Finite Group Databasebeta

← #27369407

Subgroups — #27369407

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285667C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic
3#20283796C211yesmaximal, characteristic, complemented
4#20274161C212yesmaximal, complemented
5#20274147C212yesmaximal, complemented
6#20274083C212yesmaximal, complemented
7#20274065C212yesmaximal, complemented
8#20273496C211yesmaximal, characteristic, complemented
9#20273421C211yesmaximal, characteristic
10#20273380C211yesmaximal, characteristic, complemented
11C53×#3113C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16811002C2211yescharacteristic
14#16810997C2211yescharacteristic
15#16810975C2211yescharacteristic
16#16810971C2211yescharacteristic
17#16810941—21no—
18#16810908C2212yes—
19#16810845—22no—
20#16810823C2212yes—
21#16810783—22no—
22#16810762—21no—
23Q8×#88322—22no—
24Q8×#88322C2212yes—
25Q8×#88322C2212yes—
26C4×#11719924—22no—
27#16810676—22no—
28#16810676C2211yescharacteristic
29#16810676C2211yescharacteristic
30#16810676C2211yescharacteristic
31#16810375C2212yes—
32#16810373C2212yes—
33#16810369C2212yes—
34#16810369C2212yes—
35#16810368C2212yes—
36#16810366C2212yes—
37C53×#533C2211yescharacteristic
38C2×C4×Q8×C53C2211yescharacteristic
39C53×#483—21no—
40C53×#475C2212yes—
41C53×#462C2212yes—
42C53×#451—21no—
43C53×#441—22no—
44C53×#432C2211yescharacteristic
45C2×#11719945—21no—
46C2×#11719925—22no—
47C2×#11719924—22no—
48C2×#11719924—22no—
49#13133217C2311yescharacteristic
50#13133214C2311yescharacteristic
51#13133205—21no—
52#13133191—21no—
53#13133191—21no—
54C4×D212—21no—
55C4×D212—21no—
56#13133148—22no—
57C4⋊#88322—22no—
58C4⋊#88322—22no—
59C4⋊#88322D412yes—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133141—22no—
63#13133140—22no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322—21no—
67C22⋊#88322—21no—
68C22⋊#88322D412yes—
69C4×#88322—21no—
70C4×#88322—21no—
71C4×#88322—22no—
72C4×#88322—22no—
73C4×#88322C2311yescharacteristic
74C4×#88322C2311yescharacteristic
75C22×Q8×C53D411yescharacteristic
76C2×Q16×C53—22no—
77C4⋊Q8×C53C2311yescharacteristic
78C4⋊Q8×C53C2311yescharacteristic
79#125×C53—21no—
80C22⋊Q8×C53—21no—
81C22⋊Q8×C53—21no—
82C4×Q8×C53—21no—
83C4×Q8×C53—21no—
84C4×Q8×C53—21no—
85C4×Q8×C53—21no—
86C4×Q8×C53D412yes—
87C4×Q8×C53C2312yes—
88#117×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53D411yescharacteristic
91C2×C42×C53—21no—
92C2×C42×C53C2311yescharacteristic
93#106×C53—22no—
94C4⋊C8×C53C2312yes—
95#103×C53—22no—
96#103×C53—22no—
97C22⋊C8×C53—21no—
98#97×C53—21no—
99C2×Q8×C53—24no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×Q8×C53—21no—
103C2×Q8×C53C2×D412yes—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C2×C22⋊D53—41no—
110#11719945—82no—
111C2×D212—21no—
112C2×D212—21no—
113C2×D212—21no—
114C2×C4×D53—41no—
115C2×Dic106—21no—
116Q16×C53—44no—
117C2×C8×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—41no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53C2×D412yes—
128C22⋊C4×C53—41no—
129C42×C53—22no—
130C42×C53—22no—
131C42×C53—41no—
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53#4111yescharacteristic
135C42×C53C2411yescharacteristic
136#11719925—82no—
137#11719924—44no—
138#11719924—82no—
139#11719921—41no—
140#11719921—41no—
141C4⋊Dic53—21no—
142C4⋊Dic53—21no—
143C2×#88322—22no—
144C2×#88322—22no—
145C2×#88322C2×D412yes—
146C2×#88322C2×D412yes—
147C23×C53#12111yescharacteristic
148C22×D53—41no—
149Q8×C53—24no—
150Q8×C53—22no—
151Q8×C53—42no—
152Q8×C53—42no—
153Q8×C53#13514yes—
154C2×C4×C53—24no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—22no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—81no—
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53C22×D411yescharacteristic
167C2×C4×C53C22×D411yescharacteristic
168C2×C4×C53C2×#4111yescharacteristic
169C22⋊D53—82no—
170C2×Dic53—41no—
171D212—42no—
172D212—42no—
173D212—81no—
174C4×D53—82no—
175Dic106—42no—
176C8×C53—42no—
177#88322—22no—
178#88322—22no—
179#88322—42no—
180#88322—42no—
181#19400—531nomaximal, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53C2×#12111yescharacteristic
186C2×D53—81no—
187C2×D53—81no—
188C2×D53—81no—
189C4×C53—21no—
190C4×C53—24no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—42no—
195C4×C53—81no—
196C4×C53—81no—
197C4×C53C2×#13512yes—
198C4×C53#57712yes—
199Dic53—81no—
200#3210—531no—
