Finite Group Databasebeta

← #28306278

Subgroups — #28306278

Order 13568.

598 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 108 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#20290442C211yesmaximal, characteristic
2#20289891C211yesmaximal, characteristic
3#20289588C214yesmaximal
4#20283068C211yesmaximal, characteristic
5C2×#13133274C2212yes—
6C2×#13133272—21no—
7C2×(C2×#6157)⋊C4—21no—
8C2×(C2×#6157)⋊C4—21no—
9C2×(C2×#6157)⋊C4C2212yes—
10#16811489—21no—
11C2×#13133217C411yescharacteristic, complemented
12C2×#13133197—21no—
13C2×C42×D53C2211yescharacteristic
14#16810528—21no—
15#16810522—21no—
16#16810522C412yescomplemented
17#16810480—21no—
18#16810480C2212yes—
19C53×#384C411yescharacteristic, complemented
20C23×#6157—22no—
21C22×#88324—22no—
22#13133274—22no—
23#13133274Q814yescomplemented
24#13133272—42no—
25(C2×#6157)⋊C4—24no—
26(C2×#6157)⋊C4—24no—
27(C2×#6157)⋊C4—42no—
28C22×Dic106—21no—
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53C2311yescharacteristic
32#13133217—42no—
33#13133197—44no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53C2×C412yes—
37C2×#11719921—22no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C42×D53—42no—
42#13133124—22no—
43#13133124—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C42×C53C2×C411yescharacteristic
48C23×D53C2×D411yescharacteristic
49C22×#6157—24no—
50C22×#6157—42no—
51C22×#6157—42no—
52C22×#6157—42no—
53C22×#6157—42no—
54C22×#6157—42no—
55C22×#6157—42no—
56C22×C4×C53—22no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—22no—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C2×C4×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—41no—
69C2×C4×D53—41no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×C4×D53C2×Q812yes—
73C2×C4×D53#4112yes—
74C2×Dic106—42no—
75C2×Dic106—42no—
76C2×Dic106—42no—
77C2×#88324—24no—
78C2×#88324—42no—
79C2×#88324—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—42no—
83C42×C53—22no—
84#11719921—44no—
85C4⋊Dic53—44no—
86C4⋊Dic53—42no—
87C4⋊Dic53—42no—
88C4×Dic53—22no—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—21no—
96C22×D53#12411yescharacteristic
97C22×D53C4⋊Q811yescharacteristic
98C22×D53AGL(1,ℤ/8)12yes—
99C22×D53#13712yes—
100C2×#6157—44no—
101C2×#6157—44no—
102C2×#6157—44no—
103C2×#6157—82no—
104C2×#6157—82no—
105C2×#6157—82no—
106C2×#6157—82no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53#9912yes—
117C2×C4×C53#10412yes—
118C2×C4×C53C4×Q812yes—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53C22⋊Q812yes—
131C4×D53—42no—
132C4×D53—42no—
133C4×D53—44no—
134C4×D53—44no—
135Dic106—84no—
136#88324—44no—
137#9888—531nomaximal, Sylow
138C22×C53#39111yescharacteristic
139C22×C53C2×#9911yescharacteristic
140C22×C53#41011yescharacteristic
141C22×C53#41811yescharacteristic
142C22×C53#41911yescharacteristic
143C22×C53C2×#10411yescharacteristic
144C22×C53#42111yescharacteristic
145C2×D53—21no—
146C2×D53—21no—
147C2×D53—21no—
148C2×D53—21no—
149C2×D53—22no—
150C2×D53—22no—
151C2×D53—22no—
152C2×D53—22no—
153C2×D53#48912yes—
154C2×D53#50112yes—
155#6157—84no—
156C4×C53—22no—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—82no—
160Dic53—22no—
161Dic53—41no—
162Dic53—41no—
163Dic53—82no—
164#2857—531no—
165#1749—531no—
166#1434—531no—
167#1294—534no—
168C2×C53#150611yescharacteristic
169C2×C53#150711yescharacteristic
170C2×C53#151111yescharacteristic
171C2×C53#151411yescharacteristic
172C2×C53#198611yescharacteristic
173C2×C53#199311yescharacteristic
