Finite Group Databasebeta

← #27369370

Subgroups — #27369370

Order 13568.

686 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#20285653C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3#20283796C211yesmaximal, characteristic, complemented
4#20274133C214yesmaximal, complemented
5#20274065C214yesmaximal, complemented
6#20273487C211yesmaximal, characteristic, complemented
7#20273384C211yesmaximal, characteristic
8#20273372C211yesmaximal, characteristic, complemented
9C53×#3078C211yesmaximal, characteristic, complemented
10C22×#11719924—21no—
11C2×#13133214C2211yescharacteristic
12#16810997C2212yes—
13#16810971C2212yes—
14#16810823—22no—
15#16810823—22no—
16#16810823C2214yes—
17#16810773—22no—
18Q8×#88322—22no—
19Q8×#88322C2214yescomplemented
20#16810676—22no—
21#16810676C2211yescharacteristic
22#16810676C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133137—21no—
25#16810373C2214yescomplemented
26#16810372C2212yescomplemented
27#16810369C2212yescomplemented
28#16810366C2214yes—
29C53×#533C2211yescharacteristic
30C2×C4×Q8×C53C2211yescharacteristic
31C53×#479—22no—
32C53×#477C2212yescomplemented
33C53×#475C2212yescomplemented
34C53×#428—21no—
35C53×#423C2211yescharacteristic
36C53×#419—21no—
37C22×D212—21no—
38C22×D212—21no—
39C22×D212—21no—
40C2×#11719924—24no—
41C2×#11719924—24no—
42C2×#11719924—44no—
43C22×#88322—21no—
44#13133214—21no—
45#13133214—21no—
46#13133214—21no—
47#13133214C2312yes—
48C4⋊#88322—21no—
49C4⋊#88322—24no—
50C4⋊#88322D412yes—
51C4⋊#88322C2312yes—
52C4⋊#88322C2312yes—
53#13133137—24no—
54#13133137—24no—
55C22⋊#88322—21no—
56C22⋊#88322D412yes—
57C4×#88322—21no—
58C4×#88322—24no—
59C4×#88322C2312yes—
60C22×Q8×C53D411yescharacteristic
61C2×M16×C53—21no—
62C4⋊Q8×C53C2311yescharacteristic
63C4⋊Q8×C53C2311yescharacteristic
64#125×C53—21no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—21no—
67C4×Q8×C53—21no—
68C4×Q8×C53—21no—
69C4×Q8×C53—21no—
70C4×Q8×C53—21no—
71C4×Q8×C53D412yes—
72C4×Q8×C53C2312yes—
73#117×C53—21no—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53D411yescharacteristic
76C2×C42×C53—21no—
77C2×C42×C53C2311yescharacteristic
78#107×C53—22no—
79#106×C53—22no—
80C4⋊C8×C53—21no—
81C4⋊C8×C53C2312yes—
82#103×C53—22no—
83#103×C53—22no—
84C23×D53—41no—
85C2×Q8×C53—21no—
86C2×Q8×C53—22no—
87C2×Q8×C53—21no—
88C2×Q8×C53—21no—
89C2×Q8×C53SD1614yescomplemented
90C2×Q8×C53C2×D412yes—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C2×D212—22no—
97C2×D212—22no—
98C2×D212—22no—
99C2×D212—42no—
100C2×D212—42no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—82no—
105C2×D212—82no—
106M16×C53—42no—
107C2×C8×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53C2×D412yes—
118C22⋊C4×C53—41no—
119C42×C53—22no—
120C42×C53—22no—
121C42×C53—41no—
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125C42×C53C2411yescharacteristic
126#11719924—48no—
127#11719924—84no—
128C2×#88322—22no—
129C2×#88322—22no—
130C2×#88322—42no—
131C2×#88322C2×D414yes—
132C23×C53#12111yescharacteristic
133C22×D53—42no—
134C22×D53—82no—
135C22×D53—82no—
136C22×D53—82no—
137Q8×C53—22no—
138Q8×C53—24no—
139Q8×C53—42no—
140Q8×C53—81no—
141Q8×C53C2×SD1614yes—
142C2×C4×C53—22no—
143C2×C4×C53—24no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—81no—
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53C2×SD1612yes—
154C2×C4×C53#13512yes—
155C2×C4×C53C22×D411yescharacteristic
156C2×C4×C53C22×D411yescharacteristic
157C2×C4×C53C2×#4111yescharacteristic
158D212—42no—
159D212—42no—
160D212—44no—
161D212—82no—
162D212—82no—
163D212—82no—
164D212—82no—
165C8×C53—42no—
166#88322—24no—
167#88322—42no—
168#88322—42no—
169#19396—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53#46512yes—
173C22×C53C2×#12111yescharacteristic
174C2×D53—82no—
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—164no—
178C4×C53—21no—
179C4×C53—24no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—42no—
183C4×C53—81no—
184C4×C53—81no—
185C4×C53#46512yes—
186C4×C53C22×SD1611yescharacteristic
187C4×C53C2×#13511yescharacteristic
188C4×C53#57712yes—
189#3208—531no—
190#3078D5311yescharacteristic, complemented
191C2×#460—531no—
192#2982—531no—
193#1678—534no—
194#1666—534no—
195#1570—531no—
196#1545—531no—
197#1542—531no—
198C2×C53—22no—
