Finite Group Databasebeta

← #27382229

Subgroups — #27382229

Order 13568.

1222 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810675C212yesmaximal
2C43×D53C211yesmaximal, characteristic
3#20274570C212yesmaximal, complemented
4#20274568C211yesmaximal, characteristic, complemented
5#20274478C212yesmaximal
6#20274476C212yesmaximal
7#20274459C212yesmaximal, complemented
8#97×Dic53C212yesmaximal, complemented
9C53×#1767C211yesmaximal, characteristic, complemented
10C2×C42×D53C412yes—
11C2×C42×D53C412yes—
12C2×C42×D53C2211yescharacteristic
13C2×C42×D53C2211yescharacteristic
14C2×C42×D53C2211yescharacteristic
15#16810675C4116yescomplemented
16C2×C4⋊#88322C2212yes—
17C2×#13133129C2214yes—
18C2×#13133128C2212yes—
19C2×#13133127C2214yes—
20C2×C8×Dic53C2214yes—
21C42×Dic53C412yes—
22C42×Dic53C412yes—
23C42×Dic53C2211yescharacteristic
24C42×Dic53C2211yescharacteristic
25C42×Dic53C2211yescharacteristic
26#16810378C2212yes—
27#16810377C2214yes—
28C2×C4⋊C8×C53C2212yes—
29C2×#97×C53C2212yes—
30C43×C53C2211yescharacteristic
31C53×#336C2212yes—
32C22×C4×D53C2×C412yes—
33C22×C4×D53C2×C412yes—
34C22×C4×D53C2×C411yescharacteristic
35C22×C4×D53C2×C411yescharacteristic
36C22×C4×D53C2311yescharacteristic
37C22×#88322C2314yes—
38C2×C4×Dic53C2×C412yes—
39C2×C4×Dic53C2×C412yes—
40C2×C4×Dic53C2×C412yes—
41C2×C4×Dic53C2×C412yes—
42C2×C4×Dic53C2×C412yes—
43C2×C4×Dic53C2×C412yes—
44C2×C4×Dic53C2×C412yes—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53C2×C412yes—
48C2×C4×Dic53C2311yescharacteristic
49C2×C4×Dic53C2311yescharacteristic
50C2×C4×Dic53C2311yescharacteristic
51C42×D53—22no—
52C42×D53—22no—
53C42×D53—28no—
54C42×D53—24no—
55C42×D53—28no—
56C42×D53C2×C418yes—
57C4⋊#88322C2×C418yescomplemented
58#13133129C2×C4116yes—
59#13133128—24no—
60#13133127—28no—
61C8×Dic53C2×C4116yescomplemented
62C22×C8×C53C2314yes—
63C2×C42×C53C2×C412yes—
64C2×C42×C53C2×C412yes—
65C2×C42×C53C2311yescharacteristic
66C2×C42×C53C2311yescharacteristic
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53C2×C418yescomplemented
69#97×C53—24no—
70C23×D53C22×C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C22×C412yes—
73C22×C4×C53C22×C412yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2411yescharacteristic
76C22×Dic53C22×C411yescharacteristic
77C22×Dic53C22×C412yes—
78C22×Dic53C22×C412yes—
79C22×Dic53C22×C411yescharacteristic
80C22×Dic53C22×C411yescharacteristic
81C2×C4×D53—24no—
82C2×C4×D53—24no—
83C2×C4×D53—24no—
84C2×C4×D53—24no—
85C2×C4×D53—22no—
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×C4×D53—24no—
89C2×C4×D53—24no—
90C2×C4×D53—24no—
91C2×C4×D53C4214yes—
92C2×C4×D53C4214yes—
93C2×C4×D53M1618yes—
94C2×C4×D53C22×C414yes—
95C2×C8×C53—24no—
96C2×C8×C53—24no—
97C2×C8×C53C22×C418yes—
98C42×C53—24no—
99C42×C53—24no—
100C42×C53—22no—
101C42×C53C22×C414yes—
102C42×C53#4112yes—
103C42×C53#4112yes—
104C4×Dic53—24no—
105C4×Dic53—24no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53—24no—
109C4×Dic53—24no—
110C4×Dic53—24no—
111C4×Dic53—24no—
112C4×Dic53—24no—
113C4×Dic53C4214yes—
