Finite Group Databasebeta

← #27288490

Subgroups — #27288490

Order 13568.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285490C211yesmaximal, characteristic
2#20275700C212yesmaximal
3#20275138C212yesmaximal
4#20272583C211yesmaximal, characteristic
5C53×#1850C211yesmaximal, characteristic
6C2×#13133196C2212yes—
7#16810959—22no—
8#16810864—21no—
9#16810864—21no—
10#16810781—22no—
11C2×#13133142—22no—
12#16810483C2212yes—
13#16810389—22no—
14#16810389—22no—
15#16810386—22no—
16C23×C8×C53C2211yescharacteristic
17C2×C4⋊C8×C53—21no—
18C2×C22⋊C8×C53C2211yescharacteristic
19C4×C22⋊C4×C53C2211yescharacteristic
20C53×#336—21no—
21C22×Dic106D412yes—
22C2×Dic212—21no—
23C2×Dic212—21no—
24#13133196—22no—
25#13133196—22no—
26#13133196—22no—
27#13133196—22no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53D412yes—
30C2×#11719919—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—22no—
33#13133151—21no—
34#13133151—21no—
35#13133142—22no—
36#13133142—22no—
37#13133142—44no—
38#13133124—22no—
39C23×C4×C53C2311yescharacteristic
40C22×C8×C53—21no—
41C22×C8×C53—22no—
42C22×C8×C53—42no—
43C22×C8×C53D411yescharacteristic
44C22×C8×C53D411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C42×C53—21no—
48C4⋊C8×C53—42no—
49C22⋊C8×C53—22no—
50#95×C53—21no—
51C24×C53C2×D411yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53C2×D412yes—
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53#4111yescharacteristic
59C22×C4×C53#4112yes—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—42no—
63C2×Dic106—22no—
64C2×Dic106—22no—
65C2×Dic106—42no—
66C2×Dic106—42no—
67C2×C8×C53—22no—
68C2×C8×C53—24no—
69C2×C8×C53—21no—
70C2×C8×C53—21no—
71C2×C8×C53—42no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C2×C8×C53—41no—
75C2×C8×C53—41no—
76C22⋊C4×C53—21no—
77C22⋊C4×C53—22no—
78C22⋊C4×C53Q1612yescomplemented
79C42×C53—41no—
80C42×C53—41no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C22⋊Dic53—42no—
84C22⋊Dic53—42no—
85C4⋊Dic53—22no—
86C4⋊Dic53—22no—
87C4⋊Dic53—42no—
88C4⋊Dic53—42no—
89#11719919—42no—
90#11719919—42no—
91Dic212—81no—
92Dic212—81no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53#12111yescharacteristic
96C23×C53#12112yes—
97C23×C53C2×Q1611yescharacteristic
98C23×C53#13511yescharacteristic
99C23×C53#13712yes—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53#12011yescharacteristic
113C2×C4×C53#12312yes—
114C2×C4×C53#12411yescharacteristic
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—82no—
118C2×Dic53—82no—
119C2×Dic53—82no—
120C2×Dic53—82no—
121C2×Dic53—82no—
122Dic106—44no—
123Dic106—42no—
124Dic106—42no—
125C8×C53—41no—
126C8×C53—42no—
127C8×C53—41no—
128#10566—531nomaximal, Sylow
129C22×C53—22no—
130C22×C53—21no—
131C22×C53—22no—
132C22×C53—21no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—42no—
136C22×C53#39411yescharacteristic
137C22×C53#46212yes—
138C22×C53#46412yes—
139C22×C53#46711yescharacteristic
140C22×C53#47011yescharacteristic
141C4×C53—22no—
142C4×C53—21no—
143C4×C53—42no—
144C4×C53—81no—
145C4×C53B(4,𝔽2)11yescharacteristic
146C4×C53#45811yescharacteristic
147Dic53—82no—
148Dic53—82no—
149Dic53—162no—
150#3189—531no—
151#1996—532no—
152#1892—532no—
153#1850D5311yescharacteristic
154#1289—531no—
155C2×C53—22no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—22no—
