Finite Group Databasebeta

← #28314070

Subgroups — #28314070

Order 13568.

1214 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291391C212yesmaximal
2#20289978C214yesmaximal, complemented
3#20289963C214yesmaximal
4#20289958C214yesmaximal, complemented
5C22×Q8×Dic53C211yesmaximal, characteristic
6#16811644—22no—
7#16811643C2214yes—
8#16811640—22no—
9#16811639C2214yes—
10#16811626—22no—
11#16811593C2214yes—
12C4×#11719940—22no—
13#16811468C22116yes—
14C2×Q8×Dic53—22no—
15C2×Q8×Dic53—22no—
16C2×Q8×Dic53—22no—
17C2×Q8×Dic53—22no—
18C2×Q8×Dic53C412yes—
19C2×Q8×Dic53C412yes—
20C2×Q8×Dic53C2212yes—
21C2×Q8×Dic53C2212yes—
22C22×C4⋊Dic53C411yescharacteristic
23C22×C4⋊Dic53C2212yes—
24C22×C4×Dic53C412yes—
25C22×C4×Dic53C2211yescharacteristic
26C23×Q8×C53C411yescharacteristic
27C2×#11719940—24no—
28#13133284—24no—
29C4⋊#88324—24no—
30C22⋊#88324—24no—
31C22⋊#88324C2318yes—
32#13133272—24no—
33C4×#88324—24no—
34C23×Dic53C2×C411yescharacteristic
35C23×Dic53C2×C412yes—
36C23×Dic53C2311yescharacteristic
37Q8×Dic53—24no—
38Q8×Dic53—24no—
39Q8×Dic53—24no—
40Q8×Dic53—24no—
41Q8×Dic53—24no—
42Q8×Dic53—24no—
43Q8×Dic53—24no—
44Q8×Dic53—24no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—24no—
48C2×C4⋊Dic53—24no—
49C2×C4⋊Dic53C2×C412yes—
50C2×C4⋊Dic53C2×C412yes—
51C2×C4⋊Dic53C2×C414yes—
52C2×C4⋊Dic53C2314yes—
53C2×C4×Dic53—22no—
54C2×C4×Dic53—22no—
55C2×C4×Dic53—22no—
56C2×C4×Dic53—22no—
57C2×C4×Dic53C2×C414yes—
58C2×C4×Dic53C2×C414yes—
59C2×C4×Dic53C2×C411yescharacteristic
60C2×C4×Dic53C2×C411yescharacteristic
61C2×C4×Dic53D412yes—
62C2×C4×Dic53D412yes—
63C2×C4×Dic53D412yes—
64C2×C4×Dic53D412yes—
65C2×C4×Dic53C2311yescharacteristic
66C2×C4×Dic53C2311yescharacteristic
67C22×Q8×C53—28no—
68C22×Q8×C53—22no—
69C22×Q8×C53—22no—
70C22×Q8×C53C2×C412yes—
71C22×Q8×C53C2×C412yes—
72C23×C4×C53C2×C412yes—
73C23×C4×C53C2×C411yescharacteristic
74C24×C53C22×C411yescharacteristic
75C2×Q8×C53—216no—
76C2×Q8×C53—216no—
77C2×Q8×C53—24no—
78C2×Q8×C53—24no—
79C2×Q8×C53—28no—
80C2×Q8×C53—24no—
81C2×Q8×C53M1618yescomplemented
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—28no—
85C22×C4×C53—24no—
86C22×C4×C53C22⋊C412yes—
87C22×C4×C53C22⋊C412yes—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C22×C414yes—
90C22×C4×C53C22×C411yescharacteristic
91C22×Dic53—24no—
92C22×Dic53—22no—
93C22×Dic53—22no—
94C22×Dic53—42no—
95C22×Dic53—44no—
96C22×Dic53C22⋊C411yescharacteristic
97C22×Dic53C22⋊C411yescharacteristic
98C22×Dic53C22⋊C411yescharacteristic
99C22×Dic53C22⋊C411yescharacteristic
100C22×Dic53C22×C411yescharacteristic
101C22×Dic53C22×C411yescharacteristic
102C22×Dic53C22×C414yes—
103C22×Dic53C22×C411yescharacteristic
104C22×Dic53C2×D412yes—
105C22×Dic53C2×D412yes—
106C22×Dic53C2411yescharacteristic
107#11719940—48no—
108C2×#88324—28no—
109C4⋊Dic53—24no—
110C4⋊Dic53—24no—
111C4⋊Dic53—28no—