199C2×C53#304311yescharacteristic
200C2×C53C2×#46511yescharacteristic
201C2×C53#306811yescharacteristic
202D53—162no—
203C22×SD16—1061no—
204#533C2×D5311yescharacteristic
205C2×#127—531no—
206C2×C4×Q8C2×D5311yescharacteristic
207#492—532no—
208#486—532no—
209#479—22no—
210#477C2×D5312yescomplemented
211#475C2×D5312yescomplemented
212#460—534no—
213#460—1062no—
214#460—1062no—
215#450—1062no—
216C8×Q8—534no—
217C8×Q8—1062no—
218#431—531no—
219#431—531no—
220#431—1062no—
221#428—21no—
222#423C2×D5311yescharacteristic
223C2×C4⋊C8—531no—
224#419—21no—
225C2×#102—1061no—
226#334—534no—
227#333—532no—
228#332—532no—
229#331—534no—
230C53#1939611yescharacteristic, complemented, Sylow
231C22×Q8C22⋊D5311yescharacteristic
232C22×D4—1061no—
233C22×D4—1061no—
234C22×D4—1061no—
235C2×SD16—1064no—
236C2×SD16—1064no—
237C2×SD16—2124no—
238C2×M16—21no—
239C22×C8—1061no—
240C4⋊Q8C22×D5311yescharacteristic
241C4⋊Q8C22×D5311yescharacteristic
242#127—532no—
243#127—1061no—
244#127—1061no—
245#127—1061no—
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D5312yes—
254C4×Q8C22×D5312yes—
255#117—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C22⋊D5311yescharacteristic
258C2×C42—21no—
259C2×C42C22×D5311yescharacteristic
260#107—22no—
261#106—22no—
262C4⋊C8—21no—
263C4⋊C8—532no—
264C4⋊C8—532no—
265C4⋊C8—532no—
266C4⋊C8—1064no—
267C4⋊C8—1061no—
268C4⋊C8C22×D5312yes—
269#103—22no—
270#103—22no—
271#102—1064no—
272#102—1064no—
273C22⋊C8—532no—
274C22⋊C8—1061no—
275C4×C8—532no—
276C4×C8—1064no—
277C4×C8—1061no—
278C24—2121no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8#1171992414yescomplemented
284C2×Q8C2×C22⋊D5312yes—
285C2×D4—1062no—
286C2×D4—1062no—
287C2×D4—1062no—
288C2×D4—4242no—
289C2×D4—4242no—
290C2×D4—2121no—
291C2×D4—2121no—
292C2×D4—2121no—
293C2×D4—2122no—
294C2×D4—2122no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4D4×D5311yescharacteristic
298C22×C4#1171994711yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300SD16—4244no—
301SD16—2128no—
302M16—42no—
303C2×C8—22no—
304C2×C8—534no—
305C2×C8—1062no—
306C2×C8—1062no—
307C2×C8—2122no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4C2×C22⋊D5312yes—
318C22⋊C4—41no—
319C42—22no—
320C42—22no—
321C42—41no—
322C42D4×D5311yescharacteristic
323C42#1171994711yescharacteristic
324C42C2×C22⋊D5311yescharacteristic
325C42C23×D5311yescharacteristic
326C23—4242no—
327C23—4242no—
328C23—4242no—
329C23—2122no—
330C23#1313318911yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—81no—
335Q8C2×#1171992414yes—
336D4—4242no—
337D4—4242no—
338D4—4242no—
339D4—4242no—
340D4—2122no—
341D4—2122no—
342D4—2124no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4#1313318911yescharacteristic
352C2×C4#1313318911yescharacteristic
353C2×C4#1313318911yescharacteristic
354C2×C4C2×#1171992412yes—
355C2×C4#1313323712yes—
356C2×C4C2×D4×D5311yescharacteristic
357C2×C4C22×C22⋊D5311yescharacteristic
358C2×C4C2×#1171994711yescharacteristic
359C8—42no—
360C8—1064no—
361C8—2122no—
362C8—2122no—
363C22—21no—
364C22—21no—
365C22—8484no—
366C22—4242no—
367C22—4242no—
368C22—4242no—
369C22#1681085412yes—
370C22C2×#1313318911yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—24no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4#1681085412yes—
379C4C22×#1171992411yescharacteristic
380C4C2×#1313323711yescharacteristic
381C4#1681143212yes—
382C2—22no—
383C2—8482no—
384C2#2028423811yescharacteristic
385C2C2×#1681085411yescharacteristic
386C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3078C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#3078D53
hypocenterC53#19396
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
3. #20283796C2semidirect
4. #20274133C2semidirect
5. #20274065C2semidirect
6. #20273487C2semidirect
8. #20273372C2semidirect
9. C53×#3078C2semidirect
19. Q8×#88322C22semidirect
25. #16810373C22semidirect
26. #16810372C22semidirect
27. #16810369C22semidirect
32. C53×#477C22semidirect
33. C53×#475C22semidirect
89. C2×Q8×C53SD16semidirect
190. #3078D53semidirect
210. #477C2×D53semidirect
211. #475C2×D53semidirect
230. C53#19396semidirect
283. C2×Q8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856532
C2×#168108232
#202837962
#202741332
#202740652
#202734872
#202733842
#202733722
C53×#30782
#1939653