159C2×C53#204411yescharacteristic
160C2×C53#205311yescharacteristic
161C2×C53#205611yescharacteristic
162C23×C8C2×D5311yescharacteristic
163C2×C22⋊Q8—532no—
164#484—1062no—
165#467—1061no—
166#467—1061no—
167#451—1062no—
168C2×C4⋊C8—21no—
169C2×#103—1062no—
170C2×C22⋊C8C2×D5311yescharacteristic
171#380—532no—
172C4×C22⋊C4C2×D5311yescharacteristic
173#342—1062no—
174#342—1062no—
175#340—1062no—
176#336—21no—
177C53#1056611yescharacteristic, complemented, Sylow
178C22×Q8—532no—
179C23×C4C22×D5311yescharacteristic
180C2×Q16—1061no—
181C2×Q16—1061no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—42no—
185C22×C8C22⋊D5311yescharacteristic
186C22×C8C22⋊D5311yescharacteristic
187C22⋊Q8—1062no—
188C22⋊Q8—1062no—
189C22⋊Q8—1062no—
190C22⋊Q8—1062no—
191C2×C4⋊C4—532no—
192C2×C4⋊C4—1062no—
193C2×C4⋊C4—1062no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4D21211yescharacteristic
197C2×C22⋊C4D21211yescharacteristic
198C2×C42—21no—
199#107—1061no—
200#107—1061no—
201C4⋊C8—42no—
202#103—1062no—
203#103—1062no—
204#103—2124no—
205C22⋊C8—22no—
206#95—21no—
207#95—1062no—
208C24C2×D21211yescharacteristic
209C2×Q8—1062no—
210C2×Q8—1062no—
211C2×Q8—2122no—
212C2×Q8—2122no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—1062no—
218C22×C4—1062no—
219C22×C4—2122no—
220C22×C4#1171994511yescharacteristic
221C22×C4D4×D5312yes—
222C22×C4#1171994912yes—
223C22×C4C2×C22⋊D5311yescharacteristic
224Q16—4241no—
225Q16—4241no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C4⋊C4—1062no—
236C4⋊C4—1062no—
237C4⋊C4—2122no—
238C4⋊C4—2122no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C22⋊C4—21no—
242C22⋊C4—22no—
243C22⋊C4—2122no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C22⋊C4—2122no—
247C22⋊C4Dic21212yescomplemented
248C42—41no—
249C42—41no—
250C23—41no—
251C23—41no—
252C23#1313319111yescharacteristic
253C23#1313319212yes—
254C23C2×Dic21211yescharacteristic
255C23#1313323911yescharacteristic
256C23#1313325012yes—
257Q8—2124no—
258Q8—2122no—
259Q8—2122no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—4242no—
273C2×C4—4242no—
274C2×C4—4242no—
275C2×C4—4242no—
276C2×C4—4242no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4#1313318511yescharacteristic
280C2×C4#1313320212yes—
281C2×C4#1313320611yescharacteristic
282C8—42no—
283C8—41no—
284C8—41no—
285C22—21no—
286C22—22no—
287C22—21no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22#1681054011yescharacteristic
293C22#1681083412yes—
294C22#1681084612yes—
295C22#1681086411yescharacteristic
296C22#1681088111yescharacteristic
297C4—22no—
298C4—21no—
299C4—42no—
300C4—81no—
301C4—8482no—
302C4—4242no—
303C4—4242no—
304C4#1681080911yescharacteristic
305C4#1681081211yescharacteristic
306C2—21no—
307C2—21no—
308C2—22no—
309C2—22no—
310C2#2027596611yescharacteristic
311C2#2027602111yescharacteristic
312C2#2027603811yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1850C2
socleC22×C53#394
prime radicalC24×C53C2×D4
hypercenter#1850D53
hypocenterC53#10566
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
78. C22⋊C4×C53Q16semidirect
177. C53#10566semidirect
247. C22⋊C4Dic212semidirect

Maximal subgroups

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

subgroupindex
#202854902
#202757002
#202751382
#202725832
C53×#18502
#1056653