Finite Group Databasebeta

← #27369441

Subgroups — #27369441

Order 13568.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285836C211yesmaximal, characteristic
2#20284187C211yesmaximal, characteristic
3#20283796C211yesmaximal, characteristic, complemented
4#20274161C214yesmaximal, complemented
5#20274107C214yesmaximal, complemented
6#20273489C211yesmaximal, characteristic, complemented
7#20273387C211yesmaximal, characteristic, complemented
8#20273365C211yesmaximal, characteristic
9C2×C53×#462C211yesmaximal, characteristic, complemented
10C2×#13133248—21no—
11#16811173C2211yescharacteristic
12#16811026C2212yes—
13#16810971C2212yes—
14#16810936—22no—
15#16810900—22no—
16#16810845—22no—
17#16810833C2212yes—
18#16810823C2212yes—
19Q8×#88322—22no—
20Q8×#88322C2214yes—
21#16810676—22no—
22#16810676C2211yescharacteristic
23#16810676C2211yescharacteristic
24#16810647C2211yescharacteristic
25#16810622—21no—
26#16810375C2214yes—
27#16810370C2212yes—
28#16810369C2214yes—
29#16810366C2212yes—
30C22×Q16×C53—21no—
31C2×C4⋊Q8×C53C2211yescharacteristic
32C2×C4×Q8×C53C2211yescharacteristic
33C53×#462—22no—
34C53×#462C2214yes—
35C2×C4⋊C8×C53C2211yescharacteristic
36C2×#103×C53—21no—
37C2×#11719945—21no—
38#13133248—44no—
39C2×#11719925—22no—
40C2×#11719924—22no—
41C2×#11719917—21no—
42#13133217C2311yescharacteristic
43#13133214C2311yescharacteristic
44#13133205—21no—
45#13133191—21no—
46#13133191—21no—
47C4×D212—21no—
48C4×D212—21no—
49#13133150—24no—
50C4⋊#88322—21no—
51C4⋊#88322—24no—
52C4⋊#88322D412yes—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55#13133140—22no—
56#13133137—22no—
57#13133137—24no—
58C22⋊#88322—21no—
59C22⋊#88322D412yes—
60C4×#88322—21no—
61C4×#88322—24no—
62C4×#88322C2312yes—
63C22×Q8×C53—21no—
64C22×Q8×C53D411yescharacteristic
65C2×Q16×C53—24no—
66C2×Q16×C53—44no—
67C22×C8×C53—21no—
68C4⋊Q8×C53—21no—
69C4⋊Q8×C53—21no—
70C4⋊Q8×C53—21no—
71C4⋊Q8×C53C2312yes—
72C4×Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53D412yes—
75C4×Q8×C53C2312yes—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53D411yescharacteristic
80C2×C42×C53—21no—
81C2×C42×C53C2311yescharacteristic
82C4⋊C8×C53—21no—
83C4⋊C8×C53C2312yes—
84#103×C53—24no—
85C2×Q8×C53—21no—
86C2×Q8×C53—22no—
87C2×Q8×C53—22no—
88C2×Q8×C53—41no—
89C2×Q8×C53—42no—
90C2×Q8×C53—42no—
91C2×Q8×C53D814yes—
92C2×Q8×C53C2×D412yes—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—41no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C2×C22⋊D53—41no—
100#11719945—42no—
101#11719945—42no—
102C2×D212—21no—
103C2×D212—21no—
104C2×D212—21no—
105C2×C4×D53—41no—
106C2×Dic106—21no—
107Q16×C53—48no—
108C2×C8×C53—22no—
109C2×C8×C53—42no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53C2×D412yes—
118C42×C53—22no—
119C42×C53—22no—
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C42×C53C2411yescharacteristic
124#11719925—44no—
125#11719924—44no—
126#11719921—41no—
127#11719921—41no—
128C4⋊Dic53—21no—
129C4⋊Dic53—21no—
130#11719917—42no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322C2×D414yes—
134C23×C53#12111yescharacteristic
135C22×D53—41no—
136Q8×C53—22no—
137Q8×C53—24no—
138Q8×C53—42no—
139Q8×C53—42no—
140Q8×C53—82no—
141Q8×C53C2×D814yes—
142C2×C4×C53—22no—
143C2×C4×C53—24no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—81no—
149C2×C4×C53—82no—
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53C2×D812yes—
154C2×C4×C53#13512yes—
155C2×C4×C53C22×D411yescharacteristic
156C2×C4×C53C22×D411yescharacteristic
157C2×C4×C53C2×#4111yescharacteristic
158C22⋊D53—82no—
159C2×Dic53—41no—
160D212—41no—
161D212—41no—
162D212—42no—
163D212—81no—
164C4×D53—82no—
165Dic106—41no—
166Dic106—41no—
167C8×C53—42no—
168#88322—24no—
169#88322—42no—
170#88322—42no—
171#19404—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53#46612yes—
175C22×C53C2×#12111yescharacteristic
176C2×D53—81no—
177C2×D53—81no—
178C2×D53—81no—
179C4×C53—21no—
180C4×C53—24no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53—82no—
185C4×C53#46612yes—
186C4×C53C22×D811yescharacteristic
187C4×C53C2×#13511yescharacteristic
188C4×C53#57812yes—
189Dic53—81no—
190#3226—531no—
191#3037—531no—
192C2×#462D5311yescharacteristic, complemented
