Finite Group Databasebeta

← #27280061

Subgroups — #27280061

Order 13568.

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

normal subgroups only

This group has 530 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#20283756C211yesmaximal, characteristic
2C4×#13133132C211yesmaximal, characteristic
3#20273235C212yesmaximal, complemented
4#20272567C212yesmaximal, complemented
5C53×#2638C211yesmaximal, characteristic
6#16810701—24no—
7#16810676—21no—
8C2×#13133132—21no—
9#16810587—21no—
10#16810587C2212yes—
11C4×C22⋊Dic53C2212yes—
12#16810390—21no—
13#16810360—24no—
14C2×C22⋊Q8×C53C2211yescharacteristic
15C2×C4×D4×C53C2211yescharacteristic
16C53×#398C411yescharacteristic, complemented
17C53×#397—21no—
18C53×#397C411yescharacteristic, complemented
19C53×#395C411yescharacteristic, complemented
20C53×#394—21no—
21C53×#393C411yescharacteristic, complemented
22C53×#387—21no—
23C53×#385—21no—
24C53×#381C2211yescharacteristic
25C2×#11719917—22no—
26C2×#11719917—22no—
27C2×C22⋊Dic53—22no—
28C2×C4×Dic53—22no—
29C4⋊#88322—22no—
30#13133132—24no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33C22⋊#88322D414yes—
34C4×#88322—22no—
35#13133124—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C22×D4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C22⋊Q8×C53—22no—
41C22⋊Q8×C53—22no—
42C4×D4×C53—22no—
43C4×D4×C53—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53C2×C411yescharacteristic
62#95×C53C2×C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×Q8×C53—22no—
66C2×Q8×C53—81no—
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—41no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—81no—
77C22×C4×C53—81no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4112yes—
85C22×Dic53—42no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—42no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—81no—
92C22⋊C4×C53—22no—
93C22⋊C4×C53—22no—
94C22⋊C4×C53—42no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—81no—
97C22⋊C4×C53—81no—
98C42×C53—21no—
99C42×C53—21no—
100C22⋊Dic53—24no—
101C22⋊Dic53—42no—
102C4×Dic53—82no—
103#11719917—82no—
104#11719917—82no—
105C2×#88322—24no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53#12112yes—
117Q8×C53—82no—
118D4×C53—44no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
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—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53#10412yes—
144C2×C4×C53C4×D412yes—
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12311yescharacteristic
147C2×Dic53—44no—
148C2×Dic53—82no—
149C2×Dic53—82no—
150#88322—44no—
151#9184—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—42no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53#38611yescharacteristic
165C22×C53C2×#10411yescharacteristic
166C22×C53#42111yescharacteristic
167C4×C53—22no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—22no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—81no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53—81no—
177Dic53—84no—
178#2975—531no—
179#2638D5311yescharacteristic
180C4×#99—531no—
181#1508—532no—
182#1284—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—42no—
186C2×C53—81no—
187C2×C53#187111yescharacteristic
188C2×C53#190911yescharacteristic
189C2×C53#191311yescharacteristic
190C2×C22⋊Q8C2×D5311yescharacteristic
191C2×C4×D4C2×D5311yescharacteristic
192#436—1064no—
193#431—1061no—
194C2×#99—1061no—
195#407—532no—
196#407—1061no—
197#398Dic5311yescharacteristic, complemented
198#397—21no—
199#397Dic5311yescharacteristic, complemented
200#395Dic5311yescharacteristic, complemented
201#394—21no—
202#393Dic5311yescharacteristic, complemented
203#387—21no—
204#385—21no—
205#381C2×D5311yescharacteristic
206C4×C22⋊C4—532no—
207#342—1061no—
208#328—1064no—
209C53#918411yescharacteristic, complemented, Sylow
210C22×Q8C2×Dic5311yescharacteristic
211C22×D4C22⋊D5311yescharacteristic
212C23×C4C22⋊D5311yescharacteristic
213C23×C4C22×D5311yescharacteristic
214C2×M16—1062no—
215C2×M16—1062no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C4×D4—22no—
219C4×D4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C2×Dic5311yescharacteristic
224C2×C4⋊C4C2×Dic5311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—1062no—
230C2×C22⋊C4C2×Dic5311yescharacteristic
231C2×C22⋊C4C22⋊D5311yescharacteristic
232C2×C42—1062no—
233C2×C42C22⋊D5311yescharacteristic
234C4⋊C8—1062no—
235#99—1064no—
236C22⋊C8—534no—
237C22⋊C8—1062no—
238C22⋊C8—1062no—
239C4×C8—1062no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—1062no—
246#95C2×Dic5311yescharacteristic
247#95C2×Dic5311yescharacteristic
248C24C22⋊Dic5311yescharacteristic
249C24C2×C22⋊D5311yescharacteristic
250C2×Q8—22no—
251C2×Q8—81no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—2122no—
264C22×C4C22⋊Dic5311yescharacteristic
265C22×C4C22⋊Dic5311yescharacteristic
266C22×C4C22⋊Dic5311yescharacteristic
267C22×C4D4×D5312yes—
268C22×C4#1171994712yes—
269C22×C4C22×Dic5311yescharacteristic
270C22×C4C2×C22⋊D5311yescharacteristic
271M16—4242no—
272M16—4242no—
273C2×C8—1064no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C22⋊C4—1064no—
287C22⋊C4—2122no—
288C42—21no—
289C42—21no—
290C42—4242no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C2×C22⋊Dic5311yescharacteristic
301C23#1313318912yes—
302Q8—82no—
303D4—44no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
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—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—4242no—
329C2×C4—4242no—
330C2×C4—2124no—
331C2×C4#1313314412yes—
332C2×C4D4×Dic5312yes—
333C2×C4#1313318511yescharacteristic
334C2×C4#1313319811yescharacteristic
335C8—2124no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#1681050311yescharacteristic
349C22C2×#1313314411yescharacteristic
350C22#1681064011yescharacteristic
351C4—21no—
352C4—21no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—4244no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—81no—
366C2#2027501811yescharacteristic
367C2#2027523811yescharacteristic
368C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2638C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2638D53
hypocenterC53#9184
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #20273235C2semidirect
4. #20272567C2semidirect
16. C53×#398C4semidirect
18. C53×#397C4semidirect
19. C53×#395C4semidirect
21. C53×#393C4semidirect
197. #398Dic53semidirect
199. #397Dic53semidirect
200. #395Dic53semidirect
202. #393Dic53semidirect
209. C53#9184semidirect

Maximal subgroups

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

subgroupindex
#202837562
C4×#131331322
#202732352
#202725672
C53×#26382
#918453