Finite Group Databasebeta

← #28306153

Subgroups — #28306153

Order 13568.

422 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 96 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#20290900C212yesmaximal
2#20290428C211yesmaximal, characteristic
3#20290274C211yesmaximal, characteristic
4#20290064C212yesmaximal
5#20285302C211yesmaximal, characteristic
6#16811627C2212yes—
7C4×D53⋊C8C2211yescharacteristic
8C42×#6157C2211yescharacteristic
9#16811493—24no—
10(C2×#6157)⋊C8—22no—
11#16811487—22no—
12Q8⋊#88324—24no—
13#16811178C2212yes—
14#16811003C411yescharacteristic, complemented
15#16810975C412yescomplemented
16#16810865—22no—
17#16810717—22no—
18C4×C8×D53C2211yescharacteristic
19C53×#500C411yescharacteristic, complemented
20C2×#11719936—22no—
21C2×D53⋊C8—21no—
22C2×C4×#6157—21no—
23C2×C4×#6157—21no—
24C2×C4×#6157D412yes—
25C4⋊#88324—22no—
26C4⋊#88324—22no—
27C4×#88324—21no—
28C4×#88324D412yes—
29C2×Dic212—22no—
30C2×C8×D53D412yes—
31#13133217C2×C412yes—
32#13133208—22no—
33#13133197—22no—
34#13133193—22no—
35Q8×Dic53—22no—
36#13133182—22no—
37#13133175—22no—
38C42×D53C2311yescharacteristic
39#13133140—24no—
40C4×#88322C2×C411yescharacteristic
41C8×Dic53D412yes—
42C4⋊Q8×C53C2×C412yes—
43#107×C53—22no—
44C4×C8×C53C2×C411yescharacteristic
45C22×#6157—22no—
46C2×Q8×C53—22no—
47C2×C4×D53C2×D411yescharacteristic
48C2×C4×D53C2×D411yescharacteristic
49C2×C4×D53#4111yescharacteristic
50C2×Dic106—22no—
51C2×#88324—22no—
52C2×#88324—42no—
53C4×#6157—22no—
54C4×#6157—42no—
55C4×#6157—42no—
56C4×#6157Q1614yescomplemented
57#11719936—44no—
58D53⋊C8—42no—
59C2×C8×C53C22⋊C412yes—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—22no—
62C42×C53C22×C411yescharacteristic
63#11719921—42no—
64C4⋊Dic53—22no—
65C4⋊Dic53—22no—
66C4⋊Dic53—42no—
67#11719919—42no—
68C4×Dic53—42no—
69C4×Dic53C2×D411yescharacteristic
70C4×Dic53C2×D411yescharacteristic
71C4×Dic53#4111yescharacteristic
72C2×#88322C22⋊C412yes—
73Dic212—44no—
74C8×D53—42no—
75C22×D53#12111yescharacteristic
76C2×#6157—24no—
77C2×#6157—42no—
78C2×#6157—42no—
79Q8×C53—44no—
80C2×C4×C53—42no—
81C2×C4×C53C2×C22⋊C411yescharacteristic
82C2×C4×C53C4×D411yescharacteristic
83C2×C4×C53C4×D411yescharacteristic
84C2×Dic53—42no—
85C2×Dic53#12111yescharacteristic
86C2×Dic53#12411yescharacteristic
87C2×Dic53#12711yescharacteristic
88C4×D53—22no—
89C4×D53—22no—
90C4×D53C2×Q1612yes—
91C4×D53#13512yes—
92Dic106—44no—
93#88324—42no—
94#88324—82no—
95C8×C53—22no—
96#88322—22no—
97#9848—531nomaximal, Sylow
98C22×C53#39011yescharacteristic
99C2×D53—21no—
100C2×D53—21no—
101C2×D53#46712yes—
102#6157—44no—
103C4×C53—82no—
104C4×C53#42112yes—
105C4×C53C4×Q1612yes—
106C4×C53#44412yes—
107Dic53—21no—
108Dic53—21no—
109Dic53—82no—
110Dic53#49411yescharacteristic
111Dic53#49511yescharacteristic
112#3170—531no—
113#1994—532no—
114#1744—531no—
115#1567—531no—
116#1493—532no—
117C2×C53#196311yescharacteristic
118C2×C53#196911yescharacteristic
119C2×C53#197511yescharacteristic
120D53—22no—
121#533—532no—
122#500#615711yescharacteristic, complemented
123#494—531no—
124#487—532no—
125#467—1062no—
126C4×Q16—1062no—
127#423—532no—
128C2×C4×C8—531no—
129C2×C4×C8—531no—
130C43—531no—
131#339—1064no—
132#336—1062no—
133#335—1062no—
134Q8⋊C8—1064no—
135C53#984811yescharacteristic, complemented, Sylow
136C2×Q16—1062no—
137C2×M16—1062no—
138C22×C8—532no—
139C22×C8—1061no—
140C4⋊Q8—532no—
141C4⋊Q8C2×#615712yes—
142#125—1062no—
143C22⋊Q8—1062no—
144C22⋊Q8—1062no—
145C4×Q8—1062no—
146C4×Q8—1062no—
147#117—1062no—
148C2×C42—531no—
149C2×C42—532no—
150C2×C42—1061no—
151C2×C42—1061no—
152#107—22no—
153C4⋊C8—1062no—
154C4⋊C8—1062no—
155#103—1064no—
156C4×C8—532no—
157C4×C8—532no—
158C4×C8—531no—
159C4×C8—1061no—
160C4×C8C2×#615711yescharacteristic
161C2×Q8—22no—
162C2×Q8—1062no—
163C22×C4—531no—
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—1062no—
167Q16—2124no—
168M16—2124no—
169C2×C8—532no—
170C2×C8—1062no—
171C2×C8—2122no—
172C2×C8—2122no—
173C2×C8—2122no—
174C2×C8C22⋊#615712yes—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4—2122no—
180C4⋊C4—2122no—
181C22⋊C4—2122no—
182C42—531no—
183C42—531no—
184C42—531no—
185C42—534no—
186C42—1062no—
187C42—2122no—
188C42—2122no—
189C42—2122no—
190C42C22×#615711yescharacteristic
191C23—531no—
192Q8—44no—
193Q8—2124no—
194C2×C4—42no—
195C2×C4—531no—
196C2×C4—532no—
197C2×C4—532no—
198C2×C4—531no—
199C2×C4—531no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1064no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4C2×C22⋊#615711yescharacteristic
207C2×C4D4×#615711yescharacteristic
208C2×C4D4×#615711yescharacteristic
209C8—22no—
210C8—1062no—
211C8—4242no—
212C8—2122no—
213C22—532no—
214C22—1061no—
215C22—1061no—
216C22#1681157511yescharacteristic
217C4—82no—
218C4—531no—
219C4—531no—
220C4—1061no—
221C4—1061no—
222C4—4242no—
223C4—2124no—
224C4#1681162212yes—
225C4Q16×#615712yes—
226C4#1681167012yes—
227C2—1062no—
228C2#2029085111yescharacteristic
229C2#2029086111yescharacteristic
230C2#2029087011yescharacteristic
subgroupG/H
centerC22#16811575
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#500C4
socleC22×C53#390
prime radicalC22×D53#121
cosocleC42×D53C23
hypercenter#500#6157
hypocenterC53#9848
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
14. #16811003C4semidirect
15. #16810975C4semidirect
19. C53×#500C4semidirect
56. C4×#6157Q16semidirect
122. #500#6157semidirect
135. C53#9848semidirect

Maximal subgroups

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

subgroupindex
#202909002
#202904282
#202902742
#202900642
#202853022
#984853