Finite Group Databasebeta

← #27369368

Subgroups — #27369368

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#20285658C211yesmaximal, characteristic
2#20284520C211yesmaximal, characteristic, complemented
3C2×#16810823C211yesmaximal, characteristic
4#20274131C214yesmaximal, complemented
5#20274063C214yesmaximal, complemented
6#20273486C211yesmaximal, characteristic, complemented
7#20273382C211yesmaximal, characteristic
8#20273370C211yesmaximal, characteristic, complemented
9C53×#2982C211yesmaximal, characteristic, complemented
10C22×#11719924—21no—
11#16811180C2211yescharacteristic
12C2×#13133214C2211yescharacteristic
13#16810996C2212yes—
14#16810970C2212yes—
15#16810930—22no—
16#16810918C2212yescomplemented
17#16810908C2212yescomplemented
18#16810824—22no—
19#16810823—22no—
20#16810823C2214yes—
21#16810776—22no—
22#16810663—21no—
23#16810647C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25#16810632—21no—
26C2×#13133137—21no—
27#16810374C2214yescomplemented
28#16810371C2212yescomplemented
29#16810370C2212yescomplemented
30#16810366C2214yes—
31C2×C4×Q8×C53C2211yescharacteristic
32C8×Q8×C53—22no—
33C8×Q8×C53C2214yescomplemented
34C53×#431—22no—
35C53×#431C2211yescharacteristic
36C53×#431C2211yescharacteristic
37C22×D212—21no—
38C22×D212—21no—
39C22×D212—21no—
40C2×#11719913—24no—
41C2×#11719924—24no—
42C2×#11719924—44no—
43C2×#11719917—21no—
44C22×#88322—21no—
45#13133217C2311yescharacteristic
46#13133217C2311yescharacteristic
47#13133214—21no—
48#13133214—21no—
49#13133214—21no—
50#13133214C2312yes—
51#13133209—21no—
52#13133196—21no—
53#13133196—21no—
54C4×Dic106—21no—
55C4×Dic106—21no—
56#13133176—21no—
57#13133150—22no—
58#13133148—22no—
59C4⋊#88322—21no—
60C4⋊#88322—21no—
61C4⋊#88322C2312yes—
62C4⋊#88322C2312yes—
63#13133141—22no—
64#13133141—22no—
65#13133139—24no—
66#13133137—24no—
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—
84C23×D53—41no—
85C2×Q8×C53—21no—
86C2×Q8×C53—22no—
87C2×Q8×C53SD1614yescomplemented
88C2×Q8×C53C2×D412yes—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C2×D212—22no—
95C2×D212—22no—
96C2×D212—22no—
97C2×D212—42no—
98C2×D212—42no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—82no—
103C2×D212—82no—
104C2×Dic106—21no—
105C2×Dic106—21no—
106C2×C8×C53—22no—
107C2×C8×C53C2×D414yes—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53C2×D412yes—
112C42×C53—22no—
113C42×C53—22no—
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53C2411yescharacteristic
118C22⋊Dic53—41no—
119#11719924—48no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124#11719919—41no—
125#11719919—41no—
126C4×Dic53—41no—
127#11719917—42no—
128C2×#88322—22no—
129C2×#88322—22no—
130C2×#88322—42no—
131#11719913—84no—
132C23×C53#12111yescharacteristic
133C22×D53—42no—
134C22×D53—82no—
135C22×D53—82no—
136C22×D53—82no—
137Q8×C53—22no—
138Q8×C53—24no—
139Q8×C53C2×SD1614yes—
140C2×C4×C53—22no—
141C2×C4×C53—24no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—81no—
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53C2×SD1612yes—
150C2×C4×C53#13512yes—
151C2×C4×C53C22×D411yescharacteristic
152C2×C4×C53C22×D411yescharacteristic
153C2×C4×C53C2×#4111yescharacteristic
154C2×Dic53—41no—
155C2×Dic53—41no—
156D212—42no—
157D212—42no—
158D212—44no—
159D212—82no—
160D212—82no—
161D212—82no—
162D212—82no—
163Dic106—42no—
164Dic106—81no—
165C8×C53—24no—
166C8×C53—42no—
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#46512yes—
184C4×C53C22×SD1611yescharacteristic
185C4×C53C2×#13511yescharacteristic
186C4×C53#57712yes—
187Dic53—81no—
188Dic53—81no—
189#3208—531no—
190#3078—531no—
191C2×#460—531no—
192#2982D5311yescharacteristic, complemented
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#533—531no—
205C2×#127—531no—
206C2×C4×Q8C2×D5311yescharacteristic
207#492—532no—
208#486—532no—
209#479—1062no—
210#477—532no—
211#475—532no—
212#460—534no—
213#460—1062no—
