Finite Group Databasebeta

← #28301571

Subgroups — #28301571

Order 13568.

614 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C2×#16811650C211yesmaximal, characteristic
2#20290462C211yesmaximal, characteristic
3#20290450C212yesmaximal, complemented
4#20289556C212yesmaximal, complemented
5#20273030C211yesmaximal, characteristic
6C22×D53⋊C8C2211yescharacteristic
7C23×#88324C411yescharacteristic, complemented
8#16811650—22no—
9#16811650—22no—
10C2×#13133284C411yescharacteristic, complemented
11C2×#13133275C412yes—
12C2×#13133274C2211yescharacteristic
13C2×C22⋊#88324C411yescharacteristic, complemented
14C2×C4×#88324C411yescharacteristic, complemented
15C2×C8×#6157—22no—
16C4×C22⋊#6157C2212yes—
17#16811501C412yes—
18(C2×#6157)⋊C8—22no—
19C2×#13133177C2211yescharacteristic
20C22⋊C8×D53C2212yes—
21#16810348C412yescomplemented
22#16810347C411yescharacteristic, complemented
23C53×#323C411yescharacteristic, complemented
24C2×D53⋊C8—42no—
25C2×D53⋊C8D412yes—
26C2×D53⋊C8D412yes—
27C22×#88324—21no—
28C22×#88324—21no—
29C22×#88324—42no—
30C22×#88324C2×C411yescharacteristic
31C22×#88324C2×C411yescharacteristic
32C22×#88324C2×C411yescharacteristic
33C22×#88324C2×C411yescharacteristic
34C2×C22⋊#6157C2×C414yes—
35C2×C4×#6157—22no—
36#13133284—22no—
37#13133275C818yescomplemented
38#13133274—22no—
39C22⋊#88324—22no—
40C4×#88324—21no—
41C4×#88324—21no—
42C8×#6157—44no—
43(C2×#6157)⋊C4—22no—
44C22×C22⋊D53C2×C411yescharacteristic
45C23×Dic53C2×C411yescharacteristic
46C22×C4×D53C2311yescharacteristic
47C2×C8×D53—22no—
48#13133177—22no—
49#13133177—22no—
50C2×#11719919C2×C411yescharacteristic
51C2×#11719921C2×C411yescharacteristic
52C2×C4×Dic53C2×C411yescharacteristic
53C22⋊#88322C2×C412yes—
54#13133129—22no—
55C2×C22⋊C4×C53C2×C411yescharacteristic
56C22⋊C8×C53C2×C412yes—
57C24×C53C4211yescharacteristic
58C23×D53C22×C411yescharacteristic
59C22×#6157—24no—
60C22×C4×C53C4211yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C2×C22⋊D53—22no—
63C2×C22⋊D53—22no—
64C2×C22⋊D53C2×C814yes—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53C4211yescharacteristic
70C22×Dic53C4211yescharacteristic
71C22×Dic53C22×C411yescharacteristic
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C4×D53C2×D412yes—
75C2×C4×D53#4112yes—
76C22⋊#6157—24no—
77C22⋊#6157C2×C818yescomplemented
78C2×#88324—22no—
79C2×#88324—21no—
80C2×#88324—21no—
81C2×#88324—41no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—41no—
85C2×#88324—41no—
86C2×#88324C22⋊C411yescharacteristic
87C2×#88324C22⋊C412yes—
88C2×#88324C22⋊C411yescharacteristic
89C4×#6157—42no—
90C4×#6157—42no—
91D53⋊C8—24no—
92D53⋊C8—42no—
93C2×C8×C53—22no—
94C22⋊C4×C53—22no—
95#11719921—22no—
96#11719919—22no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C2×#88322—22no—
100C8×D53—42no—
101C8×D53—42no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C4×C812yescomplemented
107C23×C53C2×C4211yescharacteristic
108C22×D53—41no—
109C22×D53—41no—
110C22×D53C4×D412yes—
111C22×D53C22×C812yes—
112C22×D53#13112yes—
113C2×#6157—28no—
114C2×#6157—44no—
115C2×#6157—44no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53C4×D412yes—
120C22⋊D53—28no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—42no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53C4×C812yescomplemented
131C2×Dic53C2×C22⋊C411yescharacteristic
132C2×Dic53#11711yescharacteristic
133C4×D53—22no—
134C4×D53—22no—
135C4×D53—42no—
136#88324—21no—
137#88324—22no—
138#88324—41no—
139#88324—41no—
140#88324#9912yescomplemented
141C8×C53—42no—
142#88322—42no—
143C8×#99—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53C4×C22⋊C411yescharacteristic
152C22×C53C2×C4×C811yescharacteristic
153C22×C53#40511yescharacteristic
154C2×D53—22no—
155C2×D53—22no—
156C2×D53—42no—
157C2×D53—42no—
158C2×D53C8×D414yes—
159#6157—48no—
160C4×C53—22no—
161C4×C53—41no—
162Dic53—22no—
163Dic53—21no—
164Dic53—41no—
165Dic53C2×#9911yescharacteristic
166Dic53#41011yescharacteristic
167C2×C8×D4—531no—
168C4×#99—531no—
169C8×C22⋊C4—532no—
