Finite Group Databasebeta

← #27297009

Subgroups — #27297009

Order 13568.

382 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 59 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#20276065C211yesmaximal, characteristic, complemented
2#20273756C211yesmaximal, characteristic, complemented
3#20273654C211yesmaximal, characteristic
4#20272742C211yesmaximal, characteristic, complemented
5#20272718C211yesmaximal, characteristic
6#20272659C211yesmaximal, characteristic, complemented
7C53×#2782C211yesmaximal, characteristic
8#16811254C2211yescharacteristic
9#16810958—21no—
10#16810774—21no—
11#16810663—21no—
12#16810392C2211yescharacteristic
13#16810383—21no—
14#16810359—21no—
15#16810359—21no—
16#16810349C2211yescharacteristic
17#16810349C2211yescharacteristic
18C53×#400—21no—
19C53×#400C2211yescharacteristic
20C53×#397—21no—
21C53×#394C2211yescharacteristic
22C53×#388—21no—
23C53×#386—21no—
24C53×#385—21no—
25C53×#385—21no—
26C53×#382C2211yescharacteristic
27C2×#11719923—41no—
28C2×#11719917—21no—
29#13133216—41no—
30#13133208—41no—
31#13133196—21no—
32#13133196—21no—
33#13133176—21no—
34#13133150—41no—
35#13133150—41no—
36#13133148—41no—
37#13133140—41no—
38#13133138—41no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22×D4×C53D411yescharacteristic
42C23×C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53C2311yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C42×C53D411yescharacteristic
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53D411yescharacteristic
61#95×C53D411yescharacteristic
62C24×C53—21no—
63C2×D4×C53—41no—
64C2×D4×C53—42no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—81no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C2×Dic106—21no—
77C2×Dic106—21no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—81no—
82C4⋊C4×C53—81no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—81no—
88C22⋊C4×C53—81no—
89C42×C53—21no—
90C42×C53—21no—
91C22⋊Dic53—41no—
92#11719923—82no—
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97#11719919—41no—
98#11719919—41no—
99C4×Dic53—41no—
100#11719917—22no—
101#11719917—41no—
102C2×#88322—41no—
103C2×#88322—41no—
104C2×#88322—41no—
105C23×C53—22no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53#12011yescharacteristic
113D4×C53—42no—
114D4×C53—81no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—84no—
137C2×C4×C53#13511yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C2×Dic53—41no—
140C2×Dic53—41no—
141Dic106—81no—
142Dic106—81no—
143#88322—42no—
144#88322—81no—
145#88322—81no—
146#12235—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—81no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53#45211yescharacteristic
157C22×C53#45611yescharacteristic
158C22×C53B(4,𝔽2)11yescharacteristic
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—81no—
164C4×C53—81no—
165C4×C53—81no—
166C4×C53—81no—
167C4×C53—81no—
168Dic53—81no—
169Dic53—81no—
170#2782D5311yescharacteristic
171#2061—531no—
172#1617—531no—
173#1599—531no—
174#1353—531no—
175#1345—531no—
176#1324—531no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—82no—
180C2×C53#160311yescharacteristic
181C2×C53#219711yescharacteristic
182C2×C53#220211yescharacteristic
183#544—531no—
184#484—1061no—
185#450—1061no—
186#428—1061no—
187#400—21no—
188#400C2×D5311yescharacteristic
189#397—21no—
190#394C2×D5311yescharacteristic
191#388—21no—
192#386—21no—
193#385—21no—
194#385—21no—
195#382C2×D5311yescharacteristic
196#343—531no—
197#339—1061no—
198#328—1061no—
199#328—1061no—
200#324—531no—
201#324—531no—
202C53#1223511yescharacteristic, complemented, Sylow
203C22×D4C22⋊D5311yescharacteristic
204C23×C4—21no—
205C2×SD16—2121no—
206C2×M16—1061no—
207C4⋊Q8—2121no—
208#125—2121no—
209C22⋊Q8—1061no—
210C22⋊Q8—1061no—
211#117—1061no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C22⋊D5311yescharacteristic
215C2×C4⋊C4C22⋊D5311yescharacteristic
216C2×C4⋊C4C22×D5311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42C22⋊D5311yescharacteristic
223#107—2121no—
224#107—2121no—
225#106—2121no—
226#103—2121no—
227#102—2121no—
228C22⋊C8—1061no—
229C22⋊C8—1061no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95C22⋊D5311yescharacteristic
237#95C22⋊D5311yescharacteristic
238C24—21no—
239C2×Q8—1061no—
240C2×Q8—1061no—
241C2×D4—41no—
242C2×D4—42no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C2×C22⋊D5311yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254SD16—4242no—
255M16—1062no—
256M16—2121no—
257C2×C8—2121no—
258C2×C8—2121no—
259C2×C8—2121no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C22⋊C4—2121no—
278C42—21no—
279C42—21no—
280C42—2121no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#1313318611yescharacteristic
289Q8—4241no—
290Q8—4241no—
291D4—42no—
292D4—81no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—84no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4#1313323711yescharacteristic
318C2×C4#1313324711yescharacteristic
319C8—4241no—
320C8—4241no—
321C8—2122no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#1681078511yescharacteristic
332C22#1681080611yescharacteristic
333C22#1681081011yescharacteristic
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—4241no—
344C4—4241no—
345C2—21no—
346C2—21no—
347C2—82no—
348C2#2027367811yescharacteristic
349C2#2027670511yescharacteristic
350C2#2027673211yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2782C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
hypercenter#2782D53
hypocenterC53#12235
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2782

Complements

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

NH ≅ G/Ntype
1. #20276065C2semidirect
2. #20273756C2semidirect
4. #20272742C2semidirect
6. #20272659C2semidirect
202. C53#12235semidirect

Maximal subgroups

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

subgroupindex
#202760652
#202737562
#202736542
#202727422
#202727182
#202726592
C53×#27822
#1223553