Finite Group Databasebeta

← #27281538

Subgroups — #27281538

Order 13568.

840 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277068C211yesmaximal, characteristic, complemented
2#20273231C212yesmaximal, complemented
3#20273042C211yesmaximal, characteristic, complemented
4#20272601C212yesmaximal, complemented
5C53×#2539C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7#16810587C2212yes—
8C22⋊C4×Dic53—24no—
9C4×C22⋊Dic53—21no—
10C4×C22⋊Dic53C2212yes—
11#16810357—24no—
12#16810347—22no—
13C2×#123×C53—21no—
14C2×#121×C53C2211yescharacteristic
15C2×#120×C53—21no—
16C22×C22⋊C4×C53C2211yescharacteristic
17C53×#399C412yescomplemented
18C53×#392C412yescomplemented
19C53×#390—22no—
20C4×C22⋊C4×C53C2211yescharacteristic
21C23×Dic53—22no—
22C2×#11719917—22no—
23C2×C22⋊Dic53—28no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—22no—
27C2×C4×Dic53—22no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30#13133124—22no—
31#13133124D414yescomplemented
32C25×C53D411yescharacteristic
33C22×D4×C53—21no—
34C22×D4×C53—22no—
35C22×D4×C53C2×C411yescharacteristic, complemented
36C23×C4×C53D411yescharacteristic
37C23×C4×C53C2311yescharacteristic
38#123×C53—44no—
39#121×C53—22no—
40#121×C53—22no—
41#120×C53—44no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53C2×C412yes—
51C2×C22⋊C4×C53D412yes—
52C2×C42×C53—21no—
53#95×C53C2×C412yescomplemented
54C24×C53—21no—
55C24×C53—24no—
56C24×C53—21no—
57C24×C53—21no—
58C24×C53—42no—
59C24×C53—81no—
60C24×C53C22⋊C412yes—
61C24×C53C2×D411yescharacteristic
62C2×D4×C53—22no—
63C2×D4×C53—44no—
64C2×D4×C53—42no—
65C2×D4×C53—42no—
66C2×D4×C53—44no—
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C2×D4×C53—82no—
70C2×D4×C53—81no—
71C22×C4×C53—22no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—42no—
76C22×C4×C53—81no—
77C22×C4×C53C22⋊C412yes—
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D412yes—
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4112yes—
82C22×Dic53—24no—
83C22×Dic53—28no—
84C22×Dic53—82no—
85C22×Dic53—82no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—44no—
88C22⋊C4×C53—24no—
89C22⋊C4×C53—24no—
90C22⋊C4×C53—24no—
91C22⋊C4×C53—44no—
92C22⋊C4×C53—44no—
93C22⋊C4×C53—42no—
94C42×C53—81no—
95C22⋊Dic53—28no—
96C22⋊Dic53—28no—
97C4×Dic53—82no—
98C4×Dic53—82no—
99#11719917—82no—
100C2×#88322—42no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—24no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—22no—
107C23×C53—22no—
108C23×C53—41no—
109C23×C53—42no—
110C23×C53—42no—
111C23×C53—42no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—42no—
115C23×C53—42no—
116C23×C53—41no—
117C23×C53—42no—
118C23×C53—81no—
119C23×C53—81no—
120C23×C53—88no—
121C23×C53#10414yes—
122C23×C53C2×C22⋊C411yescharacteristic
123C23×C53#12011yescharacteristic
124C23×C53#12311yescharacteristic
125D4×C53—82no—
126D4×C53—88no—
127D4×C53—82no—
128D4×C53—82no—
129C2×C4×C53—24no—
130C2×C4×C53—22no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53C4×D412yes—
144C2×C4×C53#12112yes—
145C2×Dic53—28no—
146C2×Dic53—44no—
147C2×Dic53—44no—
148C2×Dic53—88no—
149#88322—82no—
150#9409—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—24no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—44no—
160C22×C53—44no—
161C22×C53—44no—
162C22×C53—44no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53—82no—
166C22×C53—82no—
167C22×C53—82no—
168C22×C53—84no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53#38611yescharacteristic
173C22×C53C2×#10412yes—
174C22×C53#45312yes—
175C4×C53—22no—
176C4×C53—22no—
177C4×C53—44no—
178C4×C53—42no—
179C4×C53—81no—
180Dic53—48no—
181#2539D5311yescharacteristic
