Finite Group Databasebeta

← #27294983

Subgroups — #27294983

Order 13568.

382 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280413C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272738C212yesmaximal
4#20272714C212yesmaximal
5C53×#1820C211yesmaximal, characteristic
6C2×#13133209C411yescharacteristic, complemented
7#16810676—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810553C411yescharacteristic, complemented
11#16810548C2211yescharacteristic
12#16810518—21no—
13#16810517—21no—
14#16810507—21no—
15#16810496—21no—
16#16810469C411yescharacteristic, complemented
17#16810349—21no—
18#16810347C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#381—21no—
21C53×#342—21no—
22C53×#342C2212yes—
23#13133209—41no—
24#13133209—41no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—21no—
34C4⋊#88322—41no—
35C22⋊#88322—22no—
36C4×#88322—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45#117×C53—41no—
46#117×C53—41no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53C2311yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53D411yescharacteristic
54#95×C53—41no—
55#95×C53—41no—
56C24×C53C2×D411yescharacteristic
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—41no—
61C22×C4×C53—42no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—42no—
77C22⋊C4×C53—42no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C22⋊Dic53—42no—
83C22⋊Dic53—42no—
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86#11719919—42no—
87#11719919—42no—
88#11719919—42no—
89#11719919—41no—
90#11719919—41no—
91C4×Dic53—42no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53#13512yes—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—82no—
115C2×C4×C53—82no—
116C2×C4×C53—82no—
117C2×C4×C53#9911yescharacteristic
118C2×C4×C53#9911yescharacteristic
119C2×Dic53—22no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131#88322—82no—
132#11846—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—22no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—81no—
140C22×C53C2×#9911yescharacteristic
141C22×C53#41611yescharacteristic
142C22×C53#42111yescharacteristic
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—162no—
151Dic53—42no—
152Dic53—81no—
153Dic53—81no—
154Dic53—81no—
155#2611—531no—
156#1820D5311yescharacteristic
157#1458—531no—
158#1352—532no—
159#1344—532no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—42no—
163C2×C53#151611yescharacteristic
164C2×C53#212311yescharacteristic
165C2×C53#212611yescharacteristic
166C2×#125—531no—
167C2×#117C2×D5311yescharacteristic
168#431—1061no—
169#400—531no—
170#399—531no—
171#398—531no—
172#397—531no—
173#389—1061no—
174#388—1061no—
175#387—1061no—
176#385—1061no—
177#381—21no—
178#376—531no—
179#342—21no—
180#342C2×D5312yes—
181#324—1061no—
182#323—532no—
183C53#1184611yescharacteristic, complemented, Sylow
184C23×C4C22⋊D5311yescharacteristic
185#125—2121no—
186#125—2121no—
187#117—41no—
188#117—41no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4C4×D5311yescharacteristic
196C2×C4⋊C4D21211yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1061no—
199C2×C22⋊C4—1061no—
200C2×C22⋊C4C22⋊D5311yescharacteristic
201C2×C22⋊C4C22×D5311yescharacteristic
202C2×C42—1061no—
203C2×C42C4×D5311yescharacteristic
204C2×C42D21211yescharacteristic
205C4⋊C8—2121no—
206C22⋊C8—1062no—
207C4×C8—2121no—
208#95—41no—
209#95—41no—
210#95—531no—
211#95—531no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217C24C2×C22⋊D5311yescharacteristic
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4—1061no—
227C22×C4#1171992111yescharacteristic
228C22×C4#1171992111yescharacteristic
229C22×C4#1171992111yescharacteristic
230C22×C4#1171992111yescharacteristic
231C22×C4C2×C4×D5311yescharacteristic
232C22×C4C2×D21211yescharacteristic
233C2×C8—2122no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C4⋊C4—2122no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C4⋊C4—2121no—
242C4⋊C4—2121no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—2122no—
248C22⋊C4—2122no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—2122no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×#1171992111yescharacteristic
259C23#1313323712yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—1062no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4#1313313111yescharacteristic
291C2×C4#1313313111yescharacteristic
292C8—4242no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1313313111yescharacteristic
301C22#1681061711yescharacteristic
302C22#1681064111yescharacteristic
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—162no—
311C4—4241no—
312C4—4241no—
313C4—4241no—
314C4—2122no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2027326911yescharacteristic
319C2#2027636811yescharacteristic
320C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1820C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1820D53
hypocenterC53#11846
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
6. C2×#13133209C4semidirect
8. #16810562C4semidirect
10. #16810553C4semidirect
16. #16810469C4semidirect
183. C53#11846semidirect

Maximal subgroups

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

subgroupindex
#202804132
#202730292
#202727382
#202727142
C53×#18202
#1184653