Finite Group Databasebeta

← #27272113

Subgroups — #27272113

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277173C211yesmaximal, characteristic
2#20274857C211yesmaximal, characteristic, complemented
3#20274459C211yesmaximal, characteristic, complemented
4#20272538C211yesmaximal, characteristic, complemented
5#20272535C211yesmaximal, characteristic
6#20272535C211yesmaximal, characteristic
7C53×#1271C211yesmaximal, characteristic, complemented
8C2×#13133179C411yescharacteristic
9C2×#13133175—21no—
10C2×C8×Dic53—21no—
11#16810525C2211yescharacteristic
12#16810524C411yescharacteristic
13#16810522C411yescharacteristic
14#16810515—21no—
15#16810513—21no—
16C4⋊C4×Dic53C411yescharacteristic
17#16810470—21no—
18C42×Dic53C2211yescharacteristic
19#16810378C2211yescharacteristic
20#16810378C2211yescharacteristic
21#16810377—21no—
22#16810377—21no—
23#16810377—21no—
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×D212C2×C411yescharacteristic
28C22×C4×D53—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133179—22no—
32#13133179C814yescomplemented
33#13133175—42no—
34#13133175—42no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C8×Dic53—24no—
50#13133124—21no—
51#13133124—21no—
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C42×C53C2311yescharacteristic
58#95×C53—21no—
59C23×D53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C2×D212—21no—
71C2×D212—41no—
72C2×D212—41no—
73C2×D212C2×C812yes—
74C2×C4×D53—24no—
75C2×C4×D53—42no—
76C2×C4×D53—42no—
77C2×C8×C53—22no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C2×C8×C53—41no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53C2×C812yes—
88C42×C53—41no—
89#11719921—24no—
90#11719921—42no—
91#11719921—41no—
92#11719921—41no—
93#11719921—41no—
94#11719921—41no—
95#11719921—42no—
96C4⋊Dic53—42no—
97C4⋊Dic53—41no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—22no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105C4×Dic53—42no—
106C4×Dic53C2×C812yes—
107C4×Dic53M1612yes—
108C2×#88322—22no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322—41no—
114C23×C53C2×C22⋊C411yescharacteristic
115C22×D53—22no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—22no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53C22×C811yescharacteristic
135C2×C4×C53C2×M1611yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53#13511yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53C22⋊C812yes—
153C2×Dic53C22⋊C812yes—
154C2×Dic53#10412yes—
155D212—42no—
156D212—42no—
157C4×D53—48no—
158C8×C53—42no—
159C8×C53—81no—
160#88322—42no—
161#88322—81no—
162#7580—531nomaximal, Sylow
163C22×C53C2×C22⋊C811yescharacteristic
164C22×C53#41011yescharacteristic
165C22×C53#41311yescharacteristic
166C22×C53#41611yescharacteristic
167C22×C53#41811yescharacteristic
168C22×C53C2×#10411yescharacteristic
169C22×C53#42111yescharacteristic
170C2×D53—42no—
171C2×D53—42no—
172C2×D53—42no—
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C2×D53—81no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53—81no—
185Dic53—22no—
186Dic53—22no—
187Dic53—22no—
188Dic53—41no—
189Dic53—81no—
190#2293—531no—
191#1839—531no—
192#1744—531no—
193#1271—531no—
194#1271D5311yescharacteristic, complemented
195#1270—531no—
196#1270—531no—
197C2×C53#145811yescharacteristic
198C2×C53#147611yescharacteristic
199C2×C53#147811yescharacteristic
200C2×C53#150511yescharacteristic
201C2×C53#152111yescharacteristic
202C2×C53#153211yescharacteristic
203C2×C53#155311yescharacteristic
204D53—82no—
205C2×C4×D4—531no—
206C2×#117—1061no—
