Finite Group Databasebeta

← #28301311

Subgroups — #28301311

Order 13568.

994 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 156 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#20291159C212yesmaximal
2#20289614C214yesmaximal
3#20277094C211yesmaximal, characteristic
4C2×#13133296—22no—
5#16811566—22no—
6#16811565—22no—
7#16811565C414yescomplemented
8#16811564C2214yes—
9C4×C22⋊#6157—22no—
10C4×C22⋊#6157C414yescomplemented
11C23×C4×D53C2211yescharacteristic
12C2×#13133176C411yescharacteristic, complemented
13#16810484C412yescomplemented
14#16810480—22no—
15#16810478C2212yes—
16C4×#11719921—22no—
17C2×#117×C53C411yescharacteristic, complemented
18#13133296—44no—
19#13133296—44no—
20C2×C4⋊#6157—24no—
21C2×C22⋊#6157—24no—
22C2×C22⋊#6157—24no—
23C2×C22⋊#6157C2×C418yes—
24C2×C4×#6157—24no—
25(C2×#6157)⋊C4—24no—
26(C2×#6157)⋊C4—24no—
27C24×D53C2311yescharacteristic
28C23×Dic53C2×C411yescharacteristic
29C22×C4×D53—42no—
30C22×C4×D53—42no—
31C22×C4×D53C2×C412yes—
32C22×C4×D53C2×C412yes—
33C22×C4×D53D412yes—
34C22×C4×D53D411yescharacteristic
35C22×C4×D53D411yescharacteristic
36C22×C4×D53D412yes—
37C22×C4×D53Q811yescharacteristic
38C22×C4×D53Q811yescharacteristic
39#13133176—42no—
40C2×C4⋊Dic53—21no—
41C2×C22⋊Dic53C2×C412yes—
42C2×#11719921—24no—
43C2×C4×Dic53—21no—
44#13133124—24no—
45C23×C4×C53C2×C411yescharacteristic
46#117×C53—42no—
47C2×C4⋊C4×C53—21no—
48C2×C22⋊C4×C53C2×C412yes—
49C2×C42×C53—21no—
50C24×C53C22×C411yescharacteristic
51C23×D53—44no—
52C23×D53—41no—
53C23×D53—41no—
54C23×D53C22×C412yes—
55C23×D53C2×D411yescharacteristic
56C23×D53C2×D412yes—
57C23×D53C2×Q811yescharacteristic
58C22×#6157—28no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53C4211yescharacteristic
63C22×C4×C53C4211yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C4⋊C411yescharacteristic
67C22×C4×C53C4⋊C411yescharacteristic
68C22×Dic53—22no—
69C22×Dic53—41no—
70C22×Dic53—41no—
71C22×Dic53C4212yes—
72C22×Dic53C22⋊C412yes—
73C22×Dic53C4⋊C411yescharacteristic
74C22×Dic53C4⋊C411yescharacteristic
75C2×C4×D53—24no—
76C2×C4×D53—24no—
77C2×C4×D53—44no—
78C2×C4×D53—44no—
79C2×C4×D53—42no—
80C2×C4×D53—42no—
81C2×C4×D53—42no—
82C2×C4×D53—42no—
83C2×C4×D53—42no—
84C2×C4×D53—42no—
85C2×C4×D53C22⋊C412yes—
86C2×C4×D53C22⋊C412yes—
87C2×C4×D53C22⋊C412yes—
88C2×C4×D53C22⋊C412yes—
89C2×C4×D53C4⋊C412yes—
90C2×C4×D53C4⋊C412yes—
91C2×C4×D53C4⋊C412yes—
92C2×C4×D53C4⋊C412yes—
93C22⋊#6157—48no—
94C22⋊#6157—44no—
95C22⋊#6157—44no—
96C22⋊#6157—48no—
97C4⋊#6157—48no—
98C4×#6157—48no—
99C4⋊C4×C53—42no—
100C22⋊C4×C53—42no—
101C42×C53—42no—
102C22⋊Dic53—42no—
103#11719921—48no—
104C4⋊Dic53—42no—
105C4×Dic53—42no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—42no—
109C23×C53C2×C4211yescharacteristic
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53C2×C4⋊C411yescharacteristic
112C22×D53—22no—
