Finite Group Databasebeta

← #27279938

Subgroups — #27279938

Order 13568.

436 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283804C211yesmaximal, characteristic
2#20274545C211yesmaximal, characteristic
3#20273330C211yesmaximal, characteristic
4#20273214C211yesmaximal, characteristic
5#20272613C211yesmaximal, characteristic
6#20272572C211yesmaximal, characteristic
7C53×#3113C211yesmaximal, characteristic
8#16810754—22no—
9#16810754—21no—
10Q8×#88322—21no—
11#16810680—21no—
12#16810680C2211yescharacteristic
13#16810676—21no—
14#16810676C2211yescharacteristic
15C2×#13133141—21no—
16C2×#13133128C2211yescharacteristic
17C4×C4⋊Dic53C2211yescharacteristic
18#16810385—21no—
19#16810365—22no—
20#16810360—21no—
21#16810360—21no—
22C53×#533C2211yescharacteristic
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#483—21no—
25C53×#475C411yescharacteristic, complemented
26C53×#475C411yescharacteristic, complemented
27C53×#462C411yescharacteristic, complemented
28C53×#462C411yescharacteristic, complemented
29C53×#451—21no—
30C53×#441—21no—
31C53×#441—21no—
32C53×#432C2211yescharacteristic
33C22×#88322—21no—
34C2×C4⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C4⋊#88322—22no—
37C4⋊#88322—22no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40#13133141—24no—
41C22⋊#88322—21no—
42C22⋊#88322D411yescharacteristic
43C22⋊#88322D411yescharacteristic
44#13133128—22no—
45#13133128—21no—
46#13133128—21no—
47#13133128D412yes—
48C4×#88322—22no—
49C4×#88322—21no—
50#13133124—21no—
51C22×Q8×C53D411yescharacteristic
52C2×Q16×C53—21no—
53C2×Q16×C53—21no—
54C4⋊Q8×C53C2×C411yescharacteristic
55C4⋊Q8×C53C2×C411yescharacteristic
56#125×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53C2×C411yescharacteristic
64C4×Q8×C53C2×C411yescharacteristic
65C4×Q8×C53D411yescharacteristic
66C4×Q8×C53D411yescharacteristic
67#117×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72#106×C53—21no—
73#106×C53—21no—
74C4⋊C8×C53C2×C411yescharacteristic
75C4⋊C8×C53C2×C411yescharacteristic
76#103×C53—21no—
77#103×C53—21no—
78#103×C53—21no—
79#103×C53—21no—
80C22⋊C8×C53—21no—
81#97×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—22no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—41no—
87C2×Q8×C53C22⋊C411yescharacteristic
88C2×Q8×C53C22⋊C411yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C22×Dic53—41no—
95Q16×C53—42no—
96Q16×C53—42no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53C22⋊C411yescharacteristic
111C4⋊C4×C53C22⋊C411yescharacteristic
112C22⋊C4×C53—41no—
113C42×C53—21no—
114C42×C53—21no—
115C42×C53—21no—
116C42×C53—21no—
117C42×C53—41no—
118C42×C53C22×C411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122C4⋊Dic53—22no—
123C4⋊Dic53—41no—
124C4×Dic53—81no—
125C2×#88322—22no—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—22no—
129C2×#88322—41no—
130C2×#88322—41no—
131C23×C53#12111yescharacteristic
132Q8×C53—21no—
133Q8×C53—21no—
134Q8×C53—21no—
135Q8×C53—42no—
136Q8×C53—41no—
137Q8×C53—41no—
138Q8×C53—81no—
139Q8×C53#10412yes—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53C2×C22⋊C411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12311yescharacteristic
160C2×Dic53—42no—
161C2×Dic53—81no—
162C2×Dic53—81no—
163C8×C53—41no—
164C8×C53—41no—
165#88322—41no—
166#88322—41no—
167#88322—42no—
168#88322—42no—
169#9166—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53#38611yescharacteristic
173C22×C53#47111yescharacteristic
174C22×C53#47311yescharacteristic
175C4×C53—21no—
176C4×C53—22no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—81no—
185C4×C53—81no—
186C4×C53C2×#10411yescharacteristic
187C4×C53#42111yescharacteristic
188Dic53—82no—
189#3113D5311yescharacteristic
190#2984—531no—
191#1763—531no—
192#1531—531no—
193#1503—531no—
194#1302—531no—
195#1285—531no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53#187111yescharacteristic
199C2×C53#190511yescharacteristic
200C2×C53#190811yescharacteristic
201#533C2×D5311yescharacteristic
