Finite Group Databasebeta

← #27369374

Subgroups — #27369374

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285640C211yesmaximal, characteristic
2#20284187C211yesmaximal, characteristic
3#20283796C211yesmaximal, characteristic, complemented
4#20274143C214yesmaximal, complemented
5#20274065C214yesmaximal, complemented
6#20273485C211yesmaximal, characteristic, complemented
7#20273400C211yesmaximal, characteristic, complemented
8#20273365C211yesmaximal, characteristic
9C2×C53×#475C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11#16811173C2211yescharacteristic
12#16810997C2212yes—
13#16810973C2212yes—
14#16810845—22no—
15#16810833C2212yes—
16#16810825—22no—
17#16810823C2212yes—
18#16810762—22no—
19Q8×#88322—22no—
20Q8×#88322C2214yes—
21#16810676—22no—
22#16810676C2211yescharacteristic
23#16810676C2211yescharacteristic
24#16810647C2211yescharacteristic
25#16810622—21no—
26#16810373C2214yes—
27#16810370C2212yes—
28#16810369C2214yes—
29#16810366C2212yes—
30C2×C4⋊Q8×C53C2211yescharacteristic
31C2×C4×Q8×C53C2211yescharacteristic
32C53×#475—22no—
33C53×#475C2214yes—
34C2×#106×C53—21no—
35C2×C4⋊C8×C53C2211yescharacteristic
36C2×#103×C53—21no—
37C2×#11719945—21no—
38#13133248—44no—
39C2×#11719925—22no—
40C2×#11719924—24no—
41C2×#11719924—22no—
42C2×#11719917—21no—
43#13133217C2311yescharacteristic
44#13133214C2311yescharacteristic
45#13133205—21no—
46#13133191—21no—
47#13133191—21no—
48C4×D212—21no—
49C4×D212—21no—
50C4⋊#88322—21no—
51C4⋊#88322—24no—
52C4⋊#88322D412yes—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55#13133141—24no—
56#13133140—22no—
57#13133137—22no—
58C22⋊#88322—21no—
59C22⋊#88322D412yes—
60C4×#88322—21no—
61C4×#88322—24no—
62C4×#88322C2312yes—
63C22×Q8×C53—21no—
64C22×Q8×C53D411yescharacteristic
65C22×C8×C53—21no—
66C4⋊Q8×C53—21no—
67C4⋊Q8×C53—21no—
68C4⋊Q8×C53—21no—
69C4⋊Q8×C53C2312yes—
70C4×Q8×C53—21no—
71C4×Q8×C53—21no—
72C4×Q8×C53D412yes—
73C4×Q8×C53C2312yes—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53D411yescharacteristic
78C2×C42×C53—21no—
79C2×C42×C53C2311yescharacteristic
80#106×C53—24no—
81C4⋊C8×C53—21no—
82C4⋊C8×C53C2312yes—
83#103×C53—24no—
84C2×Q8×C53—21no—
85C2×Q8×C53—22no—
86C2×Q8×C53—22no—
87C2×Q8×C53—41no—
88C2×Q8×C53—82no—
89C2×Q8×C53SD1614yescomplemented
90C2×Q8×C53C2×D412yes—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—41no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C2×C22⋊D53—41no—
98#11719945—42no—
99#11719945—42no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212—21no—
103C2×C4×D53—41no—
104C2×Dic106—21no—
105C2×C8×C53—22no—
106C2×C8×C53—42no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53C2×D412yes—
115C42×C53—22no—
116C42×C53—22no—
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53#4111yescharacteristic
120C42×C53C2411yescharacteristic
121#11719925—44no—
122#11719924—44no—
123#11719924—84no—
124#11719921—41no—
125#11719921—41no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128#11719917—42no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322C2×D414yes—
132C23×C53#12111yescharacteristic
133C22×D53—41no—
134Q8×C53—22no—
135Q8×C53—24no—
136Q8×C53—44no—
137Q8×C53—82no—
138Q8×C53C2×SD1614yes—
139C2×C4×C53—22no—
140C2×C4×C53—24no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—81no—
146C2×C4×C53—82no—
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53C2×SD1612yes—
151C2×C4×C53#13512yes—
152C2×C4×C53C22×D411yescharacteristic
153C2×C4×C53C22×D411yescharacteristic
154C2×C4×C53C2×#4111yescharacteristic
155C22⋊D53—82no—
156C2×Dic53—41no—
157D212—41no—
158D212—41no—
159D212—42no—
160D212—81no—
161C4×D53—82no—
162Dic106—41no—
163Dic106—41no—
164C8×C53—42no—
165#88322—24no—
166#88322—42no—
167#88322—42no—
168#19397—531nomaximal, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53#46512yes—
172C22×C53C2×#12111yescharacteristic
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C4×C53—21no—
177C4×C53—24no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—42no—
181C4×C53—82no—
182C4×C53#46512yes—
183C4×C53C22×SD1611yescharacteristic
184C4×C53C2×#13511yescharacteristic
185C4×C53#57712yes—
186Dic53—81no—
187#3206—531no—
