Finite Group Databasebeta

← #27403915

Subgroups — #27403915

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×Dic53C211yesmaximal, characteristic
2#20277687C212yesmaximal, complemented
3#20275592C211yesmaximal, characteristic
4#20275338C212yesmaximal, complemented
5#20275332C212yesmaximal, complemented
6#20274876C211yesmaximal, characteristic, complemented
7#20274723C212yesmaximal, complemented
8#20274615C211yesmaximal, characteristic, complemented
9#20274612C211yesmaximal, characteristic, complemented
10#20274583C211yesmaximal, characteristic, complemented
11C53×#1972C211yesmaximal, characteristic
12C2×D4×Dic53C2212yes—
13D8×Dic53C418yescomplemented
14C2×#13133151C2211yescharacteristic
15C2×#13133150C2212yes—
16C2×#13133149C2211yescharacteristic
17C2×#13133146C2211yescharacteristic
18C2×#13133138C2212yes—
19C2×#13133138C2212yes—
20C2×#13133127C2211yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22#16810526C2212yes—
23#16810517—22no—
24#16810511—22no—
25#16810488C2212yes—
26#16810484—22no—
27C4⋊C4×Dic53C2212yes—
28C4×#11719919C2211yescharacteristic
29C4×C4⋊Dic53C2212yes—
30C22⋊C4×Dic53—22no—
31#16810386C2212yes—
32#16810385C2212yes—
33#16810378C2211yescharacteristic
34#16810377C2211yescharacteristic
35C22×D8×C53C2211yescharacteristic
36C2×#106×C53C2211yescharacteristic
37C2×#102×C53C2212yes—
38C2×#97×C53C2211yescharacteristic
39C53×#390C2212yes—
40C23×Dic53—22no—
41C22×#88322C2311yescharacteristic
42C22×#88322C2311yescharacteristic
43D4×Dic53—24no—
44D4×Dic53C2×C418yes—
45C2×C4⋊Dic53C2312yes—
46C2×C4⋊Dic53C2312yes—
47C2×#11719919D412yes—
48C2×C22⋊Dic53—22no—
49C2×C22⋊Dic53—22no—
50C2×C22⋊Dic53—22no—
51C2×C4×Dic53—22no—
52C2×C4×Dic53—22no—
53C2×C4×Dic53C2311yescharacteristic
54C2×C4×Dic53C2311yescharacteristic
55#13133151C2×C414yes—
56#13133150—42no—
57#13133149—22no—
58#13133146—22no—
59#13133138—42no—
60#13133138C2×C418yes—
61#13133127—22no—
62C8×Dic53C2×C414yes—
63#13133124—22no—
64#13133124—22no—
65#13133124—22no—
66#13133124—22no—
67#13133124D412yes—
68C22×D4×C53C2312yes—
69C2×D8×C53—22no—
70C2×D8×C53—22no—
71C2×D8×C53C2×C414yes—
72C22×C8×C53C2311yescharacteristic
73C22×C8×C53C2311yescharacteristic
74C2×C4⋊C4×C53C2312yes—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53—22no—
77C2×C42×C53C2311yescharacteristic
78#106×C53—22no—
79#102×C53—42no—
80#97×C53—22no—
81C24×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—22no—
87C2×D4×C53C22×C414yes—
88C22×C4×C53—22no—
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2411yescharacteristic
91C22×Dic53—22no—
92C22×Dic53—28no—
93C22×Dic53—22no—
94C22×Dic53—82no—
95C22×Dic53—82no—
96C22×Dic53C2×D412yes—
97C22×Dic53C2×D411yescharacteristic
98C22×Dic53C2×D411yescharacteristic
99D8×C53—28no—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53C22×C412yes—
104C2×C8×C53#4112yes—
105C2×C8×C53#4112yes—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—22no—
108C22⋊C4×C53—44no—
109C22⋊C4×C53—82no—
110C42×C53—21no—
111C42×C53—21no—
112C22⋊Dic53—28no—
113C22⋊Dic53—44no—
114C22⋊Dic53—82no—
115C4⋊Dic53—22no—
116C4⋊Dic53—22no—
117C4⋊Dic53—22no—