201#3113D5311yescharacteristic, complemented
202#3103—531no—
203#2982—531no—
204#1682—532no—
205#1680—532no—
206#1670—532no—
207#1666—532no—
208#1572—531no—
209#1553—531no—
210#1544—531no—
211C2×C53—41no—
212C2×C53#304311yescharacteristic
213C2×C53#305011yescharacteristic
214C2×C53#306911yescharacteristic
215D53—161no—
216#533C2×D5311yescharacteristic
217#532—531no—
218C2×C4×Q8C2×D5311yescharacteristic
219#494—531no—
220#492—531no—
221#487—531no—
222#486—531no—
223#483—21no—
224#481—1061no—
225#475—532no—
226#475C2×D5312yes—
227#464—1062no—
228#462C2×D5312yes—
229#460—532no—
230#452—1062no—
231#451—21no—
232#448—1061no—
233C8×Q8—532no—
234C8×Q8—532no—
235C8×Q8—1062no—
236#441—22no—
237C4×SD16—1062no—
238#432C2×D5311yescharacteristic
239#431—531no—
240#431—531no—
241#431—531no—
242#431—1062no—
243#335—532no—
244#334—532no—
245#332—532no—
246#332—532no—
247#332—532no—
248#331—532no—
249C53#1940011yescharacteristic, complemented, Sylow
250C2×#41—1061no—
251C22×Q8C22⋊D5311yescharacteristic
252C2×Q16—22no—
253C2×Q16—1062no—
254C2×SD16—1062no—
255C2×SD16—1062no—
256C4⋊Q8—531no—
257C4⋊Q8C22×D5311yescharacteristic
258C4⋊Q8C22×D5311yescharacteristic
259#127—531no—
260#125—21no—
261#124—1061no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264#121—1061no—
265#121—1061no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8C22⋊D5312yes—
271C4×Q8C22×D5312yes—
272C4×D4—1061no—
273C4×D4—1061no—
274#117—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C22⋊D5311yescharacteristic
277C2×C42—21no—
278C2×C42C22×D5311yescharacteristic
279#106—22no—
280#106—1062no—
281C4⋊C8—532no—
282C4⋊C8—532no—
283C4⋊C8—532no—
284C4⋊C8—1062no—
285C4⋊C8—1062no—
286C4⋊C8C22×D5312yes—
287#103—22no—
288#103—22no—
289#103—1062no—
290#103—1062no—
291#102—1062no—
292#102—1062no—
293C22⋊C8—21no—
294C22⋊C8—532no—
295C22⋊C8—1061no—
296C22⋊C8—1061no—
297#97—21no—
298C4×C8—531no—
299C4×C8—531no—
300C4×C8—1062no—
301C4×C8—1062no—
302C4×C8—1061no—
303C4×C8—1061no—
304#41—4242no—
305C2×Q8—24no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—1061no—
310C2×Q8C2×C22⋊D5312yes—
311C2×D4—1061no—
312C2×D4—1061no—
313C2×D4—1061no—
314C2×D4—2121no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2121no—
318C22×C4D4×D5311yescharacteristic
319C22×C4#1171994711yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321Q16—44no—
322Q16—4242no—
323SD16—4242no—
324SD16—2124no—
325C2×C8—22no—
326C2×C8—532no—
327C2×C8—532no—
328C2×C8—1062no—
329C2×C8—1062no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—1061no—
340C4⋊C4—1061no—
341C4⋊C4C2×C22⋊D5312yes—
342C22⋊C4—41no—
343C22⋊C4—2121no—
344C22⋊C4—2121no—
345C42—22no—
346C42—22no—
347C42—41no—
348C42D4×D5311yescharacteristic
349C42#1171994711yescharacteristic
350C42C2×C22⋊D5311yescharacteristic
351C42C23×D5311yescharacteristic
352C23—2121no—
353C23#1313318911yescharacteristic
354Q8—24no—
355Q8—22no—
356Q8—42no—
357Q8—42no—
358Q8—2122no—
359Q8#1313323714yes—
360D4—4241no—
361D4—4242no—
362D4—2122no—
363D4—2122no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—4242no—
374C2×C4—2121no—
375C2×C4#1313318911yescharacteristic
376C2×C4#1313318911yescharacteristic
377C2×C4#1313318911yescharacteristic
378C2×C4C2×D4×D5311yescharacteristic
379C2×C4C22×C22⋊D5311yescharacteristic
380C2×C4C2×#1171994711yescharacteristic
381C8—42no—
382C8—1062no—
383C8—1062no—
384C8—2122no—
385C8—2122no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—4241no—
390C22—4241no—
391C22—4241no—
392C22C2×#1313318911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—24no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—4241no—
402C4C2×#1313323712yes—
403C4#1681143212yes—
404C2—41no—
405C2—8481no—
406C2#2028423811yescharacteristic
407C2#2028429311yescharacteristic
408C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3113C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3113D53
hypocenterC53#19400
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
3. #20283796C2semidirect
4. #20274161C2semidirect
5. #20274147C2semidirect
6. #20274083C2semidirect
7. #20274065C2semidirect
8. #20273496C2semidirect
10. #20273380C2semidirect
11. C53×#3113C2semidirect
201. #3113D53semidirect
249. C53#19400semidirect

Maximal subgroups

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

subgroupindex
#202856672
#202847462
#202837962
#202741612
#202741472
#202740832
#202740652
#202734962
#202734212
#202733802
C53×#31132
#1940053