188C2×#475D5311yescharacteristic, complemented
189#3037—531no—
190#2982—531no—
191#1680—534no—
192#1666—534no—
193#1570—531no—
194#1549—531no—
195#1540—531no—
196C2×C53—22no—
197C2×C53#304311yescharacteristic
198C2×C53C2×#46511yescharacteristic
199C2×C53#306911yescharacteristic
200D53—161no—
201C2×#137—1061no—
202#532—531no—
203C2×C4⋊Q8C2×D5311yescharacteristic
204C2×C4×Q8C2×D5311yescharacteristic
205#492—532no—
206#487—532no—
207#475—22no—
208#475C2×D5314yes—
209#464—1062no—
210#462—532no—
211#461—1062no—
212#460—532no—
213#448—1062no—
214C8×Q8—534no—
215C8×Q8—1062no—
216#431—531no—
217#431—531no—
218#431—1062no—
219C2×#106—21no—
220#423—531no—
221C2×C4⋊C8C2×D5311yescharacteristic
222#417—1061no—
223C2×#103—21no—
224#334—534no—
225#332—534no—
226#332—532no—
227#331—532no—
228C53#1939711yescharacteristic, complemented, Sylow
229C2×#41—1061no—
230C22×Q8—21no—
231C22×Q8C22⋊D5311yescharacteristic
232#137—2124no—
233C2×Q16—1062no—
234C2×SD16—1064no—
235C2×SD16—1062no—
236C2×M16—1061no—
237C22×C8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—531no—
242C4⋊Q8C22×D5312yes—
243#127—531no—
244#124—1061no—
245#121—1061no—
246#121—1061no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C22⋊D5312yes—
250C4×Q8C22×D5312yes—
251C4×D4—1061no—
252C4×D4—1061no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C42—21no—
258C2×C42C22×D5311yescharacteristic
259#106—24no—
260C4⋊C8—21no—
261C4⋊C8—532no—
262C4⋊C8—532no—
263C4⋊C8—532no—
264C4⋊C8—1064no—
265C4⋊C8—1061no—
266C4⋊C8C22×D5312yes—
267#103—24no—
268#103—1064no—
269#103—1062no—
270#102—1062no—
271C22⋊C8—532no—
272C22⋊C8—1061no—
273C4×C8—532no—
274C4×C8—1064no—
275C4×C8—1061no—
276#41—2122no—
277#41—2122no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×Q8—1061no—
284C2×Q8#1171992414yescomplemented
285C2×Q8C2×C22⋊D5312yes—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—2121no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—2121no—
294C22×C4D4×D5311yescharacteristic
295C22×C4#1171994711yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297Q16—2124no—
298SD16—4244no—
299SD16—2124no—
300M16—2122no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8—534no—
304C2×C8—1062no—
305C2×C8—1062no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C4⋊C4C2×C22⋊D5312yes—
316C22⋊C4—2121no—
317C22⋊C4—2121no—
318C42—22no—
319C42—22no—
320C42D4×D5311yescharacteristic
321C42#1171994711yescharacteristic
322C42C2×C22⋊D5311yescharacteristic
323C42C23×D5311yescharacteristic
324C23—2121no—
325C23#1313318911yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—44no—
329Q8—82no—
330Q8—2121no—
331Q8—2121no—
332Q8C2×#1171992414yes—
333D4—4241no—
334D4—4242no—
335D4—2121no—
336D4—2121no—
337D4—2122no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—4242no—
347C2×C4—2121no—
348C2×C4#1313318911yescharacteristic
349C2×C4#1313318911yescharacteristic
350C2×C4#1313318911yescharacteristic
351C2×C4C2×#1171992412yes—
352C2×C4#1313323712yes—
353C2×C4C2×D4×D5311yescharacteristic
354C2×C4C22×C22⋊D5311yescharacteristic
355C2×C4C2×#1171994711yescharacteristic
356C8—42no—
357C8—1064no—
358C8—2122no—
359C8—2122no—
360C22—21no—
361C22—21no—
362C22—4241no—
363C22—4241no—
364C22—4241no—
365C22#1681085412yes—
366C22C2×#1313318911yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—24no—
371C4—42no—
372C4—82no—
373C4—4241no—
374C4#1681085412yes—
375C4C22×#1171992411yescharacteristic
376C4C2×#1313323711yescharacteristic
377C4#1681142912yes—
378C2—22no—
379C2—8481no—
380C2#2028423311yescharacteristic
381C2C2×#1681085411yescharacteristic
382C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#475C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×#475D53
hypocenterC53#19397
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #20283796C2semidirect
4. #20274143C2semidirect
5. #20274065C2semidirect
6. #20273485C2semidirect
7. #20273400C2semidirect
9. C2×C53×#475C2semidirect
89. C2×Q8×C53SD16semidirect
188. C2×#475D53semidirect
228. C53#19397semidirect
284. C2×Q8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856402
#202841872
#202837962
#202741432
#202740652
#202734852
#202734002
#202733652
C2×C53×#4752
#1939753