Finite Group Databasebeta

← #27393567

Subgroups — #27393567

Order 13568.

914 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C2×#16811254C211yesmaximal, characteristic
2#20283350C212yesmaximal
3#20275706C214yesmaximal, complemented
4#20275702C214yesmaximal, complemented
5#20274796C212yesmaximal
6#20274795C211yesmaximal, characteristic
7C53×#1821C211yesmaximal, characteristic
8#16811254—28no—
9C2×#13133216C412yes—
10C2×#13133216C2212yes—
11C2×#13133208C412yes—
12C2×#13133208C2212yes—
13C2×#13133196C414yes—
14C2×#13133176C2211yescharacteristic
15C2×#13133176C2211yescharacteristic
16#16810681—28no—
17#16810646C2214yes—
18#16810587C2212yes—
19M16×Dic53C2214yes—
20#16810392C2214yes—
21#16810391C2214yes—
22#16810391C2214yes—
23C22×C4⋊C4×C53C2211yescharacteristic
24C53×#407C2212yes—
25C53×#343C2214yes—
26C22×Dic106C2×C414yes—
27C2×#11719917C2314yes—
28#13133216—28no—
29#13133216—24no—
30#13133216—24no—
31#13133208—28no—
32#13133208—24no—
33#13133208—24no—
34#13133196—216no—
35#13133176—24no—
36#13133176D418yes—
37C2×C4⋊Dic53C2×C414yes—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×C4⋊Dic53C2311yescharacteristic
41C2×C4⋊Dic53C2311yescharacteristic
42C2×#11719919C2×C414yes—
43C2×#11719919C2×C414yes—
44C2×C22⋊Dic53C2×C412yes—
45C2×C22⋊Dic53C2×C412yes—
46C2×C4×Dic53C2×C412yes—
47C2×C4×Dic53C2312yes—
48#13133146—28no—
49C22⋊#88322—24no—
50#13133127—24no—
51C8×Dic53—24no—
52C23×C4×C53C2×C412yes—
53C23×C4×C53C2311yescharacteristic
54C2×M16×C53C2314yes—
55C2×C4⋊C4×C53—24no—
56C2×C4⋊C4×C53C2×C412yes—
57C2×C4⋊C4×C53C2312yes—
58C22⋊C8×C53—24no—
59C24×C53C22×C411yescharacteristic
60C22×C4×C53—24no—
61C22×C4×C53—24no—
62C22×C4×C53—82no—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C22×C414yes—
65C22×C4×C53C2×D414yes—
66C22×C4×C53C2411yescharacteristic
67C22×Dic53C22×C412yes—
68C22×Dic53C22×C412yes—
69C22×Dic53C22×C412yes—
70C22×Dic53C22×C412yes—
71C2×Dic106—28no—
72C2×Dic106—84no—
73M16×C53—28no—
74C2×C8×C53—24no—
75C4⋊C4×C53—24no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—22no—
78C22⋊Dic53—24no—
79C22⋊Dic53C22⋊C418yes—
80C4⋊Dic53—22no—
81C4⋊Dic53—22no—
82C4⋊Dic53—28no—
83C4⋊Dic53C22⋊C414yes—
84C4⋊Dic53C22⋊C414yes—
85#11719919—28no—
86#11719919—28no—
87C4×Dic53—24no—
88C4×Dic53C2×D414yes—
89C4×Dic53C2×D414yes—
90#11719917—28no—
91C2×#88322—24no—
92C23×C53—22no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53C2×C22⋊C412yes—
96C23×C53C23×C411yescharacteristic
97C2×C4×C53—22no—
98C2×C4×C53—28no—
99C2×C4×C53—24no—
100C2×C4×C53—44no—
101C2×C4×C53—84no—
102C2×C4×C53C2×C22⋊C412yes—
103C2×C4×C53C22×D412yes—
104C2×C4×C532−1+414yes—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53C2×C22⋊C414yes—
112C2×Dic53C2×C22⋊C414yes—
113Dic106—88no—
114C8×C53—44no—
115#88322—44no—
116#21832—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—22no—
120C22×C53—22no—
121C22×C53—22no—
122C22×C53—44no—
123C22×C53C22×C22⋊C411yescharacteristic
124C22×C53#51912yes—
125C4×C53—22no—
126C4×C53—44no—
127C4×C53#53614yes—
128Dic53—24no—
129Dic53—24no—
130Dic53—42no—
131Dic53—42no—
132C2×#544—531no—
