Finite Group Databasebeta

← #27282106

Subgroups — #27282106

Order 13568.

518 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 103 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#20280058C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20273044C212yesmaximal, complemented
4#20272622C212yesmaximal, complemented
5C53×#2975C211yesmaximal, characteristic
6C22×#11719919C2211yescharacteristic
7#16810587C2212yes—
8#16810556—21no—
9#16810546—21no—
10#16810507—21no—
11#16810503C411yescharacteristic
12#16810503C411yescharacteristic
13#16810496—21no—
14#16810495C411yescharacteristic
15#16810495C411yescharacteristic
16#16810483C2211yescharacteristic
17#16810349—21no—
18#16810348—24no—
19#16810347—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#436—24no—
22C53×#431—21no—
23C53×#407—21no—
24C53×#407C2212yes—
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C2×#11719917—22no—
28C2×C4⋊Dic53—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53C2×C411yescharacteristic
38C2×C22⋊Dic53C2×C411yescharacteristic
39C22⋊#88322—22no—
40C22⋊#88322—22no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×D4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53C2311yescharacteristic
49C2×M16×C53—22no—
50C2×M16×C53—22no—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C42×C53D411yescharacteristic
57C4⋊C8×C53—22no—
58C22⋊C8×C53—22no—
59C22⋊C8×C53—22no—
60C22⋊C8×C53D414yes—
61C4×C8×C53—22no—
62C24×C53C22⋊C411yescharacteristic
63C24×C53C22×C411yescharacteristic
64C2×D4×C53—22no—
65C2×D4×C53—22no—
66C2×D4×C53—41no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—81no—
70C22×C4×C53—81no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C2×D412yes—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53#4112yes—
78C22×Dic53—21no—
79C22×Dic53—22no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—22no—
84C22×Dic53—42no—
85C22×Dic53—81no—
86M16×C53—82no—
87M16×C53—82no—
88C2×C8×C53—24no—
89C4⋊C4×C53—22no—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—22no—
92C42×C53—21no—
93C42×C53—21no—
94C22⋊Dic53—42no—
95C22⋊Dic53—41no—
96C22⋊Dic53—41no—
97C22⋊Dic53—81no—
98C4⋊Dic53—81no—
99#11719919—42no—
100#11719919—41no—
101#11719919—42no—
102#11719919—41no—
103#11719919—81no—
104#11719917—82no—
105C2×#88322—42no—
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×C53C4×D412yes—
117D4×C53—44no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—81no—
130C2×C4×C53#10112yes—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12112yes—
133C2×C4×C53#12311yescharacteristic
134C2×Dic53—24no—
135C2×Dic53—22no—
136C2×Dic53—41no—
137C2×Dic53—42no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—82no—
149C8×C53—44no—
150#88322—82no—
151#9518—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—81no—
164C22×C53#38611yescharacteristic
165C22×C53C2×#10111yescharacteristic
166C22×C53#41311yescharacteristic
167C4×C53—22no—
168C4×C53—22no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—41no—
172C4×C53—41no—
173Dic53—42no—
174Dic53—81no—
175Dic53—81no—
176Dic53—81no—
177#2975D5311yescharacteristic
178#2579—531no—
179#1465—531no—
180#1463—532no—
181#1305—532no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—42no—
185C2×C53—81no—
186C2×C53#189111yescharacteristic
187C2×C53#190111yescharacteristic
188C2×C53#191311yescharacteristic
189C2×C4×D4C2×D5311yescharacteristic
190C22×C4⋊C4—531no—
191#436—24no—
192#431—21no—
193#407—21no—
194#407—532no—
195#407C2×D5312yes—
196#398—1061no—
197#396—1061no—
198#387—1061no—
199#386—531no—
200#386—531no—
201#385—531no—
202#385—531no—
203#385—1061no—
204#380—531no—
205#324—1061no—
206#324—1064no—
207#323—1061no—
208C53#951811yescharacteristic, complemented, Sylow
209C22×D4C4×D5311yescharacteristic
210C23×C4—531no—
211C23×C4—531no—
212C23×C4C22⋊D5311yescharacteristic
213C23×C4C22×D5311yescharacteristic
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—1062no—
217C4×D4—22no—
218C4×D4—22no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1062no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4D21211yescharacteristic
227C2×C22⋊C4—531no—
228C2×C22⋊C4—531no—
229C2×C22⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C22⋊C4C4×D5311yescharacteristic
232C2×C22⋊C4D21211yescharacteristic
233C2×C42C22⋊D5311yescharacteristic
234C4⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—1062no—
238C22⋊C8—1062no—
239C22⋊C8C22⋊D5314yes—
240C4×C8—22no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246C24#1171992111yescharacteristic
247C24C2×C4×D5311yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—1061no—
256C22×C4—1062no—
257C22×C4—1062no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—4241no—
262C22×C4—2122no—
263C22×C4#1171992111yescharacteristic
264C22×C4#1171992111yescharacteristic
265C22×C4#1171992111yescharacteristic
266C22×C4C2×D21211yescharacteristic
267C22×C4#1171994512yes—
268C22×C4C2×C22⋊D5312yes—
269C22×C4C2×C22⋊D5311yescharacteristic
270M16—82no—
271M16—82no—
272M16—4242no—
273C2×C8—24no—
274C2×C8—2122no—
275C4⋊C4—22no—
276C4⋊C4—4241no—
277C4⋊C4—4241no—
278C4⋊C4—2122no—
279C4⋊C4—2121no—
280C4⋊C4—2122no—
281C4⋊C4—2121no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—4241no—
285C22⋊C4—2122no—
286C22⋊C4—2121no—
287C22⋊C4—2121no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×#1171992111yescharacteristic
300C23C4×C22⋊D5312yes—
301D4—44no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—1062no—
315C2×C4—1064no—
316C2×C4—4242no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2122no—
328C2×C4—2121no—
329C2×C4#1313313512yes—
330C2×C4#1313318611yescharacteristic
331C2×C4#1313319112yes—
332C2×C4#1313320011yescharacteristic
333C8—44no—
334C8—4242no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#1681050211yescharacteristic
348C22C2×#1313313511yescharacteristic
349C22#1681060511yescharacteristic
350C4—21no—
351C4—22no—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—4241no—
357C4—4241no—
358C4—4241no—
359C4—2122no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—81no—
364C2#2027513311yescharacteristic
365C2#2027518311yescharacteristic
366C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2975C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#2975D53
hypocenterC53#9518
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #20273044C2semidirect
4. #20272622C2semidirect
208. C53#9518semidirect

Maximal subgroups

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

subgroupindex
#202800582
#202730532
#202730442
#202726222
C53×#29752
#951853