Finite Group Databasebeta

← #27278447

Subgroups — #27278447

Order 13568.

500 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 115 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#20275532C211yesmaximal, characteristic
2#20274884C212yesmaximal, complemented
3C2×#16810363C211yesmaximal, characteristic
4#20272596C212yesmaximal, complemented
5C53×#2379C211yesmaximal, characteristic
6C2×#13133150—22no—
7C2×#13133148—22no—
8C2×#13133141—21no—
9C2×#13133138—21no—
10C2×#13133128C2211yescharacteristic
11C2×#13133128C2212yes—
12#16810493C2211yescharacteristic
13#16810363C414yescomplemented
14C2×#124×C53C2211yescharacteristic
15C53×#391—21no—
16C53×#388—22no—
17C53×#386—21no—
18C53×#385—22no—
19C53×#382C2212yes—
20C22×#88322—22no—
21C22×#88322—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—22no—
24#13133150—44no—
25#13133148—44no—
26#13133141—42no—
27#13133138—42no—
28#13133128—22no—
29#13133128—22no—
30#13133128C2×C414yescomplemented
31C22×Q8×C53D411yescharacteristic
32C22×D4×C53D411yescharacteristic
33C23×C4×C53—21no—
34#124×C53—22no—
35#124×C53C2×C412yes—
36#124×C53Q812yes—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53D412yes—
39C2×C4⋊C4×C53D412yes—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—22no—
43C2×C42×C53—21no—
44C2×C42×C53C2311yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47#95×C53—22no—
48C24×C53—21no—
49C2×Q8×C53—21no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53C4⋊C412yes—
53C2×D4×C53—21no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53C4⋊C412yes—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—82no—
61C22×C4×C53—81no—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D412yes—
64C22×Dic53—41no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—81no—
68C22⋊C4×C53—24no—
69C22⋊C4×C53—24no—
70C22⋊C4×C53—82no—
71C22⋊C4×C53—81no—
72C42×C53—81no—
73C42×C53C22×C411yescharacteristic
74C42×C53C2×Q811yescharacteristic
75C42×C53#4112yes—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78C4⋊Dic53—42no—
79C4⋊Dic53—42no—
80C2×#88322—24no—
81C2×#88322—24no—
82C2×#88322—24no—
83C2×#88322—22no—
84C2×#88322—41no—
85C2×#88322—41no—
86C23×C53—22no—
87C23×C53—41no—
88C23×C53—42no—
89C23×C53—41no—
90C23×C53—42no—
91C23×C53#12011yescharacteristic
92Q8×C53—42no—
93Q8×C53—81no—
94D4×C53—42no—
95D4×C53—81no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—84no—
110C2×C4×C53C2×C4⋊C411yescharacteristic
111C2×C4×C53C4×D412yes—
112C2×C4×C53C22⋊Q812yes—
113C2×C4×C53#12311yescharacteristic
114C2×Dic53—81no—
115C2×Dic53—82no—
116#88322—28no—
117#88322—42no—
118#8910—531nomaximal, Sylow
119C22×C53—42no—
120C22×C53—44no—
121C22×C53—81no—
122C22×C53—81no—
123C22×C53—82no—
124C22×C53#38511yescharacteristic
125C22×C53#45311yescharacteristic
126C22×C53#45312yes—
127C22×C53#45511yescharacteristic
128C22×C53#45612yes—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—42no—
134C4×C53—82no—
135C4×C53—82no—
136Dic53—161no—
137#2379D5311yescharacteristic
138#1959—531no—
139#1847—532no—
140C2×#329—531no—
141#1295—532no—
142C2×C53—82no—
143C2×C53#165511yescharacteristic
144C2×C53#166011yescharacteristic
145C2×C53#166112yes—
146C2×C53#185611yescharacteristic
147C2×C53#185912yes—
148C2×#124C2×D5311yescharacteristic
149C2×#107—1062no—
150C2×#106—1062no—
151C2×#103—1061no—
152C2×#102—1061no—
153C2×#97—532no—
154C2×#97—531no—
155#391—21no—
156#388—22no—
157#386—21no—
158#385—22no—
159#384—531no—
160#382C2×D5312yes—
161#329—534no—
162C53#891011yescharacteristic, complemented, Sylow
163C22×Q8C22⋊D5311yescharacteristic
164C22×D4C22⋊D5311yescharacteristic
165C23×C4—21no—
166C22×C8—1062no—
167C22×C8—1061no—
168#124—22no—
169#124Dic10612yes—
170#124C4×D5312yes—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—1061no—
173C2×C4⋊C4—1062no—
174C2×C4⋊C4C22⋊D5312yes—
175C2×C4⋊C4C22⋊D5312yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C42—21no—
180C2×C42C22×D5311yescharacteristic
181#107—2124no—
182#106—2124no—
183#103—2122no—
184#102—2122no—
185#97—534no—
186#97—1062no—
187#97—1062no—
188#95—22no—
189#95—22no—
190#95—22no—
191C24—21no—
192C2×Q8—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8#1171991912yes—
196C2×D4—21no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4#1171991912yes—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—82no—
204C22×C4—81no—
205C22×C4—2121no—
206C22×C4C2×C22⋊D5311yescharacteristic
207C22×C4C2×C22⋊D5312yes—
208C2×C8—1064no—
209C2×C8—1064no—
210C2×C8—1064no—
211C2×C8—1062no—
212C2×C8—2121no—
213C2×C8—2121no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—81no—
217C4⋊C4—2121no—
218C4⋊C4—2121no—
219C4⋊C4—2122no—
220C4⋊C4—2122no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—82no—
224C22⋊C4—81no—
225C42—81no—
226C42C2×Dic10611yescharacteristic
227C42C2×C4×D5311yescharacteristic
228C42#1171994512yes—
229C23—22no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23#1313318611yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—42no—
238D4—81no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—4241no—
254C2×C4—4242no—
255C2×C4C2×#1171991911yescharacteristic
256C2×C4C4×C22⋊D5312yes—
257C2×C4#1313319612yes—
258C2×C4#1313320011yescharacteristic
259C8—1068no—
260C8—2122no—
261C22—42no—
262C22—44no—
263C22—81no—
264C22—81no—
265C22—82no—
266C22#1681049611yescharacteristic
267C22#1681079111yescharacteristic
268C22#1681079112yes—
269C22#1681080111yescharacteristic
270C22#1681080612yes—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—82no—
277C4—82no—
278C4—8481no—
279C2—82no—
280C2#2027400411yescharacteristic
281C2#2027402611yescharacteristic
282C2#2027403212yes—
283C2#2027492511yescharacteristic
284C2#2027494112yes—
subgroupG/H
centerC23#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2379C2
socleC23×C53#120
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2379D53
hypocenterC53#8910
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2379

Complements

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

NH ≅ G/Ntype
2. #20274884C2semidirect
4. #20272596C2semidirect
13. #16810363C4semidirect
30. #13133128C2×C4semidirect
162. C53#8910semidirect

Maximal subgroups

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

subgroupindex
#202755322
#202748842
C2×#168103632
#202725962
C53×#23792
#891053