Finite Group Databasebeta

← #28312264

Subgroups — #28312264

Order 13568.

306 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290937C211yesmaximal, characteristic
2#20290634C211yesmaximal, characteristic
3#20289797C211yesmaximal, characteristic
4#20289788C211yesmaximal, characteristic
5#20289782C211yesmaximal, characteristic
6#20289779C211yesmaximal, characteristic
7#20285244C211yesmaximal, characteristic
8D53⋊M32C2211yescharacteristic
9D53⋊M32C2211yescharacteristic
10D53⋊#108C2211yescharacteristic
11#16811633C2211yescharacteristic
12#16811516—21no—
13#16811515—21no—
14#16811505—21no—
15#16811490—21no—
16#16811253—21no—
17#16811033—21no—
18#16811031—21no—
19#16811023C411yescharacteristic, complemented
20#16811021C411yescharacteristic, complemented
21#16811020C411yescharacteristic, complemented
22#16810664—21no—
23#16810658C2211yescharacteristic
24#106×D53C2211yescharacteristic
25#16810577C2211yescharacteristic
26C53×#499C411yescharacteristic, complemented
27D53⋊M16—21no—
28#13133296—21no—
29#13133292—21no—
30#13133292Q812yescomplemented
31#13133291—21no—
32#13133291—21no—
33D53⋊C16—21no—
34D53⋊C16Q812yescomplemented
35#13133288—21no—
36#13133287—21no—
37#13133286—21no—
38#13133285—21no—
39C2×#11719912D411yescharacteristic
40C2×C8×D53C2311yescharacteristic
41#13133217—21no—
42#13133216—21no—
43#13133216—41no—
44#13133208—21no—
45#13133207—41no—
46#13133197—41no—
47#13133195—41no—
48#13133175—21no—
49#13133174D411yescharacteristic
50C4⋊C4×D53—21no—
51#13133150—21no—
52#13133150—21no—
53#13133149—21no—
54#13133149C2×C411yescharacteristic
55#13133149C2×C411yescharacteristic
56#13133148—21no—
57#13133148—21no—
58#13133128D411yescharacteristic
59#13133127D411yescharacteristic
60C8×Dic53C2×C411yescharacteristic
61#125×C53—21no—
62#107×C53—21no—
63#106×C53C2×C411yescharacteristic
64#106×C53C2×C411yescharacteristic
65C4×C8×C53C2×C411yescharacteristic
66C2×C4×D53—41no—
67C2×C4×D53C2×D411yescharacteristic
68C2×Dic106—41no—
69C2×Dic106—41no—
70C22⋊#6157—41no—
71#11719940—41no—
72C4⋊#6157—41no—
73C4×#6157—41no—
74#11719936—41no—
75D53⋊C8—41no—
76C2×C8×C53C22⋊C411yescharacteristic
77C2×C8×C53C22×C411yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C42×C53C22⋊C411yescharacteristic
83#11719921—21no—
84#11719921—41no—
85C4⋊Dic53—21no—
86C4⋊Dic53—21no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89#11719919—21no—
90#11719919—41no—
91#11719919—41no—
92#11719919—41no—
93C4×Dic53—41no—
94C4×Dic53C22⋊C411yescharacteristic
95C2×#88322C22⋊C411yescharacteristic
96C2×#88322C2×D411yescharacteristic
97#11719912—21no—
98#11719912—21no—
99C8×D53C2×Q811yescharacteristic
100C8×D53C2×Q811yescharacteristic
101C8×D53#4111yescharacteristic
102C8×D53#4111yescharacteristic
103#11719910—22no—
104#11719910—22no—
105C22×D53AGL(1,ℤ/8)11yescharacteristic
106C2×#6157—81no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53C2×C22⋊C411yescharacteristic
111C2×Dic53—21no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53AGL(1,ℤ/8)11yescharacteristic
115C4×D53—161no—
116C4×D53#12411yescharacteristic
117C4×D53C4⋊Q811yescharacteristic
118C4×D53#13711yescharacteristic
119C4×D53#13711yescharacteristic
120Dic106—161no—
121Dic106—161no—
122#88324—81no—
123C8×C53—21no—
124C8×C53C4×Q811yescharacteristic
125C8×C53C4×Q811yescharacteristic
126#88322—21no—
127#88322C22⋊Q811yescharacteristic
128#88322C22⋊Q811yescharacteristic
129#13291—531nomaximal, Sylow
130C22×C53#41811yescharacteristic
131C2×D53—41no—
132C2×D53#48911yescharacteristic
133C2×D53#50111yescharacteristic
134#6157—161no—
135C4×C53—41no—
136C4×C53—81no—
137C4×C53—81no—
