Finite Group Databasebeta

← #28309493

Subgroups — #28309493

Order 13568.

476 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290581C211yesmaximal, characteristic, complemented
2#20289957C211yesmaximal, characteristic, complemented
3#20289698C214yesmaximal, complemented
4#20279717C211yesmaximal, characteristic
5#16811640—21no—
6#16811626—21no—
7C2×#13133278—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811468—21no—
11C2×#13133214C411yescharacteristic
12C2×#13133205—21no—
13C2×C4×Dic106C2211yescharacteristic
14#16810541C412yes—
15#16810541C2212yes—
16#16810525—21no—
17#16810524—21no—
18#16810524—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719940—22no—
21#13133284—41no—
22#13133284—41no—
23C4⋊#88324—41no—
24#13133278—42no—
25C22⋊#88324—22no—
26C4×#88324—41no—
27C22×Dic106D411yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212C2×C412yes—
31#13133214—41no—
32#13133214—41no—
33#13133205—42no—
34#13133205—42no—
35C4×Dic106—41no—
36C4×Dic106—41no—
37C2×C4⋊Dic53D411yescharacteristic
38C2×#11719919C2×C411yescharacteristic
39C2×#11719919C2311yescharacteristic
40C2×#11719921—22no—
41C2×#11719921—22no—
42C2×#11719921—22no—
43C2×#11719921—22no—
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4⋊C4×C53—21no—
47C2×C42×C53—21no—
48C2×C42×C53C2×C411yescharacteristic
49#95×C53C2×C412yes—
50C23×D53—22no—
51C22×C4×C53—22no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C2×D411yescharacteristic
58C22×Dic53C2×D411yescharacteristic
59C2×D212—42no—
60C2×D212—42no—
61C2×D212—42no—
62C2×D212—42no—
63C2×D212—42no—
64C2×D212—41no—
65C2×D212—41no—
66C2×D212—42no—
67C2×D212—82no—
68C2×Dic106—41no—
69C2×Dic106—41no—
70C2×Dic106D812yes—
71C2×Dic106SD1612yes—
72#11719940—42no—
73#11719940—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C4⋊C4×C53—42no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—42no—
80#11719921—44no—
81#11719921—42no—
82#11719921—42no—
83#11719921—82no—
84#11719921—82no—
85C4⋊Dic53—41no—
86#11719919—41no—
87#11719919—41no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C23×C53C2×C22⋊C411yescharacteristic
93C22×D53—42no—
94C22×D53—42no—
95C22×D53—42no—
96C22×D53—42no—
97C22×D53—42no—
98C22×D53—42no—
99C22×D53—42no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53#10212yes—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53#10212yes—
120C2×Dic53C2×D811yescharacteristic
121C2×Dic53C2×SD1611yescharacteristic
122D212—82no—
123D212—84no—
124D212—84no—
125D212—82no—
126Dic106—42no—
127Dic106—42no—
128#88324—82no—
129#88324—82no—
130#11882—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53C2×#9911yescharacteristic
134C22×C53C2×#10211yescharacteristic
135C22×C53#42111yescharacteristic
136C2×D53—82no—
137C2×D53—82no—
138C2×D53—82no—
139C2×D53—82no—
140C2×D53—82no—
141C2×D53—82no—
142C2×D53—82no—
143C4×C53—21no—
144C4×C53—22no—
145C4×C53—41no—
146C4×C53—82no—
147C4×C53#35512yes—
148Dic53—21no—
149Dic53—22no—
150Dic53—21no—
151Dic53—41no—
152Dic53—41no—
153#2548—531no—
154#1801—531no—
155#1461—531no—
156#1345—534no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53#151911yescharacteristic
160C2×C53#212211yescharacteristic
161C2×C53C2×#35511yescharacteristic
162D53—162no—
163C2×#127—531no—
164C2×#124—1061no—
165C2×C4×Q8—531no—
166#431—1061no—
167#423—1061no—
168C2×#101—1061no—
169#394—532no—
170#394—532no—
171#390—1061no—
172#390—1061no—
173#390—1061no—
174C4×C4⋊C4#615711yescharacteristic
175#343—532no—
176#324—532no—
177#324—1061no—
178C53#1188211yescharacteristic, complemented, Sylow
179C22×Q8—531no—
180C22×D4—532no—
181C22×D4—1061no—
182C22×D4—1061no—
183C2×M16—1062no—
184#127—2121no—
185#127—2121no—
186#124—2122no—
187#124—2122no—
188C4×Q8—2121no—
189C4×Q8—2121no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—531no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4—1062no—
198C2×C42—21no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42C2×#615711yescharacteristic
202C4⋊C8—2121no—
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205#101—2122no—
206C22⋊C8—1062no—
207C4×C8—2121no—
208#95C2×#615712yes—
209C24—1062no—
210C2×Q8—532no—
211C2×Q8—532no—
212C2×Q8—2121no—
213C2×Q8—2121no—
214C2×D4—4242no—
215C2×D4—2122no—
216C2×D4—2122no—
217C2×D4—2122no—
218C2×D4—2122no—
219C2×D4—2122no—
220C2×D4—2121no—
221C2×D4—2121no—
222C2×D4—2122no—
223C22×C4—22no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4C22⋊#615711yescharacteristic
229C22×C4C22⋊#615711yescharacteristic
230C22×C4C22×#615711yescharacteristic
231M16—2122no—
232M16—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C4⋊C4—42no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C4⋊C4—2121no—
239C22⋊C4—4242no—
240C22⋊C4—4242no—
241C22⋊C4—2124no—
242C22⋊C4—2122no—
243C22⋊C4—2122no—
244C42—21no—
245C42—21no—
246C42—42no—
247C42—1061no—
248C42—1061no—
249C42—1061no—
250C42—1061no—
251C23—2122no—
252C23—2122no—
253C23—2122no—
254C23—2122no—
255C23—2122no—
256C23—2122no—
257C23—2122no—
258C23C2×C22⋊#615711yescharacteristic
259Q8—2122no—
260Q8—2122no—
261D4—4244no—
262D4—4242no—
263D4—4244no—
264D4—4242no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—532no—
274C2×C4—531no—
275C2×C4—531no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4#1313327611yescharacteristic
285C2×C4#1313327611yescharacteristic
286C2×C4#1313327912yes—
287C8—4242no—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—4242no—
292C22—4242no—
293C22—4242no—
294C22—4242no—
295C22—4242no—
296C22—4242no—
297C22—4242no—
298C22C2×#1313327611yescharacteristic
299C22C2×#1313327911yescharacteristic
300C22#1681162211yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—82no—
305C4—1061no—
306C4—1062no—
307C4—1061no—
308C4—2121no—
309C4—2121no—
310C4#1681153212yes—
311C2—21no—
312C2—21no—
313C2—8482no—
314C2#2029015211yescharacteristic
315C2#2029097611yescharacteristic
316C2C2×#1681153211yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC2×#13133214C4
cosocleC2×#11719919C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#11882
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20290581C2semidirect
2. #20289957C2semidirect
3. #20289698C2semidirect
178. C53#11882semidirect

Maximal subgroups

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

subgroupindex
#202905812
#202899572
#202896982
#202797172
#1188253