Finite Group Databasebeta

← #28309408

Subgroups — #28309408

Order 13568.

334 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 88 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#20291074C211yesmaximal, characteristic
2#20290636C211yesmaximal, characteristic
3#20289703C214yesmaximal
4#20285299C211yesmaximal, characteristic
5#16811569—21no—
6#16811544—21no—
7#16811518C2212yes—
8C4×#11719910—21no—
9#16811497—21no—
10#16811497C2212yes—
11#16811178—21no—
12#16811028C411yescharacteristic, complemented
13#16810973—21no—
14#16810861—21no—
15#16810861C2212yes—
16Q16×Dic53—21no—
17Q16×Dic53C412yescomplemented
18C4×C8×D53C2211yescharacteristic
19C53×#494C411yescharacteristic, complemented
20C2×C4⋊#6157—22no—
21C2×C4⋊#6157—41no—
22C2×C4⋊#6157—41no—
23C2×#11719910—22no—
24C2×C8×D53D411yescharacteristic
25C2×C8×D53C2311yescharacteristic
26#13133217—21no—
27#13133208—41no—
28#13133197—41no—
29#13133193—22no—
30Q8×Dic53—22no—
31#13133182—41no—
32#13133175—41no—
33C42×D53D411yescharacteristic
34#13133151—21no—
35#13133151C2×C412yes—
36#13133141—22no—
37#13133141—22no—
38C4×#88322D411yescharacteristic
39C8×Dic53C2×C411yescharacteristic
40C8×Dic53D411yescharacteristic
41C2×Q16×C53—21no—
42C2×Q16×C53C2×C412yes—
43C4⋊Q8×C53—21no—
44C4×C8×C53C2×C411yescharacteristic
45C22×#6157—42no—
46C2×Q8×C53—22no—
47C2×C4×D53D811yescharacteristic
48C2×C4×D53D811yescharacteristic
49C2×C4×D53C2×D411yescharacteristic
50C2×Dic106—41no—
51C4⋊#6157—42no—
52C4⋊#6157—42no—
53C4⋊#6157—82no—
54C4⋊#6157—82no—
55Q16×C53—42no—
56Q16×C53—42no—
57C2×C8×C53C22⋊C411yescharacteristic
58C2×C8×C53C22×C411yescharacteristic
59C4⋊C4×C53—41no—
60C4⋊C4×C53—41no—
61C42×C53C22⋊C411yescharacteristic
62#11719921—42no—
63C4⋊Dic53—22no—
64C4⋊Dic53—41no—
65C4⋊Dic53—42no—
66#11719919—42no—
67C4×Dic53—42no—
68C4×Dic53C22⋊C411yescharacteristic
69C4×Dic53SD1611yescharacteristic
70C4×Dic53SD1611yescharacteristic
71C2×#88322C22⋊C411yescharacteristic
72C2×#88322C2×D411yescharacteristic
73C8×D53—41no—
74C8×D53—41no—
75#11719910—42no—
76C22×D53C2×D811yescharacteristic
77C2×#6157—82no—
78C2×#6157—82no—
79C2×#6157—82no—
80Q8×C53—42no—
81Q8×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53#10211yescharacteristic
84C2×C4×C53#10211yescharacteristic
85C2×C4×C53C2×C22⋊C411yescharacteristic
86C2×Dic53—42no—
87C2×Dic53#10211yescharacteristic
88C2×Dic53#10211yescharacteristic
89C2×Dic53C2×SD1611yescharacteristic
90C4×D53—22no—
91C4×D53—21no—
92C4×D53—21no—
93C4×D53SD3212yes—
94C4×D53Q3212yes—
95Dic106—82no—
96C8×C53—21no—
97C8×C53—21no—
98#88322—21no—
99#88322—21no—
100#11857—531nomaximal, Sylow
101C22×C53C2×#10211yescharacteristic
102C2×D53—21no—
103C2×D53—21no—
104C2×D53C2×SD3211yescharacteristic
105C2×D53C2×Q3211yescharacteristic
106#6157—162no—
107C4×C53—82no—
108C4×C53#35812yes—
109C4×C53#36012yes—
110C4×C53#41011yescharacteristic
111C4×C53#42111yescharacteristic
112Dic53—21no—
113Dic53—21no—
114Dic53—82no—
115Dic53#35812yes—
116#3170—531no—
117#2162—531no—
