Finite Group Databasebeta

← #27279585

Subgroups — #27279585

Order 13568.

566 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 135 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#20283739C211yesmaximal, characteristic
2#20277427C211yesmaximal, characteristic
3#20273109C212yesmaximal, complemented
4#20272543C212yesmaximal, complemented
5C53×#1460C211yesmaximal, characteristic
6C2×#13133204C411yescharacteristic
7C2×C4×Dic106C2211yescharacteristic
8D4×#88322—24no—
9#16810676C2212yes—
10C2×C22⋊#88322—22no—
11#16810528C411yescharacteristic
12#16810526C411yescharacteristic
13#16810511—22no—
14#16810496—22no—
15#16810492C411yescharacteristic
16#16810469C2211yescharacteristic
17#16810353—24no—
18C2×C4×D4×C53C2211yescharacteristic
19C53×#431C2212yes—
20C53×#323—22no—
21C22×Dic106C2×C411yescharacteristic
22C22×#88322—24no—
23#13133204—22no—
24#13133204C814yescomplemented
25C4×Dic106—22no—
26C4×Dic106—22no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—22no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C4⋊#88322—22no—
36C22⋊#88322—24no—
37C22⋊#88322—22no—
38C4×#88322D414yes—
39#13133124—22no—
40#13133124—21no—
41#13133124—21no—
42C22×D4×C53C2×C411yescharacteristic
43C23×C4×C53D412yes—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53D412yes—
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C4×C8×C53—22no—
52C24×C53C22⋊C412yes—
53C2×D4×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53C2×C812yes—
56C2×D4×C53M1612yes—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—82no—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C2×Dic106—22no—
69C2×Dic106—22no—
70C2×Dic106—21no—
71C2×Dic106C2×C812yescomplemented
72C2×C8×C53—42no—
73C4⋊C4×C53—22no—
74C22⋊C4×C53—24no—
75C42×C53C2×D412yes—
76C42×C53#4112yes—
77C22⋊Dic53—28no—
78C22⋊Dic53—82no—
79C4⋊Dic53—22no—
80#11719919—41no—
81#11719919—41no—
82#11719919—82no—
83C4×Dic53—21no—
84C4×Dic53—22no—
85C4×Dic53—42no—
86C4×Dic53C2×C812yescomplemented
87C2×#88322—24no—
88C2×#88322—84no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53C22⋊C814yes—
94C23×C53C2×C22⋊C411yescharacteristic
95D4×C53—24no—
96D4×C53—24no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—44no—
101C2×C4×C53—42no—
102C2×C4×C53—82no—
103C2×C4×C53C4×D412yes—
104C2×C4×C53#12011yescharacteristic
105C2×C4×C53#12112yes—
106C2×C4×C53#12311yescharacteristic
107C2×C4×C53C22×C811yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C2×C4×C53#13112yes—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—42no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121Dic106—24no—
122Dic106—42no—
123C8×C53—82no—
124#88322—44no—
125#9099—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—24no—
129C22×C53—44no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—82no—
133C22×C53#38611yescharacteristic
134C22×C53C2×C22⋊C811yescharacteristic
135C22×C53#40811yescharacteristic
136C4×C53—22no—
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—42no—
140C4×C53C8×D412yes—
141C4×C53#43512yes—
142Dic53—42no—
143Dic53—42no—
144Dic53—81no—
145Dic53—81no—
146#2972—531no—
147#2326—531no—
148#1479—532no—
149#1460D5311yescharacteristic
150#1274—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—44no—
154C2×C53#186911yescharacteristic
155C2×C53#190911yescharacteristic
156C2×C53#191111yescharacteristic
157C2×#124—531no—
158C2×C4×Q8—531no—
159C2×C4×D4C2×D5311yescharacteristic
160C8×D4—1064no—
161#431—532no—
162#431C2×D5312yes—
163C2×C22⋊C8—1062no—
164#391—531no—
165#390—531no—
166#388—1062no—
167#385—1062no—
168#384—531no—
169#376—531no—
170Q8⋊C8—1064no—
171#323—22no—
172C53#909911yescharacteristic, complemented, Sylow
173C22×Q8—531no—
174C22×D4C4×D5311yescharacteristic
175C23×C4D21212yes—
176C22×C8—1064no—
177#124—534no—
178#124—1062no—
179C4×Q8—1062no—
180C4×Q8—1062no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—1062no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4C4×D5311yescharacteristic
187C2×C22⋊C4—1062no—
188C2×C22⋊C4—1062no—
189C2×C22⋊C4C22⋊D5312yes—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—1061no—
193C2×C42C22×D5311yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—1062no—
196C22⋊C8—22no—
197C22⋊C8—1064no—
198C22⋊C8—1062no—
199C4×C8—22no—
200C4×C8—534no—
201#95—1061no—
202#95—1061no—
203#95—1062no—
204C24#1171992112yes—
205C2×Q8—532no—
206C2×Q8—1061no—
207C2×Q8—1062no—
208C2×Q8—1062no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4C8×D5312yes—
212C2×D4#1171991212yes—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—82no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4#1171992112yes—
221C22×C4C2×C4×D5311yescharacteristic
222C22×C4C2×D21211yescharacteristic
223C22×C4C2×C22⋊D5311yescharacteristic
224C2×C8—42no—
225C2×C8—1064no—
226C2×C8—4244no—
227C4⋊C4—22no—
228C4⋊C4—1062no—
229C4⋊C4—4242no—
230C4⋊C4—2121no—
231C4⋊C4—2121no—
232C22⋊C4—24no—
233C22⋊C4—1068no—
234C22⋊C4—4242no—
235C42—532no—
236C42—1061no—
237C42—1062no—
238C42—2122no—
239C42D4×D5312yes—
240C42#1171994712yes—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23#1313312914yes—
246C23C2×#1171992111yescharacteristic
247Q8—1064no—
248Q8—2122no—
249D4—24no—
250D4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—2121no—
260C2×C4—2122no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4#1313317712yes—
269C2×C4#1313318711yescharacteristic
270C2×C4#1313318912yes—
271C2×C4#1313320111yescharacteristic
272C2×C4C2×C8×D5311yescharacteristic
273C2×C4C2×#1171991211yescharacteristic
274C2×C4#1313322512yes—
275C8—82no—
276C8—2124no—
277C22—21no—
278C22—21no—
279C22—24no—
280C22—44no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22#1681050011yescharacteristic
285C22C2×#1313312911yescharacteristic
286C22#1681059111yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—4241no—
292C4—4241no—
293C4—2122no—
294C4—2122no—
295C4#1681068612yes—
296C4#1681069112yes—
297C2—21no—
298C2—21no—
299C2—44no—
300C2#2027500511yescharacteristic
301C2#2027523511yescharacteristic
302C2#2027524511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1460C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#1460D53
hypocenterC53#9099
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #20273109C2semidirect
4. #20272543C2semidirect
24. #13133204C8semidirect
71. C2×Dic106C2×C8semidirect
86. C4×Dic53C2×C8semidirect
172. C53#9099semidirect

Maximal subgroups

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

subgroupindex
#202837392
#202774272
#202731092
#202725432
C53×#14602
#909953