114C4×Dic53C4214yes—
115C4×Dic53M1618yes—
116C4×Dic53C22×C414yes—
117C4×Dic53#4114yes—
118C2×#88322—24no—
119C2×#88322—24no—
120C2×#88322C22×C418yes—
121C23×C53C23×C411yescharacteristic
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—22no—
125C22×D53C2×C4212yes—
126C22×D53C2×M1614yes—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53C2×C4212yes—
133C2×C4×C53C2×C4212yes—
134C2×C4×C53#11714yes—
135C2×C4×C53C2×M1614yes—
136C2×C4×C53C2×M1614yes—
137C2×C4×C53C23×C412yes—
138C2×C4×C53C2×#4112yes—
139C2×C4×C53C2×#4111yescharacteristic
140C2×C4×C53C2×#4111yescharacteristic
141C2×Dic53—21no—
142C2×Dic53—22no—
143C2×Dic53—21no—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53C2×C4212yes—
148C2×Dic53C2×C4212yes—
149C2×Dic53C2×C4212yes—
150C2×Dic53#11714yes—
151C2×Dic53C2×M1614yes—
152C2×Dic53C2×M1614yes—
153C2×Dic53C2×M1614yes—
154C4×D53—216no—
155C4×D53—216no—
156C4×D53—24no—
157C4×D53—24no—
158C4×D53—28no—
159C4×D53#97116yescomplemented
160C8×C53—28no—
161#88322—28no—
162#20463—531nomaximal, Sylow
163C22×C53C22×C4211yescharacteristic
164C22×C53C2×#11711yescharacteristic
165C22×C53C2×#11711yescharacteristic
166C22×C53C22×M1612yes—
167C22×C53C22×M1612yes—
168C2×D53—22no—
169C2×D53—24no—
170C2×D53—24no—
171C2×D53—22no—
172C2×D53C2×#9714yes—
173C4×C53—22no—
174C4×C53—24no—
175C4×C53—24no—
176C4×C53C2×#9714yes—
177C4×C53C4×#4112yes—
178C4×C53C4×#4112yes—
179Dic53—24no—
180Dic53—24no—
181Dic53—22no—
182Dic53C2×#9714yes—
183Dic53C4×M1614yes—
184C2×#431—532no—
185C2×C43—531no—
186#1767—532no—
187#1767—531no—
188#1767D5311yescharacteristic, complemented
189#1749—532no—
190#1749—532no—
191#1744—532no—
192C4×#97—532no—
193C2×C53C4×#11711yescharacteristic
194C2×C53C22×#9711yescharacteristic
195C2×C53C2×C4×M1611yescharacteristic
196C2×C53#292012yes—
197C2×C53#292012yes—
198D53—24no—
199C22×C42—531no—
200C22×C42—532no—
201C22×C42—532no—
202C22×C42—531no—
203C22×C42—531no—
204#431—5316no—
205C2×C4⋊C8—532no—
206C2×C4⋊C8C2×D5312yes—
207C2×C22⋊C8—534no—
208C2×#97—534no—
209C2×#97—532no—
210C2×#97C2×D5312yes—
211C2×C4×C8—534no—
212C43—532no—
213C43—532no—
214C43—531no—
215C43—531no—
216C43—531no—
217C43C2×D5311yescharacteristic
218#336—534no—
219#336—532no—
220#336C2×D5312yes—
221C53#2046311yescharacteristic, complemented, Sylow
222C23×C4—531no—
223C23×C4—531no—
224C23×C4—531no—
225C23×C4—532no—
226C23×C4—532no—
227C22×C8—534no—
228C22×C8C22×D5314yes—
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—532no—
232C2×C42—531no—
233C2×C42—531no—
234C2×C42—538no—
235C2×C42—532no—
236C2×C42—532no—
237C2×C42—532no—
238C2×C42—532no—
239C2×C42—532no—
240C2×C42—532no—
241C2×C42—532no—
242C2×C42—531no—
243C2×C42—1068no—
244C2×C42—1064no—
245C2×C42—1068no—
246C2×C42—1062no—
247C2×C42—1062no—
248C2×C42C4×D5312yes—
249C2×C42C4×D5312yes—
250C2×C42C22×D5311yescharacteristic
251C2×C42C22×D5311yescharacteristic
252C2×C42C22×D5311yescharacteristic
253C4⋊C8—538no—
254C4⋊C8C4×D5318yescomplemented
255C22⋊C8—5316no—
256#97—24no—
257#97—1068no—
258#97—1064no—
259C4×C8—5316no—
