Finite Group Databasebeta

← #27281761

Subgroups — #27281761

Order 13568.

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

normal subgroups only

This group has 490 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#20279911C211yesmaximal, characteristic
2#20273341C212yesmaximal, complemented
3#20273056C211yesmaximal, characteristic
4#20272570C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C2×#13133209C2211yescharacteristic
7#16810680C2212yes—
8#16810546—21no—
9#16810546—21no—
10#16810526C411yescharacteristic
11#16810526C411yescharacteristic
12#16810496—21no—
13#16810496—21no—
14#16810492C411yescharacteristic
15#16810488C411yescharacteristic
16C4×C4⋊Dic53C2211yescharacteristic
17#16810364—24no—
18#16810347—21no—
19#16810347—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#435—24no—
22C53×#432C2212yes—
23C53×#407—21no—
24C2×C22⋊C8×C53—21no—
25#13133209—22no—
26#13133209—22no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40C4⋊#88322—22no—
41C22⋊#88322—22no—
42#13133128—22no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×D4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C2×M16×C53—22no—
50C22×C8×C53—22no—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—22no—
58C22⋊C8×C53—22no—
59C22⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61#97×C53D414yes—
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—21no—
70C22×C4×C53—21no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82M16×C53—82no—
83C2×C8×C53—24no—
84C2×C8×C53—82no—
85C4⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—22no—
88C42×C53C2×D412yes—
89C42×C53#4112yes—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—81no—
93C22⋊Dic53—81no—
94C4⋊Dic53—22no—
95C4⋊Dic53—22no—
96C4⋊Dic53—42no—
97#11719919—41no—
98#11719919—42no—
99#11719919—41no—
100#11719919—81no—
101#11719919—81no—
102C4×Dic53—41no—
103C4×Dic53—41no—
104C2×#88322—42no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53#10112yes—
115C23×C53C2×C22⋊C411yescharacteristic
116D4×C53—44no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53C4×D412yes—
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12112yes—
132C2×C4×C53#12311yescharacteristic
133C2×Dic53—22no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C8×C53—44no—
146#88322—82no—
147#9449—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—22no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53#38611yescharacteristic
161C22×C53C2×#10111yescharacteristic
162C22×C53#41311yescharacteristic
163C4×C53—22no—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169Dic53—42no—
170Dic53—81no—
171Dic53—81no—
172Dic53—81no—
173#2973D5311yescharacteristic
174#2566—531no—
175#1534—532no—
176#1466—531no—
177#1285—532no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53—42no—
181C2×C53—81no—
182C2×C53#188411yescharacteristic
183C2×C53#190111yescharacteristic
184C2×C53#191111yescharacteristic
185C2×#125—531no—
186C2×C4×D4C2×D5311yescharacteristic
187#435—24no—
188#432—532no—
189#432C2×D5312yes—
190#407—21no—
191C2×C22⋊C8—21no—
192#396—1061no—
193#396—1061no—
194#390—531no—
195#390—531no—
196#385—1061no—
197#385—1061no—
198#384—531no—
199#382—531no—
200C4×C4⋊C4—531no—
201#330—1064no—
202#323—1061no—
203#323—1061no—
204C53#944911yescharacteristic, complemented, Sylow
205C22×D4C4×D5311yescharacteristic
206C23×C4C22⋊D5311yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C2×M16—22no—
209C22×C8—22no—
210#125—1062no—
211#125—1062no—
212C4×D4—22no—
213C4×D4—22no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4C4×D5311yescharacteristic
222C2×C22⋊C4—1061no—
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4D21211yescharacteristic
227C2×C22⋊C4D21211yescharacteristic
228C2×C42—531no—
229C2×C42—531no—
230C2×C42C22×D5311yescharacteristic
231C4⋊C8—22no—
232C4⋊C8—1062no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—1062no—
237#97—1062no—
238#97C22⋊D5314yes—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242C24#1171992111yescharacteristic
243C24#1171992111yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4#1171992111yescharacteristic
258C22×C4#1171992111yescharacteristic
259C22×C4C2×C4×D5311yescharacteristic
260C22×C4C2×D21211yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262M16—82no—
263C2×C8—24no—
264C2×C8—82no—
265C2×C8—2122no—
266C4⋊C4—22no—
267C4⋊C4—1062no—
268C4⋊C4—1062no—
269C4⋊C4—4241no—
270C4⋊C4—4241no—
271C4⋊C4—2122no—
272C4⋊C4—2121no—
273C4⋊C4—2122no—
274C4⋊C4—2121no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—4241no—
278C22⋊C4—4241no—
279C22⋊C4—2122no—
280C22⋊C4—2122no—
281C42—2121no—
282C42—2121no—
283C42#1171994512yes—
284C42C2×C22⋊D5312yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1313313512yes—
295C23C2×#1171992111yescharacteristic
296D4—44no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—1062no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4C4×C22⋊D5312yes—
322C2×C4#1313318611yescharacteristic
323C2×C4#1313319112yes—
324C2×C4#1313320011yescharacteristic
325C8—44no—
326C8—4242no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1681050211yescharacteristic
340C22C2×#1313313511yescharacteristic
341C22#1681060511yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—4241no—
349C4—4241no—
350C4—4241no—
351C4—2122no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#2027509811yescharacteristic
357C2#2027518311yescharacteristic
358C2#2027524711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2973D53
hypocenterC53#9449
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20273341C2semidirect
4. #20272570C2semidirect
204. C53#9449semidirect

Maximal subgroups

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

subgroupindex
#202799112
#202733412
#202730562
#202725702
C53×#29732
#944953