Finite Group Databasebeta

← #27365418

Subgroups — #27365418

Order 13568.

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

normal subgroups only

This group has 938 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#20285718C212yesmaximal, complemented
2#20285264C211yesmaximal, characteristic, complemented
3#20274032C218yesmaximal, complemented
4#20273342C211yesmaximal, characteristic, complemented
5#20273313C212yesmaximal, complemented
6C2×C53×#562C211yesmaximal, characteristic
7#16811180C2211yescharacteristic
8#16811033C2214yescomplemented
9#16811022—22no—
10#16811018C2214yescomplemented
11#16810982C2214yescomplemented
12#16810842—24no—
13#16810779—22no—
14#16810774—22no—
15#16810680C2212yes—
16#16810663—22no—
17C4×#11719917C2211yescharacteristic
18#16810365C2214yescomplemented
19#16810363C2218yescomplemented
20C53×#562—22no—
21C53×#562—24no—
22C53×#562C2214yes—
23C2×#125×C53C2211yescharacteristic
24C2×#124×C53C2212yes—
25C2×#123×C53—22no—
26C2×C22⋊Q8×C53—22no—
27C2×#121×C53—22no—
28C2×#11719925—24no—
29C2×#11719923—24no—
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#13133138—24no—
41C22⋊#88322—22no—
42#13133128C2314yes—
43#13133128C2312yes—
44C4×#88322—21no—
45C22×Q8×C53D412yes—
46C22×D4×C53—22no—
47C22×D4×C53D412yes—
48C23×C4×C53—22no—
49#125×C53—22no—
50#125×C53D412yes—
51#125×C53C2312yes—
52#124×C53—22no—
53#124×C53—22no—
54#124×C53D414yes—
55#124×C53C2314yes—
56#123×C53—28no—
57#123×C53—28no—
58C22⋊Q8×C53—28no—
59C22⋊Q8×C53—28no—
60#121×C53—28no—
61#121×C53—28no—
62C2×C4⋊C4×C53—22no—
63C2×C4⋊C4×C53—22no—
64C2×C4⋊C4×C53D412yes—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—22no—
67C2×C22⋊C4×C53—22no—
68C2×C42×C53C2311yescharacteristic
69C24×C53—22no—
70C2×Q8×C53—22no—
71C2×Q8×C53—82no—
72C2×Q8×C53C2×D414yes—
73C2×D4×C53—28no—
74C2×D4×C53—22no—
75C2×D4×C53—82no—
76C2×D4×C53—84no—
77C2×D4×C53C2×D414yes—
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53—28no—
81C22×C4×C53—84no—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D412yes—
84C2×Dic106—22no—
85C4⋊C4×C53—28no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53C2×D414yes—
90C22⋊C4×C53—28no—
91C22⋊C4×C53—24no—
92C22⋊C4×C53—24no—
93C22⋊C4×C53—24no—
94C22⋊C4×C53—24no—
95C42×C53C2×D411yescharacteristic
96C42×C53C2×D412yes—
97C42×C53C2411yescharacteristic
98C22⋊Dic53—41no—
99#11719925—84no—
100#11719923—84no—
101C4⋊Dic53—22no—
102C4⋊Dic53—22no—
103#11719919—42no—
104C4×Dic53—41no—
105#11719917—24no—
106C2×#88322—22no—
107C2×#88322—24no—
108C23×C53—24no—
109C23×C53—44no—
110C23×C53—44no—
111C23×C53—44no—
112C23×C53#12011yescharacteristic
113Q8×C53—44no—
114Q8×C53—82no—
115D4×C53—44no—
116D4×C53—82no—
117D4×C53—88no—
118C2×C4×C53—22no—
119C2×C4×C53—24no—
120C2×C4×C53—24no—
121C2×C4×C53—42no—
122C2×C4×C53—44no—
123C2×C4×C53—44no—
124C2×C4×C53—88no—
125C2×C4×C53#12012yes—
126C2×C4×C53#12011yescharacteristic
127C2×C4×C53C22×D412yes—
128C2×C4×C53C22×D411yescharacteristic
129C2×Dic53—42no—
130Dic106—44no—
131#88322—24no—
132#88322—44no—
133#19006—531nomaximal, Sylow
134C22×C53—22no—
135C22×C53—44no—
136C22×C53—44no—
137C22×C53—44no—
138C22×C53—88no—
139C22×C53#45612yes—
140C22×C53C2×#12011yescharacteristic
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—22no—
144C4×C53—22no—
145C4×C53—44no—
146C4×C53—44no—
147Dic53—82no—
148C2×#562D5311yescharacteristic
