C53—24no—
52C4⋊Q8×C53—24no—
53#126×C53—28no—
54#126×C53—28no—
55#125×C53—22no—
56#125×C53—22no—
57#125×C53D414yes—
58#125×C53C2314yes—
59#124×C53—22no—
60#124×C53D412yes—
61#124×C53C2312yes—
62#123×C53—24no—
63#123×C53—24no—
64C22⋊Q8×C53—28no—
65C22⋊Q8×C53—28no—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53D412yes—
70C2×C4⋊C4×C53D412yes—
71C2×C22⋊C4×C53—22no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—22no—
74C2×C42×C53—21no—
75C2×C42×C53C2311yescharacteristic
76C24×C53—21no—
77C2×Q8×C53—24no—
78C2×Q8×C53—21no—
79C2×Q8×C53—42no—
80C2×Q8×C53—82no—
81C2×Q8×C53C2×D412yes—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53C2×D412yes—
85C22×C4×C53—22no—
86C22×C4×C53—21no—
87C22×C4×C53—22no—
88C22×C4×C53—24no—
89C22×C4×C53—82no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D412yes—
92C2×Dic106—22no—
93C4⋊C4×C53—28no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—24no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53C2×D414yes—
101C4⋊C4×C53C2×D414yes—
102C22⋊C4×C53—28no—
103C22⋊C4×C53—24no—
104C22⋊C4×C53—24no—
105C22⋊C4×C53—24no—
106C42×C53—24no—
107C42×C53C2×D412yes—
108C42×C53C2×D411yescharacteristic
109C42×C53C2411yescharacteristic
110C22⋊Dic53—41no—
111C4⋊Dic53—22no—
112C4⋊Dic53—22no—
113#11719919—42no—
114C4×Dic53—41no—
115#11719917—42no—
116C2×#88322—22no—
117C2×#88322—24no—
118C23×C53—22no—
119C23×C53—44no—
120C23×C53—42no—
121C23×C53#12011yescharacteristic
122Q8×C53—42no—
123Q8×C53—81no—
124Q8×C53—84no—
125D4×C53—42no—
126D4×C53—81no—
127C2×C4×C53—22no—
128C2×C4×C53—24no—
129C2×C4×C53—22no—
130C2×C4×C53—24no—
131C2×C4×C53—42no—
132C2×C4×C53—44no—
133C2×C4×C53—44no—
134C2×C4×C53—42no—
135C2×C4×C53—84no—
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12012yes—
138C2×C4×C53C22×D412yes—
139C2×C4×C53C22×D411yescharacteristic
140C2×Dic53—42no—
141Dic106—44no—
142#88322—44no—
143#88322—42no—
144#18999—531nomaximal, Sylow
145C22×C53—22no—
146C22×C53—42no—
147C22×C53—44no—
148C22×C53—84no—
149C22×C53#45512yes—
150C22×C53C2×#12011yescharacteristic
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—44no—
156C4×C53—42no—
157C4×C53—44no—
158Dic53—82no—
159C2×#563D5311yescharacteristic
160#3236—532no—
161#3140—531no—
162#1664—538no—
163#1533—531no—
164#1528—532no—
165C2×C53—22no—
166C2×C53—82no—
167C2×C53C2×#45511yescharacteristic
168C2×C53#302012yes—
169#563—24no—
170#563—22no—
171#563C2×D5314yes—
172#533—531no—
173C2×C4⋊Q8—21no—
174C2×#126—22no—
175C2×#125C2×D5312yes—
176C2×#124C2×D5311yescharacteristic
177C2×#123—21no—
178C2×C22⋊Q8—22no—
179#501—534no—
180#491—534no—
181#489—534no—
182#488—1062no—
183#484—1062no—
184#483—1062no—
185#479—1064no—
186#432—532no—
187#419—1061no—
188#418—1061no—
189C4×M16—531no—
190#330—538no—
191#329—534no—
192C53#1899911yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8C22⋊D5311yescharacteristic
195C22×D4C22⋊D5311yescharacteristic
196C23×C4—21no—
197C2×M16—1061no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—532no—
201#126—28no—
202#126—28no—
203#125—22no—
204#125—22no—
205#125—1061no—
206#125C22⋊D5314yes—
207#125C22×D5314yes—
208#124—22no—
209#124C22⋊D5312yes—
210#124C22×D5312yes—
211#123—24no—
212#123—24no—
213C22⋊Q8—28no—
214C22⋊Q8—28no—
215C22⋊Q8—1062no—
216C4×Q8—1062no—
217#117—1061no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C22⋊D5312yes—
222C2×C4⋊C4C22⋊D5312yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C42—21no—
227C2×C42C22×D5311yescharacteristic
228#107—1064no—
229#106—1064no—
230C4⋊C8—1062no—
231#103—1064no—
232#103—1064no—
233#103—1064no—
234#102—1064no—
235C22⋊C8—1062no—
236#97—534no—
237#97—532no—
238C4×C8—1061no—
239C24—21no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8—42no—
243C2×Q8—82no—
244C2×Q8—1062no—
245C2×Q8C2×C22⋊D5312yes—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4C2×C22⋊D5312yes—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—82no—
254C22×C4C2×C22⋊D5311yescharacteristic
255C22×C4C2×C22⋊D5312yes—
256M16—2122no—
257C2×C8—1062no—
258C2×C8—1064no—
259C4⋊C4—28no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—1062no—
267C4⋊C4—1062no—
268C4⋊C4—2122no—
269C4⋊C4C2×C22⋊D5314yes—
270C4⋊C4C2×C22⋊D5314yes—
271C22⋊C4—28no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—2121no—
276C42—24no—
277C42—2121no—
278C42C2×C22⋊D5311yescharacteristic
279C42C2×C22⋊D5312yes—
280C42C23×D5311yescharacteristic
281C23—22no—
282C23—44no—
283C23—42no—
284C23#1313318611yescharacteristic
285Q8—42no—
286Q8—81no—
287Q8—84no—
288Q8—2124no—
289D4—42no—
290D4—81no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4—2122no—
301C2×C4#1313318611yescharacteristic
302C2×C4#1313318612yes—
303C2×C4C22×C22⋊D5312yes—
304C2×C4C22×C22⋊D5311yescharacteristic
305C8—2124no—
306C8—2122no—
307C22—22no—
308C22—42no—
309C22—44no—
310C22—84no—
311C22#1681080112yes—
312C22C2×#1313318611yescharacteristic
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—42no—
319C4—44no—
320C4—4242no—
321C2—22no—
322C2—82no—
323C2C2×#1681080111yescharacteristic
324C2#2028407912yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#563C2
socleC22×C53C2×#120
prime radicalC22×D4×C53D4
hypercenterC2×#563D53
hypocenterC53#18999
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. #20285921C2semidirect
3. #20274049C2semidirect
5. #20273317C2semidirect
8. #16811033C22semidirect
9. #16810993C22semidirect
19. #16810365C22semidirect
192. C53#18999semidirect

Maximal subgroups

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

subgroupindex
#202859212
#202851012
#202740492
#202733392
#202733172
C2×C53×#5632
#1899953