202C2×C4×Q8C2×D5311yescharacteristic
203#483—21no—
204#475Dic5311yescharacteristic, complemented
205#475Dic5311yescharacteristic, complemented
206#462Dic5311yescharacteristic, complemented
207#462Dic5311yescharacteristic, complemented
208#451—21no—
209#446—1062no—
210#446—1061no—
211C8×Q8—1061no—
212#441—21no—
213#441—21no—
214#432—531no—
215#432—1061no—
216#432C2×D5311yescharacteristic
217#431—531no—
218#431—1061no—
219C2×#103—1061no—
220C2×#97—531no—
221C4×C4⋊C4—531no—
222#340—1061no—
223#330—1062no—
224#328—1061no—
225#328—1061no—
226C53#916611yescharacteristic, complemented, Sylow
227C22×Q8C22⋊D5311yescharacteristic
228C2×Q16—21no—
229C2×Q16—21no—
230C22×C8—1061no—
231C4⋊Q8C2×Dic5311yescharacteristic
232C4⋊Q8C2×Dic5311yescharacteristic
233#125—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C2×Dic5311yescharacteristic
241C4×Q8C2×Dic5311yescharacteristic
242C4×Q8C22⋊D5311yescharacteristic
243C4×Q8C22⋊D5311yescharacteristic
244#117—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4C22⋊D5311yescharacteristic
248C2×C42—21no—
249C2×C42—1061no—
250C2×C42C22×D5311yescharacteristic
251#106—21no—
252#106—21no—
253C4⋊C8—1062no—
254C4⋊C8—1062no—
255C4⋊C8—1061no—
256C4⋊C8—1061no—
257C4⋊C8C2×Dic5311yescharacteristic
258C4⋊C8C2×Dic5311yescharacteristic
259#103—21no—
260#103—21no—
261#103—21no—
262#103—21no—
263#103—1064no—
264C22⋊C8—21no—
265C22⋊C8—531no—
266C22⋊C8—531no—
267C22⋊C8—1061no—
268#97—21no—
269#97—532no—
270#97—1062no—
271#97—1061no—
272#97—1061no—
273C4×C8—1062no—
274C4×C8—1061no—
275#95—1061no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8C22⋊Dic5311yescharacteristic
282C2×Q8C22⋊Dic5311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—2121no—
286C22×C4D4×D5311yescharacteristic
287C22×C4#1171994711yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289Q16—42no—
290Q16—42no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—1062no—
294C2×C8—1061no—
295C2×C8—1062no—
296C2×C8—1061no—
297C2×C8—2121no—
298C2×C8—2121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—1062no—
311C4⋊C4—2121no—
312C4⋊C4C22⋊Dic5311yescharacteristic
313C4⋊C4C22⋊Dic5311yescharacteristic
314C22⋊C4—41no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—41no—
320C42—4241no—
321C42D4×D5311yescharacteristic
322C42#1171994711yescharacteristic
323C42C22×Dic5311yescharacteristic
324C42C2×C22⋊D5311yescharacteristic
325C23#1313318911yescharacteristic
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—41no—
331Q8—41no—
332Q8—81no—
333Q8#1313314412yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—4241no—
349C2×C4—4241no—
350C2×C4—2122no—
351C2×C4C2×C22⋊Dic5311yescharacteristic
352C2×C4D4×Dic5311yescharacteristic
353C2×C4D4×Dic5311yescharacteristic
354C2×C4#1313318511yescharacteristic
355C2×C4#1313318911yescharacteristic
356C2×C4#1313319811yescharacteristic
357C8—41no—
358C8—41no—
359C8—2121no—
360C8—2122no—
361C8—2121no—
362C8—2122no—
363C22—21no—
364C22—21no—
365C22#1681050311yescharacteristic
366C22#1681088811yescharacteristic
367C22#1681089611yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—22no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—4242no—
380C4C2×#1313314411yescharacteristic
381C4#1681064011yescharacteristic
382C2—21no—
383C2—21no—
384C2#2027501811yescharacteristic
385C2#2027521111yescharacteristic
386C2#2027523211yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3113C2
socleC22×C53#386
prime radicalC23×C53#121
cosocleC2×C42×C53C23
hypercenter#3113D53
hypocenterC53#9166
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
25. C53×#475C4semidirect
26. C53×#475C4semidirect
27. C53×#462C4semidirect
28. C53×#462C4semidirect
204. #475Dic53semidirect
205. #475Dic53semidirect
206. #462Dic53semidirect
207. #462Dic53semidirect
226. C53#9166semidirect

Maximal subgroups

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

subgroupindex
#202838042
#202745452
#202733302
#202732142
#202726132
#202725722
C53×#31132
#916653