Finite Group Databasebeta

← #28300241

Subgroups — #28300241

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291391C211yesmaximal, characteristic
2C2×C4×#11719940C211yesmaximal, characteristic
3#20289548C214yesmaximal, complemented
4C2×M16×Dic53C211yesmaximal, characteristic
5C22×#11719940—21no—
6#16811644C412yes—
7#16811643—21no—
8#16811640—21no—
9#16811639C2212yes—
10#16811626C412yes—
11#16811593—21no—
12C4×#11719940—21no—
13C4×#11719940—21no—
14C4×#11719940—21no—
15C4×#11719940C412yes—
16C4×#11719940C412yes—
17C2×#13133272—21no—
18C2×C4×#88324C2212yes—
19C22×#11719917C411yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21M16×Dic53—21no—
22M16×Dic53—21no—
23M16×Dic53C412yes—
24M16×Dic53C2212yes—
25C2×#13133127—21no—
26C2×C8×Dic53—21no—
27C22×M16×C53C411yescharacteristic
28C2×#11719940—28no—
29C2×#11719940—22no—
30C2×#11719940—22no—
31C2×#11719940—22no—
32C22×#88324—22no—
33#13133284—22no—
34C4⋊#88324C2×C414yes—
35C22⋊#88324—22no—
36C22⋊#88324—22no—
37#13133272—22no—
38#13133272—22no—
39#13133272—22no—
40C4×#88324—22no—
41C4×#88324—22no—
42C4×#88324C2×C414yes—
43C23×Dic53D411yescharacteristic
44C23×Dic53Q811yescharacteristic
45C2×#11719917—24no—
46C2×#11719917—21no—
47C2×#11719917C2×C412yes—
48C22×#88322—21no—
49C2×C4×Dic53C2×C411yescharacteristic
50C2×C4×Dic53C2×C411yescharacteristic
51C2×C4×Dic53C2×C411yescharacteristic
52C2×C4×Dic53C2×C411yescharacteristic
53C2×C4×Dic53C2×C411yescharacteristic
54C2×C4×Dic53D411yescharacteristic
55C2×C4×Dic53D411yescharacteristic
56C2×C4×Dic53D411yescharacteristic
57C2×C4×Dic53D411yescharacteristic
58C2×C4×Dic53D411yescharacteristic
59C2×C4×Dic53Q811yescharacteristic
60C2×C4×Dic53C2311yescharacteristic
61#13133127—22no—
62#13133127—22no—
63C8×Dic53—22no—
64C8×Dic53—22no—
65C23×C4×C53C2×C411yescharacteristic
66C2×M16×C53—24no—
67C2×M16×C53—21no—
68C2×M16×C53C2×C412yes—
69C22×C8×C53—21no—
70C24×C53C4⋊C411yescharacteristic
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53C4211yescharacteristic
74C22×C4×C53C4211yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C4⋊C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×Dic53—24no—
80C22×Dic53—42no—
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C22⋊C411yescharacteristic
84C22×Dic53C22⋊C411yescharacteristic
85C22×Dic53C4⋊C411yescharacteristic
86C22×Dic53C4⋊C411yescharacteristic
87C22×Dic53C4⋊C411yescharacteristic
88C22×Dic53C4⋊C411yescharacteristic
89C22×Dic53C4⋊C411yescharacteristic
90C22×Dic53C4⋊C411yescharacteristic
91C22×Dic53C2×D411yescharacteristic
92C22×Dic53C2×Q811yescharacteristic
93#11719940—28no—
94#11719940—28no—
95#11719940—44no—
96C2×#88324—28no—
97C2×#88324—24no—
98C2×#88324—24no—
99M16×C53—24no—
100M16×C53—22no—
101M16×C53M1614yescomplemented
102C2×C8×C53—24no—
103C2×C8×C53—22no—
104C4×Dic53C4211yescharacteristic
105C4×Dic53C4211yescharacteristic
106C4×Dic53C22⋊C411yescharacteristic
107C4×Dic53C22⋊C411yescharacteristic
108C4×Dic53C22⋊C411yescharacteristic
109C4×Dic53C22⋊C411yescharacteristic
110C4×Dic53C22⋊C411yescharacteristic
111C4×Dic53C22⋊C411yescharacteristic
112C4×Dic53C4⋊C411yescharacteristic
113C4×Dic53C4⋊C411yescharacteristic
114C4×Dic53C4⋊C411yescharacteristic
115C4×Dic53C4⋊C411yescharacteristic
116C4×Dic53C22×C411yescharacteristic
117C4×Dic53C22×C411yescharacteristic
118C4×Dic53C2×D411yescharacteristic
119C4×Dic53C2×D411yescharacteristic
120#11719917—24no—
121#11719917—22no—
122#11719917M1614yescomplemented
123C2×#88322—24no—
124C2×#88322—22no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—41no—
128C23×C53#9511yescharacteristic
129C23×C53#9511yescharacteristic
130C23×C53C2×C4⋊C411yescharacteristic
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53#9511yescharacteristic
139C2×C4×C53#9511yescharacteristic
140C2×C4×C53C2×C4211yescharacteristic
141C2×C4×C53C2×C22⋊C411yescharacteristic
142C2×C4×C53C2×M1612yes—
143C2×C4×C53C2×M1612yes—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—22no—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53#9511yescharacteristic
156C2×Dic53#9511yescharacteristic
157C2×Dic53#9511yescharacteristic
158C2×Dic53#9511yescharacteristic
159C2×Dic53#10414yes—
160C2×Dic53C2×C22⋊C411yescharacteristic
161C2×Dic53C2×C22⋊C411yescharacteristic
162C2×Dic53C2×C4⋊C411yescharacteristic
163C2×Dic53C2×C4⋊C411yescharacteristic
164#88324—28no—
165#88324—44no—
166C8×C53—24no—
167#88322—24no—
168#7887—531nomaximal, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—44no—
