Finite Group Databasebeta

← #27369377

Subgroups — #27369377

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#20285642C211yesmaximal, characteristic
2C2×#16810908C211yesmaximal, characteristic, complemented
3#20284187C211yesmaximal, characteristic
4#20274145C214yesmaximal, complemented
5#20274063C214yesmaximal, complemented
6#20273486C211yesmaximal, characteristic, complemented
7#20273399C211yesmaximal, characteristic, complemented
8#20273362C211yesmaximal, characteristic
9C53×#2982C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11#16811173C2211yescharacteristic
12C2×#13133217C2211yescharacteristic
13#16810996C2212yes—
14#16810974C2212yes—
15#16810908—22no—
16#16810908C2214yes—
17#16810845—22no—
18#16810833C2212yes—
19#16810829—22no—
20#16810823C2212yes—
21#16810764—22no—
22C2×#13133148—21no—
23#16810647C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25#16810622—21no—
26C2×#13133141—21no—
27#16810374C2214yes—
28#16810370C2214yes—
29#16810368C2212yes—
30#16810366C2212yes—
31C2×C4×Q8×C53C2211yescharacteristic
32C8×Q8×C53—22no—
33C8×Q8×C53C2214yes—
34C53×#431—22no—
35C53×#431C2211yescharacteristic
36C53×#431C2211yescharacteristic
37C2×#11719945—21no—
38C22×Dic106—21no—
39#13133248—44no—
40C2×#11719925—22no—
41C2×#11719913—24no—
42C2×#11719924—22no—
43C2×#11719917—21no—
44C22×#88322—21no—
45#13133217—21no—
46#13133217—21no—
47#13133217—21no—
48#13133217C2311yescharacteristic
49#13133217C2312yes—
50#13133214C2311yescharacteristic
51#13133205—21no—
52#13133191—21no—
53#13133191—21no—
54C4×D212—21no—
55C4×D212—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58#13133148—24no—
59C4⋊#88322—21no—
60C4⋊#88322—21no—
61C4⋊#88322C2312yes—
62C4⋊#88322C2312yes—
63#13133142—24no—
64#13133141—24no—
65#13133140—22no—
66#13133137—22no—
67C22×Q8×C53D411yescharacteristic
68C4×Q8×C53—21no—
69C4×Q8×C53—21no—
70C4×Q8×C53D412yes—
71C4×Q8×C53C2312yes—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C42×C53—21no—
75C2×C42×C53C2311yescharacteristic
76C4⋊C8×C53—24no—
77C4⋊C8×C53D412yes—
78C4⋊C8×C53C2312yes—
79C22⋊C8×C53—21no—
80C22⋊C8×C53D412yes—
81C4×C8×C53—24no—
82C4×C8×C53—21no—
83C4×C8×C53C2312yes—
84C2×Q8×C53—21no—
85C2×Q8×C53—22no—
86C2×Q8×C53SD1614yescomplemented
87C2×Q8×C53C2×D412yes—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C2×C22⋊D53—41no—
94C22×Dic53—41no—
95#11719945—42no—
96#11719945—42no—
97C2×D212—21no—
98C2×D212—21no—
99C2×D212—21no—
100C2×C4×D53—41no—
101C2×Dic106—21no—
102C2×Dic106—22no—
103C2×Dic106—41no—
104C2×Dic106—82no—
105C2×C8×C53—22no—
106C2×C8×C53C2×D414yes—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53C2×D412yes—
111C42×C53—22no—
112C42×C53—22no—
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53#4111yescharacteristic
116C42×C53C2411yescharacteristic
117#11719925—44no—
118#11719924—44no—
119#11719921—41no—
120#11719921—41no—
121C4⋊Dic53—22no—
122C4⋊Dic53—21no—
123C4⋊Dic53—22no—
124C4⋊Dic53—21no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127#11719917—42no—
128C2×#88322—22no—
129C2×#88322—22no—
130C2×#88322—42no—
131#11719913—84no—
132C23×C53#12111yescharacteristic
133C22×D53—41no—
134Q8×C53—22no—
135Q8×C53—24no—
136Q8×C53C2×SD1614yes—
137C2×C4×C53—22no—
138C2×C4×C53—24no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53C2×SD1612yes—
147C2×C4×C53#13512yes—
148C2×C4×C53C22×D411yescharacteristic
149C2×C4×C53C22×D411yescharacteristic
150C2×C4×C53C2×#4111yescharacteristic
151C22⋊D53—82no—
152C2×Dic53—42no—
153C2×Dic53—41no—
154C2×Dic53—82no—
155D212—41no—
156D212—41no—
157D212—42no—
158D212—81no—
159C4×D53—82no—
160Dic106—41no—
161Dic106—41no—
162Dic106—44no—
163Dic106—82no—
164C8×C53—24no—
165C8×C53—42no—
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#46512yes—
182C4×C53C22×SD1611yescharacteristic
183C4×C53C2×#13511yescharacteristic
184C4×C53#57712yes—
185Dic53—82no—
186Dic53—81no—
187#3206—531no—
188C2×#475—531no—
189#3037—531no—
190#2982D5311yescharacteristic, complemented
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⋊Q8—531no—
