Finite Group Databasebeta

← #28304355

Subgroups — #28304355

Order 13568.

546 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 110 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#20291610C211yesmaximal, characteristic, complemented
2#20290016C212yesmaximal
3#20289959C211yesmaximal, characteristic, complemented
4#20289539C212yesmaximal, complemented
5#20277423C211yesmaximal, characteristic
6#16811672—24no—
7#16811640—22no—
8#16811639C2212yes—
9#16811639C2212yes—
10#16811471—24no—
11#16811469—22no—
12C2×#13133205C411yescharacteristic
13C2×#13133182C2211yescharacteristic
14C2×#13133179C2211yescharacteristic
15#16810528C411yescharacteristic
16#16810524C411yescharacteristic
17#16810513—22no—
18#16810494—22no—
19#16810469C2211yescharacteristic
20C53×#384C411yescharacteristic
21#13133284—24no—
22C4⋊#88324—22no—
23C4⋊#88324—22no—
24C22⋊#88324—22no—
25C22⋊#88324—22no—
26C4×#88324—24no—
27C4×#88324—22no—
28C4×#88324D414yes—
29C22×Dic106C2×C411yescharacteristic
30C22×D212C2×C411yescharacteristic
31C22×C4×D53—21no—
32#13133205—22no—
33#13133205C814yescomplemented
34#13133182—22no—
35#13133182—22no—
36#13133179—22no—
37#13133179—22no—
38C2×#11719919D412yes—
39C2×#11719921—22no—
40C2×#11719921—22no—
41C2×#11719921—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D412yes—
44C2×C4×Dic53C2311yescharacteristic
45#13133124—22no—
46#13133124—21no—
47#13133124—21no—
48C2×C4⋊C4×C53—22no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C23×D53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—21no—
58C22×Dic53C22⋊C412yes—
59C22×Dic53C2×D411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C2×D212—22no—
62C2×D212—22no—
63C2×D212—21no—
64C2×D212C2×C812yes—
65C2×C4×D53—42no—
66C2×C4×D53—82no—
67C2×Dic106—22no—
68C2×Dic106—22no—
69C2×Dic106C2×C812yes—
70C2×Dic106M1612yes—
71C2×#88324—24no—
72C2×#88324—42no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—82no—
76C42×C53—21no—
77C42×C53C2×C812yes—
78#11719921—28no—
79#11719921—41no—
80#11719921—41no—
81#11719921—82no—
82#11719919—24no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53C2×D412yes—
87C4×Dic53#4112yes—
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—22no—
90C22×D53—42no—
91C22×D53—42no—
92C22×D53—41no—
93C22×D53—41no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53C4×D412yes—
102C2×C4×C53C22×C811yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C2×C4×C53#13112yes—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53C22⋊C814yes—
113C2×Dic53#12011yescharacteristic
114C2×Dic53#12112yes—
115C2×Dic53#12311yescharacteristic
116D212—24no—
117D212—42no—
118C4×D53—84no—
119Dic106—24no—
120Dic106—24no—
121#88324—44no—
122#88324—82no—
123#9102—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53#38611yescharacteristic
127C22×C53C2×C22⋊C811yescharacteristic
128C22×C53#40811yescharacteristic
129C2×D53—44no—
130C2×D53—42no—
131C2×D53—82no—
132C2×D53—81no—
133C2×D53—81no—
134C4×C53—42no—
135C4×C53—42no—
136C4×C53—81no—
137C4×C53C8×D412yes—
138C4×C53#43512yes—
139Dic53—24no—
140Dic53—22no—
141Dic53—21no—
142Dic53—21no—
143Dic53—81no—
144#2982—531no—
145#2326—531no—
146#1478—532no—
147#1461—531no—
148#1274—532no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53#186911yescharacteristic
152C2×C53#191011yescharacteristic
153C2×C53#191211yescharacteristic
154D53—82no—
155C2×#124—531no—
156C2×C4×Q8—531no—
157C2×C4×D4—531no—
158C8×Q8—1064no—
159#431—532no—
160#431—532no—
161#431—1062no—
162#391—531no—
163#390—531no—
164#388—1062no—
165#385—1062no—
166#384#615711yescharacteristic
167#376—531no—
168D4⋊C8—1064no—
169#324—1062no—
170C53#910211yescharacteristic, complemented, Sylow
171C22×Q8—531no—
172C22×D4—531no—
173C23×C4—1061no—
174#124—534no—
175#124—1062no—
176C4×Q8—1062no—
177C4×Q8—1062no—
178C4×D4—1062no—
179C4×D4—1062no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4C2×#615711yescharacteristic
183C2×C4⋊C4C2×#615711yescharacteristic
184C2×C22⋊C4—1062no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1061no—
187C2×C42—531no—
188C2×C42—532no—
189C2×C42—531no—
190C2×C42C2×#615711yescharacteristic
191C4⋊C8—1062no—
192C4⋊C8—1062no—
193C4⋊C8—1064no—
194C22⋊C8—1062no—
195C22⋊C8—1062no—
196C4×C8—534no—
197C4×C8—1062no—
198C4×C8—1064no—
199#95—1061no—
200#95—1061no—
201#95—1062no—
202C24—1061no—
203C2×Q8—532no—
204C2×Q8—532no—
205C2×Q8—1062no—
206C2×Q8—1062no—
207C2×D4—532no—
208C2×D4—1062no—
209C2×D4—1061no—
210C2×D4—1062no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—532no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—1061no—
217C22×C4—4242no—
218C22×C4—2122no—
219C22×C4C22⋊#615712yes—
220C22×C4C22×#615711yescharacteristic
221C2×C8—1064no—
222C2×C8—2122no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—82no—
226C4⋊C4—1064no—
227C22⋊C4—1068no—
228C22⋊C4—4242no—
229C22⋊C4—2121no—
230C22⋊C4—2121no—
231C42—21no—
232C42—532no—
233C42—532no—
234C42—1062no—
235C42—1062no—
236C42—1062no—
237C42D53⋊C812yes—
238C23—1062no—
239C23—2122no—
240C23—2122no—
241C23—2121no—
242C23—2121no—
243C23C2×C22⋊#615711yescharacteristic
244Q8—1064no—
245Q8—1064no—
246D4—1064no—
247D4—2122no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—531no—
256C2×C4—532no—
257C2×C4—531no—
258C2×C4—534no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—4244no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4D4×#615712yes—
268C2×C4C2×D53⋊C811yescharacteristic
269C2×C4D53⋊M1611yescharacteristic
270C2×C4#1313330512yes—
271C8—4242no—
272C8—2124no—
273C22—21no—
274C22—21no—
275C22—4242no—
276C22—4241no—
277C22—4241no—
278C22—2124no—
279C22—2122no—
280C22#1681157111yescharacteristic
281C22D53⋊(C22⋊C8)11yescharacteristic
282C22#1681159811yescharacteristic
283C4—42no—
284C4—42no—
285C4—81no—
286C4—1064no—
287C4—1062no—
288C4—1061no—
289C4—1061no—
290C4—4241no—
291C4#1681165112yes—
292C4#1681165312yes—
293C2—21no—
294C2—21no—
295C2—4242no—
296C2#2029069111yescharacteristic
297C2#2029075811yescharacteristic
298C2#2029076211yescharacteristic
subgroupG/H
centerC22#16811571
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#9102
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
1. #20291610C2semidirect
3. #20289959C2semidirect
4. #20289539C2semidirect
33. #13133205C8semidirect
170. C53#9102semidirect

Maximal subgroups

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

subgroupindex
#202916102
#202900162
#202899592
#202895392
#202774232
#910253