170#1459—531no—
171#1282—532no—
172C2×C53—21no—
173C2×C53—21no—
174C2×C53—42no—
175C2×C53C8×C22⋊C411yescharacteristic
176C2×C53C4×#9911yescharacteristic
177C2×C53#175711yescharacteristic
178D53—24no—
179C23×C8—531no—
180C23×C8—531no—
181C2×C4×D4—531no—
182C8×D4—1062no—
183C8×D4—1062no—
184C2×C4⋊C8—531no—
185C2×#99—532no—
186C2×C22⋊C8—532no—
187C2×C22⋊C8—531no—
188C2×C22⋊C8—531no—
189C2×C4×C8—531no—
190C2×C4×C8—1062no—
191C4×C22⋊C4—532no—
192#342—532no—
193#336—1062no—
194#324—532no—
195#323—531no—
196#323#615711yescharacteristic, complemented
197C53C8×#9911yescharacteristic, complemented, Sylow
198C22×D4—531no—
199C23×C4—531no—
200C23×C4—531no—
201C22×C8—531no—
202C22×C8—532no—
203C22×C8—532no—
204C22×C8—531no—
205C22×C8—531no—
206C22×C8—531no—
207C22×C8—1062no—
208C22×C8—1061no—
209C22×C8—1061no—
210C22×C8—2122no—
211C22×C8—2122no—
212C4×D4—1062no—
213C4×D4—1062no—
214C2×C4⋊C4—531no—
215C2×C22⋊C4—534no—
216C2×C22⋊C4—531no—
217C2×C22⋊C4C2×#615711yescharacteristic
218C2×C42—531no—
219C2×C42—1062no—
220C4⋊C8—1062no—
221#99—538no—
222C22⋊C8—532no—
223C22⋊C8—1062no—
224C22⋊C8—1062no—
225C22⋊C8—1062no—
226C22⋊C8C2×#615712yes—
227C4×C8—1061no—
228C4×C8—1061no—
229C4×C8—2124no—
230#95—1062no—
231C24—531no—
232C24C4×#615711yescharacteristic
233C2×D4—534no—
234C2×D4—1062no—
235C2×D4—1062no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—532no—
239C22×C4—532no—
240C22×C4—531no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—1064no—
244C22×C4—2121no—
245C22×C4—2121no—
246C22×C4—2121no—
247C22×C4—2121no—
248C22×C4C4×#615711yescharacteristic
249C22×C4C22×#615711yescharacteristic
250C2×C8—22no—
251C2×C8—531no—
252C2×C8—531no—
253C2×C8—532no—
254C2×C8—1062no—
255C2×C8—1064no—
256C2×C8—1061no—
257C2×C8—1061no—
258C2×C8—1062no—
259C2×C8—2122no—
260C2×C8—2122no—
261C2×C8—2122no—
262C2×C8—2121no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2121no—
266C2×C8—2121no—
267C4⋊C4—1062no—
268C22⋊C4—22no—
269C22⋊C4—538no—
270C22⋊C4—1062no—
271C22⋊C4—1064no—
272C42—1061no—
273C42—1061no—
274C42—2122no—
275C42—2122no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—532no—
281C23—532no—
282C23—532no—
283C23—2121no—
284C23—2121no—
285C23C8×#615712yescomplemented
286C23C2×C4×#615711yescharacteristic
287D4—1068no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—531no—
292C2×C4—531no—
293C2×C4—532no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1068no—
301C2×C4—1061no—
302C2×C4—2122no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2122no—
306C2×C4—2124no—
307C2×C4—2124no—
308C2×C4—2121no—
309C2×C4D4×#615712yes—
310C8—42no—
311C8—532no—
312C8—1062no—
313C8—1061no—
314C8—2122no—
315C8—2121no—
316C8—2121no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—534no—
325C22—1062no—
326C22—1062no—
327C22—2122no—
328C22—2122no—
329C22C22⋊C4×#615711yescharacteristic
330C22C2×C8×#615711yescharacteristic
331C22#1681158911yescharacteristic
332C4—22no—
333C4—41no—
334C4—531no—
335C4—531no—
336C4—1062no—
337C4—1061no—
338C4—2128no—
339C4—2121no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—1064no—
344C2#2029045011yescharacteristic
345C2#99×#615711yescharacteristic
346C2#2029047211yescharacteristic
subgroupG/H
centerC22C22⋊C4×#6157
commutatorC22×C53C2×C4×C8
FrattiniC22×C4C22×#6157
FittingC53×#323C4
socleC22×C53C4×C22⋊C4
prime radicalC22×C22⋊D53C2×C4
cosocleC22×C4×D53C23
hypercenter#323#6157
hypocenterC53C8×#99
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
3. #20290450C2semidirect
4. #20289556C2semidirect
7. C23×#88324C4semidirect
10. C2×#13133284C4semidirect
13. C2×C22⋊#88324C4semidirect
14. C2×C4×#88324C4semidirect
21. #16810348C4semidirect
22. #16810347C4semidirect
23. C53×#323C4semidirect
37. #13133275C8semidirect
77. C22⋊#6157C2×C8semidirect
106. C23×C53C4×C8semidirect
130. C2×Dic53C4×C8semidirect
140. #88324#99semidirect
196. #323#6157semidirect
197. C53C8×#99semidirect
285. C23C8×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168116502
#202904622
#202904502
#202895562
#202730302
C8×#9953