Finite Group Databasebeta

← #27281791

Subgroups — #27281791

Order 13568.

554 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 107 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#20279668C211yesmaximal, characteristic
2#20273433C212yesmaximal, complemented
3#20273056C211yesmaximal, characteristic
4#20272572C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133196—21no—
8C2×#13133196—21no—
9#16810680C2212yes—
10#16810559—21no—
11#16810559—21no—
12#16810526C412yes—
13#16810492C412yes—
14C4×C4⋊Dic53C2211yescharacteristic
15#16810370—24no—
16#16810347—21no—
17#16810347—21no—
18C2×C4×D4×C53C2211yescharacteristic
19C53×#435—24no—
20C53×#432C2212yes—
21C53×#407—21no—
22C2×C22⋊C8×C53—21no—
23C22×Dic106—21no—
24#13133217—22no—
25#13133217—22no—
26#13133196—44no—
27#13133196—44no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C412yes—
31C2×#11719919—22no—
32C2×#11719919—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C4×Dic53C2×C412yes—
36C4⋊#88322—22no—
37C22⋊#88322—22no—
38#13133128—22no—
39#13133124—21no—
40C22×D4×C53C2×C411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C2×M16×C53—22no—
44C22×C8×C53—22no—
45C4×D4×C53—22no—
46C4×D4×C53—22no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53C2311yescharacteristic
51C4⋊C8×C53D414yes—
52C22⋊C8×C53—22no—
53C22⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55#97×C53—22no—
56C24×C53C22⋊C411yescharacteristic
57C24×C53C22⋊C411yescharacteristic
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—41no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—22no—
73C22×Dic53—22no—
74C2×Dic106—42no—
75C2×Dic106—42no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78M16×C53—82no—
79C2×C8×C53—24no—
80C2×C8×C53—82no—
81C4⋊C4×C53—22no—
82C22⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C42×C53C2×D412yes—
85C42×C53#4112yes—
86C22⋊Dic53—44no—
87C22⋊Dic53—44no—
88C4⋊Dic53—24no—
89C4⋊Dic53—24no—
90C4⋊Dic53—42no—
91#11719919—44no—
92#11719919—44no—
93C4×Dic53—42no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53#10112yes—
105C23×C53C2×C22⋊C411yescharacteristic
106D4×C53—44no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—81no—
118C2×C4×C53—81no—
119C2×C4×C53C4×D412yes—
120C2×C4×C53#12011yescharacteristic
121C2×C4×C53#12112yes—
122C2×C4×C53#12311yescharacteristic
123C2×Dic53—24no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129Dic106—82no—
130Dic106—82no—
131C8×C53—44no—
132#88322—82no—
133#9455—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—22no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53#38611yescharacteristic
147C22×C53C2×#10111yescharacteristic
148C22×C53#41311yescharacteristic
149C4×C53—21no—
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—41no—
153C4×C53—41no—
154C4×C53#45312yes—
155Dic53—44no—
156Dic53—82no—
157#2973D5311yescharacteristic
158#2542—531no—
159#1556—532no—
160#1466—531no—
161#1285—532no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—42no—
165C2×C53—81no—
166C2×C53#188711yescharacteristic
167C2×C53#190111yescharacteristic
168C2×C53#190911yescharacteristic
169C2×C4⋊Q8—531no—
170C2×C22⋊Q8—1061no—
171C2×C22⋊Q8—1061no—
172C2×C4×D4C2×D5311yescharacteristic
173#435—24no—
174#432—532no—
175#432C2×D5312yes—
176#407—21no—
177C2×C22⋊C8—21no—
178#399—1061no—
179#399—1061no—
180#390—532no—
181#384—532no—
182C4×C4⋊C4—531no—
183#332—1064no—
184#323—1061no—
185#323—1061no—
186C53#945511yescharacteristic, complemented, Sylow
187C22×Q8—1061no—
188C22×D4C4×D5311yescharacteristic
189C23×C4C22⋊D5311yescharacteristic
190C23×C4C22⋊D5311yescharacteristic
191C2×M16—22no—
192C22×C8—22no—
193C4⋊Q8—1062no—
194C4⋊Q8—1062no—
195C22⋊Q8—2124no—
196C22⋊Q8—2124no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—532no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1062no—
203C2×C4⋊C4—1062no—
204C2×C4⋊C4C4×D5311yescharacteristic
205C2×C22⋊C4—1062no—
206C2×C22⋊C4—1062no—
207C2×C22⋊C4D21211yescharacteristic
208C2×C22⋊C4D21211yescharacteristic
209C2×C42—532no—
210C2×C42C22×D5311yescharacteristic
211C4⋊C8—1062no—
212C4⋊C8C22⋊D5314yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—1062no—
217#97—22no—
218#97—1062no—
219#95—1061no—
220C24#1171992111yescharacteristic
221C24#1171992111yescharacteristic
222C2×Q8—2122no—
223C2×Q8—2122no—
224C2×Q8—2121no—
225C2×Q8—2121no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1062no—
236C22×C4—1062no—
237C22×C4#1171992111yescharacteristic
238C22×C4#1171992111yescharacteristic
239C22×C4C2×C4×D5311yescharacteristic
240C22×C4C2×D21211yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—82no—
243C2×C8—24no—
244C2×C8—82no—
245C2×C8—2122no—
246C4⋊C4—22no—
247C4⋊C4—1064no—
248C4⋊C4—1064no—
249C4⋊C4—2124no—
250C4⋊C4—2124no—
251C4⋊C4—2122no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—2124no—
255C22⋊C4—2124no—
256C42—2122no—
257C42#1171994512yes—
258C42C2×C22⋊D5312yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#1313313512yes—
269C23C2×#1171992111yescharacteristic
270Q8—4242no—
271Q8—4242no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—1064no—
286C2×C4—2122no—
287C2×C4—2122no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4—2122no—
291C2×C4C4×C22⋊D5312yes—
292C2×C4#1313318611yescharacteristic
293C2×C4#1313319112yes—
294C2×C4#1313320011yescharacteristic
295C8—44no—
296C8—4242no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#1681050211yescharacteristic
310C22C2×#1313313511yescharacteristic
311C22#1681060511yescharacteristic
312C4—22no—
313C4—21no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—4242no—
318C4—2124no—
319C4#1681079112yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#2027511311yescharacteristic
325C2#2027518311yescharacteristic
326C2#2027523711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2973D53
hypocenterC53#9455
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20273433C2semidirect
4. #20272572C2semidirect
186. C53#9455semidirect

Maximal subgroups

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

subgroupindex
#202796682
#202734332
#202730562
#202725722
C53×#29732
#945553