Finite Group Databasebeta

← #27365993

Subgroups — #27365993

Order 13568.

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

normal subgroups only

This group has 806 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#20285913C212yesmaximal
2#20285063C211yesmaximal, characteristic
3#20274043C214yesmaximal, complemented
4#20274017C214yesmaximal, complemented
5#20273353C211yesmaximal, characteristic, complemented
6#20273330C212yesmaximal, complemented
7C53×#3556C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16811033C2214yes—
10#16810993C2212yes—
11#16810989C2212yes—
12#16810989C2214yes—
13#16810985—22no—
14#16810947—24no—
15#16810910—22no—
16#16810900—22no—
17#16810680C2212yes—
18#16810617—22no—
19C2×#13133128C2211yescharacteristic
20#16810365C2218yes—
21#16810365C2214yes—
22C53×#564C2214yes—
23C53×#557—22no—
24C53×#557—22no—
25C53×#557—22no—
26C53×#541—22no—
27C53×#541—24no—
28C53×#533C2211yescharacteristic
29C53×#533C2212yes—
30C2×#11719945—21no—
31C22×#88322—21no—
32#13133217C2311yescharacteristic
33#13133214C2311yescharacteristic
34#13133205—21no—
35#13133191—22no—
36C4×D212—22no—
37#13133150—24no—
38#13133148—24no—
39C4⋊#88322—22no—
40#13133140—24no—
41#13133137—24no—
42#13133137—24no—
43#13133137—24no—
44C22⋊#88322—22no—
45#13133128—21no—
46#13133128C2314yes—
47#13133128C2312yes—
48C4⋊Q8×C53—24no—
49C4⋊Q8×C53—22no—
50C4⋊Q8×C53D412yes—
51C4⋊Q8×C53D412yes—
52C4⋊Q8×C53D412yes—
53#126×C53—28no—
54#125×C53—24no—
55#125×C53—22no—
56#125×C53—28no—
57#125×C53—28no—
58#125×C53—28no—
59#125×C53C2314yes—
60#125×C53C2312yes—
61C22⋊Q8×C53—22no—
62C22⋊Q8×C53—22no—
63C22⋊Q8×C53—22no—
64C4×Q8×C53—24no—
65C4×Q8×C53—24no—
66C4×Q8×C53—24no—
67C4×Q8×C53—22no—
68C4×Q8×C53—22no—
69C4×Q8×C53—22no—
70#117×C53D412yes—
71#117×C53D414yes—
72C2×C42×C53C2311yescharacteristic
73C2×Q8×C53—22no—
74C2×Q8×C53—22no—
75C2×Q8×C53—22no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D412yes—
78C2×C22⋊D53—41no—
79#11719945—44no—
80C2×D212—22no—
81C2×D212—21no—
82C2×C4×D53—41no—
83C2×Dic106—21no—
84C4⋊C4×C53—28no—
85C4⋊C4×C53—28no—
86C4⋊C4×C53—28no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—24no—
93C4⋊C4×C53C2×D414yes—
94C4⋊C4×C53C2×D414yes—
95C4⋊C4×C53C2×D414yes—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—24no—
98C42×C53—28no—
99C42×C53—24no—
100C42×C53—22no—
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D412yes—
103C42×C53C2411yescharacteristic
104#11719921—42no—
105C4⋊Dic53—22no—
106C2×#88322—22no—
107C2×#88322—24no—
108C2×#88322—42no—
109C23×C53#12011yescharacteristic
110C22×D53—41no—
111Q8×C53—44no—
112Q8×C53—44no—
113Q8×C53—44no—
114C2×C4×C53—22no—
115C2×C4×C53—24no—
116C2×C4×C53—24no—
117C2×C4×C53—24no—
118C2×C4×C53—42no—
119C2×C4×C53#12012yes—
120C2×C4×C53#12011yescharacteristic
121C2×C4×C53C22×D411yescharacteristic
122C2×C4×C53C22×D412yes—
123C22⋊D53—82no—
124C2×Dic53—41no—
125D212—44no—
126D212—42no—
127C4×D53—82no—
128Dic106—42no—
129#88322—44no—
130#88322—42no—
131#19063—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—22no—
134C22×C53C2×#12011yescharacteristic
135C2×D53—81no—
136C2×D53—82no—
137C4×C53—21no—
138C4×C53—22no—
139C4×C53—22no—
140C4×C53—44no—
141C4×C53—44no—
142C4×C53—44no—
143C4×C53#45412yes—
144Dic53—81no—
