Finite Group Databasebeta

← #27359000

Subgroups — #27359000

Order 13568.

870 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 191 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#20288525C211yesmaximal, characteristic
2#20285868C212yesmaximal
3#20279927C211yesmaximal, characteristic
4#20273759C218yesmaximal
5#20273215C212yesmaximal
6C53×#1461C211yesmaximal, characteristic
7Q8×Dic106—22no—
8#16811352—22no—
9#16811349—24no—
10#16811254—21no—
11#16811254C2214yes—
12#16811180—22no—
13#16811180C2212yes—
14C2×#13133217C2211yescharacteristic
15#16811101—21no—
16#16811028—22no—
17#16810974—22no—
18#16810959C2218yes—
19#16810919—24no—
20#16810562C2214yes—
21#16810530—21no—
22#16810529—22no—
23#16810488—21no—
24C4×C4⋊Dic53C2211yescharacteristic
25#16810361C2218yes—
26C2×C4×Q8×C53C2211yescharacteristic
27C53×#431C2212yes—
28C53×#324C2214yes—
29C22×Dic106—21no—
30#13133217—22no—
31#13133217—22no—
32#13133217—22no—
33#13133217—22no—
34#13133217D412yes—
35#13133217D412yes—
36#13133216—24no—
37#13133216—28no—
38#13133209—24no—
39#13133209—22no—
40#13133208—24no—
41#13133208—28no—
42#13133196—22no—
43#13133196—24no—
44#13133196C2314yes—
45Q8×Dic53—24no—
46Q8×Dic53—24no—
47C4×Dic106—24no—
48C4×Dic106—24no—
49C4×Dic106—22no—
50C4×Dic106D414yes—
51#13133176—24no—
52#13133176—22no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53C2312yes—
56C2×C4×Dic53—21no—
57#13133151—28no—
58#13133142—28no—
59#13133124—24no—
60#13133124—24no—
61#13133124C2312yes—
62C22×Q8×C53D411yescharacteristic
63C4×Q8×C53—22no—
64C4×Q8×C53—22no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C4⋊C4×C53C2312yes—
67C2×C42×C53D412yes—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—22no—
70C22⋊C8×C53C2314yes—
71C4×C8×C53—22no—
72C2×Q8×C53—22no—
73C2×Q8×C53—22no—
74C2×Q8×C53Q1614yes—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D412yes—
77C22×C4×C53C2×D412yes—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2411yescharacteristic
80C22×Dic53—22no—
81C22×Dic53—22no—
82C2×Dic106—22no—
83C2×Dic106—24no—
84C2×Dic106—44no—
85C2×Dic106—41no—
86C2×Dic106—41no—
87C2×Dic106C2×D414yes—
88C2×C8×C53—24no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—24no—
91C42×C53—22no—
92C42×C53—22no—
93C42×C53C2×D412yes—
94C42×C53C2×D412yes—
95C22⋊Dic53—24no—
96C4⋊Dic53—24no—
97C4⋊Dic53—24no—
98C4⋊Dic53—24no—
99C4⋊Dic53—22no—
100C4⋊Dic53—22no—
101C4⋊Dic53—22no—
102C4⋊Dic53—24no—
103C4⋊Dic53—42no—
104C4⋊Dic53—42no—
105C4⋊Dic53C2×D414yes—
106#11719919—28no—
107#11719919—24no—
108#11719919—24no—
109C4×Dic53—28no—
110C4×Dic53—24no—
111C4×Dic53—42no—
112C23×C53C22×D411yescharacteristic
113Q8×C53—28no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53#12011yescharacteristic
121C2×C4×C53#12012yes—
122C2×C4×C53#12011yescharacteristic
123C2×C4×C53C2×Q1614yes—
124C2×C4×C53C2×Q1612yes—
125C2×C4×C53#13712yes—
126C2×C4×C53C22×D412yes—
127C2×Dic53—24no—
128C2×Dic53—24no—
129C2×Dic53—44no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134Dic106—44no—
135Dic106—44no—
136Dic106—84no—
137Dic106—82no—
138Dic106—82no—
139C8×C53—44no—
140#18411—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53C2×#12011yescharacteristic
144C22×C53C22×Q1611yescharacteristic
145C22×C53C2×#13711yescharacteristic
146C4×C53—24no—
147C4×C53—21no—
148C4×C53—22no—
149C4×C53—42no—
150C4×C53#45114yes—
151C4×C53#45412yes—
152Dic53—44no—
153Dic53—44no—
154Dic53—82no—
155Dic53—82no—
156#3502—531no—
157#3230—532no—
158#2568—531no—
159#1617—538no—
160#1503—532no—
161#1461D5311yescharacteristic