118C4⋊Dic53C22×C414yes—
119#11719919—42no—
120C4×Dic53—21no—
121C4×Dic53—21no—
122C4×Dic53—82no—
123C4×Dic53—82no—
124C4×Dic53C22×C412yes—
125C4×Dic53#4112yes—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322C22×C412yes—
132C23×C53—24no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53C22×D411yescharacteristic
140D4×C53—24no—
141D4×C53—24no—
142D4×C53—42no—
143D4×C53—42no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53C23×C411yescharacteristic
151C2×C4×C53C2×#4111yescharacteristic
152C2×C4×C53C2×#4111yescharacteristic
153C2×C4×C53C2×#4111yescharacteristic
154C2×C4×C53H(5,𝔽2)11yescharacteristic
155C2×C4×C532−1+411yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—24no—
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53—22no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—88no—
168C2×Dic53C4×D412yes—
169C2×Dic53C4×D412yes—
170C8×C53—41no—
171C8×C53#11714yes—
172#88322—22no—
173#88322—41no—
174#22816—531nomaximal, Sylow
175C22×C53—44no—
176C22×C53—44no—
177C22×C53—44no—
178C22×C53—82no—
179C22×C53—82no—
180C22×C53—82no—
181C22×C53—82no—
182C22×C53C2×C4×D411yescharacteristic
183C22×C53#54711yescharacteristic
184C22×C53#55011yescharacteristic
185C22×C53#57512yes—
186C22×C53#57612yes—
187C4×C53—42no—
188C4×C53—82no—
189C4×C53C2×#11711yescharacteristic
190C4×C53#52011yescharacteristic
191C4×C53#55511yescharacteristic
192C4×C53#55611yescharacteristic
193Dic53—22no—
194Dic53—22no—
195Dic53—42no—
196Dic53—44no—
197Dic53—82no—
198C2×C4×D8—531no—
199#2357—532no—
200#1972—531no—
201#1972D5311yescharacteristic
202#1925—532no—
203#1925—532no—
204#1846—531no—
205#1804—532no—
206#1776—531no—
207#1776—531no—
208#1770—531no—
209C2×C53—84no—
210C2×C53#235411yescharacteristic
211C2×C53#294311yescharacteristic
212C2×C53#294811yescharacteristic
213C2×C53#335512yes—
214C2×C53#340312yes—
215C22×D8C2×D5311yescharacteristic
216C2×C4×D4—532no—
217C4×D8—538no—
218C2×#107—531no—
219C2×#107—532no—
220C2×#106—531no—
221C2×#106C2×D5311yescharacteristic
222C2×C4⋊C8—531no—
223C2×#102—532no—
224C2×#102—532no—
225C2×#102C2×D5312yes—
226C2×#97—531no—
227C2×#97C2×D5311yescharacteristic
228C2×C4×C8—531no—
229#390—532no—
230#390C2×D5312yes—
231#388—1062no—
232#388—1062no—
233#382—532no—
234#381—1062no—
235C4×C4⋊C4—531no—
236C4×C4⋊C4—532no—
237C4×C4⋊C4—532no—
238C4×C22⋊C4—1062no—
239#340—532no—
240#340—532no—
241#336—531no—
242#336—531no—
243C53#2281611yescharacteristic, complemented, Sylow
244C22×D4C22×D5312yes—
245C23×C4—1062no—
246C2×D8—22no—
247C2×D8—22no—
248C2×D8C4×D5314yes—
249C22×C8—531no—
250C22×C8—531no—
251C22×C8C22×D5311yescharacteristic
252C22×C8C22×D5311yescharacteristic
253C4×D4—538no—
254C4×D4—1064no—
255C2×C4⋊C4—532no—
256C2×C4⋊C4—532no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4C22×D5312yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—1062no—
262C2×C22⋊C4—1062no—
263C2×C22⋊C4—1062no—
264C2×C42—531no—
265C2×C42—531no—
266C2×C42—1062no—
267C2×C42—1062no—
268C2×C42C22×D5311yescharacteristic
269#107—534no—
270#107—2122no—
271#106—22no—
272#106—1062no—
273C4⋊C8—1062no—
274#102—42no—
275#102—538no—
