Finite Group Databasebeta

← #27280919

Subgroups — #27280919

Order 13568.

428 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283366C211yesmaximal, characteristic, complemented
2#20273326C211yesmaximal, characteristic, complemented
3#20273149C211yesmaximal, characteristic, complemented
4#20273132C211yesmaximal, characteristic, complemented
5#20272613C211yesmaximal, characteristic, complemented
6#20272586C211yesmaximal, characteristic, complemented
7C53×#3115C211yesmaximal, characteristic
8#16810684—22no—
9#16810680—22no—
10#16810680C2211yescharacteristic
11#16810676C2211yescharacteristic
12#16810650—21no—
13#16810590—21no—
14C4×#11719917C2211yescharacteristic
15C2×#13133128C2211yescharacteristic
16#16810363—22no—
17Q8⋊#88322—21no—
18D4⋊#88322—22no—
19D4⋊#88322—21no—
20C53×#532C2211yescharacteristic
21C4×#41×C53C2211yescharacteristic
22C53×#482—21no—
23C53×#475C411yescharacteristic
24C53×#474C411yescharacteristic
25C53×#460C411yescharacteristic
26C53×#459C411yescharacteristic
27C53×#449—21no—
28C53×#442—21no—
29C53×#440—21no—
30C53×#432C2211yescharacteristic
31C2×#11719917—21no—
32C22×#88322—21no—
33C4⋊#88322—24no—
34C4⋊#88322—21no—
35C4⋊#88322—21no—
36C22⋊#88322—24no—
37C22⋊#88322—21no—
38C22⋊#88322—21no—
39#13133128—21no—
40#13133128—21no—
41#13133128—21no—
42#13133128D412yes—
43C4×#88322—21no—
44C4×#88322D412yes—
45C2×#41×C53—21no—
46C2×#41×C53D411yescharacteristic
47C2×SD16×C53—21no—
48C2×D8×C53—21no—
49C4⋊Q8×C53C2×C411yescharacteristic
50#127×C53C2×C411yescharacteristic
51#124×C53—21no—
52#121×C53—21no—
53#121×C53—21no—
54C4×Q8×C53C2×C411yescharacteristic
55C4×Q8×C53D411yescharacteristic
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53C2×C411yescharacteristic
61C4×D4×C53D411yescharacteristic
62#117×C53—21no—
63#117×C53D411yescharacteristic
64C2×C42×C53—21no—
65C2×C42×C53C2311yescharacteristic
66#107×C53—21no—
67#106×C53—21no—
68C4⋊C8×C53C2×C411yescharacteristic
69C4⋊C8×C53C2×C411yescharacteristic
70#103×C53—21no—
71#102×C53—21no—
72#102×C53—21no—
73#102×C53—21no—
74C22⋊C8×C53—21no—
75#97×C53—21no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—82no—
79C2×Q8×C53—21no—
80C2×Q8×C53C22⋊C411yescharacteristic
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—41no—
86C2×D4×C53C22⋊C411yescharacteristic
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×C4×C53#4111yescharacteristic
94SD16×C53—42no—
95D8×C53—42no—
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53C22⋊C411yescharacteristic
103C4⋊C4×C53C22⋊C411yescharacteristic
104C22⋊C4×C53—21no—
105C22⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C42×C53—21no—
110C42×C53—21no—
111C42×C53—21no—
112C42×C53C22×C411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116#11719917—41no—
117#11719917—41no—
118C2×#88322—22no—
119C2×#88322—22no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322—41no—
124C23×C53—21no—
125C23×C53—41no—
126C23×C53#12311yescharacteristic
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—81no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—41no—
135D4×C53—41no—
136D4×C53—81no—
137D4×C53—82no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×C4×C53—82no—
150C2×C4×C53C2×C22⋊C411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C8×C53—41no—
157C8×C53—41no—
158#88322—42no—
159#88322—42no—
160#88322—81no—
161#88322—81no—
162#9313—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53#38611yescharacteristic
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—81no—
182C4×C53#42111yescharacteristic
183C4×C53#42111yescharacteristic
184#3115D5311yescharacteristic
185#2923—531no—
186#1530—531no—
187#1488—531no—
188#1484—531no—
189#1302—531no—
190#1291—531no—
191C2×C53—41no—
192C2×C53—81no—
193C2×C53—161no—
