Finite Group Databasebeta

← #27369371

Subgroups — #27369371

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
2#20284516C211yesmaximal, characteristic, complemented
3C2×#16810824C211yesmaximal, characteristic
4#20283795C211yesmaximal, characteristic, complemented
5#20274134C214yesmaximal, complemented
6#20274066C214yesmaximal, complemented
7#20273485C211yesmaximal, characteristic, complemented
8#20273382C211yesmaximal, characteristic
9C53×#1542C211yesmaximal, characteristic, complemented
10C22×#11719913—21no—
11C2×#13133214C2211yescharacteristic
12C2×C4×Dic106C2211yescharacteristic
13#16810997C2212yes—
14#16810971C2212yes—
15#16810927—22no—
16#16810915C2212yescomplemented
17#16810905C2212yescomplemented
18#16810824—22no—
19#16810824C2214yes—
20#16810823—22no—
21#16810773—22no—
22C8×Dic106—22no—
23#16810747C2214yescomplemented
24#16810676C2211yescharacteristic
25#16810673—22no—
26#16810666—21no—
27#16810647C2211yescharacteristic
28#16810631—21no—
29C2×#13133139—21no—
30#16810373C2214yescomplemented
31#16810366C2214yes—
32C53×#533C2211yescharacteristic
33C53×#431C2211yescharacteristic
34C2×C4⋊C8×C53C2211yescharacteristic
35C53×#333C2212yescomplemented
36C53×#332C2212yescomplemented
37C22×Dic106D411yescharacteristic
38C22×D212—21no—
39C22×D212—21no—
40C22×D212—21no—
41C2×#11719913—24no—
42C2×#11719913—44no—
43C2×#11719924—24no—
44C2×#11719917—21no—
45#13133214—21no—
46#13133214—21no—
47#13133214—21no—
48#13133214C2312yes—
49C4×Dic106—21no—
50C4×Dic106—21no—
51C4×Dic106D412yes—
52C4×Dic106C2312yes—
53C2×C4⋊Dic53D411yescharacteristic
54C2×#11719919—21no—
55C2×C4×Dic53—21no—
56#13133150—22no—
57#13133148—22no—
58C4⋊#88322—21no—
59C4⋊#88322D412yes—
60C4⋊#88322C2312yes—
61#13133146—24no—
62#13133140—22no—
63#13133140—22no—
64#13133139—24no—
65#13133137—24no—
66C22⋊#88322D412yes—
67C4×#88322C2312yes—
68C8×Dic53—24no—
69C22×C8×C53—21no—
70C4⋊Q8×C53C2311yescharacteristic
71C4⋊Q8×C53C2311yescharacteristic
72#125×C53—21no—
73C22⋊Q8×C53—21no—
74C22⋊Q8×C53—21no—
75C4×Q8×C53—21no—
76C4×Q8×C53—21no—
77#117×C53—21no—
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53—21no—
80C4⋊C8×C53C2312yes—
81C4⋊C8×C53C2312yes—
82C22⋊C8×C53—21no—
83C4×C8×C53—21no—
84C23×D53—41no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×Dic53—21no—
91C22×Dic53—21no—
92C2×D212—22no—
93C2×D212—22no—
94C2×D212—22no—
95C2×D212—42no—
96C2×D212—42no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—82no—
101C2×D212—82no—
102C2×Dic106—21no—
103C2×Dic106—22no—
104C2×Dic106SD1614yescomplemented
105C2×Dic106C2×D412yes—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53—42no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C42×C53—41no—
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53#4111yescharacteristic
120C42×C53C2411yescharacteristic
121#11719924—84no—
122C4⋊Dic53—21no—
123C4⋊Dic53C2×D412yes—
124#11719919—22no—
125#11719919—22no—
126C4×Dic53—22no—
127C4×Dic53—22no—
128#11719917—42no—
129C2×#88322—22no—
130C2×#88322C2×D414yes—
131#11719913—48no—
132C23×C53#12111yescharacteristic
133C22×D53—42no—
134C22×D53—82no—
135C22×D53—82no—
136C22×D53—82no—
137Q8×C53—42no—
138Q8×C53—81no—
139C2×C4×C53—22no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
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
151C2×Dic53—22no—
152C2×Dic53—24no—
153C2×Dic53—22no—
154C2×Dic53—81no—
155D212—42no—
156D212—42no—
157D212—44no—
158D212—82no—
159D212—82no—
160D212—82no—
161D212—82no—
162Dic106—22no—
163Dic106—24no—
164Dic106C2×SD1614yes—
165C8×C53—42no—
166C8×C53—42no—
167#88322—24no—
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—21no—
180C4×C53—21no—
181C4×C53—81no—
182C4×C53—81no—
183C4×C53#46512yes—
184C4×C53C22×SD1611yescharacteristic
185C4×C53C2×#13511yescharacteristic
186C4×C53#57712yes—
187Dic53—24no—
188Dic53—42no—
189#3208—531no—
190#3078—531no—
191C2×#460—531no—
192#2982—531no—
193#1678—534no—