133#2921—532no—
134#1998—534no—
135#1997—534no—
136#1822—532no—
137#1821—531no—
138#1821D5311yescharacteristic
139C2×C53—22no—
140C2×C53—42no—
141C2×C53#231011yescharacteristic
142C2×C53#288811yescharacteristic
143C2×C53#288911yescharacteristic
144#544—1068no—
145C2×C4⋊Q8—532no—
146C2×C4⋊Q8—532no—
147C2×#125—532no—
148C2×#125—532no—
149C2×C22⋊Q8—534no—
150C2×#117—531no—
151C2×#117—531no—
152C22×C4⋊C4C2×D5311yescharacteristic
153#433—1068no—
154#423—534no—
155#407—532no—
156#407C2×D5312yes—
157C4×M16—534no—
158#343—534no—
159#343—534no—
160#343—534no—
161#343C2×D5314yes—
162C53#2183211yescharacteristic, complemented, Sylow
163C22×Q8—534no—
164C23×C4C4×D5312yes—
165C23×C4C22×D5311yescharacteristic
166C2×M16—534no—
167C2×M16C22×D5314yes—
168C4⋊Q8—1064no—
169C4⋊Q8—1064no—
170C4⋊Q8—1068no—
171#125—1064no—
172#125—1064no—
173#125—1068no—
174C22⋊Q8—10616no—
175#117—538no—
176#117—1064no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—534no—
181C2×C4⋊C4—534no—
182C2×C4⋊C4—534no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4C4×D5312yes—
186C2×C4⋊C4C22×D5312yes—
187C2×C22⋊C4—532no—
188C2×C22⋊C4—532no—
189C2×C42—532no—
190C2×C42—532no—
191C4⋊C8—1068no—
192C22⋊C8—24no—
193C22⋊C8—1064no—
194#97—1064no—
195C4×C8—1064no—
196C24C2×C4×D5311yescharacteristic
197C2×Q8—1068no—
198C2×Q8—4244no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—82no—
202C22×C4—532no—
203C22×C4—532no—
204C22×C4—532no—
205C22×C4—532no—
206C22×C4C2×C4×D5311yescharacteristic
207C22×C4C2×C4×D5314yes—
208C22×C4D4×D5314yes—
209C22×C4C23×D5311yescharacteristic
210M16—28no—
211M16—1068no—
212C2×C8—24no—
213C2×C8—1064no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—534no—
218C4⋊C4—534no—
219C4⋊C4—1068no—
220C4⋊C4—1068no—
221C4⋊C4—1068no—
222C4⋊C4—1062no—
223C4⋊C4—1062no—
224C22⋊C4—538no—
225C22⋊C4—1064no—
226C42—534no—
227C42—534no—
228C42—1064no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23C22⋊C4×D5312yes—
233C23C22×C4×D5311yescharacteristic
234Q8—4248no—
235C2×C4—22no—
236C2×C4—28no—
237C2×C4—24no—
238C2×C4—44no—
239C2×C4—84no—
240C2×C4—534no—
241C2×C4—534no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4C22⋊C4×D5312yes—
249C2×C4C2×D4×D5312yes—
250C2×C4#1313326414yes—
251C8—44no—
252C8—2124no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—44no—
259C22C2×C22⋊C4×D5311yescharacteristic
260C22#1681110012yes—
261C4—22no—
262C4—44no—
263C4—1064no—
264C4—1064no—
265C4—2122no—
266C4—2122no—
267C4#1681119514yes—
268C2—22no—
269C2—42no—
270C2#2027730011yescharacteristic
271C2#2028319111yescharacteristic
272C2#2028319511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1821C2
socleC22×C53C22×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC22×C4×C53C24
hypercenter#1821D53
hypocenterC53#21832
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #20275706C2semidirect
4. #20275702C2semidirect
162. C53#21832semidirect

Maximal subgroups

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

subgroupindex
C2×#168112542
#202833502
#202757062
#202757022
#202747962
#202747952
C53×#18212
#2183253