113C22×D53—22no—
114C22×D53—24no—
115C22×D53—24no—
116C22×D53—24no—
117C22×D53—42no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—48no—
124C22×D53—48no—
125C22×D53C2×C22⋊C412yes—
126C22×D53C2×C4⋊C412yes—
127C2×#6157—48no—
128C2×#6157—48no—
129C2×#6157—48no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—44no—
137C2×C4×C53—42no—
138C2×C4×C53#9511yescharacteristic
139C2×C4×C53#9511yescharacteristic
140C2×C4×C53#9511yescharacteristic
141C2×C4×C53#9511yescharacteristic
142C2×C4×C53#9912yes—
143C2×C4×C53#9912yes—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—44no—
150C2×Dic53—42no—
151C2×Dic53#9512yes—
152C2×Dic53#9512yes—
153C4×D53—24no—
154C4×D53—24no—
155C4×D53—24no—
156C4×D53—24no—
157C4×D53—42no—
158C4×D53—42no—
159C4×D53—42no—
160C4×D53—42no—
161#8100—531nomaximal, Sylow
162C22×C53—22no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—42no—
168C22×C53—42no—
169C22×C53C2×#9511yescharacteristic
170C22×C53C2×#9911yescharacteristic
171C22×C53C2×#9911yescharacteristic
172C2×D53—22no—
173C2×D53—22no—
174C2×D53—22no—
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—44no—
178C2×D53—44no—
179C2×D53—44no—
180C2×D53—48no—
181C2×D53—44no—
182C2×D53#41014yes—
183#6157—88no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—82no—
187C4×C53#34212yes—
188C4×C53#34212yes—
189Dic53—22no—
190Dic53—22no—
191Dic53—82no—
192#2280—531no—
193#2280—532no—
194#1304—534no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—41no—
198C2×C53—41no—
199C2×C53C2×#34211yescharacteristic
200C2×C53#174012yes—
201D53—24no—
202D53—42no—
203C24×C4—531no—
204C2×#117—531no—
205C2×#117—1062no—
206C2×#117#615711yescharacteristic, complemented
207#381—532no—
208#381—1062no—
209#380—534no—
210#380—1062no—
211#380—1062no—
212#379—534no—
213#379—532no—
214C4×C22⋊C4—534no—
215C4×C22⋊C4—1062no—
216C4×C22⋊C4—1062no—
217C53#810011yescharacteristic, complemented, Sylow
218C25—531no—
219C23×C4—531no—
220C23×C4—532no—
221C23×C4—532no—
222C23×C4—532no—
223C23×C4—531no—
224C23×C4—531no—
225C23×C4—531no—
226C23×C4—531no—
227C23×C4—532no—
228C23×C4—2122no—
229C23×C4—2122no—
230C23×C4C2×#615711yescharacteristic
231#117—42no—
232#117—2124no—
233#117—2124no—
234#117—2122no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—1064no—
237C2×C4⋊C4—1061no—
238C2×C22⋊C4—538no—
239C2×C22⋊C4—532no—
240C2×C22⋊C4—1064no—
241C2×C22⋊C4—1064no—
242C2×C22⋊C4—1064no—
243C2×C22⋊C4C2×#615712yes—
244C2×C42—21no—
245C2×C42—1064no—
246C2×C42—1061no—
247#95—1064no—
248#95—1064no—
249#95—1064no—
250C24—532no—
251C24—531no—
252C24—532no—
253C24—531no—
254C24—2121no—
255C24—2121no—
256C24—2124no—
257C24C22×#615711yescharacteristic
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—532no—
262C22×C4—532no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—532no—
266C22×C4—532no—
267C22×C4—532no—
268C22×C4—532no—
