Finite Group Databasebeta

← #27277643

Subgroups — #27277643

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274703C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272594C212yesmaximal, complemented
4#20272582C212yesmaximal, complemented
5C53×#2349C211yesmaximal, characteristic
6C2×#13133132—21no—
7C2×C22⋊#88322C2212yes—
8#16810483C2212yes—
9#16810390—21no—
10#16810349—21no—
11#16810347—21no—
12C2×C4×D4×C53—21no—
13C2×C4×D4×C53C2211yescharacteristic
14C53×#389—21no—
15C53×#385—21no—
16C53×#385—21no—
17C53×#382C411yescharacteristic, complemented
18C53×#380C411yescharacteristic, complemented
19C53×#380C2211yescharacteristic
20C4×C4⋊C4×C53C411yescharacteristic, complemented
21C2×#95×C53C411yescharacteristic, complemented
22C2×#95×C53C2211yescharacteristic
23C22×#88322—22no—
24C2×C4⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26#13133132—42no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—22no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53—21no—
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35C4×D4×C53—22no—
36C4×D4×C53—22no—
37C4×D4×C53—44no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C42×C53—21no—
48C2×C42×C53C2×C411yescharacteristic
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53C2×C411yescharacteristic
53#95×C53C2×C411yescharacteristic
54#95×C53C2×C411yescharacteristic
55#95×C53C2×C411yescharacteristic
56C24×C53C22⋊C411yescharacteristic
57C24×C53C2×D411yescharacteristic
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53SD1612yes—
61C2×D4×C53Q1612yes—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53—81no—
73C22×C4×C53—81no—
74C22×C4×C53—81no—
75C22×C4×C53—81no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4112yes—
82C22×C4×C53#4112yes—
83C22×Dic53—42no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—81no—
88C4⋊C4×C53—81no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—42no—
96C22⋊Dic53—42no—
97C22⋊Dic53—42no—
98C4⋊Dic53—82no—
99C2×#88322—42no—
100C2×#88322—82no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#10312yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C23×C53#12312yes—
110C23×C53C2×SD1611yescharacteristic
111C23×C53C2×Q1611yescharacteristic
112C23×C53AGL(1,ℤ/8)12yes—
113D4×C53—24no—
114D4×C53—24no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53—84no—
140C2×C4×C53—84no—
141C2×C4×C53#10312yes—
142C2×C4×C53#11711yescharacteristic
143C2×C4×C53#11711yescharacteristic
144C2×C4×C53#12312yes—
145C2×Dic53—82no—
146C2×Dic53—82no—
147C2×Dic53—82no—
148#88322—82no—
149#8761—531nomaximal, Sylow
150C22×C53—22no—
151C22×C53—21no—
152C22×C53—22no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—42no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53#38111yescharacteristic
163C22×C53C2×#10311yescharacteristic
164C22×C53#41811yescharacteristic
165C22×C53#48112yes—
166C22×C53#48412yes—
167C4×C53—22no—
168C4×C53—22no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—42no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53—81no—
177Dic53—162no—
178#2349D5311yescharacteristic
179#1799—531no—
180#1459—531no—
181#1294—532no—
182#1289—532no—
183C2×C53—22no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—22no—
187C2×C53—42no—
188C2×C53#182411yescharacteristic
189C2×C53#182711yescharacteristic
190C2×C53#182811yescharacteristic
191C2×C4×D4—21no—
192C2×C4×D4C2×D5311yescharacteristic
193C2×#99—1061no—
194C2×C22⋊C8—532no—
195#389—21no—
196#385—21no—
197#385—21no—
198#382Dic5311yescharacteristic, complemented
199#380—532no—
200#380Dic5311yescharacteristic, complemented
201#380C2×D5311yescharacteristic
202C4×C4⋊C4Dic5311yescharacteristic, complemented
203C2×#95Dic5311yescharacteristic, complemented
204C2×#95C2×D5311yescharacteristic
205#342—1061no—
206#324—1061no—
207#323—1061no—
208C53#876111yescharacteristic, complemented, Sylow
209C22×D4C22⋊D5311yescharacteristic
210C23×C4—21no—
211C23×C4—21no—
212C23×C4C2×Dic5311yescharacteristic
213C23×C4C22×D5311yescharacteristic
214C22×C8—1062no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4C22⋊D5311yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—1062no—
227C2×C22⋊C4C22⋊D5311yescharacteristic
228C2×C22⋊C4C22⋊D5311yescharacteristic
229C2×C42—21no—
230C2×C42C2×Dic5311yescharacteristic
231#99—2122no—
232C22⋊C8—1062no—
233C22⋊C8—1062no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—1062no—
238#95C2×Dic5311yescharacteristic
239#95C2×Dic5311yescharacteristic
240#95C2×Dic5311yescharacteristic
241#95C2×Dic5311yescharacteristic
242C24C22⋊Dic5311yescharacteristic
243C24C2×C22⋊D5311yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4#1171992412yes—
247C2×D4#1171992512yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—2122no—
263C22×C4C22⋊Dic5311yescharacteristic
264C22×C4C22⋊Dic5311yescharacteristic
265C22×C4C22⋊Dic5311yescharacteristic
266C22×C4#1171994512yes—
267C22×C4#1171994512yes—
268C22×C4C22×Dic5311yescharacteristic
269C22×C4C2×C22⋊D5311yescharacteristic
270C2×C8—4242no—
271C2×C8—2122no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—4242no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—2122no—
283C22⋊C4—2122no—
284C42—21no—
285C42—21no—
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#1313314112yes—
294C23C2×C22⋊Dic5311yescharacteristic
295C23#1313320012yes—
296C23C2×#1171992411yescharacteristic
297C23C2×#1171992511yescharacteristic
298C23#1313324312yes—
299D4—24no—
300D4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—4242no—
328C2×C4—4242no—
329C2×C4—4242no—
330C2×C4#1313314112yes—
331C2×C4#1313317611yescharacteristic
332C2×C4#1313317611yescharacteristic
333C2×C4#1313320012yes—
334C8—4242no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22#1681048611yescharacteristic
348C22C2×#1313314111yescharacteristic
349C22#1681062811yescharacteristic
350C22#1681094112yes—
351C22#1681095812yes—
352C4—22no—
353C4—22no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—8482no—
363C2—21no—
364C2—22no—
365C2—21no—
366C2—22no—
367C2—42no—
368C2#2027480511yescharacteristic
369C2#2027481911yescharacteristic
370C2#2027482311yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2349C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2349D53
hypocenterC53#8761
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #20272594C2semidirect
4. #20272582C2semidirect
17. C53×#382C4semidirect
18. C53×#380C4semidirect
20. C4×C4⋊C4×C53C4semidirect
21. C2×#95×C53C4semidirect
198. #382Dic53semidirect
200. #380Dic53semidirect
202. C4×C4⋊C4Dic53semidirect
203. C2×#95Dic53semidirect
208. C53#8761semidirect

Maximal subgroups

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

subgroupindex
#202747032
#202730332
#202725942
#202725822
C53×#23492
#876153