Finite Group Databasebeta

← #27278092

Subgroups — #27278092

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277333C211yesmaximal, characteristic
2#20274910C212yesmaximal
3#20272968C211yesmaximal, characteristic
4#20272595C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic
6C2×#13133176—21no—
7C22×C4⋊Dic53C2211yescharacteristic
8C2×#13133149—22no—
9#16810522—22no—
10#16810522—22no—
11#16810483—21no—
12#16810483C2212yes—
13C2×#13133124—21no—
14C2×#13133124—21no—
15C2×#13133124C2211yescharacteristic
16#16810386—22no—
17C22×C4⋊C4×C53C2211yescharacteristic
18C2×C22⋊C8×C53C2212yes—
19C53×#324C414yescomplemented
20C23×Dic53—21no—
21C23×Dic53—21no—
22#13133176—42no—
23#13133176—42no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53—42no—
30C2×C4⋊Dic53D412yes—
31C2×C4⋊Dic53D412yes—
32C2×C22⋊Dic53—22no—
33C2×C4×Dic53—22no—
34#13133149—44no—
35#13133124—22no—
36#13133124—22no—
37#13133124—22no—
38#13133124—42no—
39#13133124—42no—
40C23×C4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C23×C4×C53C2311yescharacteristic
43C22×C8×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53—22no—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C4⋊C4×C53Q812yes—
48C22⋊C8×C53—22no—
49C22⋊C8×C53C2×C414yes—
50C24×C53C2×D411yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53C4⋊C412yes—
62C22×C4×C53C4⋊C412yes—
63C22×C4×C53SD1612yes—
64C22×C4×C53SD1612yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×C4×C53C2×Q811yescharacteristic
68C22×C4×C53#4112yes—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C2×C8×C53—24no—
86C2×C8×C53—42no—
87C2×C8×C53—42no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—24no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—24no—
93C4⋊Dic53—42no—
94C4⋊Dic53—42no—
95C4⋊Dic53—42no—
96C4⋊Dic53—42no—
97C4⋊Dic53—42no—
98C4⋊Dic53—81no—
99C4⋊Dic53—81no—
100C4×Dic53—44no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53C2×C4⋊C411yescharacteristic
106C23×C53#12011yescharacteristic
107C23×C53#12311yescharacteristic
108C23×C53C2×SD1611yescharacteristic
109C23×C53C2×SD1611yescharacteristic
110C23×C53AGL(1,ℤ/8)11yescharacteristic
111C23×C53#13711yescharacteristic
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—81no—
128C2×C4×C53#10614yescomplemented
129C2×C4×C53C4×D412yes—
130C2×C4×C53C22⋊Q812yes—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150C2×Dic53—81no—
151C8×C53—44no—
152#8857—531nomaximal, Sylow
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53#38511yescharacteristic
166C22×C53C2×#10611yescharacteristic
167C22×C53#42811yescharacteristic
168C22×C53#44811yescharacteristic
169C22×C53#45011yescharacteristic
170C22×C53#45411yescharacteristic
171C22×C53#45611yescharacteristic
172C22×C53B(4,𝔽2)11yescharacteristic
173C22×C53#45811yescharacteristic
174C22×C53#48111yescharacteristic
175C22×C53#48311yescharacteristic
176C4×C53—24no—
177C4×C53—22no—
178C4×C53—22no—
179C4×C53—42no—
180Dic53—82no—
181Dic53—82no—
182Dic53—82no—
183Dic53—82no—
184#2316—531no—
185#1853—532no—
186C2×#324D5311yescharacteristic
187#1441—531no—
188#1294—532no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53#159911yescharacteristic
194C2×C53#160411yescharacteristic
195C2×C53#161111yescharacteristic
196C2×C53#161611yescharacteristic
197C2×C53#185311yescharacteristic
198C2×C53#185711yescharacteristic
199C2×C53#186711yescharacteristic
200C2×#117—1061no—
201C22×C4⋊C4—531no—
202C22×C4⋊C4C2×D5311yescharacteristic
203C2×#106—1062no—
204C2×C22⋊C8C2×D5312yes—
205#389—1062no—
206#389—1062no—
207#380—532no—
208#380—1061no—
209C2×#95—531no—
210C2×#95—1061no—
211C2×#95—1061no—
212#340—1062no—
213#324Dic5314yescomplemented
214C53#885711yescharacteristic, complemented, Sylow
215C23×C4—1061no—
216C23×C4—1061no—
217C23×C4D21211yescharacteristic
218C23×C4D21211yescharacteristic
219C23×C4C22×D5311yescharacteristic
220C22×C8—22no—
221#117—2122no—
222#117—2122no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—532no—
226C2×C4⋊C4—532no—
227C2×C4⋊C4—1062no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—2122no—
233C2×C4⋊C4Dic10612yes—
234C2×C4⋊C4C2×Dic5312yes—
235C2×C22⋊C4—1062no—
236C2×C42—1062no—
237#106—2124no—
238C22⋊C8—22no—
239C22⋊C8C2×Dic5314yes—
240#95—1062no—
241#95—1062no—
242#95—1062no—
243#95—2122no—
244#95—2122no—
245C24C2×D21211yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—1062no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—2121no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4—2121no—
266C22×C4—2121no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4#1171991312yes—
273C22×C4#1171991312yes—
274C22×C4C4⋊Dic5312yes—
275C22×C4C4⋊Dic5312yes—
276C22×C4C2×Dic10611yescharacteristic
277C22×C4D4×D5312yes—
278C22×C4#1171994712yes—
279C22×C4C22×Dic5311yescharacteristic
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—1064no—
286C4⋊C4—4241no—
287C4⋊C4—4241no—
288C4⋊C4—2122no—
289C4⋊C4—2122no—
290C4⋊C4—2122no—
291C4⋊C4—2122no—
292C4⋊C4—2122no—
293C22⋊C4—2122no—
294C22⋊C4—2122no—
295C42—2124no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23C2×C4⋊Dic5311yescharacteristic
301C23#1313318711yescharacteristic
302C23#1313320111yescharacteristic
303C23C2×#1171991311yescharacteristic
304C23C2×#1171991311yescharacteristic
305C23#1313324511yescharacteristic
306C23#1313325011yescharacteristic
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—4241no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—4241no—
329C2×C4—4241no—
330C2×C4—4241no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4#1313314914yescomplemented
344C2×C4D4×Dic5312yes—
345C2×C4#1313319412yes—
346C8—44no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22#1681049811yescharacteristic
360C22C2×#1313314911yescharacteristic
361C22#1681066511yescharacteristic
362C22#1681076411yescharacteristic
363C22#1681077611yescharacteristic
364C22#1681079811yescharacteristic
365C22#1681080811yescharacteristic
366C22#1681081111yescharacteristic
367C22#1681081411yescharacteristic
368C22#1681094211yescharacteristic
369C22#1681095411yescharacteristic
370C4—22no—
371C4—22no—
372C4—24no—
373C4—42no—
374C4—4242no—
375C4—4242no—
376C4—4242no—
377C4—4242no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2#2027365711yescharacteristic
383C2#2027368611yescharacteristic
384C2#2027372511yescharacteristic
385C2#2027375411yescharacteristic
386C2#2027491011yescharacteristic
387C2#2027493211yescharacteristic
388C2#2027499811yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#8857
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
19. C53×#324C4semidirect
128. C2×C4×C53#106semidirect
213. #324Dic53semidirect
214. C53#8857semidirect
343. C2×C4#13133149semidirect

Maximal subgroups

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

subgroupindex
#202773332
#202749102
#202729682
#202725952
C2×C53×#3242
#885753