Finite Group Databasebeta

← #27281558

Subgroups — #27281558

Order 13568.

720 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277313C211yesmaximal, characteristic, complemented
2#20273231C212yesmaximal, complemented
3#20273042C211yesmaximal, characteristic, complemented
4#20272601C212yesmaximal, complemented
5C53×#2539C211yesmaximal, characteristic
6#16810587C2212yes—
7#16810516—24no—
8#16810483—21no—
9#16810479—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
21C2×#11719917—22no—
22C2×C4⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—22no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—22no—
30#13133124D414yescomplemented
31C25×C53D411yescharacteristic
32C22×D4×C53—21no—
33C22×D4×C53—22no—
34C22×D4×C53C2×C411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37#123×C53—44no—
38#121×C53—22no—
39#121×C53—22no—
40#120×C53—44no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53C2×C412yes—
50C2×C22⋊C4×C53D412yes—
51C2×C42×C53—21no—
52#95×C53C2×C412yes—
53C24×C53—21no—
54C24×C53—21no—
55C24×C53—21no—
56C24×C53—42no—
57C24×C53—42no—
58C24×C53—81no—
59C24×C53C22⋊C412yes—
60C24×C53C2×D411yescharacteristic
61C2×D4×C53—22no—
62C2×D4×C53—44no—
63C2×D4×C53—42no—
64C2×D4×C53—42no—
65C2×D4×C53—44no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C2×D4×C53—82no—
69C2×D4×C53—81no—
70C22×C4×C53—22no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—42no—
75C22×C4×C53—81no—
76C22×C4×C53C22⋊C412yes—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D412yes—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4112yes—
81C22×Dic53—24no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—44no—
84C22⋊C4×C53—24no—
85C22⋊C4×C53—24no—
86C22⋊C4×C53—24no—
87C22⋊C4×C53—44no—
88C22⋊C4×C53—44no—
89C22⋊C4×C53—42no—
90C42×C53—81no—
91C22⋊Dic53—44no—
92C22⋊Dic53—82no—
93C22⋊Dic53—82no—
94C4⋊Dic53—82no—
95C4×Dic53—82no—
96#11719917—82no—
97C2×#88322—42no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—41no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53—42no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—42no—
112C23×C53—42no—
113C23×C53—41no—
114C23×C53—42no—
115C23×C53—81no—
116C23×C53—81no—
117C23×C53—88no—
118C23×C53C2×C22⋊C411yescharacteristic
119C23×C53#12011yescharacteristic
120C23×C53#12311yescharacteristic
121D4×C53—82no—
122D4×C53—88no—
123D4×C53—82no—
124D4×C53—82no—
125C2×C4×C53—24no—
126C2×C4×C53—22no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—81no—
139C2×C4×C53C4×D412yes—
140C2×C4×C53#12112yes—
141C2×Dic53—44no—
142C2×Dic53—44no—
143C2×Dic53—44no—
144#88322—82no—
145#9413—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—44no—
154C22×C53—44no—
155C22×C53—42no—
156C22×C53—42no—
157C22×C53—42no—
158C22×C53—82no—
159C22×C53—84no—
160C22×C53—82no—
161C22×C53—82no—
162C22×C53—84no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53#38611yescharacteristic
167C22×C53#42112yes—
168C22×C53#45312yes—
169C4×C53—22no—
170C4×C53—22no—
171C4×C53—44no—
172C4×C53—42no—
173C4×C53—81no—
174Dic53—84no—
175#2539D5311yescharacteristic
176#2312—531no—
177#1507—532no—
178#1462—531no—
179#1297—532no—
180C2×C53—22no—
181C2×C53—22no—
182C2×C53—41no—
183C2×C53—82no—
184C2×C53—161no—
185C2×C53#188711yescharacteristic
186C2×C53#189711yescharacteristic
187C2×C53#191311yescharacteristic
188C2×#123—21no—
189C2×#121C2×D5311yescharacteristic
190C2×#120—21no—
191C22×C22⋊C4C2×D5311yescharacteristic
192#407—532no—
193#399Dic5312yescomplemented
194#392Dic5312yescomplemented
195#390—22no—
196#388—1064no—
197#380—1061no—
198#379—1061no—
199C4×C22⋊C4—532no—
200C4×C22⋊C4C2×D5311yescharacteristic
201#327—1064no—
202#323—1062no—
203C53#941311yescharacteristic, complemented, Sylow
204C25C22⋊D5311yescharacteristic
205C22×D4—21no—
206C22×D4—22no—
207C22×D4C2×Dic5311yescharacteristic
208C23×C4C22⋊D5311yescharacteristic
209C23×C4C22×D5311yescharacteristic
210C2×M16—1062no—
211#123—44no—
212#121—22no—
213#121—22no—
214#120—44no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—1062no—
217C2×C4⋊C4C2×Dic5311yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—1062no—
225C2×C22⋊C4—1062no—
226C2×C22⋊C4—1062no—
227C2×C22⋊C4C2×Dic5312yes—
228C2×C22⋊C4C22⋊D5312yes—
229C2×C42—21no—
230C2×C42—1062no—
231C22⋊C8—1062no—
232C22⋊C8—1062no—
233#95—534no—
234#95—1062no—
235#95C2×Dic5312yes—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—42no—
240C24—42no—
241C24—81no—
242C24C22⋊Dic5312yes—
243C24C2×C22⋊D5311yescharacteristic
244C2×D4—22no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—82no—
252C2×D4—81no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—1064no—
260C22×C4C22⋊Dic5312yes—
261C22×C4D4×D5312yes—
262C22×C4#1171994712yes—
263C22×C4C22×Dic5311yescharacteristic
264C22×C4C2×C22⋊D5311yescharacteristic
265M16—4242no—
266C2×C8—2122no—
267C4⋊C4—22no—
268C4⋊C4—44no—
269C4⋊C4—4242no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C22⋊C4—4242no—
277C22⋊C4—4242no—
278C22⋊C4—2124no—
279C42—81no—
280C42—4242no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—41no—
297C23—42no—
298C23—81no—
299C23—81no—
300C23—88no—
301C23C2×C22⋊Dic5311yescharacteristic
302C23#1313318511yescharacteristic
303C23#1313319811yescharacteristic
304D4—82no—
305D4—88no—
306D4—82no—
307D4—82no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—2124no—
323C2×C4—2124no—
324C2×C4—2124no—
325C2×C4D4×Dic5312yes—
326C2×C4#1313318912yes—
327C8—4242no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—84no—
342C22—82no—
343C22—82no—
344C22—84no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#1681050311yescharacteristic
349C22#1681064012yes—
350C22#1681079012yes—
351C4—22no—
352C4—22no—
353C4—44no—
354C4—42no—
355C4—81no—
356C4—4244no—
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—161no—
362C2#2027511411yescharacteristic
363C2#2027516411yescharacteristic
364C2#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#9413
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. #20277313C2semidirect
2. #20273231C2semidirect
3. #20273042C2semidirect
4. #20272601C2semidirect
17. C53×#399C4semidirect
18. C53×#392C4semidirect
30. #13133124D4semidirect
193. #399Dic53semidirect
194. #392Dic53semidirect
203. C53#9413semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202732312
#202730422
#202726012
C53×#25392
#941353