194#1666—534no—
195#1570—531no—
196#1545—531no—
197#1542D5311yescharacteristic, complemented
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×Q8—531no—
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—534no—
217C8×Q8—1062no—
218#431—531no—
219#431—1062no—
220#431C2×D5311yescharacteristic
221#428—1061no—
222#423—531no—
223C2×C4⋊C8C2×D5311yescharacteristic
224#419—1061no—
225C2×#102—1061no—
226#334—534no—
227#333C2×D5312yescomplemented
228#332C2×D5312yescomplemented
229#331—534no—
230C53#1939611yescharacteristic, complemented, Sylow
231C22×Q8—531no—
232C22×D4—1061no—
233C22×D4—1061no—
234C22×D4—1061no—
235C2×SD16—1064no—
236C2×SD16—1064no—
237C2×SD16—2124no—
238C2×M16—1061no—
239C22×C8—21no—
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—532no—
252C4×Q8—532no—
253C4×Q8—1061no—
254C4×Q8—1061no—
255#117—21no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—1061no—
258C2×C42—1061no—
259C2×C42C22×D5311yescharacteristic
260#107—1062no—
261#106—1062no—
262C4⋊C8—21no—
263C4⋊C8—532no—
264C4⋊C8—532no—
265C4⋊C8—1064no—
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⋊C8—532no—
275C4×C8—21no—
276C4×C8—532no—
277C4×C8—1064no—
278C24—2121no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—534no—
282C2×Q8—532no—
283C2×Q8—1061no—
284C2×Q8—1062no—
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—1061no—
296C22×C4—1061no—
297C22×C4#1171994511yescharacteristic
298C22×C4D4×D5311yescharacteristic
299C22×C4D4×D5311yescharacteristic
300SD16—4244no—
301SD16—2128no—
302M16—2122no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—42no—
306C2×C8—534no—
307C2×C8—1062no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—532no—
315C4⋊C4—1061no—
316C4⋊C4—1062no—
317C4⋊C4—1062no—
318C22⋊C4—41no—
319C42—41no—
320C42—1062no—
321C42—1062no—
322C42#1171994511yescharacteristic
323C42D4×D5311yescharacteristic
324C42D4×D5311yescharacteristic
325C42C23×D5311yescharacteristic
326C23—4242no—
327C23—4242no—
328C23—4242no—
329C23—2122no—
330C23#1313319011yescharacteristic
331Q8—42no—
332Q8—81no—
333Q8—534no—
334Q8—1064no—
335Q8—1062no—
336D4—4242no—
337D4—4242no—
338D4—4242no—
339D4—4242no—
340D4—2122no—
341D4—2122no—
342D4—2124no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—1064no—
348C2×C4—1062no—
349C2×C4—1062no—
350C2×C4—4241no—
351C2×C4#1313319011yescharacteristic
352C2×C4#1313319011yescharacteristic
353C2×C4#1313319011yescharacteristic
354C2×C4SD16×D5312yes—
355C2×C4#1313324012yes—
356C2×C4C2×D4×D5311yescharacteristic
357C2×C4C2×D4×D5311yescharacteristic
358C2×C4C2×#1171994511yescharacteristic
359C8—42no—
360C8—42no—
361C8—1064no—
362C8—2122no—
363C22—21no—
364C22—21no—
365C22—8484no—
366C22—4242no—
367C22—4242no—
368C22—4242no—
369C22#1681085512yes—
370C22C2×#1313319011yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—81no—
375C4—81no—
376C4—1064no—
377C4—2122no—
378C4#1681085512yes—
379C4C2×SD16×D5311yescharacteristic
380C4C2×#1313324011yescharacteristic
381C4#1681143012yes—
382C2—22no—
383C2—8482no—
384C2#2028423911yescharacteristic
385C2C2×#1681085511yescharacteristic
386C2#2028444611yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1542C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#1542D53
hypocenterC53#19396
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
2. #20284516C2semidirect
4. #20283795C2semidirect
5. #20274134C2semidirect
6. #20274066C2semidirect
7. #20273485C2semidirect
9. C53×#1542C2semidirect
16. #16810915C22semidirect
17. #16810905C22semidirect
23. #16810747C22semidirect
30. #16810373C22semidirect
35. C53×#333C22semidirect
36. C53×#332C22semidirect
104. C2×Dic106SD16semidirect
197. #1542D53semidirect
227. #333C2×D53semidirect
228. #332C2×D53semidirect
230. C53#19396semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202845162
C2×#168108242
#202837952
#202741342
#202740662
#202734852
#202733822
C53×#15422
#1939653