Finite Group Databasebeta

← #28308671

Subgroups — #28308671

Order 13568.

320 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 82 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#20291073C211yesmaximal, characteristic, complemented
2C2×#16811486C211yesmaximal, characteristic, complemented
3#20289671C212yesmaximal, complemented
4#20289670C212yesmaximal, complemented
5#20285286C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7#16811544—21no—
8C22⋊#11719910—21no—
9#16811509C2212yes—
10#16811486—21no—
11#16811486C2212yes—
12#16811159—21no—
13#16811024C411yescharacteristic
14#16811008C411yescharacteristic
15#16810979—21no—
16#16810859C411yescharacteristic
17#16810854—21no—
18Q16×Dic53C2211yescharacteristic
19#16810708—21no—
20#16810705C2211yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22C53×#467C411yescharacteristic
23C22×#88324—41no—
24C2×#11719910—22no—
25C4⋊#88324—22no—
26C4⋊#88324—41no—
27C2×#11719924—21no—
28C2×D424C2×C411yescharacteristic
29C22×#88322D411yescharacteristic
30#13133212—41no—
31#13133208—21no—
32#13133206—21no—
33#13133201—41no—
34#13133191—21no—
35Q8×Dic53—21no—
36#13133179—21no—
37#13133177—41no—
38#13133175—41no—
39C2×C4×Dic53D411yescharacteristic
40#13133151C2×C411yescharacteristic
41#13133148—21no—
42#13133141—21no—
43#13133139—21no—
44#13133137—21no—
45C8×Dic53C2×C411yescharacteristic
46C8×Dic53D411yescharacteristic
47C8×Dic53D411yescharacteristic
48C8×Dic53C2311yescharacteristic
49C2×Q16×C53C2×C411yescharacteristic
50C22×C8×C53C2×C411yescharacteristic
51C22⋊Q8×C53—21no—
52#107×C53C2×C411yescharacteristic
53#103×C53—21no—
54C2×Q8×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C2×C22⋊D53—41no—
57C22×Dic53D811yescharacteristic
58C22×Dic53SD1611yescharacteristic
59C2×D212—21no—
60C2×C4×D53—41no—
61C2×#88324—42no—
62C2×#88324—82no—
63Q16×C53—22no—
64C2×C8×C53—22no—
65C2×C8×C53C22⋊C411yescharacteristic
66C2×C8×C53C22×C411yescharacteristic
67C4⋊C4×C53—21no—
68C4⋊C4×C53—41no—
69C22⋊C4×C53—41no—
70#11719924—42no—
71#11719921—41no—
72#11719921—41no—
73#11719921—41no—
74#11719921—41no—
75C4⋊Dic53—21no—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78#11719919—41no—
79C4×Dic53—41no—
80C4×Dic53C22⋊C411yescharacteristic
81C4×Dic53D811yescharacteristic
82C4×Dic53SD1611yescharacteristic
83C4×Dic53C2×D411yescharacteristic
84C2×#88322—41no—
85C2×#88322C22⋊C411yescharacteristic
86C2×#88322C2×D411yescharacteristic
87D424—22no—
88#11719910—42no—
89C23×C53#10211yescharacteristic
90C22×D53—41no—
91Q8×C53—41no—
92Q8×C53—41no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53#10211yescharacteristic
98C2×C4×C53C2×C22⋊C411yescharacteristic
99C22⋊D53—82no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—41no—
104C2×Dic53#10211yescharacteristic
105C2×Dic53#10211yescharacteristic
106C2×Dic53C2×D811yescharacteristic
107C2×Dic53C2×SD1611yescharacteristic
108D212—41no—
109D212—41no—
110C4×D53—81no—
111C4×D53—81no—
112#88324—82no—
113C8×C53—21no—
114C8×C53#13112yes—
115#88322—21no—
116#88322—21no—
117#11648—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53#35912yes—
121C22×C53C2×#10211yescharacteristic
122C2×D53—81no—
123C2×D53—81no—
124C2×D53—81no—
125C4×C53—21no—
126C4×C53—81no—
127C4×C53—81no—
128C4×C53#40811yescharacteristic
129C4×C53#42111yescharacteristic
130Dic53—21no—
131Dic53—21no—
