Finite Group Databasebeta

← #27300098

Subgroups — #27300098

Order 13568.

480 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 93 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#20277429C211yesmaximal, characteristic
2C2×#16810419C212yesmaximal
3#20274709C212yesmaximal, complemented
4#20272842C211yesmaximal, characteristic, complemented
5C53×#1394C211yesmaximal, characteristic, complemented
6C2×#13133206C2211yescharacteristic
7C2×Q8×Dic53C2211yescharacteristic
8C2×#13133179C411yescharacteristic
9#16810646—22no—
10C2×#13133136C2212yes—
11#16810525C411yescharacteristic
12#16810512—22no—
13#16810497—22no—
14#16810492C411yescharacteristic
15#16810469C411yescharacteristic
16#16810419—24no—
17#16810391—22no—
18C2×#101×C53C2212yes—
19C53×#391C2211yescharacteristic
20C22×D212C2×C411yescharacteristic
21C22×C4×D53—21no—
22C2×#11719917—22no—
23#13133206—22no—
24#13133206D412yes—
25#13133206Q812yes—
26Q8×Dic53—42no—
27#13133179—42no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53D412yes—
30C2×#11719919—22no—
31C2×#11719921—22no—
32C2×#11719921—21no—
33C2×#11719921—22no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37#13133146—44no—
38#13133136—22no—
39#13133136—22no—
40#13133124—21no—
41#13133124—21no—
42C22×Q8×C53C2311yescharacteristic
43C2×M16×C53—22no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46#101×C53—22no—
47#101×C53—22no—
48#95×C53—22no—
49C23×D53—21no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53C2×D411yescharacteristic
53C2×Q8×C53C2×Q811yescharacteristic
54C2×Q8×C53#4112yes—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22×C411yescharacteristic
58C22×C4×C53C2×D412yes—
59C22×Dic53—21no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C412yes—
62C22×Dic53C22⋊C411yescharacteristic
63C2×D212—21no—
64C2×D212—41no—
65C2×D212—41no—
66C2×D212C4⋊C412yes—
67C2×C4×D53—42no—
68C2×C4×D53—82no—
69M16×C53—42no—
70M16×C53—42no—
71C2×C8×C53—42no—
72C4⋊C4×C53—41no—
73C42×C53—41no—
74#11719921—24no—
75#11719921—24no—
76#11719921—42no—
77#11719921—82no—
78C4⋊Dic53—41no—
79C4⋊Dic53—42no—
80#11719919—82no—
81C4×Dic53—21no—
82C4×Dic53—24no—
83C4×Dic53—22no—
84C4×Dic53C4⋊C412yes—
85#11719917—42no—
86#11719917—42no—
87C2×#88322—42no—
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—22no—
90C22×D53—42no—
91C22×D53—42no—
92C22×D53—41no—
93C22×D53—41no—
94Q8×C53—41no—
95Q8×C53—41no—
96Q8×C53—42no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—42no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53C2×C4⋊C411yescharacteristic
107C2×C4×C53#11711yescharacteristic
108C2×C4×C53#12112yes—
109C2×C4×C53C22⋊Q812yes—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—22no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—42no—
118C2×Dic53#9912yes—
119D212—42no—
120D212—42no—
121C4×D53—84no—
122C8×C53—82no—
123#88322—82no—
124#12803—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53#38011yescharacteristic
128C22×C53C2×#9911yescharacteristic
129C22×C53#41011yescharacteristic
130C2×D53—44no—
131C2×D53—42no—
132C2×D53—82no—
133C2×D53—81no—
134C2×D53—81no—
135C4×C53—22no—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—22no—
139C4×C53—81no—
140C4×C53—81no—
141Dic53—22no—
142Dic53—42no—
143Dic53—41no—
144Dic53—42no—
145#2326—531no—
146C2×#355—532no—
147#1801—532no—
148#1394—531no—