118#1834—531no—
119#1348—534no—
120C2×C53#151711yescharacteristic
121C2×C53C2×#35811yescharacteristic
122C2×C53C2×#36011yescharacteristic
123D53—21no—
124D53—21no—
125#533—1061no—
126#500—531no—
127#494#615711yescharacteristic, complemented
128#487—1061no—
129#467—532no—
130#467—1061no—
131C4×Q16—532no—
132C4×Q16—1061no—
133C2×C4×C8—531no—
134#384—1061no—
135C4⋊C16—1061no—
136C22⋊C16—532no—
137C4×C16—1061no—
138#340—532no—
139#340—1061no—
140C53#1185711yescharacteristic, complemented, Sylow
141C2×Q16—21no—
142C2×Q16C2×#615712yes—
143C22×C8—531no—
144C22×C8—531no—
145C4⋊Q8—21no—
146C4⋊Q8—1061no—
147#125—2121no—
148C22⋊Q8—1062no—
149C22⋊Q8—2121no—
150C4×Q8—1062no—
151C4×Q8—2121no—
152#117—2121no—
153C2×C4⋊C4—1062no—
154C2×C4⋊C4—2121no—
155C2×C4⋊C4—2121no—
156C2×C42—531no—
157C2×C16—1062no—
158#107—532no—
159#107—1061no—
160#103—1062no—
161#103—1062no—
162C4×C8—531no—
163C4×C8—531no—
164C4×C8—531no—
165C4×C8C2×#615711yescharacteristic
166C2×Q8—22no—
167C2×Q8—2121no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—531no—
171C22×C4—2122no—
172Q16—42no—
173Q16—42no—
174C2×C8—531no—
175C2×C8—531no—
176C2×C8—2121no—
177C2×C8—2121no—
178C2×C8C22⋊#615711yescharacteristic
179C2×C8C22×#615711yescharacteristic
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—1062no—
183C4⋊C4—4242no—
184C4⋊C4—4242no—
185C4⋊C4—2122no—
186C4⋊C4—2122no—
187C4⋊C4—2122no—
188C4⋊C4—2122no—
189C4⋊C4—2121no—
190C22⋊C4—2122no—
191C42—531no—
192C42—531no—
193C42—531no—
194C42—2122no—
195C42C22⋊#615711yescharacteristic
196C16—2122no—
197C23—531no—
198Q8—42no—
199Q8—42no—
200Q8—4242no—
201C2×C4—42no—
202C2×C4—531no—
203C2×C4—532no—
204C2×C4—532no—
205C2×C4—531no—
206C2×C4—531no—
207C2×C4—1062no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—4242no—
211C2×C4—4242no—
212C2×C4—4242no—
213C2×C4—2122no—
214C2×C4#1313327911yescharacteristic
215C2×C4#1313327911yescharacteristic
216C2×C4C2×C22⋊#615711yescharacteristic
217C8—21no—
218C8—21no—
219C8—1061no—
220C8—1061no—
221C22—531no—
222C22—531no—
223C22—1061no—
224C22—1061no—
225C22C2×#1313327911yescharacteristic
226C4—82no—
227C4—532no—
228C4—8482no—
229C4—1061no—
230C4—1061no—
231C4—4242no—
232C4#1681153612yes—
233C4#1681154012yes—
234C4#1681160311yescharacteristic
235C4#1681162111yescharacteristic
236C2—1061no—
237C2—1061no—
238C2#2029014411yescharacteristic
239C2C2×#1681153611yescharacteristic
240C2C2×#1681154011yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#494C4
socleC22×C53C2×#102
prime radicalC22×D53C2×D8
cosocleC2×C8×D53C23
hypercenter#494#6157
hypocenterC53#11857
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #494

Complements

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

NH ≅ G/Ntype
12. #16811028C4semidirect
17. Q16×Dic53C4semidirect
19. C53×#494C4semidirect
127. #494#6157semidirect
140. C53#11857semidirect

Maximal subgroups

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

subgroupindex
#202910742
#202906362
#202897032
#202852992
#1185753