260C24—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—534no—
264C22×C4—538no—
265C22×C4—531no—
266C22×C4—532no—
267C22×C4—532no—
268C22×C4—534no—
269C22×C4—534no—
270C22×C4—1064no—
271C22×C4—1064no—
272C22×C4—1064no—
273C22×C4—1064no—
274C22×C4—1064no—
275C22×C4—1064no—
276C22×C4—1064no—
277C22×C4—1062no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4C2×C4×D5311yescharacteristic
281C22×C4C2×C4×D5312yes—
282C22×C4C2×C4×D5312yes—
283C22×C4C2×C4×D5311yescharacteristic
284C22×C4C23×D5311yescharacteristic
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—538no—
288C2×C8—1064no—
289C2×C8—1064no—
290C2×C8C2×C4×D5318yes—
291C42—24no—
292C42—24no—
293C42—22no—
294C42—534no—
295C42—534no—
296C42—538no—
297C42—534no—
298C42—534no—
299C42—1064no—
300C42—1064no—
301C42—1064no—
302C42—1064no—
303C42—1064no—
304C42—1064no—
305C42—1064no—
306C42—1061no—
307C42—1061no—
308C42C2×C4×D5314yes—
309C42#1171994712yes—
310C42#1171994912yes—
311C23—532no—
312C23—534no—
313C23—1061no—
314C23—1061no—
315C23—1062no—
316C23C22×C4×D5311yescharacteristic
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—532no—
323C2×C4—532no—
324C2×C4—534no—
325C2×C4—534no—
326C2×C4—5316no—
327C2×C4—532no—
328C2×C4—534no—
329C2×C4—534no—
330C2×C4—1068no—
331C2×C4—10616no—
332C2×C4—10616no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1062no—
336C2×C4—1062no—
337C2×C4—1062no—
338C2×C4—1062no—
339C2×C4—1064no—
340C2×C4—1064no—
341C2×C4C42×D5312yes—
342C2×C4C42×D5312yes—
343C2×C4#1313317514yes—
344C2×C4M16×D5314yes—
345C2×C4M16×D5314yes—
346C2×C4C22×C4×D5312yes—
347C2×C4#41×D5312yes—
348C2×C4C2×#1171994711yescharacteristic
349C2×C4C2×#1171994911yescharacteristic
350C8—28no—
351C8—1068no—
352C22—534no—
353C22—1062no—
354C22—1064no—
355C22—1064no—
356C22—1062no—
357C22C2×C42×D5311yescharacteristic
358C22#117×D5311yescharacteristic
359C22C2×#1313317511yescharacteristic
360C22C2×M16×D5312yes—
361C22C2×M16×D5312yes—
362C4—24no—
363C4—24no—
364C4—22no—
365C4—534no—
366C4—534no—
367C4—1062no—
368C4—1064no—
369C4—1064no—
370C4#97×D5314yes—
371C4C4×#1171994712yes—
372C4C4×#1171994912yes—
373C2—1064no—
374C2C4×#1313317511yescharacteristic
375C2C2×#97×D5311yescharacteristic
376C2C4×M16×D5311yescharacteristic
377C2#2028334512yes—
378C2#2028334712yes—
subgroupG/H
center, FrattiniC22×C4C23×D53
commutatorC22×C53C22×C42
FittingC53×#1767C2
socleC23×C53C23×C4
prime radicalC23×D53C22×C4
cosocleC22×C4×C53C24
hypercenter#1767D53
hypocenterC53#20463
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1767

Complements

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

NH ≅ G/Ntype
3. #20274570C2semidirect
4. #20274568C2semidirect
7. #20274459C2semidirect
8. #97×Dic53C2semidirect
9. C53×#1767C2semidirect
15. #16810675C4semidirect
57. C4⋊#88322C2×C4semidirect
61. C8×Dic53C2×C4semidirect
68. C4⋊C8×C53C2×C4semidirect
159. C4×D53#97semidirect
188. #1767D53semidirect
221. C53#20463semidirect
254. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106752
C43×D532
#202745702
#202745682
#202744782
#202744762
#202744592
#97×Dic532
C53×#17672
#2046353