194C2×C53#188111yescharacteristic
195C2×C53#189711yescharacteristic
196C2×C53#190311yescharacteristic
197#532C2×D5311yescharacteristic
198C4×#41C2×D5311yescharacteristic
199#482—21no—
200#475Dic5311yescharacteristic
201#474Dic5311yescharacteristic
202#460Dic5311yescharacteristic
203#459Dic5311yescharacteristic
204#449—21no—
205#442—21no—
206#440—21no—
207#433—1062no—
208#432—531no—
209#432—1062no—
210#432C2×D5311yescharacteristic
211#431—531no—
212#424—1061no—
213#408—1061no—
214C4×M16—531no—
215C2×#97—531no—
216#329—1062no—
217Q8⋊C8—1061no—
218D4⋊C8—1062no—
219D4⋊C8—1061no—
220C53#931311yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41C22⋊D5311yescharacteristic
223C2×SD16—21no—
224C2×D8—21no—
225C2×M16—1061no—
226C22×C8—1061no—
227C4⋊Q8C2×Dic5311yescharacteristic
228#127C2×Dic5311yescharacteristic
229#124—21no—
230#121—21no—
231#121—21no—
232C4×Q8C2×Dic5311yescharacteristic
233C4×Q8C22⋊D5311yescharacteristic
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C2×Dic5311yescharacteristic
239C4×D4C22⋊D5311yescharacteristic
240#117—21no—
241#117C22⋊D5311yescharacteristic
242C2×C42—21no—
243C2×C42C22×D5311yescharacteristic
244#107—21no—
245#106—21no—
246C4⋊C8—1064no—
247C4⋊C8—1061no—
248C4⋊C8—1061no—
249C4⋊C8C2×Dic5311yescharacteristic
250C4⋊C8C2×Dic5311yescharacteristic
251#103—21no—
252#102—21no—
253#102—21no—
254#102—21no—
255C22⋊C8—21no—
256C22⋊C8—1064no—
257C22⋊C8—1061no—
258C22⋊C8—1061no—
259#97—21no—
260#97—532no—
261#97—1061no—
262#97—1061no—
263#97—1061no—
264C4×C8—532no—
265C4×C8—1061no—
266#41—41no—
267#41—41no—
268#41—82no—
269C2×Q8—21no—
270C2×Q8C22⋊Dic5311yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4C22⋊Dic5311yescharacteristic
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4#1171994711yescharacteristic
282C22×C4#1171994711yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284SD16—42no—
285D8—42no—
286M16—2121no—
287M16—2121no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—1062no—
291C2×C8—1062no—
292C2×C8—2121no—
293C2×C8—2121no—
294C2×C8—2121no—
295C2×C8—2121no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4C22⋊Dic5311yescharacteristic
301C4⋊C4C22⋊Dic5311yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42D4×D5311yescharacteristic
311C42D4×D5311yescharacteristic
312C42C22×Dic5311yescharacteristic
313C42C2×C22⋊D5311yescharacteristic
314C23—21no—
315C23—41no—
316C23#1313319811yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—81no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—41no—
325D4—41no—
326D4—81no—
327D4—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4C2×C22⋊Dic5311yescharacteristic
341C2×C4D4×Dic5311yescharacteristic
342C2×C4D4×Dic5311yescharacteristic
343C2×C4#1313318511yescharacteristic
344C2×C4#1313318911yescharacteristic
345C2×C4#1313318911yescharacteristic
346C8—41no—
347C8—41no—
348C8—4241no—
349C8—4241no—
350C8—2122no—
351C8—2122no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#1681050311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—81no—
371C4#1681064011yescharacteristic
372C4#1681064011yescharacteristic
373C2—41no—
374C2—81no—
375C2—161no—
376C2#2027507911yescharacteristic
377C2#2027516411yescharacteristic
378C2#2027519711yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3115C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3115D53
hypocenterC53#9313
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #20283366C2semidirect
2. #20273326C2semidirect
3. #20273149C2semidirect
4. #20273132C2semidirect
5. #20272613C2semidirect
6. #20272586C2semidirect
220. C53#9313semidirect

Maximal subgroups

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

subgroupindex
#202833662
#202733262
#202731492
#202731322
#202726132
#202725862
C53×#31152
#931353