145#3556D5311yescharacteristic, complemented
146#3235—532no—
147#3135—531no—
148#1663—534no—
149#1658—534no—
150#1537—531no—
151#1531—532no—
152C2×C53—41no—
153C2×C53C2×#45411yescharacteristic
154C2×C53#302012yes—
155D53—161no—
156#564C2×D5314yes—
157#557—22no—
158#557—22no—
159#557—22no—
160#541—22no—
161#541—24no—
162#533C2×D5311yescharacteristic
163#533C2×D5312yes—
164#532—531no—
165#501—534no—
166#491—532no—
167#490—532no—
168#490—534no—
169#489—1062no—
170#482—1064no—
171#476—1062no—
172#474—1062no—
173#432—532no—
174#416—1062no—
175C2×#97—531no—
176#330—534no—
177#330—538no—
178C53#1906311yescharacteristic, complemented, Sylow
179C2×#41—1061no—
180C22×C8—1061no—
181C4⋊Q8—24no—
182C4⋊Q8—22no—
183C4⋊Q8—531no—
184C4⋊Q8C22⋊D5312yes—
185C4⋊Q8C22⋊D5312yes—
186C4⋊Q8C22⋊D5312yes—
187#127—531no—
188#126—28no—
189#125—24no—
190#125—22no—
191#125—28no—
192#125—28no—
193#125—28no—
194#125C22×D5312yes—
195#125C22×D5314yes—
196#124—1061no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200#121—1062no—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×D4—1062no—
208#117C22⋊D5312yes—
209#117C22⋊D5314yes—
210C2×C42C22×D5311yescharacteristic
211#107—1064no—
212#106—1064no—
213C4⋊C8—1062no—
214#103—1064no—
215#102—1064no—
216#102—1064no—
217#102—1064no—
218C22⋊C8—1062no—
219#97—532no—
220#97—534no—
221#97—1061no—
222#41—2124no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—1061no—
227C2×D4—1062no—
228C2×D4—1061no—
229C2×D4—2121no—
230C22×C4—2121no—
231C22×C4C2×C22⋊D5311yescharacteristic
232C22×C4C2×C22⋊D5312yes—
233C2×C8—1062no—
234C2×C8—1064no—
235C2×C8—2122no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C4⋊C4—28no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—1062no—
246C4⋊C4C2×C22⋊D5314yes—
247C4⋊C4C2×C22⋊D5314yes—
248C4⋊C4C2×C22⋊D5314yes—
249C22⋊C4—22no—
250C22⋊C4—24no—
251C22⋊C4—2122no—
252C42—28no—
253C42—24no—
254C42—22no—
255C42C2×C22⋊D5311yescharacteristic
256C42C2×C22⋊D5312yes—
257C42C23×D5311yescharacteristic
258C23—2121no—
259C23#1313318611yescharacteristic
260Q8—44no—
261Q8—44no—
262Q8—44no—
263Q8—2122no—
264D4—4242no—
265D4—2124no—
266D4—2122no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—4242no—
273C2×C4—2121no—
274C2×C4#1313318612yes—
275C2×C4#1313318611yescharacteristic
276C2×C4C22×C22⋊D5312yes—
277C2×C4C22×C22⋊D5311yescharacteristic
278C8—2124no—
279C8—2122no—
280C22—21no—
281C22—22no—
282C22—4241no—
283C22—4242no—
284C22C2×#1313318611yescharacteristic
285C4—21no—
286C4—22no—
287C4—22no—
288C4—44no—
289C4—44no—
290C4—44no—
291C4—4241no—
292C4#1681079612yes—
293C2—41no—
294C2—8481no—
295C2C2×#1681079611yescharacteristic
296C2#2028407912yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3556C2
socleC22×C53C2×#120
prime radical#16811173C22
hypercenter#3556D53
hypocenterC53#19063
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

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

NH ≅ G/Ntype
3. #20274043C2semidirect
4. #20274017C2semidirect
5. #20273353C2semidirect
6. #20273330C2semidirect
7. C53×#3556C2semidirect
145. #3556D53semidirect
178. C53#19063semidirect

Maximal subgroups

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

subgroupindex
#202859132
#202850632
#202740432
#202740172
#202733532
#202733302
C53×#35562
#1906353