149#1394D5311yescharacteristic, complemented
150C2×C53—21no—
151C2×C53—21no—
152C2×C53#179911yescharacteristic
153C2×C53#213311yescharacteristic
154C2×C53#213411yescharacteristic
155D53—82no—
156C2×#124—531no—
157C2×C4×Q8—531no—
158C2×C4×D4—531no—
159#423—1062no—
160C2×#101—532no—
161C2×#101C2×D5312yes—
162#391C2×D5311yescharacteristic
163#390—531no—
164#388—1062no—
165#385—1062no—
166#384—531no—
167#376—531no—
168#355—1064no—
169#343—1062no—
170C53#1280311yescharacteristic, complemented, Sylow
171C22×Q8C22×D5311yescharacteristic
172C22×D4—531no—
173C23×C4—1061no—
174C2×M16—22no—
175C2×M16—1062no—
176#124—532no—
177#124—532no—
178#124—1062no—
179C4×Q8—2122no—
180C4×D4—2122no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—532no—
183C2×C4⋊C4—1062no—
184C2×C4⋊C4C4×D5311yescharacteristic
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1062no—
187C2×C22⋊C4—1061no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—532no—
191C2×C42C4×D5311yescharacteristic
192C4⋊C8—2124no—
193#101—22no—
194#101—22no—
195#101—1062no—
196#101—1062no—
197#95—22no—
198#95—1061no—
199#95—1061no—
200C24—1061no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8#1171994512yes—
204C2×Q8D4×D5311yescharacteristic
205C2×Q8Q8×D5311yescharacteristic
206C2×D4—532no—
207C2×D4—1061no—
208C2×D4—2121no—
209C2×D4—2121no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—532no—
215C22×C4—1061no—
216C22×C4—4242no—
217C22×C4—2122no—
218C22×C4C2×C4×D5311yescharacteristic
219C22×C4D4×D5312yes—
220M16—42no—
221M16—42no—
222M16—2122no—
223M16—2122no—
224C2×C8—42no—
225C2×C8—2122no—
226C4⋊C4—41no—
227C4⋊C4—4242no—
228C4⋊C4—2121no—
229C4⋊C4—2122no—
230C22⋊C4—1064no—
231C22⋊C4—1064no—
232C22⋊C4—4242no—
233C22⋊C4—2122no—
234C42—41no—
235C42—532no—
236C42—1064no—
237C42—1062no—
238C42—1061no—
239C23—1062no—
240C23—2122no—
241C23—2122no—
242C23—2121no—
243C23—2121no—
244C23C22⋊C4×D5311yescharacteristic
245Q8—42no—
246Q8—41no—
247Q8—41no—
248D4—2122no—
249D4—2122no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—532no—
260C2×C4—1062no—
261C2×C4—1061no—
262C2×C4—1062no—
263C2×C4—4244no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2122no—
268C2×C4—2121no—
269C2×C4C4⋊C4×D5311yescharacteristic
270C2×C4#1313317411yescharacteristic
271C2×C4#1313319012yes—
272C2×C4#1313319512yes—
273C8—82no—
274C8—4242no—
275C22—21no—
276C22—21no—
277C22—4242no—
278C22—4241no—
279C22—4241no—
280C22—2124no—
281C22—2122no—
282C22#1681048211yescharacteristic
283C22#99×D5311yescharacteristic
284C22#1681059511yescharacteristic
285C4—22no—
286C4—21no—
287C4—21no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—1062no—
292C4—2122no—
293C4—2121no—
294C4—2122no—
295C2—21no—
296C2—21no—
297C2—4242no—
298C2#2027470211yescharacteristic
299C2#2027640311yescharacteristic
300C2#2027640511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#1394C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC22×Q8×C53C23
hypercenter#1394D53
hypocenterC53#12803
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #20274709C2semidirect
4. #20272842C2semidirect
5. C53×#1394C2semidirect
149. #1394D53semidirect
170. C53#12803semidirect

Maximal subgroups

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

subgroupindex
#202774292
C2×#168104192
#202747092
#202728422
C53×#13942
#1280353