Finite Group Databasebeta

← #28308842

Subgroups — #28308842

Order 13568.

452 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 88 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#20290420C211yesmaximal, characteristic
2#20289948C211yesmaximal, characteristic
3#20289699C214yesmaximal, complemented
4#20280482C211yesmaximal, characteristic
5C2×D53⋊M16—21no—
6#16811639—21no—
7#16811503—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811467—21no—
11C2×#13133193—21no—
12C2×#13133174C2211yescharacteristic
13#16810554—21no—
14#16810554C2212yes—
15#16810549—21no—
16#16810549C412yes—
17#16810530—21no—
18#16810489C411yescharacteristic
19C2×C4⋊Q8×C53C411yescharacteristic
20C2×#11719936—41no—
21D53⋊M16—24no—
22D53⋊M16—42no—
23C2×#11719940—22no—
24C2×D53⋊C8—41no—
25C4⋊#88324—41no—
26C22⋊#88324—22no—
27C4×#88324—41no—
28C22×C4×D53D411yescharacteristic
29#13133193—42no—
30#13133193—42no—
31#13133174—41no—
32#13133174—41no—
33C2×C4⋊Dic53—21no—
34C2×#11719919—22no—
35C2×#11719919D411yescharacteristic
36C2×#11719919C2311yescharacteristic
37C2×#11719921—22no—
38C2×#11719921C2×C411yescharacteristic
39C2×#11719921D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—21no—
42#13133124—22no—
43#13133124—22no—
44#13133124C2×C412yes—
45C22×Q8×C53—21no—
46C4⋊Q8×C53—41no—
47C4⋊Q8×C53—41no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53C2×C412yes—
50C2×C42×C53C2×C411yescharacteristic
51C23×D53C22⋊C411yescharacteristic
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—42no—
55C2×Q8×C53—42no—
56C22×C4×C53—22no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—22no—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C2×C4×D53—21no—
65C2×C4×D53—21no—
66C2×C4×D53—22no—
67C2×C4×D53D812yes—
68C2×C4×D53SD1612yes—
69#11719940—24no—
70#11719940—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73#11719936—44no—
74D53⋊C8—44no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C42×C53—21no—
78C42×C53—21no—
79#11719921—41no—
80#11719921—41no—
81#11719921—44no—
82C4⋊Dic53—42no—
83#11719919—41no—
84#11719919—41no—
85#11719919—42no—
86#11719919—42no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53#10212yes—
95Q8×C53—82no—
96Q8×C53—82no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#9911yescharacteristic
106C2×C4×C53#10212yes—
107C2×C4×C53#10412yes—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53C2×D811yescharacteristic
118C2×Dic53C2×SD1611yescharacteristic
119C4×D53—22no—
120C4×D53—22no—
121C4×D53—22no—
122C4×D53—22no—
123#88324—44no—
124#88324—82no—
125#11700—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53C2×#9911yescharacteristic
129C22×C53C2×#10211yescharacteristic
130C22×C53C2×#10411yescharacteristic
131C2×D53—22no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—42no—
135C2×D53—81no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53—82no—
139C4×C53#32712yes—
140C4×C53#32712yes—
141Dic53—21no—
142Dic53—21no—
143Dic53—41no—
144Dic53—41no—
145Dic53—82no—
146#2621—531no—
147#1742—531no—
148#1458—531no—
149#1346—534no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53C2×#32711yescharacteristic
153C2×C53#212311yescharacteristic
154C2×C53#213511yescharacteristic
155D53—42no—
156C22×M16—1061no—
157C2×C4⋊Q8#615711yescharacteristic
158C2×C22⋊Q8—1061no—
159C2×#117—531no—
160#431—1061no—
161#398—532no—
162#398—1061no—
163#397—532no—
164#397—1061no—
165#391—1061no—
166#382—531no—
167#343—532no—
168#343—1061no—
169#324—532no—
170#323—1061no—
171C53#1170011yescharacteristic, complemented, Sylow
172C22×Q8—21no—
173C23×C4—531no—
174C2×M16—1062no—
175C2×M16—1064no—
176C2×M16—2122no—
177C2×M16—2121no—
178C22×C8—2121no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C22⋊Q8—2122no—
182C22⋊Q8—2122no—
183#117—2121no—
184#117—2121no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—531no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—1062no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4C2×#615712yes—
191C2×C22⋊C4—531no—
192C2×C22⋊C4—531no—
193C2×C22⋊C4—1062no—
194C2×C42—531no—
195C2×C42C2×#615711yescharacteristic
196C4⋊C8—2121no—
197C22⋊C8—1062no—
198C4×C8—2121no—
199#95—532no—
200#95—1062no—
201#95—1062no—
202#95—1061no—
203C24—531no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—42no—
208C22×C4—22no—
209C22×C4—531no—
210C22×C4—532no—
211C22×C4—532no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—1061no—
215C22×C4—1061no—
216C22×C4—1062no—
217C22×C4—1062no—
218C22×C4C22⋊#615711yescharacteristic
219C22×C4C22⋊#615711yescharacteristic
220C22×C4C22×#615711yescharacteristic
221M16—1064no—
222M16—2122no—
223M16—2124no—
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2124no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—2121no—
230C4⋊C4—2121no—
231C4⋊C4—2122no—
232C4⋊C4—2122no—
233C4⋊C4—2122no—
234C22⋊C4—2121no—
235C22⋊C4—2121no—
236C22⋊C4—2124no—
237C42—21no—
238C42—21no—
239C42—1061no—
240C42—1061no—
241C23—532no—
242C23—1061no—
243C23—1061no—
244C23—2121no—
245C23—2121no—
246C23C2×C22⋊#615711yescharacteristic
247Q8—82no—
248Q8—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—531no—
257C2×C4—531no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4—2122no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4#1313327511yescharacteristic
272C2×C4#1313327511yescharacteristic
273C2×C4#1313327912yes—
274C2×C4#1313328112yes—
275C8—4242no—
276C8—2124no—
277C22—21no—
278C22—21no—
279C22—1062no—
280C22—4241no—
281C22—2122no—
282C22—2121no—
283C22—2121no—
284C22C2×#1313327511yescharacteristic
285C22C2×#1313327911yescharacteristic
286C22C2×#1313328111yescharacteristic
287C4—41no—
288C4—41no—
289C4—82no—
290C4—1061no—
291C4—1061no—
292C4—4242no—
293C4—2121no—
294C4—2121no—
295C4#1681147412yes—
296C4#1681147412yes—
297C2—21no—
298C2—21no—
299C2—2122no—
300C2C2×#1681147411yescharacteristic
301C2#2029097711yescharacteristic
302C2#2029101711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC23×D53C22⋊C4
cosocleC2×#11719919C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11700
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289699C2semidirect
171. C53#11700semidirect

Maximal subgroups

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

subgroupindex
#202904202
#202899482
#202896992
#202804822
#1170053