Finite Group Databasebeta

← #27279590

Subgroups — #27279590

Order 13568.

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

normal subgroups only

This group has 498 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#20283796C211yesmaximal, characteristic
2#20277431C211yesmaximal, characteristic
3#20273109C212yesmaximal
4#20272543C212yesmaximal
5C53×#1461C211yesmaximal, characteristic
6C2×#13133215C411yescharacteristic
7C2×C4×Dic106C2211yescharacteristic
8Q8×#88322—24no—
9#16810676—22no—
10#16810676C2212yes—
11#16810530C411yescharacteristic
12#16810528C411yescharacteristic
13#16810507—22no—
14#16810504—22no—
15#16810488C411yescharacteristic
16#16810469C2211yescharacteristic
17#16810353—24no—
18C2×C4×Q8×C53C2211yescharacteristic
19C53×#431C2212yes—
20C53×#324—22no—
21C22×Dic106C2×C411yescharacteristic
22#13133215—22no—
23#13133215C814yescomplemented
24C4×Dic106—22no—
25C4×Dic106—22no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C4×Dic53—21no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32C4⋊#88322—24no—
33C4⋊#88322—22no—
34C22⋊#88322—22no—
35C4×#88322—24no—
36C4×#88322D414yes—
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40#13133124—21no—
41#13133124—21no—
42C22×Q8×C53C2×C411yescharacteristic
43C4×Q8×C53—22no—
44C4×Q8×C53—22no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53D412yes—
47C2×C42×C53D412yes—
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C4×C8×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—22no—
54C2×Q8×C53C2×C812yes—
55C2×Q8×C53M1612yes—
56C22×C4×C53C22⋊C412yes—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C2×Dic106—22no—
66C2×Dic106—22no—
67C2×Dic106—21no—
68C2×Dic106C2×C812yes—
69C2×C8×C53—42no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—24no—
72C42×C53—22no—
73C42×C53—22no—
74C42×C53C2×D412yes—
75C42×C53#4112yes—
76C4⋊Dic53—22no—
77#11719919—28no—
78#11719919—41no—
79#11719919—41no—
80C4×Dic53—21no—
81C4×Dic53—22no—
82C4×Dic53—42no—
83C4×Dic53C2×C812yes—
84C2×#88322—24no—
85C23×C53C2×C22⋊C411yescharacteristic
86Q8×C53—24no—
87Q8×C53—24no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53C22⋊C814yes—
95C2×C4×C53C4×D412yes—
96C2×C4×C53#12011yescharacteristic
97C2×C4×C53#12112yes—
98C2×C4×C53#12311yescharacteristic
99C2×C4×C53C22×C811yescharacteristic
100C2×C4×C53C2×M1611yescharacteristic
101C2×C4×C53#13112yes—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113Dic106—24no—
114Dic106—42no—
115C8×C53—82no—
116#88322—44no—
117#9100—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53#38611yescharacteristic
121C22×C53C2×C22⋊C811yescharacteristic
122C22×C53#40811yescharacteristic
123C4×C53—24no—
124C4×C53—22no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—42no—
128C4×C53C8×D412yes—
129C4×C53#43512yes—
130Dic53—42no—
131Dic53—42no—
132Dic53—81no—
133Dic53—81no—
134#2982—531no—
135#2327—531no—
136#1479—532no—
137#1461D5311yescharacteristic
138#1274—532no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#186911yescharacteristic
142C2×C53#190911yescharacteristic
143C2×C53#191211yescharacteristic
144C2×C4⋊Q8—531no—
145C2×C4×Q8—531no—
146C2×C4×Q8C2×D5311yescharacteristic
147C8×Q8—1064no—
148#431—532no—
149#431—1062no—
150#431C2×D5312yes—
151#391—531no—
152#391—531no—
153#387—1062no—
154#387—1062no—
155#382—531no—
156#376—531no—
157Q8⋊C8—1064no—
158#324—22no—
159C53#910011yescharacteristic, complemented, Sylow
160C22×Q8—531no—
161C22×Q8C4×D5311yescharacteristic
162C4⋊Q8—534no—
163C4⋊Q8—1062no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—1062no—
167C4×Q8—1062no—
168C2×C4⋊C4—531no—
169C2×C4⋊C4—1062no—
170C2×C4⋊C4—1061no—
171C2×C4⋊C4C4×D5311yescharacteristic
172C2×C4⋊C4C22⋊D5312yes—
173C2×C42—531no—
174C2×C42—531no—
175C2×C42—1061no—
176C2×C42D21212yes—
177C2×C42C22×D5311yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—1064no—
180C4⋊C8—1062no—
181C22⋊C8—22no—
182C22⋊C8—1062no—
183C4×C8—22no—
184C4×C8—534no—
185C4×C8—1064no—
186#95—1061no—
187#95—1061no—
188#95—1062no—
189#95—1062no—
190#95—1062no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—532no—
194C2×Q8—1061no—
195C2×Q8—1062no—
196C2×Q8—1062no—
197C2×Q8C8×D5312yes—
198C2×Q8#1171991212yes—
199C22×C4—1061no—
200C22×C4—1061no—
201C22×C4—1061no—
202C22×C4—1061no—
203C22×C4#1171992112yes—
204C22×C4#1171992112yes—
205C22×C4C2×C4×D5311yescharacteristic
206C22×C4C2×D21211yescharacteristic
207C22×C4C2×C22⋊D5311yescharacteristic
208C2×C8—42no—
209C2×C8—1064no—
210C4⋊C4—22no—
211C4⋊C4—24no—
212C4⋊C4—1068no—
213C4⋊C4—1062no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C42—22no—
217C42—22no—
218C42—532no—
219C42—1061no—
220C42—1062no—
221C42—2122no—
222C42D4×D5312yes—
223C42#1171994712yes—
224C23C2×#1171992111yescharacteristic
225Q8—24no—
226Q8—24no—
227Q8—1064no—
228Q8—2122no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—2121no—
238C2×C4—2121no—
239C2×C4—2122no—
240C2×C4—2121no—
241C2×C4—2121no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4#1313312914yes—
247C2×C4#1313317712yes—
248C2×C4#1313318711yescharacteristic
249C2×C4#1313318912yes—
250C2×C4#1313320111yescharacteristic
251C2×C4C2×C8×D5311yescharacteristic
252C2×C4C2×#1171991211yescharacteristic
253C2×C4#1313322512yes—
254C8—82no—
255C8—2124no—
256C22—21no—
257C22—21no—
258C22#1681050011yescharacteristic
259C22C2×#1313312911yescharacteristic
260C22#1681059111yescharacteristic
261C4—22no—
262C4—21no—
263C4—21no—
264C4—24no—
265C4—42no—
266C4—4241no—
267C4—4241no—
268C4—2122no—
269C4—2122no—
270C4#1681068612yes—
271C4#1681069112yes—
272C2—21no—
273C2—21no—
274C2#2027500511yescharacteristic
275C2#2027523511yescharacteristic
276C2#2027525011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1461C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1461D53
hypocenterC53#9100
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
23. #13133215C8semidirect
159. C53#9100semidirect

Maximal subgroups

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

subgroupindex
#202837962
#202774312
#202731092
#202725432
C53×#14612
#910053