204C2×C4×Q8C2×D5311yescharacteristic
205#492—532no—
206#487—532no—
207#475—534no—
208#475—1062no—
209#464—1062no—
210#462—532no—
211#461—1062no—
212#460—532no—
213#448—1062no—
214C8×Q8—22no—
215C8×Q8C2×D5314yes—
216#431—22no—
217#431C2×D5311yescharacteristic
218#431C2×D5311yescharacteristic
219C2×#106—1061no—
220#423—531no—
221C2×C4⋊C8—531no—
222#417—1061no—
223C2×#103—1061no—
224#334—534no—
225#332—532no—
226#332—534no—
227#331—532no—
228C53#1939711yescharacteristic, complemented, Sylow
229C2×#41—1061no—
230C22×Q8—1061no—
231C22×Q8D21211yescharacteristic
232#137—2124no—
233C2×Q16—1062no—
234C2×SD16—1062no—
235C2×SD16—1064no—
236C2×M16—1061no—
237C22×C8—1061no—
238C4⋊Q8—532no—
239C4⋊Q8—531no—
240C4⋊Q8—1061no—
241C4⋊Q8—1061no—
242C4⋊Q8—1061no—
243#127—531no—
244#124—1061no—
245#121—1061no—
246#121—1061no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8D21212yes—
250C4×Q8C22×D5312yes—
251C4×D4—1061no—
252C4×D4—1061no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4D21211yescharacteristic
257C2×C42—21no—
258C2×C42C22×D5311yescharacteristic
259#106—1064no—
260C4⋊C8—24no—
261C4⋊C8—532no—
262C4⋊C8—532no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C4⋊C8C22⋊D5312yes—
266C4⋊C8C22×D5312yes—
267#103—1064no—
268#103—1064no—
269#103—1062no—
270#102—1062no—
271C22⋊C8—21no—
272C22⋊C8C22⋊D5312yes—
273C4×C8—24no—
274C4×C8—21no—
275C4×C8C22×D5312yes—
276#41—2122no—
277#41—2122no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—1062no—
281C2×Q8—1061no—
282C2×Q8—4242no—
283C2×Q8—2121no—
284C2×Q8#1171991314yescomplemented
285C2×Q8C2×D21212yes—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—2121no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4C2×D21211yescharacteristic
295C22×C4#1171994511yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297Q16—2124no—
298SD16—4244no—
299SD16—2124no—
300M16—2122no—
301C2×C8—22no—
302C2×C8—1062no—
303C2×C8—1062no—
304C2×C8—2122no—
305C2×C8C2×C22⋊D5314yes—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—1061no—
310C4⋊C4—1062no—
311C4⋊C4—1062no—
312C4⋊C4—1061no—
313C4⋊C4—2121no—
314C4⋊C4—2121no—
315C4⋊C4C2×D21212yes—
316C22⋊C4—2121no—
317C22⋊C4—2121no—
318C42—22no—
319C42—22no—
320C42C2×D21211yescharacteristic
321C42#1171994511yescharacteristic
322C42C2×C22⋊D5311yescharacteristic
323C42C23×D5311yescharacteristic
324C23—2121no—
325C23#1313319111yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—4242no—
329Q8—2121no—
330Q8—2121no—
331Q8—2124no—
332Q8C2×#1171991314yes—
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—81no—
344C2×C4—4242no—
345C2×C4—4242no—
346C2×C4—2121no—
347C2×C4—2122no—
348C2×C4#1313319111yescharacteristic
349C2×C4#1313319111yescharacteristic
350C2×C4#1313319111yescharacteristic
351C2×C4C2×#1171991312yes—
352C2×C4#1313323912yes—
353C2×C4C22×C22⋊D5311yescharacteristic
354C2×C4C22×D21211yescharacteristic
355C2×C4C2×#1171994511yescharacteristic
356C8—24no—
357C8—42no—
358C8—2122no—
359C8—2122no—
360C22—21no—
361C22—21no—
362C22—4241no—
363C22—4241no—
364C22—4241no—
365C22#1681085212yes—
366C22C2×#1313319111yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—24no—
371C4—42no—
372C4—4242no—
373C4—4241no—
374C4#1681085212yes—
375C4C22×#1171991311yescharacteristic
376C4C2×#1313323911yescharacteristic
377C4#1681143112yes—
378C2—22no—
379C2—8481no—
380C2#2028423611yescharacteristic
381C2C2×#1681085211yescharacteristic
382C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2982C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#2982D53
hypocenterC53#19397
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#16810908C2semidirect
4. #20274145C2semidirect
5. #20274063C2semidirect
6. #20273486C2semidirect
7. #20273399C2semidirect
9. C53×#2982C2semidirect
86. C2×Q8×C53SD16semidirect
190. #2982D53semidirect
228. C53#19397semidirect
284. C2×Q8#11719913semidirect

Maximal subgroups

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

subgroupindex
#202856422
C2×#168109082
#202841872
#202741452
#202740632
#202734862
#202733992
#202733622
C53×#29822
#1939753