112C4⋊Dic53—28no—
113C4×Dic53—24no—
114C4×Dic53—24no—
115C4×Dic53—24no—
116C4×Dic53—24no—
117C4×Dic53C22⋊C412yes—
118C4×Dic53C22⋊C412yes—
119C4×Dic53C22⋊C412yes—
120C4×Dic53C22⋊C412yes—
121C4×Dic53C2×D412yes—
122C4×Dic53C2×D412yes—
123C4×Dic53C2×D412yes—
124C4×Dic53C2×D412yes—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—22no—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53C2×C22⋊C411yescharacteristic
130C23×C53C23×C411yescharacteristic
131Q8×C53—216no—
132Q8×C53—216no—
133C2×C4×C53—28no—
134C2×C4×C53—28no—
135C2×C4×C53—24no—
136C2×C4×C53—22no—
137C2×C4×C53—24no—
138C2×C4×C53—24no—
139C2×C4×C53—24no—
140C2×C4×C53C2×C22⋊C412yes—
141C2×C4×C53C2×C22⋊C412yes—
142C2×C4×C53C2×M1618yes—
143C2×C4×C53C2×M1614yes—
144C2×Dic53—24no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—22no—
150C2×Dic53—24no—
151C2×Dic53—41no—
152C2×Dic53—42no—
153C2×Dic53—41no—
154C2×Dic53—44no—
155C2×Dic53—44no—
156C2×Dic53—44no—
157C2×Dic53#10114yes—
158C2×Dic53C2×C22⋊C411yescharacteristic
159C2×Dic53C2×C22⋊C411yescharacteristic
160C2×Dic53C2×C22⋊C412yes—
161C2×Dic53C2×C22⋊C411yescharacteristic
162C2×Dic53C2×C22⋊C411yescharacteristic
163C2×Dic53C22×D412yes—
164#88324—48no—
165#17365—531nomaximal, Sylow
166C22×C53—28no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53#41012yes—
174C22×C53C2×#10112yes—
175C22×C53C22×C22⋊C411yescharacteristic
176C22×C53C22×M1612yes—
177C4×C53—22no—
178C4×C53—28no—
179C4×C53#40714yes—
180Dic53—22no—
181Dic53—22no—
182Dic53—21no—
183Dic53—21no—
184Dic53—44no—
185Dic53C2×#10112yes—
186Dic53C2×#10112yes—
187C22×C4×Q8—531no—
188#2920—532no—
189#1467—534no—
190#1463—534no—
191#1461—534no—
192C2×C53—22no—
193C2×C53—22no—
194C2×C53—22no—
195C2×C53C2×#40711yescharacteristic
196C2×C53C2×#41011yescharacteristic
197C2×C53C22×#10111yescharacteristic
198C23×Q8#615711yescharacteristic
199C2×C4×Q8—532no—
200C2×C4×Q8—532no—
201C2×C4×Q8—532no—
202C2×C4×Q8—532no—
203C2×C4×Q8—1062no—
204C2×C4×Q8—1062no—
205C2×C4×Q8—1062no—
206C2×C4×Q8—1062no—
207C22×C4⋊C4—532no—
208C22×C4⋊C4—531no—
209C22×C42—531no—
210C22×C42—532no—
211#432—534no—
212#432—1062no—
213#431—534no—
214#431—1062no—
215#423—1062no—
216#407—534no—
217C4×M16—1062no—
218#324—5316no—
219C53#1736511yescharacteristic, complemented, Sylow
220C22×Q8—28no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8C2×#615712yes—
224C22×Q8C2×#615712yes—
225C23×C4—531no—
226C23×C4—531no—
227C23×C4—532no—
228C23×C4C2×#615712yes—
229C23×C4C2×#615711yescharacteristic
230C2×M16—1064no—
231C4×Q8—1064no—
232C4×Q8—1064no—
233C4×Q8—1064no—
234C4×Q8—1064no—
235C4×Q8—1064no—
236C4×Q8—1064no—
237C4×Q8—1064no—
238C4×Q8—1064no—
239C2×C4⋊C4—534no—
240C2×C4⋊C4—534no—
241C2×C4⋊C4—532no—
242C2×C4⋊C4—532no—
243C2×C4⋊C4—1064no—
244C2×C4⋊C4—1064no—
245C2×C4⋊C4—1062no—
246C2×C4⋊C4—1062no—
247C2×C42—532no—
248C2×C42—532no—
249C2×C42—532no—
