Finite Group Databasebeta

← #27281827

Subgroups — #27281827

Order 13568.

700 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277301C211yesmaximal, characteristic
2#20274639C211yesmaximal, characteristic
3#20273248C212yesmaximal
4#20272568C212yesmaximal
5C53×#2434C211yesmaximal, characteristic
6C2×#13133176—21no—
7C2×#13133136—22no—
8#16810587C2212yes—
9#16810530—24no—
10#16810486—21no—
11#16810486C2212yes—
12#16810360—24no—
13C23×Q8×C53C2211yescharacteristic
14C2×C4⋊Q8×C53C412yescomplemented
15C2×C22⋊Q8×C53C2211yescharacteristic
16C53×#395—22no—
17C53×#391C412yescomplemented
18C53×#385—21no—
19C53×#385—21no—
20C4×C22⋊C4×C53C2211yescharacteristic
21C2×#11719917—22no—
22#13133176—44no—
23C2×C4⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—22no—
26#13133136—44no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—24no—
30#13133124—22no—
31#13133124D414yes—
32C22×Q8×C53—22no—
33C22×Q8×C53—22no—
34C22×Q8×C53—44no—
35C22×Q8×C53C2×C411yescharacteristic
36C22×Q8×C53C2×C412yes—
37C22×Q8×C53D412yes—
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41C4⋊Q8×C53—44no—
42C22⋊Q8×C53—22no—
43C22⋊Q8×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53—22no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C42×C53C2×C412yes—
51#95×C53—22no—
52#95×C53—22no—
53#95×C53—21no—
54C24×C53C2×D411yescharacteristic
55C2×Q8×C53—24no—
56C2×Q8×C53—24no—
57C2×Q8×C53—24no—
58C2×Q8×C53—24no—
59C2×Q8×C53—22no—
60C2×Q8×C53—42no—
61C2×Q8×C53—42no—
62C2×Q8×C53—48no—
63C2×Q8×C53—44no—
64C2×Q8×C53—44no—
65C2×Q8×C53—42no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—24no—
69C22×C4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53C22⋊C412yes—
77C22×C4×C53C22⋊C412yes—
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D412yes—
81C22×C4×C53#4112yes—
82C22×Dic53—24no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—44no—
85C4⋊C4×C53—44no—
86C4⋊C4×C53—42no—
87C22⋊C4×C53—24no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53—42no—
91C22⋊Dic53—44no—
92C4⋊Dic53—44no—
93C4×Dic53—44no—
94#11719917—44no—
95C2×#88322—42no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100C23×C53#12011yescharacteristic
101C23×C53#12311yescharacteristic
102Q8×C53—48no—
103Q8×C53—48no—
104Q8×C53—82no—
105C2×C4×C53—24no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—24no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53#9914yes—
127C2×C4×C53C4×D412yes—
128C2×C4×C53#12112yes—
129C2×Dic53—44no—
130C2×Dic53—44no—
131C2×Dic53—44no—
132#88322—82no—
133#9462—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—44no—
142C22×C53#38611yescharacteristic
143C22×C53C2×#9912yes—
144C22×C53#45612yes—
145C4×C53—22no—
146C4×C53—24no—
147C4×C53—22no—
148C4×C53—44no—
149C4×C53—42no—
150C4×C53—81no—
151C4×C53—81no—
152Dic53—84no—
153#2434D5311yescharacteristic
154#2310—531no—
155#1784—531no—
156#1511—532no—
157#1284—532no—
158C2×C53—22no—
159C2×C53—22no—
160C2×C53—41no—
161C2×C53#188611yescharacteristic
162C2×C53#189811yescharacteristic
163C2×C53#191211yescharacteristic
164C23×Q8C2×D5311yescharacteristic
165C2×C4⋊Q8Dic5312yescomplemented
166C2×C22⋊Q8C2×D5311yescharacteristic
167C2×#117—1061no—
168C2×#101—1062no—
169#407—532no—
170#395—22no—
171#391—1064no—
172#391Dic5312yescomplemented
173#385—21no—
174#385—21no—
175#381—532no—
176#381—1061no—
177C4×C22⋊C4C2×D5311yescharacteristic
178#328—1064no—
179C53#946211yescharacteristic, complemented, Sylow
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×Q8C2×Dic5311yescharacteristic
184C22×Q8C2×Dic5312yes—
185C22×Q8C22⋊D5312yes—
186C23×C4C22⋊D5311yescharacteristic
187C23×C4C22⋊D5311yescharacteristic
188C23×C4C22×D5311yescharacteristic
189C2×M16—1062no—
190C4⋊Q8—44no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193#117—2124no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—1062no—
198C2×C4⋊C4C2×Dic5311yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—1062no—
202C2×C42—1062no—
203C2×C42C2×Dic5312yes—
204#101—2124no—
205C22⋊C8—1062no—
206C22⋊C8—1062no—
207#95—21no—
208#95—22no—
209#95—22no—
210#95—534no—
211#95—1064no—
212#95—1062no—
213C24C2×C22⋊D5311yescharacteristic
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—22no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1064no—
236C22×C4C22⋊Dic5312yes—
237C22×C4C22⋊Dic5312yes—
238C22×C4D4×D5312yes—
239C22×C4#1171994712yes—
240C22×C4C22×Dic5311yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—2124no—
243C2×C8—2122no—
244C4⋊C4—22no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—2124no—
249C22⋊C4—24no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—2124no—
253C42—42no—
254C42—2124no—
255C23—22no—
256C23—22no—
257C23—41no—
258C23C2×C22⋊Dic5311yescharacteristic
259C23#1313318511yescharacteristic
260C23#1313319811yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—82no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—2124no—
286C2×C4—2124no—
287C2×C4—2124no—
288C2×C4#1313313214yes—
289C2×C4D4×Dic5312yes—
290C2×C4#1313318912yes—
291C8—4242no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#1681050311yescharacteristic
301C22C2×#1313313212yes—
302C22#1681080512yes—
303C4—22no—
304C4—24no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—4244no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#2027510911yescharacteristic
315C2#2027516911yescharacteristic
316C2#2027525311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2434C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2434D53
hypocenterC53#9462
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
14. C2×C4⋊Q8×C53C4semidirect
17. C53×#391C4semidirect
165. C2×C4⋊Q8Dic53semidirect
172. #391Dic53semidirect
179. C53#9462semidirect

Maximal subgroups

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

subgroupindex
#202773012
#202746392
#202732482
#202725682
C53×#24342
#946253