269C22×C4—532no—
270C22×C4—532no—
271C22×C4—532no—
272C22×C4—532no—
273C22×C4—1064no—
274C22×C4—1064no—
275C22×C4—1068no—
276C22×C4—1062no—
277C22×C4—2124no—
278C22×C4—2124no—
279C22×C4—2122no—
280C22×C4—2122no—
281C22×C4—2122no—
282C22×C4—2122no—
283C22×C4—2122no—
284C22×C4—2122no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4C4×#615711yescharacteristic
288C22×C4C4×#615711yescharacteristic
289C22×C4C4⋊#615711yescharacteristic
290C22×C4C4⋊#615711yescharacteristic
291C22×C4C22⋊#615711yescharacteristic
292C22×C4C22⋊#615711yescharacteristic
293C4⋊C4—42no—
294C4⋊C4—2122no—
295C4⋊C4—2128no—
296C22⋊C4—42no—
297C22⋊C4—2128no—
298C22⋊C4—2128no—
299C22⋊C4—2124no—
300C22⋊C4—2124no—
301C22⋊C4—2122no—
302C22⋊C4—2128no—
303C42—42no—
304C42—2128no—
305C42—2122no—
306C23—21no—
307C23—21no—
308C23—42no—
309C23—532no—
310C23—532no—
311C23—1064no—
312C23—1064no—
313C23—1064no—
314C23—1062no—
315C23—1062no—
316C23—2122no—
317C23—2121no—
318C23—2121no—
319C23—2122no—
320C23—2122no—
321C23—2122no—
322C23—2128no—
323C23—2128no—
324C23C2×C4×#615711yescharacteristic
325C23C2×C22⋊#615711yescharacteristic
326C23C2×C4⋊#615711yescharacteristic
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—532no—
336C2×C4—532no—
337C2×C4—1062no—
338C2×C4—1062no—
339C2×C4—1064no—
340C2×C4—1064no—
341C2×C4—1064no—
342C2×C4—1064no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2124no—
351C2×C4—2128no—
352C2×C4—2128no—
353C2×C4—2128no—
354C2×C4—2122no—
355C2×C4(C2×#6157)⋊C411yescharacteristic
356C2×C4(C2×#6157)⋊C411yescharacteristic
357C2×C4(C2×#6157)⋊C411yescharacteristic
358C2×C4(C2×#6157)⋊C411yescharacteristic
359C2×C4#1313327512yes—
360C2×C4#1313327612yes—
361C22—22no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22—534no—
369C22—1062no—
370C22—1062no—
371C22—1062no—
372C22—1062no—
373C22—1062no—
374C22—2124no—
375C22—2124no—
376C22—2124no—
377C22—2128no—
378C22—2124no—
379C22C2×(C2×#6157)⋊C411yescharacteristic
380C22C2×#1313327511yescharacteristic
381C22C2×#1313327611yescharacteristic
382C4—21no—
383C4—21no—
384C4—82no—
385C4—1062no—
386C4—1062no—
387C4—4242no—
388C4—4248no—
389C4#1681150112yes—
390C4#1681150112yes—
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—1064no—
396C2—2122no—
397C2C2×#1681150111yescharacteristic
398C2#2029041312yes—
subgroupG/H
centerC2×C4(C2×#6157)⋊C4
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#95
prime radical, cosocleC24×D53C23
hypercenterC2×#117#6157
hypocenterC53#8100
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
7. #16811565C4semidirect
10. C4×C22⋊#6157C4semidirect
12. C2×#13133176C4semidirect
13. #16810484C4semidirect
17. C2×#117×C53C4semidirect
206. C2×#117#6157semidirect
217. C53#8100semidirect

Maximal subgroups

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

subgroupindex
#202911592
#202896142
#202770942
#810053