138C4×C53#39111yescharacteristic
139C4×C53#41911yescharacteristic
140Dic53—41no—
141Dic53—81no—
142Dic53—81no—
143Dic53#48911yescharacteristic
144Dic53#50111yescharacteristic
145#3163—531no—
146#2110—531no—
147#1833—531no—
148#1389—531no—
149#1386—531no—
150#1384—531no—
151#1383—531no—
152C2×C53—21no—
153C2×C53#199311yescharacteristic
154D53—21no—
155D53—21no—
156#544—1061no—
157C2×M32—531no—
158C2×M32—531no—
159#501—1061no—
160#501—1061no—
161#499—531no—
162#499#615711yescharacteristic, complemented
163#498—531no—
164#498—531no—
165C2×#108—531no—
166#428—1061no—
167#427—531no—
168#427—531no—
169C2×#106—531no—
170#405—531no—
171#347—1061no—
172#347—1061no—
173#344—1061no—
174H(3,ℤ/4)—1061no—
175C53#1329111yescharacteristic, complemented, Sylow
176C2×M16—531no—
177C2×M16—1061no—
178C22×C8—531no—
179C4⋊Q8—1061no—
180C4⋊Q8—1061no—
181C4⋊Q8—2121no—
182#125—21no—
183#125—1061no—
184#125—2121no—
185C22⋊Q8—2121no—
186C22⋊Q8—2121no—
187#117—531no—
188#117—1061no—
189#117—1061no—
190C2×C4⋊C4—1061no—
191M32—532no—
192M32—1061no—
193M32—1061no—
194M32—1061no—
195C2×C16—532no—
196C2×C16—1061no—
197#108—1061no—
198#108—1061no—
199#107—21no—
200#107—1061no—
201#107—1061no—
202#107—1061no—
203#106—531no—
204#106—531no—
205#106—1061no—
206#106—1061no—
207#106—1061no—
208#106—1061no—
209#106C2×#615711yescharacteristic
210#106C2×#615711yescharacteristic
211#97—531no—
212#97—531no—
213C4×C8—531no—
214C4×C8C2×#615711yescharacteristic
215C2×Q8—2121no—
216C2×Q8—2121no—
217C22×C4—531no—
218C22×C4—2121no—
219M16—1061no—
220M16—1061no—
221M16—2121no—
222M16—2121no—
223C2×C8—531no—
224C2×C8—531no—
225C2×C8—531no—
226C2×C8—531no—
227C2×C8—531no—
228C2×C8—531no—
229C2×C8—2121no—
230C2×C8C22⋊#615711yescharacteristic
231C2×C8C22×#615711yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—1061no—
237C4⋊C4—1061no—
238C4⋊C4—1061no—
239C4⋊C4—2121no—
240C4⋊C4—2121no—
241C4⋊C4—2121no—
242C4⋊C4—2121no—
243C4⋊C4—2121no—
244C4⋊C4—2121no—
245C22⋊C4—1061no—
246C22⋊C4—2121no—
247C22⋊C4—2121no—
248C42—531no—
249C42—2121no—
250C42—2121no—
251C42C22⋊#615711yescharacteristic
252C16—1062no—
253C16—1062no—
254C23—531no—
255Q8—8481no—
256Q8—8481no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—531no—
261C2×C4—531no—
262C2×C4—531no—
263C2×C4—531no—
264C2×C4—531no—
265C2×C4—8481no—
266C2×C4—1061no—
267C2×C4—4241no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4C2×C22⋊#615711yescharacteristic
271C8—21no—
272C8—531no—
273C8—531no—
274C8—1061no—
275C8—4241no—
276C8Q8×#615711yescharacteristic
277C8Q8×#615711yescharacteristic
278C22—531no—
279C22—531no—
280C22—2121no—
281C22#1681161511yescharacteristic
282C4—41no—
283C4—81no—
284C4—81no—
285C4—531no—
286C4—531no—
287C4—8481no—
288C4—4241no—
289C4—4241no—
290C4—2121no—
291C4(C2×#6157)⋊Q811yescharacteristic
292C4#1681161611yescharacteristic
293C2—21no—
294C2—1061no—
295C2—1061no—
296C2#2029089911yescharacteristic
subgroupG/H
centerC2#20290899
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#499C4
socleC2×C53#1993
prime radicalC22×D53AGL(1,ℤ/8)
cosocleC2×C8×D53C23
hypercenter#499#6157
hypocenterC53#13291
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #499

Complements

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

NH ≅ G/Ntype
19. #16811023C4semidirect
20. #16811021C4semidirect
21. #16811020C4semidirect
26. C53×#499C4semidirect
30. #13133292Q8semidirect
34. D53⋊C16Q8semidirect
162. #499#6157semidirect
175. C53#13291semidirect

Maximal subgroups

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

subgroupindex
#202909372
#202906342
#202897972
#202897882
#202897822
#202897792
#202852442
#1329153