Finite Group Databasebeta

← #28306965

Subgroups — #28306965

Order 13568.

762 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 170 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#20291153C211yesmaximal, characteristic
2#20290418C211yesmaximal, characteristic
3#20290394C212yesmaximal
4C2×#16811490C212yesmaximal
5#20277340C211yesmaximal, characteristic
6C2×D53⋊M16C2212yes—
7C2×#13133296C2212yes—
8C4⋊C4×#6157—24no—
9C4×C22⋊#6157—22no—
10#16811503—22no—
11#16811493—24no—
12#16811490C418yescomplemented
13C2×#13133175C2211yescharacteristic
14C2×#13133174C2211yescharacteristic
15C2×C4⋊C4×D53C2211yescharacteristic
16#16810520—22no—
17#16810497—22no—
18#16810493C411yescharacteristic, complemented
19#16810492C411yescharacteristic, complemented
20#16810488C411yescharacteristic, complemented
21C53×#382C411yescharacteristic, complemented
22C2×#11719936—22no—
23D53⋊M16—22no—
24D53⋊M16—22no—
25D53⋊M16—22no—
26D53⋊M16C2×C414yes—
27C2×#11719940—22no—
28C2×D53⋊C8—22no—
29#13133296—22no—
30#13133296—22no—
31#13133296C2×C414yes—
32#13133296Q814yes—
33C2×C4⋊#6157—22no—
34C2×C22⋊#6157—22no—
35C2×C4×#6157—24no—
36C2×C4×#6157D414yes—
37(C2×#6157)⋊C4—24no—
38C22×C4×D53D412yes—
39C22×C4×D53C2311yescharacteristic
40#13133175—22no—
41#13133175—22no—
42#13133174—22no—
43#13133174C2×C414yes—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53C2×C411yescharacteristic
46C2×C4⋊Dic53C2×C411yescharacteristic
47C2×#11719919—22no—
48C2×#11719919—21no—
49C2×#11719919D412yes—
50C4⋊C4×D53—22no—
51C4⋊C4×D53—22no—
52C2×#11719921—21no—
53C2×#11719921—21no—
54C2×C4×Dic53—21no—
55C2×C4×Dic53C2×C411yescharacteristic
56#13133124—22no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C42×C53C2×C411yescharacteristic
60#95×C53—22no—
61C23×D53C2×D411yescharacteristic
62C22×#6157—24no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53C22⋊C412yes—
70C22×Dic53C2×D411yescharacteristic
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—82no—
74C2×C4×D53C22×C412yes—
75C2×C4×D53C2×D412yes—
76C2×C4×D53C2×Q812yes—
77C2×C4×D53#4112yes—
78C2×C4×D53#4112yes—
79C2×C4×D53#4112yes—
80C22⋊#6157—24no—
81C22⋊#6157—24no—
82#11719940—24no—
83#11719940—42no—
84C2×#88324—42no—
85C4⋊#6157—24no—
86C4⋊#6157—24no—
87C4×#6157—24no—
88C4×#6157—84no—
89C4×#6157C4⋊C418yescomplemented
90#11719936—24no—
91#11719936—42no—
92D53⋊C8—24no—
93D53⋊C8—42no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C42×C53—21no—
97C42×C53C4212yescomplemented
98#11719921—24no—
99#11719921—41no—
100#11719921—41no—
101C4⋊Dic53—22no—
102C4⋊Dic53—22no—
103C4⋊Dic53—82no—
104#11719919—24no—
105#11719919—24no—
106#11719919—82no—
107C4×Dic53—21no—
108C4×Dic53—42no—
109C4×Dic53C4212yescomplemented
110C23×C53C2×C22⋊C411yescharacteristic
111C22×D53—41no—
112C22×D53—41no—
113C22×D53C4×D412yes—
114C22×D53#12011yescharacteristic
115C22×D53C22⋊Q812yes—
116C22×D53#12311yescharacteristic
117C2×#6157—28no—
118C2×#6157—44no—
119C2×#6157—44no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—41no—
127C2×C4×C53C2×C4211yescharacteristic
128C2×C4×C53#11711yescharacteristic
129C2×C4×C53C4×D412yes—
130C2×C4×C53C4×Q812yes—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—41no—
138C2×Dic53C4×D412yes—
139C2×Dic53#12112yes—
140C2×Dic53#12411yescharacteristic
141C2×Dic53C4⋊Q811yescharacteristic
142C4×D53—44no—
143C4×D53—82no—
144C4×D53C2×C4⋊C412yes—
145C4×D53#11712yes—
146C4×D53C4×D412yes—
147C4×D53C4×Q812yes—
148C4×D53C22⋊Q812yes—
149C4×D53#12312yes—
150C4×D53#12512yes—
151C4×D53#12612yes—
152#88324—44no—
153#10888—531nomaximal, Sylow
154C22×C53—21no—
155C22×C53—21no—
156C22×C53C4×C22⋊C411yescharacteristic
157C22×C53#38611yescharacteristic
158C22×C53#39111yescharacteristic
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—21no—
163C2×D53—42no—
164C2×D53—42no—
165C2×D53#38512yes—
166C2×D53#38712yes—
167#6157—48no—
168C4×C53—42no—
169C4×C53—42no—
170C4×C53—81no—
171C4×C53C4×C4⋊C412yes—
172C4×C53#38712yes—
173Dic53—42no—
174Dic53—42no—
175Dic53—81no—
176Dic53#38812yes—
177Dic53#38912yes—
178#2317—531no—
179#2274—531no—
180C2×#343—531no—
181#1733—532no—
182C2×H(3,ℤ/4)—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53#143611yescharacteristic
