Finite Group Databasebeta

← #27278471

Subgroups — #27278471

Order 13568.

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

normal subgroups only

This group has 468 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#20274848C212yesmaximal, complemented
3C2×#16810363C211yesmaximal, characteristic
4#20272596C212yesmaximal, complemented
5C53×#2379C211yesmaximal, characteristic
6C2×#13133141—21no—
7C2×#13133138—21no—
8C2×#13133128C2211yescharacteristic
9C2×#13133128C2212yes—
10#16810493C2211yescharacteristic
11#16810385—22no—
12#16810385—22no—
13#16810363C414yescomplemented
14C2×#124×C53C2211yescharacteristic
15C53×#391—21no—
16C53×#388—22no—
17C53×#386—21no—
18C53×#385—22no—
19C53×#382C2212yes—
20C22×#88322—21no—
21C22×#88322—22no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—22no—
24#13133141—42no—
25#13133138—42no—
26#13133128—22no—
27#13133128—22no—
28#13133128C2×C414yescomplemented
29C22×Q8×C53D411yescharacteristic
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32#124×C53—22no—
33#124×C53C2×C412yes—
34#124×C53Q812yes—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53D412yes—
37C2×C4⋊C4×C53D412yes—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—22no—
41C2×C42×C53—21no—
42C2×C42×C53C2311yescharacteristic
43#95×C53—22no—
44#95×C53—22no—
45#95×C53—22no—
46C24×C53—21no—
47C2×Q8×C53—21no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53C4⋊C412yes—
51C2×D4×C53—21no—
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53C4⋊C412yes—
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—81no—
59C22×C4×C53—82no—
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D412yes—
62C22×Dic53—41no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—81no—
66C22⋊C4×C53—24no—
67C22⋊C4×C53—24no—
68C22⋊C4×C53—82no—
69C22⋊C4×C53—81no—
70C42×C53—81no—
71C42×C53C22×C411yescharacteristic
72C42×C53C2×Q811yescharacteristic
73C42×C53#4112yes—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76C4⋊Dic53—44no—
77C2×#88322—22no—
78C2×#88322—24no—
79C2×#88322—42no—
80C2×#88322—41no—
81C2×#88322—42no—
82C2×#88322—41no—
83C23×C53—22no—
84C23×C53—41no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—41no—
88C23×C53#12011yescharacteristic
89Q8×C53—42no—
90Q8×C53—81no—
91D4×C53—42no—
92D4×C53—81no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—84no—
107C2×C4×C53C2×C4⋊C411yescharacteristic
108C2×C4×C53C4×D412yes—
109C2×C4×C53C22⋊Q812yes—
110C2×C4×C53#12311yescharacteristic
111C2×Dic53—81no—
112C2×Dic53—82no—
113#88322—42no—
114#88322—44no—
115#8914—531nomaximal, Sylow
116C22×C53—42no—
117C22×C53—44no—
118C22×C53—81no—
119C22×C53—81no—
120C22×C53—82no—
121C22×C53#38511yescharacteristic
122C22×C53#45311yescharacteristic
123C22×C53#45412yes—
124C22×C53#45511yescharacteristic
125C22×C53#45512yes—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—42no—
131C4×C53—82no—
132C4×C53—82no—
133Dic53—161no—
134#2379D5311yescharacteristic
135#1959—531no—
136#1836—532no—
137C2×#329—531no—
138#1295—532no—
139C2×C53—82no—
140C2×C53#165711yescharacteristic
141C2×C53#165911yescharacteristic
142C2×C53#166212yes—
143C2×C53#185611yescharacteristic
144C2×C53#186012yes—
145C2×#124C2×D5311yescharacteristic
146C2×#103—1061no—
147C2×#102—1061no—
148C2×#97—532no—
149C2×#97—531no—
150#391—21no—
151#388—22no—
152#386—21no—
153#385—22no—
154#384—531no—
155#382C2×D5312yes—
156#340—1062no—
157#340—1062no—
158#329—534no—
159C53#891411yescharacteristic, complemented, Sylow
160C22×Q8C22⋊D5311yescharacteristic
161C22×D4C22⋊D5311yescharacteristic
162C23×C4—21no—
163C22×C8—1062no—
164C22×C8—1061no—
165#124—22no—
166#124Dic10612yes—
167#124C4×D5312yes—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—1062no—
170C2×C4⋊C4—1061no—
171C2×C4⋊C4C22⋊D5312yes—
172C2×C4⋊C4C22⋊D5312yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C42—21no—
177C2×C42C22×D5311yescharacteristic
178#103—2122no—
179#102—2122no—
180#97—534no—
181#97—1062no—
182#97—1062no—
183#95—22no—
184#95—22no—
185#95—22no—
186C24—21no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8#1171991912yes—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4#1171991912yes—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—81no—
199C22×C4—82no—
200C22×C4—2121no—
201C22×C4C2×C22⋊D5311yescharacteristic
202C22×C4C2×C22⋊D5312yes—
203C2×C8—1062no—
204C2×C8—1064no—
205C2×C8—2122no—
206C2×C8—2122no—
207C2×C8—2121no—
208C2×C8—2121no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C4⋊C4—2124no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—81no—
218C22⋊C4—82no—
219C42—81no—
220C42C2×Dic10611yescharacteristic
221C42C2×C4×D5311yescharacteristic
222C42#1171994512yes—
223C23—22no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23#1313318611yescharacteristic
229Q8—42no—
230Q8—81no—
231D4—42no—
232D4—81no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4—4241no—
248C2×C4—4242no—
249C2×C4C2×#1171991911yescharacteristic
250C2×C4C4×C22⋊D5312yes—
251C2×C4#1313319612yes—
252C2×C4#1313320011yescharacteristic
253C8—2124no—
254C8—2122no—
255C22—44no—
256C22—42no—
257C22—81no—
258C22—81no—
259C22—82no—
260C22#1681049611yescharacteristic
261C22#1681079111yescharacteristic
262C22#1681079612yes—
263C22#1681080112yes—
264C22#1681080111yescharacteristic
265C4—22no—
266C4—22no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—82no—
271C4—82no—
272C4—8481no—
273C2—82no—
274C2#2027401311yescharacteristic
275C2#2027402111yescharacteristic
276C2#2027403812yes—
277C2#2027492511yescharacteristic
278C2#2027494812yes—
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#8914
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. #20274848C2semidirect
4. #20272596C2semidirect
13. #16810363C4semidirect
28. #13133128C2×C4semidirect
159. C53#8914semidirect

Maximal subgroups

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

subgroupindex
#202755322
#202748482
C2×#168103632
#202725962
C53×#23792
#891453