Finite Group Databasebeta

← #27366026

Subgroups — #27366026

Order 13568.

784 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 171 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#20285929C212yesmaximal
2#20285237C211yesmaximal, characteristic
3#20274043C218yesmaximal, complemented
4#20273353C211yesmaximal, characteristic
5#20273330C212yesmaximal
6C53×#3556C211yesmaximal, characteristic
7#16811180C2211yescharacteristic
8#16811033C2214yes—
9#16811033C2214yes—
10#16811029—22no—
11#16810993C2214yes—
12#16810964—24no—
13#16810915—22no—
14#16810905—22no—
15#16810680C2212yes—
16#16810659—22no—
17C2×#13133128C2211yescharacteristic
18#16810365C2218yes—
19#16810365C2214yes—
20C53×#564C2214yes—
21C53×#557—22no—
22C53×#557—22no—
23C53×#557—22no—
24C53×#541—22no—
25C53×#541—24no—
26C53×#533C2211yescharacteristic
27C53×#533C2212yes—
28C22×#88322—21no—
29#13133217C2312yes—
30#13133209—21no—
31#13133196—22no—
32C4×Dic106—22no—
33#13133176—21no—
34#13133150—24no—
35#13133150—24no—
36#13133148—24no—
37#13133148—24no—
38C4⋊#88322—22no—
39#13133140—24no—
40#13133140—24no—
41C22⋊#88322—22no—
42#13133128—21no—
43#13133128C2312yes—
44#13133128C2314yes—
45C4⋊Q8×C53—24no—
46C4⋊Q8×C53—22no—
47C4⋊Q8×C53D412yes—
48C4⋊Q8×C53D412yes—
49C4⋊Q8×C53D412yes—
50#126×C53—28no—
51#125×C53—24no—
52#125×C53—22no—
53#125×C53—28no—
54#125×C53—28no—
55#125×C53—28no—
56#125×C53C2312yes—
57#125×C53C2314yes—
58C22⋊Q8×C53—22no—
59C22⋊Q8×C53—22no—
60C22⋊Q8×C53—22no—
61C4×Q8×C53—24no—
62C4×Q8×C53—24no—
63C4×Q8×C53—24no—
64C4×Q8×C53—22no—
65C4×Q8×C53—22no—
66C4×Q8×C53—22no—
67#117×C53D412yes—
68#117×C53D414yes—
69C2×C42×C53C2311yescharacteristic
70C2×Q8×C53—22no—
71C2×Q8×C53—22no—
72C2×Q8×C53—22no—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D412yes—
75C2×Dic106—22no—
76C4⋊C4×C53—28no—
77C4⋊C4×C53—28no—
78C4⋊C4×C53—28no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—24no—
83C4⋊C4×C53—24no—
84C4⋊C4×C53—24no—
85C4⋊C4×C53C2×D414yes—
86C4⋊C4×C53C2×D414yes—
87C4⋊C4×C53C2×D414yes—
88C22⋊C4×C53—22no—
89C22⋊C4×C53—24no—
90C42×C53—28no—
91C42×C53—24no—
92C42×C53—22no—
93C42×C53C2×D411yescharacteristic
94C42×C53C2×D412yes—
95C42×C53C2411yescharacteristic
96C22⋊Dic53—41no—
97C4⋊Dic53—22no—
98C4⋊Dic53—22no—
99#11719919—42no—
100C4×Dic53—41no—
101C2×#88322—24no—
102C2×#88322—22no—
103C2×#88322—24no—
104C23×C53#12011yescharacteristic
105Q8×C53—44no—
106Q8×C53—44no—
107Q8×C53—44no—
108C2×C4×C53—22no—
109C2×C4×C53—24no—
110C2×C4×C53—24no—
111C2×C4×C53—24no—
112C2×C4×C53—42no—
113C2×C4×C53#12012yes—
114C2×C4×C53#12011yescharacteristic
115C2×C4×C53C22×D412yes—
116C2×C4×C53C22×D411yescharacteristic
117C2×Dic53—42no—
118Dic106—44no—
119#88322—24no—
120#88322—44no—
121#19067—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—22no—
124C22×C53C2×#12011yescharacteristic
125C4×C53—21no—
126C4×C53—22no—
127C4×C53—22no—
128C4×C53—44no—
129C4×C53—44no—
130C4×C53—44no—