214#460—1062no—
215#450—1062no—
216C8×Q8—22no—
217C8×Q8C2×D5314yescomplemented
218#431—22no—
219#431C2×D5311yescharacteristic
220#431C2×D5311yescharacteristic
221#428—1061no—
222#423—531no—
223C2×C4⋊C8—531no—
224#419—1061no—
225C2×#102—1061no—
226#334—534no—
227#333—532no—
228#332—532no—
229#331—534no—
230C53#1939611yescharacteristic, complemented, Sylow
231C22×Q8D21211yescharacteristic
232C22×D4—1061no—
233C22×D4—1061no—
234C22×D4—1061no—
235C2×SD16—1064no—
236C2×SD16—1064no—
237C2×SD16—2124no—
238C2×M16—1061no—
239C22×C8—1061no—
240C4⋊Q8—531no—
241C4⋊Q8—531no—
242#127—532no—
243#127—1061no—
244#127—1061no—
245#127—1061no—
246#125—1061no—
247C22⋊Q8—1061no—
248C22⋊Q8—1061no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—1061no—
252C4×Q8—1061no—
253C4×Q8D21212yes—
254C4×Q8C22×D5312yes—
255#117—1061no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D21211yescharacteristic
258C2×C42—21no—
259C2×C42C22×D5311yescharacteristic
260#107—1062no—
261#106—1062no—
262C4⋊C8—24no—
263C4⋊C8—532no—
264C4⋊C8—532no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8C22⋊D5312yes—
268C4⋊C8C22×D5312yes—
269#103—1062no—
270#103—1062no—
271#102—1064no—
272#102—1064no—
273C22⋊C8—21no—
274C22⋊C8C22⋊D5312yes—
275C4×C8—24no—
276C4×C8—21no—
277C4×C8C22×D5312yes—
278C24—2121no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—1061no—
282C2×Q8—1061no—
283C2×Q8#1171991314yescomplemented
284C2×Q8C2×D21212yes—
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×C4C2×D21211yescharacteristic
298C22×C4#1171994511yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300SD16—4244no—
301SD16—2128no—
302M16—2122no—
303C2×C8—22no—
304C2×C8—1062no—
305C2×C8—1062no—
306C2×C8—2122no—
307C2×C8C2×C22⋊D5314yes—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C4⋊C4C2×D21212yes—
318C22⋊C4—2121no—
319C42—22no—
320C42—22no—
321C42—2121no—
322C42C2×D21211yescharacteristic
323C42#1171994511yescharacteristic
324C42C2×C22⋊D5311yescharacteristic
325C42C23×D5311yescharacteristic
326C23—4242no—
327C23—4242no—
328C23—4242no—
329C23—2122no—
330C23#1313319111yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—4241no—
334Q8—2122no—
335Q8C2×#1171991314yes—
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—81no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4#1313319111yescharacteristic
352C2×C4#1313319111yescharacteristic
353C2×C4#1313319111yescharacteristic
354C2×C4C2×#1171991312yes—
355C2×C4#1313323912yes—
356C2×C4C22×C22⋊D5311yescharacteristic
357C2×C4C22×D21211yescharacteristic
358C2×C4C2×#1171994511yescharacteristic
359C8—24no—
360C8—42no—
361C8—2122no—
362C8—2122no—
363C22—21no—
364C22—21no—
365C22—8484no—
366C22—4242no—
367C22—4242no—
368C22—4242no—
369C22#1681085212yes—
370C22C2×#1313319111yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—24no—
375C4—42no—
376C4—4241no—
377C4—4241no—
378C4#1681085212yes—
379C4C22×#1171991311yescharacteristic
380C4C2×#1313323911yescharacteristic
381C4#1681143112yes—
382C2—22no—
383C2—8482no—
384C2#2028423611yescharacteristic
385C2C2×#1681085211yescharacteristic
386C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2982C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#2982D53
hypocenterC53#19396
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. #20284520C2semidirect
4. #20274131C2semidirect
5. #20274063C2semidirect
6. #20273486C2semidirect
8. #20273370C2semidirect
9. C53×#2982C2semidirect
16. #16810918C22semidirect
17. #16810908C22semidirect
27. #16810374C22semidirect
28. #16810371C22semidirect
29. #16810370C22semidirect
33. C8×Q8×C53C22semidirect
87. C2×Q8×C53SD16semidirect
192. #2982D53semidirect
217. C8×Q8C2×D53semidirect
230. C53#19396semidirect
283. C2×Q8#11719913semidirect

Maximal subgroups

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

subgroupindex
#202856582
#202845202
C2×#168108232
#202741312
#202740632
#202734862
#202733822
#202733702
C53×#29822
#1939653