193#2982—531no—
194#1682—534no—
195#1674—534no—
196#1571—531no—
197#1546—531no—
198#1540—531no—
199C2×C53—22no—
200C2×C53#304411yescharacteristic
201C2×C53C2×#46611yescharacteristic
202C2×C53#306911yescharacteristic
203D53—161no—
204C2×#137—1061no—
205C22×Q16—21no—
206#532—531no—
207C2×C4⋊Q8C2×D5311yescharacteristic
208C2×C4×Q8C2×D5311yescharacteristic
209#500—532no—
210#486—532no—
211#480—1062no—
212#474—1062no—
213#464—1062no—
214#462—22no—
215#462—532no—
216#462C2×D5314yes—
217#460—532no—
218C8×Q8—534no—
219C8×Q8—1062no—
220#431—531no—
221#431—531no—
222#431—1062no—
223#423—531no—
224C2×C4⋊C8C2×D5311yescharacteristic
225#417—1061no—
226C2×#103—21no—
227#335—534no—
228#332—534no—
229#332—532no—
230#331—532no—
231C53#1940411yescharacteristic, complemented, Sylow
232C2×#41—1061no—
233C22×Q8—21no—
234C22×Q8C22⋊D5311yescharacteristic
235#137—2124no—
236C2×Q16—24no—
237C2×Q16—44no—
238C2×Q16—1062no—
239C2×SD16—1062no—
240C2×M16—1061no—
241C22×C8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—531no—
246C4⋊Q8C22×D5312yes—
247#127—531no—
248#124—1061no—
249#121—1061no—
250#121—1061no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D5312yes—
254C4×Q8C22×D5312yes—
255C4×D4—1061no—
256C4×D4—1061no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22⋊D5311yescharacteristic
261C2×C42—21no—
262C2×C42C22×D5311yescharacteristic
263#107—1064no—
264C4⋊C8—21no—
265C4⋊C8—532no—
266C4⋊C8—532no—
267C4⋊C8—532no—
268C4⋊C8—1064no—
269C4⋊C8—1061no—
270C4⋊C8C22×D5312yes—
271#103—24no—
272#103—1062no—
273#102—1064no—
274#102—1062no—
275C22⋊C8—532no—
276C22⋊C8—1061no—
277C4×C8—532no—
278C4×C8—1064no—
279C4×C8—1061no—
280#41—2122no—
281#41—2122no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—1061no—
289C2×Q8#1171992214yes—
290C2×Q8C2×C22⋊D5312yes—
291C2×D4—1061no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—2121no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—2121no—
299C22×C4D4×D5311yescharacteristic
300C22×C4#1171994711yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302Q16—48no—
303Q16—2124no—
304SD16—2124no—
305M16—2122no—
306C2×C8—22no—
307C2×C8—42no—
308C2×C8—534no—
309C2×C8—1062no—
310C2×C8—1062no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—1061no—
319C4⋊C4—1061no—
320C4⋊C4C2×C22⋊D5312yes—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C42—22no—
324C42—22no—
325C42D4×D5311yescharacteristic
326C42#1171994711yescharacteristic
327C42C2×C22⋊D5311yescharacteristic
328C42C23×D5311yescharacteristic
329C23—2121no—
330C23#1313318911yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—42no—
335Q8—82no—
336Q8—2121no—
337Q8—2121no—
338Q8C2×#1171992214yes—
339D4—4241no—
340D4—4242no—
341D4—2121no—
342D4—2121no—
343D4—2122no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—4242no—
353C2×C4—2121no—
354C2×C4#1313318911yescharacteristic
355C2×C4#1313318911yescharacteristic
356C2×C4#1313318911yescharacteristic
357C2×C4C2×#1171992212yes—
358C2×C4#1313323712yes—
359C2×C4C2×D4×D5311yescharacteristic
360C2×C4C22×C22⋊D5311yescharacteristic
361C2×C4C2×#1171994711yescharacteristic
362C8—42no—
363C8—1064no—
364C8—2122no—
365C8—2122no—
366C22—21no—
367C22—21no—
368C22—4241no—
369C22—4241no—
370C22—4241no—
371C22#1681085712yes—
372C22C2×#1313318911yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—24no—
377C4—42no—
378C4—82no—
379C4—4241no—
380C4#1681085712yes—
381C4C22×#1171992211yescharacteristic
382C4C2×#1313323711yescharacteristic
383C4#1681143612yes—
384C2—22no—
385C2—8481no—
386C2#2028424211yescharacteristic
387C2C2×#1681085711yescharacteristic
388C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#462C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×#462D53
hypocenterC53#19404
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. #20283796C2semidirect
4. #20274161C2semidirect
5. #20274107C2semidirect
6. #20273489C2semidirect
7. #20273387C2semidirect
9. C2×C53×#462C2semidirect
192. C2×#462D53semidirect
231. C53#19404semidirect

Maximal subgroups

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

subgroupindex
#202858362
#202841872
#202837962
#202741612
#202741072
#202734892
#202733872
#202733652
C2×C53×#4622
#1940453