131C4×C53#45612yes—
132Dic53—82no—
133#3556D5311yescharacteristic
134#3237—532no—
135#3162—531no—
136#1663—538no—
137#1537—531no—
138#1531—532no—
139C2×C53—41no—
140C2×C53C2×#45611yescharacteristic
141C2×C53#302012yes—
142#564C2×D5314yes—
143#557—22no—
144#557—22no—
145#557—22no—
146#541—22no—
147#541—24no—
148#533—531no—
149#533C2×D5311yescharacteristic
150#533C2×D5312yes—
151#501—534no—
152#501—534no—
153#501—1062no—
154#491—534no—
155#485—1064no—
156#477—1062no—
157#475—1062no—
158#432—532no—
159#427—1062no—
160C2×#97—531no—
161#330—534no—
162#330—538no—
163C53#1906711yescharacteristic, complemented, Sylow
164C22×C8—1061no—
165C4⋊Q8—24no—
166C4⋊Q8—22no—
167C4⋊Q8—532no—
168C4⋊Q8C22⋊D5312yes—
169C4⋊Q8C22⋊D5312yes—
170C4⋊Q8C22⋊D5312yes—
171#126—28no—
172#125—24no—
173#125—22no—
174#125—28no—
175#125—28no—
176#125—28no—
177#125—1061no—
178#125C22×D5314yes—
179#125C22×D5312yes—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—1062no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—1062no—
191#117—1061no—
192#117C22⋊D5312yes—
193#117C22⋊D5314yes—
194C2×C42C22×D5311yescharacteristic
195#107—1064no—
196#107—1064no—
197#106—1064no—
198#106—1064no—
199C4⋊C8—1062no—
200#103—1064no—
201#103—1064no—
202C22⋊C8—1062no—
203#97—532no—
204#97—534no—
205#97—1061no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—1062no—
210C22×C4C2×C22⋊D5311yescharacteristic
211C22×C4C2×C22⋊D5312yes—
212C2×C8—1064no—
213C2×C8—1062no—
214C2×C8—1064no—
215C4⋊C4—28no—
216C4⋊C4—28no—
217C4⋊C4—28no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—1062no—
225C4⋊C4—1062no—
226C4⋊C4—2122no—
227C4⋊C4C2×C22⋊D5314yes—
228C4⋊C4C2×C22⋊D5314yes—
229C4⋊C4C2×C22⋊D5314yes—
230C22⋊C4—22no—
231C22⋊C4—24no—
232C22⋊C4—2121no—
233C42—28no—
234C42—24no—
235C42—22no—
236C42—2121no—
237C42C2×C22⋊D5311yescharacteristic
238C42C2×C22⋊D5312yes—
239C42C23×D5311yescharacteristic
240C23#1313318611yescharacteristic
241Q8—44no—
242Q8—44no—
243Q8—44no—
244Q8—2124no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—2122no—
251C2×C4#1313318612yes—
252C2×C4#1313318611yescharacteristic
253C2×C4C22×C22⋊D5311yescharacteristic
254C2×C4C22×C22⋊D5312yes—
255C8—1064no—
256C8—2124no—
257C22—21no—
258C22—22no—
259C22C2×#1313318611yescharacteristic
260C4—21no—
261C4—22no—
262C4—22no—
263C4—44no—
264C4—44no—
265C4—44no—
266C4—4242no—
267C4#1681080612yes—
268C2—41no—
269C2C2×#1681080611yescharacteristic
270C2#2028407912yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3556C2
socleC22×C53C2×#120
prime radicalC23×C53#120
hypercenter#3556D53
hypocenterC53#19067
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

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

NH ≅ G/Ntype
3. #20274043C2semidirect
163. C53#19067semidirect

Maximal subgroups

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

subgroupindex
#202859292
#202852372
#202740432
#202733532
#202733302
C53×#35562
#1906753