Finite Group Databasebeta

← #28304368

Subgroups — #28304368

Order 13568.

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

normal subgroups only

This group has 522 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#20291611C211yesmaximal, characteristic, complemented
2#20290014C212yesmaximal
3#20289958C211yesmaximal, characteristic, complemented
4#20289539C212yesmaximal, complemented
5#20277428C211yesmaximal, characteristic
6#16811674—24no—
7#16811643—22no—
8#16811639C2212yes—
9#16811639C2212yes—
10D4⋊#88324—24no—
11#16811468—22no—
12C2×Q8×Dic53C2211yescharacteristic
13C2×D4×Dic53C2211yescharacteristic
14#16810530C411yescharacteristic
15#16810526C411yescharacteristic
16#16810516—22no—
17#16810498—22no—
18#16810493C411yescharacteristic
19#16810469C2211yescharacteristic
20C2×#124×C53C411yescharacteristic
21#13133284—24no—
22C4⋊#88324—22no—
23C4⋊#88324—22no—
24C22⋊#88324—22no—
25C22⋊#88324—22no—
26#13133272—24no—
27C4×#88324—22no—
28C4×#88324D414yes—
29C23×Dic53—21no—
30Q8×Dic53—22no—
31Q8×Dic53—22no—
32D4×Dic53—22no—
33D4×Dic53—22no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53D412yes—
38C2×C22⋊Dic53—22no—
39C2×C22⋊Dic53—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53D412yes—
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×C53—21no—
53C2×Q8×C53—22no—
54C2×Q8×C53—21no—
55C2×Q8×C53—21no—
56C2×Q8×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—21no—
59C2×D4×C53—22no—
60C2×D4×C53M1612yes—
61C22×C4×C53—21no—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—42no—
67C22×Dic53—82no—
68C22×Dic53C22⋊C412yes—
69C22×Dic53C2×D411yescharacteristic
70C22×Dic53C2×D411yescharacteristic
71C2×#88324—24no—
72C2×#88324—42no—
73C22⋊C4×C53—28no—
74C42×C53—21no—
75C42×C53M1612yes—
76C22⋊Dic53—41no—
77C22⋊Dic53—41no—
78C22⋊Dic53—82no—
79C4⋊Dic53—22no—
80C4⋊Dic53—24no—
81C4⋊Dic53—22no—
82C4⋊Dic53—82no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53C2×D412yes—
87C4×Dic53#4112yes—
88C23×C53—22no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53C2×C22⋊C411yescharacteristic
94Q8×C53—42no—
95Q8×C53—42no—
96D4×C53—24no—
97D4×C53—42no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53C4×D412yes—
105C2×C4×C53C2×M1611yescharacteristic
106C2×C4×C53C2×M1611yescharacteristic
107C2×C4×C53#13112yes—
108C2×Dic53—22no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—42no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—84no—
118C2×Dic53#12011yescharacteristic
119C2×Dic53#12112yes—
120C2×Dic53#12311yescharacteristic
121#88324—44no—
122#88324—82no—
123#9106—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—44no—
127C22×C53—42no—
128C22×C53—82no—
129C22×C53—81no—
130C22×C53—81no—
131C22×C53#38611yescharacteristic
132C22×C53#40711yescharacteristic
133C22×C53#40811yescharacteristic
134C4×C53—42no—
135C4×C53—42no—
136C4×C53#43512yes—
137C4×C53#43612yes—
138Dic53—22no—
139Dic53—21no—
140Dic53—21no—
141Dic53—42no—
142Dic53—81no—
143Dic53—81no—
144#2983—531no—
145#2326—531no—
146#1478—532no—
147#1461—531no—
148#1274—532no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—82no—
152C2×C53#187011yescharacteristic
153C2×C53#191011yescharacteristic
154C2×C53#191211yescharacteristic
155C2×#124#615711yescharacteristic
156C2×C4×Q8—531no—
157C2×C4×D4—531no—
158#446—1064no—
159#432—1062no—
160#431—532no—
161#431—532no—
162#391—531no—
163#390—531no—
164#388—1062no—
165#385—1062no—
166#384—531no—
167#376—531no—
168D4⋊C8—1064no—
169#324—1062no—
170C53#910611yescharacteristic, complemented, Sylow
171C22×Q8C2×#615711yescharacteristic
172C22×D4C2×#615711yescharacteristic
173C23×C4—1061no—
174#124—22no—
175#124—22no—
176C4×Q8—1062no—
177C4×Q8—1062no—
178C4×D4—1062no—
179C4×D4—1062no—
180C2×C4⋊C4—532no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—1062no—
184C2×C22⋊C4—22no—
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—
196#97—1064no—
197C4×C8—534no—
198C4×C8—1062no—
199#95—1061no—
200#95—1061no—
201#95—1062no—
202C24—21no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—22no—
210C2×D4#1171994012yes—
211C22×C4—21no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—532no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4—4242no—
218C22×C4—2122no—
219C22×C4C22⋊#615712yes—
220C22×C4C22×#615711yescharacteristic
221C2×C8—1064no—
222C2×C8—2122no—
223C4⋊C4—1064no—
224C4⋊C4—1062no—
225C4⋊C4—1062no—
226C4⋊C4—4242no—
227C22⋊C4—28no—
228C22⋊C4—4242no—
229C22⋊C4—2121no—
230C22⋊C4—2121no—
231C42—21no—
232C42—532no—
233C42—532no—
234C42—1062no—
235C42—1062no—
236C42—1062no—
237C42#1171994012yes—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23C2×C22⋊#615711yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—24no—
247D4—42no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—531no—
255C2×C4—532no—
256C2×C4—531no—
257C2×C4—1062no—
258C2×C4—4244no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4—2122no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2122no—
267C2×C4D4×#615712yes—
268C2×C4C2×#1171994011yescharacteristic
269C2×C4C2×#1171994011yescharacteristic
270C2×C4#1313330412yes—
271C8—4242no—
272C8—2124no—
273C22—21no—
274C22—21no—
275C22—44no—
276C22—42no—
277C22—82no—
278C22—81no—
279C22—81no—
280C22#1681157111yescharacteristic
281C22#1681159311yescharacteristic
282C22#1681159611yescharacteristic
283C4—42no—
284C4—42no—
285C4—1062no—
286C4—1061no—
287C4—1061no—
288C4—4241no—
289C4—4241no—
290C4—2122no—
291C4#1681165412yes—
292C4#1681165812yes—
293C2—21no—
294C2—21no—
295C2—82no—
296C2#2029069211yescharacteristic
297C2#2029075911yescharacteristic
298C2#2029076311yescharacteristic
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#9106
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
1. #20291611C2semidirect
3. #20289958C2semidirect
4. #20289539C2semidirect
170. C53#9106semidirect

Maximal subgroups

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

subgroupindex
#202916112
#202900142
#202899582
#202895392
#202774282
#910653