149#3215—532no—
150#3166—531no—
151#1661—538no—
152#1534—531no—
153#1527—532no—
154C2×C53—22no—
155C2×C53—84no—
156C2×C53C2×#45611yescharacteristic
157C2×C53#301912yes—
158#562—22no—
159#562—24no—
160#562C2×D5314yes—
161#533—531no—
162C2×#125C2×D5311yescharacteristic
163C2×#124C2×D5312yes—
164C2×#123—22no—
165C2×C22⋊Q8—22no—
166C2×#121—22no—
167#501—534no—
168#499—1062no—
169#497—534no—
170#489—534no—
171#464—1064no—
172#451—1062no—
173#450—1062no—
174#432—532no—
175#428—1062no—
176C4×M16—531no—
177#330—534no—
178#329—538no—
179C53#1900611yescharacteristic, complemented, Sylow
180C22×Q8C22⋊D5312yes—
181C22×D4—22no—
182C22×D4C22⋊D5312yes—
183C23×C4—22no—
184C2×Q16—1064no—
185C2×SD16—1064no—
186C2×M16—1061no—
187C4⋊Q8—532no—
188#125—22no—
189#125—1061no—
190#125C22⋊D5312yes—
191#125C22×D5312yes—
192#124—22no—
193#124—22no—
194#124C22⋊D5314yes—
195#124C22×D5314yes—
196#123—28no—
197#123—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—1062no—
201#121—28no—
202#121—28no—
203C4×Q8—1062no—
204#117—1061no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C22⋊D5312yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C42C22×D5311yescharacteristic
212#107—1064no—
213#106—1064no—
214C4⋊C8—1062no—
215#103—1064no—
216#102—1064no—
217C22⋊C8—1062no—
218#97—532no—
219#97—534no—
220C4×C8—1061no—
221C24—22no—
222C2×Q8—22no—
223C2×Q8—82no—
224C2×Q8—1062no—
225C2×Q8C2×C22⋊D5314yes—
226C2×D4—28no—
227C2×D4—22no—
228C2×D4—82no—
229C2×D4—84no—
230C2×D4C2×C22⋊D5314yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—84no—
235C22×C4C2×C22⋊D5311yescharacteristic
236C22×C4C2×C22⋊D5312yes—
237Q16—4244no—
238SD16—4244no—
239M16—1064no—
240C2×C8—1062no—
241C2×C8—1064no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—1062no—
247C4⋊C4—1062no—
248C4⋊C4—2122no—
249C4⋊C4C2×C22⋊D5314yes—
250C22⋊C4—28no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—2121no—
256C42—2121no—
257C42C2×C22⋊D5311yescharacteristic
258C42C2×C22⋊D5312yes—
259C42C23×D5311yescharacteristic
260C23—24no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23#1313318611yescharacteristic
265Q8—44no—
266Q8—82no—
267Q8—2124no—
268D4—44no—
269D4—82no—
270D4—88no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—88no—
278C2×C4—2122no—
279C2×C4#1313318611yescharacteristic
280C2×C4#1313318612yes—
281C2×C4C22×C22⋊D5311yescharacteristic
282C2×C4C22×C22⋊D5312yes—
283C8—1064no—
284C8—2124no—
285C22—22no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—88no—
290C22#1681080612yes—
291C22C2×#1313318611yescharacteristic
292C4—21no—
293C4—21no—
294C4—22no—
295C4—22no—
296C4—44no—
297C4—44no—
298C4—4242no—
299C2—22no—
300C2—84no—
301C2C2×#1681080611yescharacteristic
302C2#2028407212yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×C53×#562C2
socleC22×C53C2×#120
hypercenterC2×#562D53
hypocenterC53#19006
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #20285718C2semidirect
2. #20285264C2semidirect
3. #20274032C2semidirect
4. #20273342C2semidirect
5. #20273313C2semidirect
8. #16811033C22semidirect
10. #16811018C22semidirect
11. #16810982C22semidirect
18. #16810365C22semidirect
19. #16810363C22semidirect
179. C53#19006semidirect

Maximal subgroups

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

subgroupindex
#202857182
#202852642
#202740322
#202733422
#202733132
C2×C53×#5622
#1900653