Finite Group Databasebeta

← #27366177

Subgroups — #27366177

Order 13568.

824 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 179 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#20285262C212yesmaximal
2#20285097C211yesmaximal, characteristic
3#20274043C218yesmaximal, complemented
4#20273310C212yesmaximal
5#20273310C211yesmaximal, characteristic
6C53×#3556C211yesmaximal, characteristic
7C2×#13133217C2211yescharacteristic
8#16811033C2218yes—
9#16811026—22no—
10#16811019—22no—
11#16810993C2214yes—
12#16810975—22no—
13#16810666—24no—
14#16810631—22no—
15C4×#11719917C2212yes—
16C4×#11719917C2211yescharacteristic
17#16810365C2218yes—
18#16810365C2214yes—
19C53×#564C2214yes—
20C53×#557—22no—
21C53×#557—22no—
22C53×#557—22no—
23C53×#541—22no—
24C53×#541—24no—
25C53×#533C2212yes—
26C53×#533C2211yescharacteristic
27C22×Dic106—21no—
28C2×#11719917—22no—
29C2×#11719917—21no—
30#13133217—21no—
31#13133217—22no—
32#13133217C2312yes—
33C2×C4⋊Dic53—22no—
34#13133150—28no—
35#13133148—28no—
36#13133140—28no—
37#13133128C2314yes—
38#13133128C2312yes—
39C4×#88322—22no—
40C4×#88322—21no—
41C4⋊Q8×C53—22no—
42C4⋊Q8×C53—24no—
43C4⋊Q8×C53D412yes—
44C4⋊Q8×C53D412yes—
45C4⋊Q8×C53D412yes—
46#126×C53—28no—
47#125×C53—22no—
48#125×C53—24no—
49#125×C53—28no—
50#125×C53—28no—
51#125×C53—28no—
52#125×C53C2314yes—
53#125×C53C2312yes—
54C22⋊Q8×C53—22no—
55C22⋊Q8×C53—22no—
56C22⋊Q8×C53—22no—
57C4×Q8×C53—24no—
58C4×Q8×C53—24no—
59C4×Q8×C53—24no—
60C4×Q8×C53—22no—
61C4×Q8×C53—22no—
62C4×Q8×C53—22no—
63#117×C53D412yes—
64#117×C53D414yes—
65C2×C42×C53C2311yescharacteristic
66C2×Q8×C53—22no—
67C2×Q8×C53—22no—
68C2×Q8×C53—22no—
69C22×C4×C53C2×D412yes—
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—41no—
72C2×Dic106—22no—
73C2×Dic106—44no—
74C2×Dic106—41no—
75C4⋊C4×C53—28no—
76C4⋊C4×C53—28no—
77C4⋊C4×C53—28no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—24no—
82C4⋊C4×C53—24no—
83C4⋊C4×C53—24no—
84C4⋊C4×C53C2×D414yes—
85C4⋊C4×C53C2×D414yes—
86C4⋊C4×C53C2×D414yes—
87C22⋊C4×C53—22no—
88C22⋊C4×C53—24no—
89C42×C53—28no—
90C42×C53—22no—
91C42×C53—24no—
92C42×C53C2×D412yes—
93C42×C53C2×D411yescharacteristic
94C42×C53C2411yescharacteristic
95C4⋊Dic53—24no—
96C4⋊Dic53—42no—
97#11719917—28no—
98#11719917—42no—
99C2×#88322—24no—
100C2×#88322—22no—
101C23×C53#12011yescharacteristic
102Q8×C53—44no—
103Q8×C53—44no—
104Q8×C53—44no—
105C2×C4×C53—22no—
106C2×C4×C53—24no—
107C2×C4×C53—24no—
108C2×C4×C53—24no—
109C2×C4×C53—24no—
110C2×C4×C53#12011yescharacteristic
111C2×C4×C53#12012yes—
112C2×C4×C53C22×D411yescharacteristic
113C2×C4×C53C22×D412yes—
114C2×Dic53—42no—
115C2×Dic53—82no—
116Dic106—44no—
117Dic106—82no—
118#88322—28no—
119#88322—42no—
120#19084—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—22no—
123C22×C53C2×#12011yescharacteristic
124C4×C53—21no—
125C4×C53—22no—
126C4×C53—44no—