182#2273—531no—
183#1507—532no—
184#1462—531no—
185#1297—532no—
186C2×C53—22no—
187C2×C53—22no—
188C2×C53—41no—
189C2×C53—44no—
190C2×C53—161no—
191C2×C53#188711yescharacteristic
192C2×C53#189611yescharacteristic
193C2×C53#191311yescharacteristic
194C2×#123—21no—
195C2×#121C2×D5311yescharacteristic
196C2×#120—21no—
197C22×C22⋊C4—1061no—
198C22×C22⋊C4C2×D5311yescharacteristic
199#407—532no—
200#399Dic5312yescomplemented
201#392Dic5312yescomplemented
202#390—22no—
203C4×C22⋊C4—532no—
204C4×C22⋊C4—1064no—
205C4×C22⋊C4—1061no—
206C4×C22⋊C4C2×D5311yescharacteristic
207#327—1064no—
208#323—1062no—
209C53#940911yescharacteristic, complemented, Sylow
210C25C22⋊D5311yescharacteristic
211C22×D4—21no—
212C22×D4—22no—
213C22×D4C2×Dic5311yescharacteristic, complemented
214C23×C4—1062no—
215C23×C4C22⋊D5311yescharacteristic
216C23×C4C22×D5311yescharacteristic
217C2×M16—1062no—
218#123—44no—
219#121—22no—
220#121—22no—
221#120—44no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C2×Dic5311yescharacteristic, complemented
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—1068no—
231C2×C22⋊C4—1062no—
232C2×C22⋊C4—1062no—
233C2×C22⋊C4C2×Dic5312yes—
234C2×C22⋊C4C22⋊D5312yes—
235C2×C42—21no—
236C2×C42—1062no—
237C2×C42—1062no—
238C22⋊C8—1062no—
239C22⋊C8—1062no—
240#95—534no—
241#95—1062no—
242#95C2×Dic5312yescomplemented
243C24—21no—
244C24—24no—
245C24—21no—
246C24—21no—
247C24—42no—
248C24—81no—
249C24C22⋊Dic5312yes—
250C24C2×C22⋊D5311yescharacteristic
251C2×D4—22no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—82no—
259C2×D4—81no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—1068no—
267C22×C4—1064no—
268C22×C4—4242no—
269C22×C4—4242no—
270C22×C4C22⋊Dic5312yes—
271C22×C4D4×D5312yes—
272C22×C4#1171994712yes—
273C22×C4C22×Dic5311yescharacteristic
274C22×C4C2×C22⋊D5311yescharacteristic
275M16—4242no—
276C2×C8—2122no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—1068no—
286C22⋊C4—1068no—
287C42—81no—
288C42—4242no—
289C42—4242no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—42no—
307C23—81no—
308C23—81no—
309C23—88no—
310C23#1313314414yes—
311C23C2×C22⋊Dic5311yescharacteristic
312C23#1313318511yescharacteristic
313C23#1313319811yescharacteristic
314D4—82no—
315D4—88no—
316D4—82no—
317D4—82no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—1068no—
333C2×C4—4248no—
334C2×C4—2124no—
335C2×C4—2124no—
336C2×C4D4×Dic5312yes—
337C2×C4#1313318912yes—
338C8—4242no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—21no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—84no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#1681050311yescharacteristic
361C22C2×#1313314412yes—
362C22#1681079012yes—
363C4—22no—
364C4—22no—
365C4—44no—
366C4—42no—
367C4—81no—
368C4—2128no—
369C2—22no—
370C2—22no—
371C2—41no—
372C2—44no—
373C2—161no—
374C2#2027511411yescharacteristic
375C2#2027515911yescharacteristic
376C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2539C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#2539D53
hypocenterC53#9409
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. #20277068C2semidirect
2. #20273231C2semidirect
3. #20273042C2semidirect
4. #20272601C2semidirect
17. C53×#399C4semidirect
18. C53×#392C4semidirect
31. #13133124D4semidirect
35. C22×D4×C53C2×C4semidirect
43. C2×C4⋊C4×C53C2×C4semidirect
53. #95×C53C2×C4semidirect
200. #399Dic53semidirect
201. #392Dic53semidirect
209. C53#9409semidirect
213. C22×D4C2×Dic53semidirect
223. C2×C4⋊C4C2×Dic53semidirect
242. #95C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202770682
#202732312
#202730422
#202726012
C53×#25392
#940953