Finite Group Databasebeta

← #27277976

Subgroups — #27277976

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283823C211yesmaximal, characteristic
2#20274867C211yesmaximal, characteristic, complemented
3C4×#13133145C211yesmaximal, characteristic
4#20273468C211yesmaximal, characteristic, complemented
5#20273103C211yesmaximal, characteristic
6#20272564C211yesmaximal, characteristic, complemented
7C53×#1276C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9C8×Dic106—21no—
10C8×D212—21no—
11#16810686—22no—
12#16810675—21no—
13C2×C4⋊#88322—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C4⋊C8×D53—21no—
17C2×#13133145C411yescharacteristic
18C2×#13133145C411yescharacteristic
19C4×#11719917C2211yescharacteristic
20#16810383C411yescharacteristic
21#16810383C411yescharacteristic
22#16810375—21no—
23#16810373—21no—
24#16810355—21no—
25#16810352—21no—
26C53×#431C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C43×C53C2211yescharacteristic
29C2×#11719945C2×C411yescharacteristic
30C2×#11719917C2×C411yescharacteristic
31C2×#11719917C2×C411yescharacteristic
32C22×#88322—21no—
33C22×#88322D412yes—
34C2×C8×D53—22no—
35C4×Dic106D411yescharacteristic
36C4×Dic106Q811yescharacteristic
37C4×D212D411yescharacteristic
38C4×D212Q811yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53—21no—
41#13133176C2×C411yescharacteristic
42#13133174—21no—
43C42×D53—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—44no—
49#13133146—22no—
50#13133145—22no—
51#13133145C814yescomplemented
52#13133129—22no—
53#13133128—21no—
54C4×#88322—21no—
55C8×Dic53—22no—
56C2×C42×C53—21no—
57C2×C42×C53—21no—
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53C2×C411yescharacteristic
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53D411yescharacteristic
63C4⋊C8×C53D411yescharacteristic
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53#4111yescharacteristic
73C2×C22⋊D53—21no—
74#11719945—21no—
75#11719945—21no—
76#11719945C2×C812yes—
77#11719945M1612yes—
78C2×D212C4⋊C411yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×Dic106C4⋊C411yescharacteristic
83C2×C8×C53—21no—
84C2×C8×C53—21no—
85C2×C8×C53—41no—
86C42×C53—22no—
87C42×C53—22no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—42no—
92C42×C53C2×C812yescomplemented
93C42×C53C2×D411yescharacteristic
94C42×C53C2×D411yescharacteristic
95C42×C53C2×Q811yescharacteristic
96C42×C53#4111yescharacteristic
97C22⋊Dic53—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53C4⋊C411yescharacteristic
101C4⋊Dic53C4⋊C411yescharacteristic
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107#11719917—21no—
108#11719917—21no—
109#11719917—21no—
110#11719917C2×C812yescomplemented
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—22no—
114C2×#88322—42no—
115C2×#88322—42no—
116C2×#88322—41no—
117C8×D53—82no—
118C23×C53C4×D411yescharacteristic
119C22×D53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—22no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53C2×C4⋊C411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53C22⋊Q811yescharacteristic
133C2×C4×C53C22⋊Q811yescharacteristic
134C2×C4×C53#12311yescharacteristic
135C2×C4×C53C22×C811yescharacteristic
136C2×C4×C53C2×M1611yescharacteristic
137C2×C4×C53#13111yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C22⋊D53—22no—
140C22⋊D53—22no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144D212—21no—
145D212C4⋊C812yescomplemented
146C4×D53—22no—
147C4×D53—22no—
148C4×D53—41no—
149C4×D53—41no—
150C4×D53—81no—
151Dic106—21no—
152Dic106C4⋊C812yescomplemented
153C8×C53—41no—
154C8×C53—41no—
155C8×C53—81no—
156#88322—22no—
157#88322—41no—
158#88322—42no—
159#88322—81no—
160#8839—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53#38511yescharacteristic
164C22×C53C8×D411yescharacteristic
165C22×C53#43511yescharacteristic
166C2×D53—22no—
167C2×D53—41no—
168C2×D53—41no—
169C4×C53—24no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—42no—
174C4×C53C2×C4⋊C811yescharacteristic
175C4×C53#42411yescharacteristic
176C4×C53C8×D411yescharacteristic
177C4×C53#43511yescharacteristic
178C4×C53#45311yescharacteristic
179C4×C53#45611yescharacteristic
180Dic53—22no—
181Dic53—41no—
182Dic53—41no—
183#2988—531no—
184#1842—531no—
185C4×#104—531no—
186#1565—531no—
187#1477—531no—
188#1283—531no—