127C4×C53—44no—
128C4×C53—44no—
129C4×C53#45412yes—
130C4×C53#45614yes—
131Dic53—82no—
132#3556D5311yescharacteristic
133#3165—532no—
134#3139—531no—
135#1663—538no—
136#1526—531no—
137#1526—532no—
138C2×C53—41no—
139C2×C53C2×#45411yescharacteristic
140C2×C53C2×#45612yes—
141#564C2×D5314yes—
142#557—22no—
143#557—22no—
144#557—22no—
145#541—22no—
146#541—24no—
147#533C2×D5311yescharacteristic
148#533C2×D5312yes—
149C2×C4⋊Q8—531no—
150#501—538no—
151#500—1062no—
152#498—1062no—
153#491—534no—
154#487—1062no—
155#428—1064no—
156#419—1062no—
157C4×M16—531no—
158C4×M16—532no—
159#330—538no—
160#330—534no—
161C53#1908411yescharacteristic, complemented, Sylow
162C22×Q8—1061no—
163C2×M16—1062no—
164C2×M16—1061no—
165C4⋊Q8—22no—
166C4⋊Q8—24no—
167C4⋊Q8—532no—
168C4⋊Q8—1062no—
169C4⋊Q8—1061no—
170C4⋊Q8C22⋊D5312yes—
171C4⋊Q8C22⋊D5312yes—
172C4⋊Q8C22⋊D5312yes—
173#126—28no—
174#125—22no—
175#125—24no—
176#125—28no—
177#125—28no—
178#125—28no—
179#125C22×D5312yes—
180#125C22×D5314yes—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190#117C22⋊D5312yes—
191#117C22⋊D5314yes—
192C2×C4⋊C4—1062no—
193C2×C42C22×D5311yescharacteristic
194#107—1068no—
195#106—1068no—
196#103—1068no—
197#97—532no—
198#97—534no—
199C4×C8—1061no—
200C4×C8—1062no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—1062no—
205C2×Q8—2124no—
206C2×Q8—2121no—
207C22×C4—2121no—
208C22×C4C2×C22⋊D5312yes—
209C22×C4C2×C22⋊D5311yescharacteristic
210M16—1068no—
211M16—2122no—
212C2×C8—1064no—
213C2×C8—1062no—
214C4⋊C4—28no—
215C4⋊C4—28no—
216C4⋊C4—28no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—1064no—
224C4⋊C4—2122no—
225C4⋊C4C2×C22⋊D5314yes—
226C4⋊C4C2×C22⋊D5314yes—
227C4⋊C4C2×C22⋊D5314yes—
228C22⋊C4—22no—
229C22⋊C4—24no—
230C42—28no—
231C42—22no—
232C42—24no—
233C42C2×C22⋊D5312yes—
234C42C2×C22⋊D5311yescharacteristic
235C42C23×D5311yescharacteristic
236C23#1313318611yescharacteristic
237Q8—44no—
238Q8—44no—
239Q8—44no—
240Q8—4242no—
241Q8—2124no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—4242no—
248C2×C4—2122no—
249C2×C4#1313318611yescharacteristic
250C2×C4#1313318612yes—
251C2×C4C22×C22⋊D5312yes—
252C2×C4C22×C22⋊D5311yescharacteristic
253C8—1068no—
254C8—2122no—
255C22—21no—
256C22—22no—
257C22C2×#1313318611yescharacteristic
258C4—21no—
259C4—22no—
260C4—44no—
261C4—44no—
262C4—44no—
263C4—4242no—
264C4#1681079612yes—
265C4#1681080614yes—
266C2—41no—
267C2C2×#1681079611yescharacteristic
268C2C2×#1681080612yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3556C2
socleC22×C53C2×#120
prime radicalC23×C53#120
hypercenter#3556D53
hypocenterC53#19084
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
161. C53#19084semidirect

Maximal subgroups

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

subgroupindex
#202852622
#202850972
#202740432
#202733102
C53×#35562
#1908453