Finite Group Databasebeta

← #28304356

Subgroups — #28304356

Order 13568.

538 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 100 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#20291577C211yesmaximal, characteristic
2#20290017C212yesmaximal, complemented
3#20289955C211yesmaximal, characteristic
4#20289539C212yesmaximal, complemented
5#20277428C211yesmaximal, characteristic
6#16811658—24no—
7#16811639C2212yes—
8#16811639C2212yes—
9#16811593—22no—
10Q8⋊#88324—24no—
11#16811466—22no—
12C2×Q8×Dic53C2211yescharacteristic
13C2×D4×Dic53C2211yescharacteristic
14#16810530C411yescharacteristic
15#16810526C411yescharacteristic
16#16810516—22no—
17#16810498—22no—
18#16810493C411yescharacteristic
19#16810469C2211yescharacteristic
20C2×#124×C53C411yescharacteristic
21C2×#11719940—24no—
22C4⋊#88324—22no—
23C4⋊#88324—22no—
24C22⋊#88324—24no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27C4×#88324—22no—
28C4×#88324D414yes—
29C23×Dic53D412yes—
30Q8×Dic53—22no—
31Q8×Dic53—22no—
32D4×Dic53—22no—
33D4×Dic53—22no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C22⋊Dic53—22no—
39C2×C22⋊Dic53D412yes—
40C2×C4×Dic53—21no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43#13133124—22no—
44#13133124—21no—
45#13133124—21no—
46C22×Q8×C53C2×C411yescharacteristic
47C22×D4×C53C2×C411yescharacteristic
48#124×C53—22no—
49#124×C53—22no—
50C2×C22⋊C4×C53—22no—
51C2×C42×C53C2×C411yescharacteristic
52C24×C53C22⋊C412yes—
53C2×Q8×C53—21no—
54C2×Q8×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53M1612yescomplemented
57C2×D4×C53—22no—
58C2×D4×C53—21no—
59C2×D4×C53—21no—
60C2×D4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—82no—
69C22×Dic53C22⋊C412yes—
70C22×Dic53C2×D411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72#11719940—84no—
73C2×#88324—24no—
74C2×#88324—42no—
75C22⋊C4×C53—28no—
76C42×C53—21no—
77C42×C53M1612yescomplemented
78C22⋊Dic53—24no—
79C22⋊Dic53—82no—
80C4⋊Dic53—22no—
81C4⋊Dic53—22no—
82C4⋊Dic53—41no—
83C4⋊Dic53—41no—
84C4⋊Dic53—82no—
85C4×Dic53—22no—
86C4×Dic53—42no—
87C4×Dic53C2×D412yes—
88C4×Dic53#4112yes—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—24no—
96Q8×C53—42no—
97D4×C53—42no—
98D4×C53—42no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53C4×D412yes—
106C2×C4×C53C2×M1611yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53#13112yes—
109C2×Dic53—22no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—44no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—82no—
120C2×Dic53#12011yescharacteristic
121C2×Dic53#12112yes—
122C2×Dic53#12311yescharacteristic
123#88324—44no—
124#88324—82no—
125#9103—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—44no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—82no—
133C22×C53#38611yescharacteristic
134C22×C53#40711yescharacteristic
135C22×C53#40811yescharacteristic
136C4×C53—42no—
137C4×C53—42no—
138C4×C53#43512yes—
139C4×C53#43612yes—
140Dic53—22no—
141Dic53—21no—
142Dic53—21no—
143Dic53—42no—
144Dic53—81no—
145Dic53—81no—
146#2975—531no—
147#2326—531no—
148#1479—532no—
149#1460—531no—
150#1274—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—82no—
154C2×C53#187011yescharacteristic
155C2×C53#190911yescharacteristic
156C2×C53#191111yescharacteristic
157C2×#124#615711yescharacteristic
158C2×C4×Q8—531no—
159C2×C4×D4—531no—
160#436—1064no—
161#431—532no—
162#431—532no—
163#407—1062no—
164#391—531no—
165#390—531no—
166#388—1062no—
167#385—1062no—
168#384—531no—
169#376—531no—
170Q8⋊C8—1064no—
171#323—1062no—
172C53#910311yescharacteristic, complemented, Sylow
173C22×Q8C2×#615711yescharacteristic
174C22×D4C2×#615711yescharacteristic
175C23×C4—532no—
176C2×M16—1064no—
177#124—22no—
178#124—22no—
179C4×Q8—1062no—
180C4×Q8—1062no—
181C4×D4—1062no—
182C4×D4—1062no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—1062no—
186C2×C4⋊C4—1061no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—532no—
189C2×C22⋊C4—1062no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—1061no—
193C2×C42C2×#615711yescharacteristic
194C4⋊C8—1062no—
195C4⋊C8—1062no—
196C22⋊C8—1062no—
197C22⋊C8—1064no—
198C22⋊C8—1062no—
199C4×C8—534no—
200C4×C8—1062no—
201#95—1061no—
202#95—1061no—
203#95—1062no—
204C24C22⋊#615712yes—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8#1171994012yescomplemented
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—532no—
218C22×C4—1062no—
219C22×C4—1062no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4—4242no—
223C22×C4C22×#615711yescharacteristic
224M16—4244no—
225C2×C8—1064no—
226C2×C8—2122no—
227C4⋊C4—1062no—
228C4⋊C4—1062no—
229C4⋊C4—4242no—
230C4⋊C4—2121no—
231C4⋊C4—2121no—
232C22⋊C4—28no—
233C22⋊C4—1064no—
234C22⋊C4—4242no—
235C42—21no—
236C42—532no—
237C42—532no—
238C42—1062no—
239C42—2122no—
240C42#1171994012yescomplemented
241C23—22no—
242C23—22no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23C2×C22⋊#615711yescharacteristic
247Q8—24no—
248Q8—42no—
249D4—42no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—531no—
258C2×C4—532no—
259C2×C4—531no—
260C2×C4—1062no—
261C2×C4—4242no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2124no—
265C2×C4—2121no—
266C2×C4—2122no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2122no—
271C2×C4D4×#615712yes—
272C2×C4C2×#1171994011yescharacteristic
273C2×C4C2×#1171994011yescharacteristic
274C2×C4#1313330412yes—
275C8—4242no—
276C8—2124no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22#1681157111yescharacteristic
285C22#1681159311yescharacteristic
286C22#1681159611yescharacteristic
287C4—42no—
288C4—42no—
289C4—1062no—
290C4—1061no—
291C4—1061no—
292C4—4241no—
293C4—4241no—
294C4—2122no—
295C4#1681165412yes—
296C4#1681165812yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2029069211yescharacteristic
301C2#2029075611yescharacteristic
302C2#2029076011yescharacteristic
subgroupG/H
centerC22#16811571
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#124×C53C4
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×#124#6157
hypocenterC53#9103
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #20290017C2semidirect
4. #20289539C2semidirect
56. C2×Q8×C53M16semidirect
77. C42×C53M16semidirect
172. C53#9103semidirect
208. C2×Q8#11719940semidirect
240. C42#11719940semidirect

Maximal subgroups

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

subgroupindex
#202915772
#202900172
#202899552
#202895392
#202774282
#910353