250C2×C42—532no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—531no—
254C2×C42—531no—
255C2×C42—534no—
256C2×C42—534no—
257C2×C42—1062no—
258C2×C42—1062no—
259C2×C42—1062no—
260C2×C42—1062no—
261C4⋊C8—1064no—
262C4⋊C8—1064no—
263C22⋊C8—538no—
264C22⋊C8—1064no—
265#97—1064no—
266C4×C8—1064no—
267C24C22×#615711yescharacteristic
268C2×Q8—216no—
269C2×Q8—216no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—28no—
273C2×Q8—24no—
274C2×Q8#1171994018yescomplemented
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—532no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—532no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—534no—
290C22×C4—1064no—
291C22×C4—1062no—
292C22×C4—1062no—
293C22×C4—2122no—
294C22×C4—2124no—
295C22×C4C22⋊#615712yes—
296C22×C4C22⋊#615712yes—
297C22×C4C22×#615711yescharacteristic
298C22×C4C22×#615714yes—
299C22×C4C22×#615711yescharacteristic
300M16—2128no—
301C2×C8—1068no—
302C4⋊C4—1068no—
303C4⋊C4—1068no—
304C4⋊C4—1064no—
305C4⋊C4—1064no—
306C42—532no—
307C42—532no—
308C42—532no—
309C42—532no—
310C42—532no—
311C42—532no—
312C42—532no—
313C42—532no—
314C42—1064no—
315C42—1064no—
316C42—1064no—
317C42—1064no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23C2×C22⋊#615711yescharacteristic
322C23C2×C22⋊#615711yescharacteristic
323C23C23×#615711yescharacteristic
324Q8—216no—
325Q8—216no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—534no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—531no—
337C2×C4—531no—
338C2×C4—532no—
339C2×C4—532no—
340C2×C4—1062no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4—1062no—
344C2×C4—1064no—
345C2×C4—1062no—
346C2×C4—1064no—
347C2×C4—2121no—
348C2×C4—2122no—
349C2×C4—2121no—
350C2×C4—2124no—
351C2×C4—2124no—
352C2×C4—2124no—
353C2×C4C2×C22⋊#615712yes—
354C2×C4C2×C22⋊#615712yes—
355C2×C4C2×#1171994018yes—
356C2×C4C2×#1171994014yes—
357C8—2128no—
358C22—28no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22#1681160412yes—
366C22D53⋊#10112yes—
367C22C22×C22⋊#615711yescharacteristic
368C22C22×#1171994012yes—
369C4—22no—
370C4—28no—
371C4—532no—
372C4—532no—
373C4—1062no—
374C4—1062no—
375C4—1061no—
376C4—1061no—
377C4—2124no—
378C4#1681159314yes—
379C2—22no—
380C2—22no—
381C2—22no—
382C2C2×#1681159311yescharacteristic
383C2C2×#1681160411yescharacteristic
384C2C2×D53⋊#10111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC23×Q8×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC22×Dic53C24
hypercenterC23×Q8#6157
hypocenterC53#17365
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #20289978C2semidirect
4. #20289958C2semidirect
81. C2×Q8×C53M16semidirect
219. C53#17365semidirect
274. C2×Q8#11719940semidirect

Maximal subgroups

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

subgroupindex
#202913912
#202899782
#202899632
#202899582
C22×Q8×Dic532
#1736553