Finite Group Databasebeta

← #28314235

Subgroups — #28314235

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291391C212yesmaximal
2#20289987C218yesmaximal, complemented
3#20289965C212yesmaximal
4#20289963C212yesmaximal
5C22×#13133207C211yesmaximal, characteristic
6#16811644—22no—
7#16811643C2214yes—
8#16811640—22no—
9#16811639C2214yes—
10#16811626—22no—
11#16811593C2214yes—
12C4×#11719940—22no—
13#16811469C2218yes—
14#16811468C2218yes—
15C2×#13133207—22no—
16C2×#13133207—22no—
17C2×#13133207—22no—
18C2×#13133207—22no—
19C2×#13133207C414yes—
20C2×#13133207C2214yes—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×#11719919C411yescharacteristic
23C22×#11719919C411yescharacteristic
24C22×#11719919C411yescharacteristic
25C22×#11719919C2211yescharacteristic
26C22×C4×Dic53C2211yescharacteristic
27C22×C4⋊C4×C53C411yescharacteristic
28C2×#11719940—24no—
29#13133284—24no—
30C4⋊#88324—24no—
31C22⋊#88324—24no—
32C22⋊#88324C2318yes—
33#13133272—24no—
34C4×#88324—24no—
35C23×Dic53C2×C411yescharacteristic
36C23×Dic53C2×C411yescharacteristic
37C23×Dic53C2×C411yescharacteristic
38C23×Dic53C2311yescharacteristic
39#13133207—24no—
40#13133207—24no—
41#13133207—24no—
42#13133207—24no—
43#13133207—24no—
44#13133207—24no—
45#13133207—24no—
46#13133207—24no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53C2×C412yes—
50C2×C4⋊Dic53C2312yes—
51C2×#11719919—21no—
52C2×#11719919—21no—
53C2×#11719919—22no—
54C2×#11719919—22no—
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×#11719919—21no—
58C2×#11719919—22no—
59C2×#11719919—21no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919C2×C414yes—
63C2×#11719919C2×C414yes—
64C2×#11719919C2×C414yes—
65C2×#11719919C2×C412yes—
66C2×#11719919C2312yes—
67C2×C4×Dic53C2×C412yes—
68C2×C4×Dic53D412yes—
69C2×C4×Dic53D412yes—
70C2×C4×Dic53D412yes—
71C2×C4×Dic53D412yes—
72C2×C4×Dic53C2312yes—
73C23×C4×C53C2×C411yescharacteristic
74C23×C4×C53C2×C411yescharacteristic
75C23×C4×C53C2×C411yescharacteristic
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53C2×C414yes—
80C24×C53C22×C411yescharacteristic
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22⋊C412yes—
91C22×C4×C53C22×C412yes—
92C22×C4×C53C22×C412yes—
93C22×C4×C53C22×C412yes—
94C22×Dic53—24no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—21no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—42no—
102C22×Dic53C22⋊C411yescharacteristic
103C22×Dic53C22⋊C412yes—
104C22×Dic53C22⋊C411yescharacteristic
105C22×Dic53C22×C412yes—
106C22×Dic53C22×C412yes—
107C22×Dic53C22×C412yes—
108C22×Dic53C22×C411yescharacteristic
109C22×Dic53C2×D412yes—
110C22×Dic53C2×D412yes—
111C22×Dic53C2411yescharacteristic
112#11719940—48no—
113C2×#88324—28no—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—24no—
116C4⋊Dic53—24no—
117C4⋊Dic53—24no—
118#11719919—24no—
119#11719919—24no—
120#11719919—24no—
121#11719919—24no—
122#11719919—24no—
123#11719919—24no—
124#11719919—24no—
125#11719919—24no—
126C4×Dic53C22⋊C412yes—
127C4×Dic53C22⋊C412yes—
128C4×Dic53C22⋊C412yes—
129C4×Dic53C22⋊C412yes—
130C4×Dic53C2×D412yes—
131C4×Dic53C2×D412yes—
132C4×Dic53C2×D412yes—
133C4×Dic53C2×D412yes—
134C23×C53—22no—
135C23×C53—22no—
136C23×C53—22no—
137C23×C53C2×C22⋊C411yescharacteristic
138C23×C53C2×C22⋊C411yescharacteristic
139C23×C53C23×C411yescharacteristic
140C2×C4×C53—24no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—24no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53C2×C22⋊C412yes—
153C2×C4×C53C2×C22⋊C412yes—
154C2×Dic53—24no—
155C2×Dic53—22no—
156C2×Dic53—24no—
157C2×Dic53—24no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—24no—
161C2×Dic53—41no—
162C2×Dic53—42no—
163C2×Dic53—41no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53#10114yes—
170C2×Dic53C2×C22⋊C412yes—
171C2×Dic53C2×C22⋊C412yes—
172C2×Dic53C2×C22⋊C412yes—
173C2×Dic53C22×D412yes—
174#88324—48no—
175#17394—531nomaximal, Sylow
176C22×C53—28no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53#41012yes—
184C22×C53C2×#10112yes—
185C22×C53C22×C22⋊C411yescharacteristic
186C22×C53#56812yes—
187C4×C53—22no—
188C4×C53—22no—
189C4×C53—42no—