174C2×C53#199611yescharacteristic
175D53—22no—
176D53—22no—
177C2×C4⋊Q8—531no—
178C2×C22⋊Q8—1061no—
179C22×C42—531no—
180C2×C22⋊C8—532no—
181C2×#97—1061no—
182#391—1061no—
183#389—532no—
184#389—1061no—
185#384#615711yescharacteristic, complemented
186#380—532no—
187#380—1061no—
188C2×#95—532no—
189C2×#95—1061no—
190C2×#95—1061no—
191#336—1061no—
192C53#988811yescharacteristic, complemented, Sylow
193C22×Q8—1061no—
194C23×C4—531no—
195C23×C4—531no—
196C23×C4—531no—
197C23×C4—1062no—
198C22×C8—1062no—
199C4⋊Q8—2122no—
200C22⋊Q8—2124no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—532no—
204C2×C4⋊C4—1062no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4C2×#615712yes—
207C2×C22⋊C4—1062no—
208C2×C42—531no—
209C2×C42—531no—
210C2×C42—531no—
211C2×C42—2122no—
212C2×C42C2×#615711yescharacteristic
213C22⋊C8—534no—
214C22⋊C8—1062no—
215#97—2122no—
216#95—1062no—
217#95—1062no—
218#95—1064no—
219#95—1064no—
220#95—2122no—
221C24—531no—
222C2×Q8—2122no—
223C2×Q8—2122no—
224C2×Q8—2122no—
225C22×C4—22no—
226C22×C4—531no—
227C22×C4—532no—
228C22×C4—532no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4—1062no—
235C22×C4—1064no—
236C22×C4—1062no—
237C22×C4—2121no—
238C22×C4—2121no—
239C22×C4—2121no—
240C22×C4—2121no—
241C22×C4—2122no—
242C22×C4—2122no—
243C22×C4—2122no—
244C22×C4—2122no—
245C22×C4—2122no—
246C22×C4—2122no—
247C22×C4C22⋊#615711yescharacteristic
248C22×C4C22⋊#615711yescharacteristic
249C22×C4C22×#615711yescharacteristic
250C2×C8—1064no—
251C2×C8—2122no—
252C2×C8—2122no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—2122no—
257C4⋊C4—2124no—
258C4⋊C4—2122no—
259C22⋊C4—2124no—
260C42—22no—
261C42—1062no—
262C42—2121no—
263C42—2121no—
264C23—531no—
265C23—531no—
266C23—532no—
267C23—532no—
268C23—1061no—
269C23—1061no—
270C23—1061no—
271C23—1061no—
272C23C2×C22⋊#615711yescharacteristic
273Q8—4244no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—532no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—4242no—
293C2×C4—4242no—
294C2×C4—4242no—
295C2×C4—4242no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2124no—
299C2×C4—2124no—
300C2×C4—2124no—
301C2×C4—2124no—
302C2×C4—2124no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2122no—
306C2×C4#1313327612yes—
307C2×C4#1313328212yes—
308C2×C4Q8×#615712yes—
309C8—2124no—
310C22—532no—
311C22—532no—
312C22—1062no—
313C22—1062no—
314C22—1061no—
315C22—1061no—
316C22—1062no—
317C22—1062no—
318C22—1061no—
319C22—1061no—
320C22(C2×#6157)⋊Q811yescharacteristic
321C22C2×#1313327611yescharacteristic
322C22#1681160311yescharacteristic
323C22#1681161511yescharacteristic
324C22#1681161611yescharacteristic
325C22C2×#1313328211yescharacteristic
326C22#1681162111yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—82no—
331C4—1062no—
332C4—4244no—
333C4—4242no—
334C4—2121no—
335C4—2121no—
336C2—1062no—
337C2—1062no—
338C2#2029010911yescharacteristic
339C2#2029011211yescharacteristic
340C2#2029012311yescharacteristic
341C2#2029013511yescharacteristic
342C2#2029089011yescharacteristic
343C2#2029089911yescharacteristic
344C2#2029090511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenter#384#6157
hypocenterC53#9888
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
11. C2×#13133217C4semidirect
16. #16810522C4semidirect
19. C53×#384C4semidirect
23. #13133274Q8semidirect
185. #384#6157semidirect
192. C53#9888semidirect

Maximal subgroups

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

subgroupindex
#202904422
#202898912
#202895882
#202830682
#988853