Finite Group Databasebeta

← #27272256

Subgroups — #27272256

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277198C211yesmaximal, characteristic
2#20274477C211yesmaximal, characteristic
3#20272539C212yesmaximal, complemented
4#20272537C212yesmaximal, complemented
5C53×#1797C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7C2×#13133176C411yescharacteristic
8C2×C22⋊#88322—21no—
9C2×#13133128—21no—
10#16810521C411yescharacteristic
11#16810521C411yescharacteristic
12#16810521C2211yescharacteristic
13#16810521C2211yescharacteristic
14#16810496—21no—
15#16810496—21no—
16C4×#11719919—21no—
17C4×#11719919—21no—
18#16810378C2212yes—
19C22×C42×C53C2211yescharacteristic
20C2×C4⋊C8×C53—21no—
21C2×C22⋊C8×C53—21no—
22C53×#336C2212yes—
23C22×#88322—22no—
24#13133176—22no—
25#13133176—22no—
26#13133176—42no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C22⋊#88322—24no—
40#13133128—42no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C22×C8×C53—22no—
49C2×C42×C53—42no—
50C2×C42×C53C2×C411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53C2311yescharacteristic
54C4⋊C8×C53—42no—
55C22⋊C8×C53—42no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C2×C8×C53—42no—
79C2×C8×C53—42no—
80C2×C8×C53—42no—
81C42×C53—22no—
82C42×C53—22no—
83C42×C53—41no—
84C42×C53—41no—
85C22⋊Dic53—24no—
86C22⋊Dic53—42no—
87C4⋊Dic53—21no—
88C4⋊Dic53—21no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91C4⋊Dic53M1612yes—
92C4⋊Dic53M1612yes—
93#11719919—42no—
94#11719919—42no—
95#11719919—42no—
96#11719919—41no—
97#11719919—41no—
98C4×Dic53—24no—
99C4×Dic53—42no—
100C2×#88322—24no—
101C2×#88322—42no—
102C2×#88322—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53#10112yes—
110C23×C53C2×C22⋊C411yescharacteristic
111C2×C4×C53—24no—
112C2×C4×C53—24no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—44no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53#9912yes—
130C2×C4×C53#10412yes—
131C2×C4×C53C2×M1611yescharacteristic
132C2×C4×C53C2×M1611yescharacteristic
133C2×C4×C53#13512yes—
134C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C53#13711yescharacteristic
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C8×C53—82no—
149#88322—44no—
150#7613—531nomaximal, Sylow
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53#40711yescharacteristic
159C22×C53C2×#9911yescharacteristic
160C22×C53C2×#10111yescharacteristic
161C22×C53#41611yescharacteristic
162C22×C53#41711yescharacteristic
163C22×C53C2×#10411yescharacteristic
164C22×C53#42111yescharacteristic
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—22no—
169C4×C53—22no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—41no—
173Dic53—42no—
174Dic53—42no—
175Dic53—81no—
176Dic53—81no—
177#2296—531no—
178#1797D5311yescharacteristic
179#1749—531no—
180#1271—532no—
181#1271—532no—
182C2×C53—42no—
183C2×C53#146311yescharacteristic
184C2×C53#148811yescharacteristic
185C2×C53#148911yescharacteristic
186C2×C53#150111yescharacteristic
187C2×C53#151311yescharacteristic
188C2×C53#153111yescharacteristic
189C2×C53#155211yescharacteristic
190C2×#117—531no—
191C2×#117—531no—
192C22×C42C2×D5311yescharacteristic
193C2×C4⋊C8—21no—
194C2×C22⋊C8—21no—
195C2×C22⋊C8—1061no—
196C2×#97—1061no—
197#389—531no—
198#389—531no—
199#389—531no—
200#389—531no—
201#385—1061no—
202#385—1061no—
203C4×C4⋊C4—1061no—
204C4×C4⋊C4—1061no—
205#336—532no—
206#336C2×D5312yes—
207C53#761311yescharacteristic, complemented, Sylow
208C23×C4C4×D5311yescharacteristic
209C23×C4D21211yescharacteristic
210C23×C4C22⋊D5311yescharacteristic
211C22×C8—22no—
212C22×C8—1062no—
213#117—1062no—
214#117—1062no—
215#117—2122no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C42—42no—
227C2×C42—1061no—
228C2×C42—1061no—
229C2×C42C4×D5311yescharacteristic
230C2×C42D21211yescharacteristic
231C2×C42C22⋊D5311yescharacteristic
232C2×C42C22×D5311yescharacteristic
233C4⋊C8—42no—
234C22⋊C8—42no—
235C22⋊C8—1064no—
236#97—2122no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241C24#1171992111yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4#1171992111yescharacteristic
258C22×C4#1171992111yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4C2×C4×D5311yescharacteristic
261C22×C4C2×D21211yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—1064no—
267C2×C8—2122no—
268C2×C8—2122no—
269C4⋊C4—532no—
270C4⋊C4—532no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C4⋊C4—2122no—
276C4⋊C4—2122no—
277C4⋊C4—2122no—
278C4⋊C4—2121no—
279C4⋊C4—2121no—
280C22⋊C4—1064no—
281C22⋊C4—2122no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—1064no—
287C42—2122no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1313313512yes—
295C23C2×#1171992111yescharacteristic
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—1062no—
315C2×C4—1062no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4#1313313112yes—
327C2×C4#1313314312yes—
328C2×C4M16×D5311yescharacteristic
329C2×C4M16×D5311yescharacteristic
330C2×C4#1313323912yes—
331C2×C4#1313324311yescharacteristic
332C2×C4#1313324811yescharacteristic
333C8—82no—
334C8—2124no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22#1681058611yescharacteristic
343C22C2×#1313313111yescharacteristic
344C22C2×#1313313511yescharacteristic
345C22#1681062011yescharacteristic
346C22#1681062211yescharacteristic
347C22C2×#1313314311yescharacteristic
348C22#1681064111yescharacteristic
349C4—22no—
350C4—22no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—4241no—
358C4—4241no—
359C4—2122no—
360C4—2122no—
361C2—42no—
362C2#2027304411yescharacteristic
363C2#2027315011yescharacteristic
364C2#2027315211yescharacteristic
365C2#2027320711yescharacteristic
366C2#2027325611yescharacteristic
367C2#2027333111yescharacteristic
368C2#2027341311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1797C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1797D53
hypocenterC53#7613
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20272539C2semidirect
4. #20272537C2semidirect
207. C53#7613semidirect

Maximal subgroups

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

subgroupindex
#202771982
#202744772
#202725392
#202725372
C53×#17972
#761353