190C4×C53—42no—
191Dic53—22no—
192Dic53—22no—
193Dic53—22no—
194Dic53—42no—
195Dic53—42no—
196Dic53C2×#10114yes—
197C22×#125—531no—
198#2920—532no—
199#1470—538no—
200#1463—532no—
201#1463—532no—
202C2×C53—22no—
203C2×C53—22no—
204C2×C53—22no—
205C2×C53#287911yescharacteristic
206C2×C53C2×#41011yescharacteristic
207C2×C53C22×#10111yescharacteristic
208C2×#125—534no—
209C2×#125—534no—
210C2×#125—1062no—
211C2×#125—1062no—
212C2×#125—1062no—
213C2×#125—1062no—
214C22×C4⋊C4—531no—
215C22×C4⋊C4—531no—
216C22×C4⋊C4—531no—
217C22×C4⋊C4—531no—
218C22×C4⋊C4—531no—
219C22×C4⋊C4#615711yescharacteristic
220C22×C42—531no—
221#432—534no—
222#432—1062no—
223#431—534no—
224#431—1062no—
225#423—1062no—
226#407—534no—
227C4×M16—1062no—
228#324—538no—
229#324—538no—
230C53#1739411yescharacteristic, complemented, Sylow
231C23×C4—531no—
232C23×C4—531no—
233C23×C4—531no—
234C23×C4—531no—
235C23×C4C2×#615711yescharacteristic
236C23×C4C2×#615711yescharacteristic
237C23×C4C2×#615711yescharacteristic
238C2×M16—1064no—
239#125—1064no—
240#125—1064no—
241#125—1064no—
242#125—1064no—
243#125—1064no—
244#125—1064no—
245#125—1064no—
246#125—1064no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—532no—
251C2×C4⋊C4—532no—
252C2×C4⋊C4—534no—
253C2×C4⋊C4—534no—
254C2×C4⋊C4—532no—
255C2×C4⋊C4—534no—
256C2×C4⋊C4—532no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1062no—
260C2×C4⋊C4—1062no—
261C2×C4⋊C4—1061no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4—1062no—
264C2×C4⋊C4—1062no—
265C2×C4⋊C4—1062no—
266C2×C4⋊C4—1062no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4—1062no—
270C2×C4⋊C4C2×#615714yes—
271C2×C42—532no—
272C2×C42—532no—
273C2×C42—532no—
274C2×C42—532no—
275C2×C42—532no—
276C2×C42—532no—
277C4⋊C8—1064no—
278C4⋊C8—1064no—
279C22⋊C8—538no—
280C22⋊C8—1064no—
281#97—1064no—
282C4×C8—1064no—
283C24C22×#615711yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—531no—
292C22×C4—532no—
293C22×C4—531no—
294C22×C4—532no—
295C22×C4—532no—
296C22×C4—532no—
297C22×C4—531no—
298C22×C4—531no—
299C22×C4—532no—
300C22×C4—532no—
301C22×C4—1064no—
302C22×C4—1061no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4—2122no—
307C22×C4—2122no—
308C22×C4—2122no—
309C22×C4C22⋊#615711yescharacteristic
310C22×C4C22⋊#615711yescharacteristic
311C22×C4C22⋊#615712yes—
312C22×C4C22×#615712yes—
313C22×C4C22×#615712yes—
314C22×C4C22×#615712yes—
315M16—2128no—
316C2×C8—1068no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—1064no—
320C4⋊C4—1064no—
321C4⋊C4—1064no—
322C4⋊C4—1064no—
323C4⋊C4—1064no—
324C4⋊C4—1064no—
325C4⋊C4—1064no—
326C4⋊C4—1064no—
327C4⋊C4—1064no—
328C4⋊C4—1064no—
329C42—532no—
330C42—532no—
331C42—532no—
332C42—532no—
333C42—532no—
334C42—532no—
335C42—532no—
336C42—532no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23C2×C22⋊#615711yescharacteristic
341C23C2×C22⋊#615711yescharacteristic
342C23C23×#615711yescharacteristic
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—534no—
356C2×C4—532no—
357C2×C4—532no—
358C2×C4—532no—
359C2×C4—532no—
360C2×C4—1064no—
361C2×C4—1064no—
362C2×C4—1064no—
363C2×C4—1062no—
364C2×C4—1062no—
365C2×C4—1062no—
366C2×C4—1064no—
367C2×C4—2121no—
368C2×C4—2122no—
369C2×C4—2121no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4C2×C22⋊#615712yes—
376C2×C4C2×C22⋊#615712yes—
377C8—2128no—
378C22—28no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22#1681160212yes—
386C22D53⋊#10112yes—
387C22C22×C22⋊#615711yescharacteristic
388C22#1681170112yes—
389C4—22no—
390C4—22no—
391C4—42no—
392C4—42no—
393C4—534no—
394C4—1062no—
395C4—1062no—
396C4—1062no—
397C4—2122no—
398C4—2122no—
399C2—22no—
400C2—22no—
401C2—22no—
402C2#2029129911yescharacteristic
403C2C2×#1681160211yescharacteristic
404C2C2×D53⋊#10111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC22×Dic53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53#17394
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289987C2semidirect
230. C53#17394semidirect

Maximal subgroups

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

subgroupindex
#202913912
#202899872
#202899652
#202899632
C22×#131332072
#1739453