207C2×C4×C8—1061no—
208#390—531no—
209#390—531no—
210#389—531no—
211#389C2×D5311yescharacteristic
212#388—1061no—
213#388—1061no—
214C4×C4⋊C4—531no—
215C4×C22⋊C4—1061no—
216C43—531no—
217#336—531no—
218#336—531no—
219#336—1061no—
220#336—1061no—
221#336—1061no—
222#336C2×D5311yescharacteristic
223#336C2×D5311yescharacteristic
224C53#758011yescharacteristic, complemented, Sylow
225C22×D4—531no—
226C23×C4—1061no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—1061no—
230C22×C8—1061no—
231C4×D4—534no—
232C4×D4—1062no—
233#117—2122no—
234#117—2122no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4C4×D5311yescharacteristic
239C2×C4⋊C4C4×D5311yescharacteristic
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C42—531no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—1061no—
252C2×C42C22×D5311yescharacteristic
253C4×C8—1064no—
254#95—21no—
255#95—1061no—
256#95—1061no—
257C24—1061no—
258C2×D4—532no—
259C2×D4—1061no—
260C2×D4—2121no—
261C2×D4—2121no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—1064no—
269C22×C4—1061no—
270C22×C4—2122no—
271C22×C4—2122no—
272C22×C4C2×C4×D5311yescharacteristic
273C22×C4D4×D5311yescharacteristic
274C22×C4D4×D5311yescharacteristic
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—1062no—
282C2×C8—2121no—
283C2×C8—2121no—
284C2×C8—2121no—
285C2×C8—2121no—
286C2×C8—2121no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—2121no—
292C4⋊C4—2122no—
293C4⋊C4C8×D5312yes—
294C22⋊C4—1064no—
295C22⋊C4—2122no—
296C22⋊C4—2122no—
297C22⋊C4—2121no—
298C22⋊C4—2121no—
299C22⋊C4—2121no—
300C22⋊C4—2121no—
301C42—41no—
302C42—532no—
303C42—532no—
304C42—1062no—
305C42—1061no—
306C42—1061no—
307C42—1061no—
308C42—1061no—
309C42—2121no—
310C42—2121no—
311C42—2122no—
312C23—1062no—
313C23—2121no—
314C23—2121no—
315C23—2121no—
316C23—2121no—
317C23—2121no—
318C23—2121no—
319C23C22⋊C4×D5311yescharacteristic
320D4—2122no—
321D4—2122no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—532no—
335C2×C4—532no—
336C2×C4—532no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—2128no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4C2×C8×D5311yescharacteristic
351C2×C4M16×D5311yescharacteristic
352C2×C4#1313323811yescharacteristic
353C2×C4#1313324011yescharacteristic
354C2×C4#1313324111yescharacteristic
355C2×C4#1313324411yescharacteristic
356C8—42no—
357C8—81no—
358C8—4241no—
359C8—2122no—
360C22—4241no—
361C22—4241no—
362C22—4241no—
363C22—4241no—
364C22—2122no—
365C22—2122no—
366C22—2122no—
367C22C22⋊C8×D5311yescharacteristic
368C22#1681059511yescharacteristic
369C22#1681060611yescharacteristic
370C22#1681061811yescharacteristic
371C22#1681062611yescharacteristic
372C22#104×D5311yescharacteristic
373C22#1681063811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—81no—
382C4—1062no—
383C4—1062no—
384C4—1062no—
385C4—4241no—
386C4—2121no—
387C2—4242no—
388C2#2027302811yescharacteristic
389C2#2027309611yescharacteristic
390C2#2027310411yescharacteristic
391C2#2027322111yescharacteristic
392C2#2027329111yescharacteristic
393C2#2027333411yescharacteristic
394C2#2027341911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7580
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
2. #20274857C2semidirect
3. #20274459C2semidirect
4. #20272538C2semidirect
7. C53×#1271C2semidirect
32. #13133179C8semidirect
194. #1271D53semidirect
224. C53#7580semidirect

Maximal subgroups

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

subgroupindex
#202771732
#202748572
#202744592
#202725382
#202725352
C53×#12712
#758053