276#102—2122no—
277#97—22no—
278#97—1062no—
279C4×C8—534no—
280#95—532no—
281#95—1062no—
282#95—1062no—
283#95—1062no—
284#95—1062no—
285C24—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4C2×C4×D5314yes—
292C22×C4—22no—
293C22×C4—531no—
294C22×C4—531no—
295C22×C4—532no—
296C22×C4—1068no—
297C22×C4—1062no—
298C22×C4—1062no—
299C22×C4—4242no—
300C22×C4—4242no—
301C22×C4D4×D5312yes—
302C22×C4C23×D5311yescharacteristic
303D8—28no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—532no—
308C2×C8—1061no—
309C2×C8—1061no—
310C2×C8—1061no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8C2×C4×D5312yes—
314C2×C8#1171994712yes—
315C2×C8#1171994712yes—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—534no—
319C4⋊C4—1062no—
320C4⋊C4—1062no—
321C4⋊C4—1062no—
322C4⋊C4—2122no—
323C22⋊C4—44no—
324C22⋊C4—82no—
325C22⋊C4—1068no—
326C22⋊C4—4242no—
327C22⋊C4—2124no—
328C42—21no—
329C42—21no—
330C42—532no—
331C42—532no—
332C42—1061no—
333C42—1061no—
334C42—4242no—
335C42—4242no—
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23C2×D4×D5311yescharacteristic
344D4—24no—
345D4—24no—
346D4—42no—
347D4—42no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—532no—
355C2×C4—532no—
356C2×C4—1062no—
357C2×C4—1062no—
358C2×C4—1062no—
359C2×C4—1064no—
360C2×C4—1062no—
361C2×C4—1062no—
362C2×C4—4248no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4—2122no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4C22×C4×D5311yescharacteristic
369C2×C4#41×D5311yescharacteristic
370C2×C4C2×#1171994711yescharacteristic
371C2×C4C2×#1171994711yescharacteristic
372C2×C4#1313326211yescharacteristic
373C2×C4#1313326411yescharacteristic
374C8—41no—
375C8—1062no—
376C8—2121no—
377C8#1313317314yes—
378C22—44no—
379C22—44no—
380C22—44no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22C4×D4×D5311yescharacteristic
386C22#1681127311yescharacteristic
387C22#1681129311yescharacteristic
388C22#1681141912yes—
389C22#1681142712yes—
390C4—42no—
391C4—82no—
392C4—1062no—
393C4—1062no—
394C4—4242no—
395C4—2122no—
396C4—2124no—
397C4C2×#1313317311yescharacteristic
398C4#1681110211yescharacteristic
399C4#1681133711yescharacteristic
400C4#1681134211yescharacteristic
401C2—84no—
402C2#2027766311yescharacteristic
403C2#2028351211yescharacteristic
404C2#2028356411yescharacteristic
405C2#2028731612yes—
406C2#2028780012yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1972C2
socleC23×C53C22×D4
prime radicalC22×D8×C53C22
cosocleC22×C4×C53C24
hypercenter#1972D53
hypocenterC53#22816
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1972

Complements

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

NH ≅ G/Ntype
2. #20277687C2semidirect
4. #20275338C2semidirect
5. #20275332C2semidirect
6. #20274876C2semidirect
7. #20274723C2semidirect
8. #20274615C2semidirect
9. #20274612C2semidirect
10. #20274583C2semidirect
13. D8×Dic53C4semidirect
243. C53#22816semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic532
#202776872
#202755922
#202753382
#202753322
#202748762
#202747232
#202746152
#202746122
#202745832
C53×#19722
#2281653