189#1276D5311yescharacteristic, complemented
190C2×C53—21no—
191C2×C53—21no—
192C2×C53#185011yescharacteristic
193C2×C53#185911yescharacteristic
194C2×C53#186111yescharacteristic
195D53—42no—
196C4×#41—531no—
197C8×Q8—1061no—
198C8×D4—1061no—
199C8×D4—1062no—
200#431—1061no—
201#431C2×D5311yescharacteristic
202#431C2×D5311yescharacteristic
203C2×C4⋊C8—531no—
204C2×C4⋊C8—531no—
205C2×C4⋊C8—1061no—
206C2×C4⋊C8—1062no—
207C2×#104—531no—
208C2×#104—531no—
209C4×M16—531no—
210C43C2×D5311yescharacteristic
211#339—531no—
212#339—531no—
213#335—1061no—
214#334—1061no—
215Q8⋊C8—1061no—
216D4⋊C8—1061no—
217C53#883911yescharacteristic, complemented, Sylow
218C2×#41—531no—
219C2×M16—531no—
220C2×M16—531no—
221C22×C8—532no—
222C22×C8—1062no—
223C22×C8—1061no—
224C4×Q8—531no—
225C4×Q8—531no—
226C4×D4—531no—
227C4×D4—531no—
228C4×D4—1061no—
229C4×D4—1061no—
230#117—531no—
231#117—1061no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—1061no—
235C2×C42C4×D5311yescharacteristic
236C2×C42C4×D5311yescharacteristic
237C2×C42C22×D5311yescharacteristic
238C4⋊C8—21no—
239C4⋊C8—1062no—
240C4⋊C8—1061no—
241C4⋊C8—1061no—
242C4⋊C8—1061no—
243C4⋊C8—1061no—
244C4⋊C8—2124no—
245C4⋊C8D21211yescharacteristic
246C4⋊C8D21211yescharacteristic
247#104—534no—
248#104—1062no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—1062no—
252#97—1061no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—1062no—
256C4×C8—1061no—
257#41—532no—
258#41—532no—
259#41—1061no—
260#41—1061no—
261C2×Q8—531no—
262C2×D4—531no—
263C2×D4—1061no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4C2×C4×D5311yescharacteristic
270C22×C4D4×D5311yescharacteristic
271C22×C4#1171994711yescharacteristic
272M16—532no—
273M16—1061no—
274M16—1061no—
275M16—1061no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—1061no—
280C2×C8—1061no—
281C2×C8—1062no—
282C2×C8—4242no—
283C2×C8—2122no—
284C2×C8—2121no—
285C2×C8—2122no—
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C42—22no—
294C42—22no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—42no—
299C42—1061no—
300C42—1061no—
301C42—1061no—
302C42C8×D5312yescomplemented
303C42C2×D21211yescharacteristic
304C42#1171994511yescharacteristic
305C42D4×D5311yescharacteristic
306C42Q8×D5311yescharacteristic
307C23—1061no—
308C23#1313317711yescharacteristic
309Q8—532no—
310Q8—1061no—
311D4—532no—
312D4—1062no—
313D4—1062no—
314D4—1061no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—1061no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—4241no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4C4⋊C4×D5311yescharacteristic
333C2×C4C4×D21211yescharacteristic
334C2×C4#1313318711yescharacteristic
335C2×C4#1313319511yescharacteristic
336C2×C4#1313319711yescharacteristic
337C2×C4#1313319911yescharacteristic
338C2×C4C2×C8×D5311yescharacteristic
339C2×C4M16×D5311yescharacteristic
340C2×C4#1313322311yescharacteristic
341C2×C4#1313322511yescharacteristic
342C8—41no—
343C8—41no—
344C8—81no—
345C8—1062no—
346C8—4241no—
347C8—2122no—
348C8—2121no—
349C22—21no—
350C22—21no—
351C22—1062no—
352C22—2121no—
353C22—2121no—
354C22#1681049711yescharacteristic
355C22#1681068611yescharacteristic
356C22#1681069711yescharacteristic
357C4—24no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—1062no—
363C4—2121no—
364C4—2121no—
365C4C4⋊C8×D5311yescharacteristic
366C4#1681064811yescharacteristic
367C4C8×D21211yescharacteristic
368C4#1681069611yescharacteristic
369C4#1681079211yescharacteristic
370C4#1681080711yescharacteristic
371C2—21no—
372C2—21no—
373C2—2122no—
374C2#2027489411yescharacteristic
375C2#2027494211yescharacteristic
376C2#2027495611yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#385
prime radicalC2×#11719945C2×C4
cosocleC2×C42×C53C23
hypercenter#1276D53
hypocenterC53#8839
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #20274867C2semidirect
4. #20273468C2semidirect
6. #20272564C2semidirect
7. C53×#1276C2semidirect
51. #13133145C8semidirect
92. C42×C53C2×C8semidirect
110. #11719917C2×C8semidirect
145. D212C4⋊C8semidirect
152. Dic106C4⋊C8semidirect
189. #1276D53semidirect
217. C53#8839semidirect
302. C42C8×D53semidirect

Maximal subgroups

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

subgroupindex
#202838232
#202748672
C4×#131331452
#202734682
#202731032
#202725642
C53×#12762
#883953