177C22×C53#33912yes—
178C22×C53C2×#9511yescharacteristic
179C22×C53#42312yes—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53C4×M1612yes—
183C4×C53#40711yescharacteristic
184C4×C53#40711yescharacteristic
185Dic53—21no—
186Dic53—21no—
187Dic53—21no—
188Dic53—21no—
189Dic53—21no—
190Dic53#33912yes—
191Dic53C2×#10412yes—
192Dic53#43012yes—
193#2920—531no—
194C2×C4×M16—531no—
195C2×C4×M16—531no—
196#1278—534no—
197C2×C53—22no—
198C2×C53—22no—
199C2×C53—41no—
200C2×C53#172811yescharacteristic
201C2×C53C2×#33911yescharacteristic
202C2×C53#173811yescharacteristic
203C22×M16—531no—
204C22×M16—1061no—
205C22×M16#615711yescharacteristic
206C22×C42—531no—
207#432—532no—
208#432—1061no—
209#431—532no—
210#431—1061no—
211#423—532no—
212#407—1061no—
213C4×M16—532no—
214C4×M16—532no—
215C4×M16—532no—
216C4×M16—532no—
217C4×M16—1061no—
218C4×M16—1061no—
219C4×M16—1061no—
220C4×M16—1061no—
221C4×M16—1061no—
222C2×#97—1061no—
223C2×#97—1061no—
224C2×C4×C8—532no—
225C2×C4×C8—1061no—
226C53#788711yescharacteristic, complemented, Sylow
227C23×C4—531no—
228C23×C4—531no—
229C23×C4C2×#615711yescharacteristic
230C2×M16—21no—
231C2×M16—24no—
232C2×M16—532no—
233C2×M16—1064no—
234C2×M16—1062no—
235C2×M16—1062no—
236C2×M16—1068no—
237C2×M16—1061no—
238C2×M16—1062no—
239C2×M16C2×#615712yes—
240C22×C8—21no—
241C22×C8—1062no—
242C22×C8—1061no—
243C2×C42—531no—
244C2×C42—531no—
245C2×C42—531no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—531no—
254C2×C42—531no—
255C4⋊C8—534no—
256C4⋊C8—1062no—
257C22⋊C8—1062no—
258C22⋊C8—1062no—
259#97—1062no—
260#97—1062no—
261#97—1062no—
262#97—1062no—
263#97—1062no—
264C4×C8—534no—
265C4×C8—1062no—
266C4×C8—1062no—
267C4×C8—1062no—
268C4×C8—1062no—
269C24C4⋊#615711yescharacteristic
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—1064no—
285C22×C4—2122no—
286C22×C4C4×#615711yescharacteristic
287C22×C4C4×#615711yescharacteristic
288C22×C4C4⋊#615711yescharacteristic
289C22×C4C22⋊#615711yescharacteristic
290C22×C4C22⋊#615711yescharacteristic
291C22×C4C22×#615711yescharacteristic
292M16—24no—
293M16—22no—
294M16—534no—
295M16—1064no—
296M16—1068no—
297M16—1068no—
298M16—1062no—
299M16—2124no—
300M16#1171994014yescomplemented
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—1064no—
304C2×C8—1064no—
305C2×C8—1068no—
306C2×C8—1062no—
307C2×C8—1064no—
308C42—531no—
309C42—531no—
310C42—531no—
311C42—531no—
312C42—531no—
313C42—531no—
314C42—531no—
315C42—531no—
316C42—531no—
317C42—531no—
318C42—531no—
319C42—531no—
320C42—531no—
321C42—531no—
322C42—531no—
323C42—531no—
324C23—22no—
325C23—22no—
326C23—41no—
327C23(C2×#6157)⋊C411yescharacteristic
328C23(C2×#6157)⋊C411yescharacteristic
329C23C2×C4⋊#615711yescharacteristic
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—531no—
338C2×C4—531no—
339C2×C4—531no—
340C2×C4—534no—
341C2×C4—531no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—1062no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—1062no—
350C2×C4—1062no—
351C2×C4—1062no—
352C2×C4—1062no—
353C2×C4—1062no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4(C2×#6157)⋊C411yescharacteristic
358C2×C4(C2×#6157)⋊C411yescharacteristic
359C2×C4C2×C4×#615711yescharacteristic
360C2×C4C2×C22⋊#615711yescharacteristic
361C2×C4C2×#1171994012yes—
362C2×C4D53⋊M1612yes—
363C8—24no—
364C8—1068no—
365C8—1064no—
366C8—2124no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—44no—
375C22#1681149512yes—
376C22C2×(C2×#6157)⋊C411yescharacteristic
377C22#1681162612yes—
378C4—21no—
379C4—21no—
380C4—532no—
381C4—532no—
382C4—532no—
383C4—1061no—
384C4—1061no—
385C4—1061no—
386C4—1061no—
387C4—1061no—
388C4C4×#1171994012yes—
389C4#1681159311yescharacteristic
390C4#1681159511yescharacteristic
391C2—22no—
392C2—22no—
393C2—41no—
394C2#2029037811yescharacteristic
395C2C2×#1681149511yescharacteristic
396C2#2029040811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC22×M16×C53C4
socleC22×C53C2×#95
prime radicalC24×C53C4⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×M16#6157
hypocenterC53#7887
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. #20289548C2semidirect
101. M16×C53M16semidirect
122. #11719917M16semidirect
226. C53#7887semidirect
300. M16#11719940semidirect

Maximal subgroups

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

subgroupindex
#202913912
C2×C4×#117199402
#202895482
C2×M16×Dic532
#788753