162C2×C53—21no—
163C2×C53—21no—
164C2×C53C2×#45111yescharacteristic
165C2×C53C2×#45411yescharacteristic
166C2×C53#302811yescharacteristic
167Q82—1062no—
168#557—1064no—
169#557—1062no—
170#544—534no—
171#544—1061no—
172#533—532no—
173#533—1062no—
174C2×C4⋊Q8—531no—
175#519—1061no—
176C2×C4×Q8C2×D5311yescharacteristic
177#500—1062no—
178#487—1062no—
179#484—538no—
180#477—1064no—
181#431C2×D5312yes—
182#400—534no—
183#391—1061no—
184#391—1062no—
185#382—1061no—
186C4×C4⋊C4—531no—
187#328—538no—
188#324C2×D5314yes—
189C53#1841111yescharacteristic, complemented, Sylow
190C22×Q8—1061no—
191C22×Q8D21211yescharacteristic
192C4⋊Q8—532no—
193C4⋊Q8—532no—
194C4⋊Q8—1062no—
195C4⋊Q8—1062no—
196C4⋊Q8—1068no—
197C4⋊Q8—1064no—
198C4⋊Q8—1062no—
199C4⋊Q8—1062no—
200#125—1064no—
201#125—1068no—
202#125—1064no—
203#125—1062no—
204C22⋊Q8—534no—
205C22⋊Q8—1062no—
206C22⋊Q8—1064no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—534no—
210C4×Q8—1064no—
211C4×Q8—1064no—
212C4×Q8—1064no—
213C4×Q8—1064no—
214C4×Q8—1062no—
215#117—1062no—
216#117—1064no—
217C2×C4⋊C4—532no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4D21211yescharacteristic
221C2×C4⋊C4C22×D5312yes—
222C2×C42—1061no—
223C2×C42D21212yes—
224C2×C42C22×D5311yescharacteristic
225#107—1068no—
226C4⋊C8—22no—
227#103—1068no—
228C22⋊C8C22×D5314yes—
229C4×C8—22no—
230#95—532no—
231#95—1064no—
232#95—1064no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—534no—
236C2×Q8—1064no—
237C2×Q8—1062no—
238C2×Q8—2124no—
239C2×Q8—2121no—
240C2×Q8—2121no—
241C2×Q8Dic21214yes—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4C2×D21211yescharacteristic
245C22×C4C2×D21212yes—
246C22×C4C2×D21212yes—
247C22×C4C2×D21211yescharacteristic
248C22×C4C23×D5311yescharacteristic
249C2×C8—24no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—534no—
253C4⋊C4—1064no—
254C4⋊C4—1064no—
255C4⋊C4—1064no—
256C4⋊C4—1068no—
257C4⋊C4—1062no—
258C4⋊C4—1062no—
259C4⋊C4—1064no—
260C4⋊C4—1062no—
261C4⋊C4—1064no—
262C4⋊C4—1064no—
263C4⋊C4—2122no—
264C4⋊C4—2122no—
265C22⋊C4—1064no—
266C42—22no—
267C42—22no—
268C42—1068no—
269C42—1064no—
270C42—2122no—
271C42D4×D5312yes—
272C42D4×D5312yes—
273C23C22×D21211yescharacteristic
274Q8—28no—
275Q8—4242no—
276Q8—4244no—
277Q8—4242no—
278Q8—2124no—
279Q8—2124no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—1064no—
287C2×C4—1064no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4—2124no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4#1313318711yescharacteristic
294C2×C4#1313318712yes—
295C2×C4#1313318711yescharacteristic
296C2×C4C2×Dic21214yes—
297C2×C4C2×Dic21212yes—
298C2×C4#1313325012yes—
299C2×C4C2×D4×D5312yes—
300C8—44no—
301C22—21no—
302C22—21no—
303C22C2×#1313318711yescharacteristic
304C22C22×Dic21211yescharacteristic
305C22C2×#1313325011yescharacteristic
306C4—24no—
307C4—21no—
308C4—22no—
309C4—42no—
310C4—4242no—
311C4—4242no—
312C4—2124no—
313C4—2124no—
314C4#1681078114yes—
315C4#1681079812yes—
316C2—21no—
317C2—21no—
318C2C2×#1681078111yescharacteristic
319C2C2×#1681079811yescharacteristic
320C2#2028412211yescharacteristic
subgroupG/H
centerC22C2×#13133187
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1461C2
socleC22×C53C2×#120
prime radicalC23×C53C22×D4
hypercenter#1461D53
hypocenterC53#18411
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
189. C53#18411semidirect

Maximal subgroups

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

subgroupindex
#202885252
#202858682
#202799272
#202737592
#202732152
C53×#14612
#1841153