186C2×C53#191111yescharacteristic
187C2×C53#191211yescharacteristic
188D53—22no—
189D53—22no—
190C22×M16—532no—
191C2×#117—532no—
192C2×#117—531no—
193C2×#117—531no—
194C22×C4⋊C4—531no—
195#389—1062no—
196#385—1062no—
197#384—531no—
198#384—531no—
199#382—531no—
200#382#615711yescharacteristic, complemented
201C4×C4⋊C4—1064no—
202C4×C22⋊C4—1062no—
203#343—1062no—
204#339—1064no—
205H(3,ℤ/4)—538no—
206C53#1088811yescharacteristic, complemented, Sylow
207C23×C4—531no—
208C23×C4—532no—
209C2×M16—534no—
210C2×M16—1062no—
211C2×M16—1062no—
212C2×M16—1062no—
213C2×M16—1062no—
214C2×M16—1062no—
215C22×C8—1062no—
216#117—534no—
217#117—534no—
218#117—534no—
219#117—1062no—
220#117—1062no—
221#117—1062no—
222#117—1062no—
223#117—1062no—
224C2×C4⋊C4—532no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—1062no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1062no—
231C2×C4⋊C4—1062no—
232C2×C4⋊C4—1062no—
233C2×C4⋊C4C2×#615711yescharacteristic
234C2×C4⋊C4C2×#615711yescharacteristic
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1062no—
237C2×C22⋊C4—1061no—
238C2×C42—534no—
239C2×C42—531no—
240C2×C42—1064no—
241C2×C42—1061no—
242C2×C42C2×#615711yescharacteristic
243#95—22no—
244#95—1062no—
245#95—1064no—
246C24—531no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—531no—
250C22×C4—532no—
251C22×C4—532no—
252C22×C4—532no—
253C22×C4—532no—
254C22×C4—532no—
255C22×C4—532no—
256C22×C4—532no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—1064no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—4242no—
263C22×C4C22⋊#615712yes—
264C22×C4C22×#615711yescharacteristic
265M16—1064no—
266M16—1064no—
267M16—2122no—
268M16—2122no—
269C2×C8—1064no—
270C2×C8—2122no—
271C2×C8—2122no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—1064no—
275C4⋊C4—1064no—
276C4⋊C4—1064no—
277C4⋊C4—1064no—
278C4⋊C4—1062no—
279C4⋊C4—1062no—
280C4⋊C4—4242no—
281C4⋊C4—4242no—
282C22⋊C4—1064no—
283C22⋊C4—1064no—
284C22⋊C4—1064no—
285C22⋊C4—2121no—
286C22⋊C4—2121no—
287C42—21no—
288C42—538no—
289C42—532no—
290C42—1064no—
291C42—1061no—
292C42—4244no—
293C42—2122no—
294C42C4×#615712yescomplemented
295C23—531no—
296C23—531no—
297C23—532no—
298C23—532no—
299C23—2121no—
300C23—2121no—
301C23C2×C22⋊#615711yescharacteristic
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—532no—
310C2×C4—532no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—532no—
314C2×C4—532no—
315C2×C4—532no—
316C2×C4—532no—
317C2×C4—532no—
318C2×C4—532no—
319C2×C4—532no—
320C2×C4—532no—
321C2×C4—1068no—
322C2×C4—1062no—
323C2×C4—1062no—
324C2×C4—4242no—
325C2×C4—2124no—
326C2×C4—2122no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2124no—
331C2×C4—2124no—
332C2×C4—2122no—
333C2×C4C2×C4×#615711yescharacteristic
334C2×C4#1313329611yescharacteristic
335C2×C4D4×#615712yes—
336C2×C4Q8×#615712yes—
337C8—2124no—
338C22—21no—
339C22—21no—
340C22—532no—
341C22—532no—
342C22—1061no—
343C22—1061no—
344C22—1061no—
345C22—1061no—
346C22—2122no—
347C22—2122no—
348C22C4×C22⋊#615711yescharacteristic
349C22#1681157111yescharacteristic
350C22(C2×#6157)⋊Q811yescharacteristic
351C4—42no—
352C4—42no—
353C4—81no—
354C4—532no—
355C4—532no—
356C4—4241no—
357C4—2122no—
358C4—2128no—
359C4—2122no—
360C4C4⋊C4×#615712yes—
361C4#1681157212yes—
362C2—21no—
363C2—21no—
364C2—1062no—
365C2—1062no—
366C2#2028989411yescharacteristic
367C2#2029076011yescharacteristic
368C2#2029076311yescharacteristic
subgroupG/H
centerC22#16811571
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC22×C53#386
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenter#382#6157
hypocenterC53#10888
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
12. #16811490C4semidirect
18. #16810493C4semidirect
19. #16810492C4semidirect
20. #16810488C4semidirect
21. C53×#382C4semidirect
89. C4×#6157C4⋊C4semidirect
97. C42×C53C42semidirect
109. C4×Dic53C42semidirect
200. #382#6157semidirect
206. C53#10888semidirect
294. C42C4×#6157semidirect

Maximal subgroups

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

subgroupindex
#202911532
#202904182
#202903942
C2×#168114902
#202773402
#1088853