Finite Group Databasebeta

← #27285539

Subgroups — #27285539

Order 13568.

458 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 85 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#20285921C211yesmaximal, characteristic
2#20285257C211yesmaximal, characteristic
3#20273432C211yesmaximal, characteristic
4#20273317C211yesmaximal, characteristic
5#20273226C211yesmaximal, characteristic
6#20272608C211yesmaximal, characteristic
7C53×#2642C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16811033—21no—
10#16811029—22no—
11#16811026—21no—
12#16811019—21no—
13#16810993—21no—
14#16810958—21no—
15#16810953—21no—
16#16810927—22no—
17#16810680C2211yescharacteristic
18#16810680C2211yescharacteristic
19#16810663—22no—
20C4×#11719917C2211yescharacteristic
21#16810369—22no—
22#16810365—22no—
23#16810359—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×#125×C53C2211yescharacteristic
26C53×#400—22no—
27C53×#400—22no—
28C53×#391—21no—
29C53×#391—21no—
30C53×#376C2211yescharacteristic
31C2×#11719917—21no—
32#13133217D411yescharacteristic
33#13133217D411yescharacteristic
34#13133209—21no—
35#13133196—21no—
36#13133196—21no—
37C4×Dic106—21no—
38C4×Dic106—21no—
39#13133176—21no—
40#13133150—24no—
41#13133150—22no—
42#13133148—24no—
43#13133148—22no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133140—22no—
47#13133140—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50#13133128—21no—
51#13133128—21no—
52#13133128D412yes—
53C4×#88322—21no—
54C22×Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59#125×C53—21no—
60#125×C53—21no—
61#125×C53—21no—
62#125×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C4⋊C4×C53D412yes—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71#95×C53—24no—
72#95×C53—24no—
73#95×C53—21no—
74#95×C53D412yes—
75C2×Q8×C53—44no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C22×C4×C53—22no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C2×Dic106—21no—
86C2×Dic106—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C42×C53—42no—
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53#4111yescharacteristic
104C22⋊Dic53—41no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109#11719919—41no—
110#11719919—41no—
111C4×Dic53—41no—
112#11719917—24no—
113C2×#88322—22no—
114C2×#88322—22no—
115C2×#88322—41no—
116C23×C53#12011yescharacteristic
117Q8×C53—82no—
118Q8×C53—82no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—44no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12111yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12711yescharacteristic
137C2×Dic53—41no—
138C2×Dic53—41no—
139Dic106—42no—
140Dic106—81no—
141#88322—24no—
142#88322—42no—
143#88322—81no—
144#10103—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—21no—
147C22×C53#39211yescharacteristic
148C22×C53#45612yes—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—82no—
155C4×C53—82no—
156C4×C53—81no—
157C4×C53—81no—
158C4×C53#45412yes—
159Dic53—81no—
160Dic53—81no—
161#3236—531no—
162#3165—531no—
163#2642D5311yescharacteristic
164#1556—531no—
165#1528—531no—
166#1506—531no—
167#1300—531no—
168C2×C53—22no—
169C2×C53#200311yescharacteristic
170C2×C53#200511yescharacteristic
171C2×C53#200911yescharacteristic
172#533—531no—
173C2×C4⋊Q8C2×D5311yescharacteristic
174C2×#125C2×D5311yescharacteristic
175#501—1062no—
176#501—1061no—
177#500—1061no—
178#498—1061no—
179#491—1061no—
180#484—1061no—
181#483—1061no—
182#479—1062no—
183#432—531no—
184#432—531no—
185#428—1062no—
186C4×M16—531no—
187#400—22no—
188#400—22no—
189#391—21no—
190#391—21no—
191#376C2×D5311yescharacteristic
192#332—1062no—
193#330—1062no—
194#328—1062no—
195C53#1010311yescharacteristic, complemented, Sylow
196C22×Q8—21no—
197C2×M16—1061no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—531no—
203C4⋊Q8—531no—
204#125—21no—
205#125—21no—
206#125—21no—
207#125—21no—
208#125—1061no—
209C22⋊Q8—1061no—
210C22⋊Q8—1061no—
211C4×Q8—1061no—
212C4×Q8—1061no—
213#117—1061no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C22⋊D5311yescharacteristic
218C2×C4⋊C4C22⋊D5311yescharacteristic
219C2×C4⋊C4C22⋊D5312yes—
220C2×C42—21no—
221C2×C42C22×D5311yescharacteristic
222#107—1064no—
223#107—1062no—
224#106—1064no—
225#106—1062no—
226C4⋊C8—532no—
227C4⋊C8—1061no—
228#103—1062no—
229#103—1062no—
230C22⋊C8—1061no—
231C22⋊C8—1061no—
232#97—532no—
233#97—1061no—
234#97—1061no—
235C4×C8—1061no—
236#95—24no—
237#95—24no—
238#95—21no—
239#95C22⋊D5312yes—
240C2×Q8—44no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—1061no—
244C2×Q8—1061no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4C2×C22⋊D5311yescharacteristic
250C22×C4C2×C22⋊D5311yescharacteristic
251C22×C4C2×C22⋊D5311yescharacteristic
252M16—1064no—
253C2×C8—1062no—
254C2×C8—1062no—
255C2×C8—2121no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C22⋊C4—2121no—
275C42—42no—
276C42—2121no—
277C42D4×D5311yescharacteristic
278C42D4×D5311yescharacteristic
279C42D4×D5311yescharacteristic
280C42#1171994711yescharacteristic
281C23#1313318611yescharacteristic
282Q8—82no—
283Q8—82no—
284Q8—4241no—
285Q8—2122no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4#1313318511yescharacteristic
301C2×C4#1313318511yescharacteristic
302C2×C4#1313318811yescharacteristic
303C2×C4#1313318911yescharacteristic
304C2×C4#1313318911yescharacteristic
305C2×C4#1313321311yescharacteristic
306C8—1064no—
307C8—4241no—
308C8—2122no—
309C22—21no—
310C22—21no—
311C22#1681053111yescharacteristic
312C22#1681080612yes—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—82no—
319C4—82no—
320C4—81no—
321C4—81no—
322C4—4241no—
323C4—4241no—
324C4#1681079512yes—
325C2—22no—
326C2#2027573711yescharacteristic
327C2#2027574911yescharacteristic
328C2#2027577511yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2642C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#2642D53
hypocenterC53#10103
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2642

Complements

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

NH ≅ G/Ntype
195. C53#10103semidirect

Maximal subgroups

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

subgroupindex
#202859212
#202852572
#202734322
#202733172
#202732262
#202726082
C53×#26422
#1010353