132Dic53—21no—
133Dic53—81no—
134Dic53#50911yescharacteristic
135Dic53#51011yescharacteristic
136#3168—531no—
137#2162—531no—
138C2×#335—531no—
139#1332—532no—
140#1331—532no—
141C2×C53—22no—
142C2×C53#148011yescharacteristic
143C2×C53#213911yescharacteristic
144C2×C53C2×#35911yescharacteristic
145D53—161no—
146#529—1061no—
147#499—531no—
148#495—531no—
149#488—1061no—
150#467#615711yescharacteristic
151#466—531no—
152#465—1061no—
153C4×Q16—531no—
154C4×SD16—1061no—
155C4×D8—531no—
156C2×C4⋊C8—1061no—
157C2×C4×C8—531no—
158C4⋊C16—1061no—
159C22⋊C16—1061no—
160C4×C16—532no—
161#335—532no—
162#335—1061no—
163C53#1164811yescharacteristic, complemented, Sylow
164C2×Q16C2×#615711yescharacteristic
165C2×SD16—1061no—
166C2×D8—531no—
167C22×C8—531no—
168C22×C8—2121no—
169C22×C8C2×#615711yescharacteristic
170#126—2121no—
171#125—1061no—
172#124—1061no—
173#123—2121no—
174C22⋊Q8—21no—
175#121—1061no—
176C4×Q8—1061no—
177C4×D4—1061no—
178C4×D4—2121no—
179#117—2121no—
180C2×C42—531no—
181C2×C16—1062no—
182#107—531no—
183#107C2×#615711yescharacteristic
184#106—1061no—
185C4⋊C8—1062no—
186C4⋊C8—2121no—
187#103—21no—
188#103—1061no—
189#102—1061no—
190#102—1061no—
191C4×C8—531no—
192C4×C8—531no—
193C4×C8—531no—
194C4×C8—531no—
195C2×Q8—21no—
196C2×D4—1061no—
197C2×D4—2121no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—2121no—
201C22×C4C22⋊#615711yescharacteristic
202Q16—22no—
203SD16—2122no—
204D8—1062no—
205C2×C8—22no—
206C2×C8—531no—
207C2×C8—531no—
208C2×C8—4242no—
209C2×C8—2121no—
210C2×C8—2122no—
211C2×C8C22⋊#615711yescharacteristic
212C2×C8C22×#615711yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—41no—
215C4⋊C4—1061no—
216C4⋊C4—2121no—
217C4⋊C4—2121no—
218C4⋊C4—2121no—
219C22⋊C4—41no—
220C22⋊C4—2121no—
221C22⋊C4—2121no—
222C22⋊C4—2121no—
223C22⋊C4—2121no—
224C42—531no—
225C42—531no—
226C42—531no—
227C42—531no—
228C42—2121no—
229C16—2122no—
230C23—2121no—
231C23#1313327911yescharacteristic
232Q8—41no—
233Q8—41no—
234D4—4242no—
235D4—2121no—
236D4—2121no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—531no—
242C2×C4—531no—
243C2×C4—531no—
244C2×C4—531no—
245C2×C4—1061no—
246C2×C4—1061no—
247C2×C4—1062no—
248C2×C4—4241no—
249C2×C4—4241no—
250C2×C4—2121no—
251C2×C4#1313327911yescharacteristic
252C2×C4C2×C22⋊#615711yescharacteristic
253C8—21no—
254C8—1061no—
255C8—1061no—
256C8—4242no—
257C8#1313330412yes—
258C22—21no—
259C22—21no—
260C22—4241no—
261C22—4241no—
262C22—4241no—
263C22#1681153812yes—
264C22C2×#1313327911yescharacteristic
265C4—21no—
266C4—81no—
267C4—81no—
268C4—531no—
269C4—531no—
270C4—1061no—
271C4—1061no—
272C4—1061no—
273C4—4241no—
274C4#1681159611yescharacteristic
275C4#1681162011yescharacteristic
276C2—22no—
277C2—8481no—
278C2#2029002011yescharacteristic
279C2#2029102311yescharacteristic
280C2C2×#1681153811yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#467C4
socleC22×C53C2×#102
prime radical#16810859C4
cosocleC8×Dic53C23
hypercenter#467#6157
hypocenterC53#11648
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #20291073C2semidirect
2. C2×#16811486C2semidirect
3. #20289671C2semidirect
4. #20289670C2semidirect
163. C53#11648semidirect

Maximal subgroups

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

subgroupindex
#202910732
C2×#168114862
#202896712
#202896702
#202852862
#1164853