Finite Group Databasebeta

← #28305223

Subgroups — #28305223

Order 13568.

432 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 82 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
2#20290244C212yesmaximal
3#20289978C211yesmaximal, characteristic
4#20289566C212yesmaximal
5#20279932C211yesmaximal, characteristic
6#16811674—24no—
7#16811643—21no—
8#16811643C2212yes—
9#16811643C2212yes—
10#16811639—21no—
11#16811482—24no—
12#16811468—21no—
13#16811468—21no—
14C2×#13133216C2211yescharacteristic
15C2×Q8×Dic53C2211yescharacteristic
16#16810565—21no—
17#16810565—21no—
18#16810530C411yescharacteristic
19#16810527—21no—
20#16810527—21no—
21#16810489C411yescharacteristic
22#16810488C411yescharacteristic
23C4×#11719919C2211yescharacteristic
24C53×#391C411yescharacteristic
25#13133284—22no—
26#13133284—22no—
27#13133284D414yes—
28C4⋊#88324—22no—
29C22⋊#88324—22no—
30C22⋊#88324—22no—
31#13133272—22no—
32#13133272—22no—
33#13133272—22no—
34C4×#88324—22no—
35C22×Dic106—21no—
36#13133216—22no—
37#13133216—22no—
38Q8×Dic53—22no—
39Q8×Dic53—22no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×C4⋊Dic53D411yescharacteristic
43C2×C4⋊Dic53D411yescharacteristic
44C2×#11719919—21no—
45C2×#11719919—21no—
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C2×C4×Dic53C2311yescharacteristic
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C22×Q8×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C42×C53C2×C411yescharacteristic
60#95×C53—21no—
61#95×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—22no—
64C2×Q8×C53—41no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C22×Dic53C2×D411yescharacteristic
76C2×Dic106—44no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×#88324—24no—
80C2×#88324—42no—
81C4⋊C4×C53—22no—
82C42×C53—41no—
83C4⋊Dic53—22no—
84C4⋊Dic53—22no—
85C4⋊Dic53—22no—
86C4⋊Dic53—22no—
87#11719919—42no—
88#11719919—42no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C4×Dic53—21no—
93C4×Dic53—41no—
94C4×Dic53C2×D412yes—
95C4×Dic53#4112yes—
96C23×C53C2×C22⋊C411yescharacteristic
97Q8×C53—44no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53#10112yes—
111C2×C4×C53C4×D412yes—
112C2×Dic53—22no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53#12011yescharacteristic
124C2×Dic53#12112yes—
125C2×Dic53#12311yescharacteristic
126Dic106—82no—
127Dic106—82no—
128#88324—44no—
129#88324—82no—
130#9456—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53#38611yescharacteristic
134C22×C53C2×#10111yescharacteristic
135C22×C53#41311yescharacteristic
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—41no—
139C4×C53—81no—
140C4×C53—81no—
141Dic53—21no—
142Dic53—22no—
143Dic53—41no—
144Dic53—41no—
145Dic53—81no—
146Dic53—81no—
147Dic53#45412yes—
148#2983—531no—
149#2568—531no—
150#1556—532no—
151#1467—531no—
152#1285—532no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53#188811yescharacteristic
156C2×C53#190111yescharacteristic
157C2×C53#190911yescharacteristic
158C2×C4⋊Q8—531no—
159C2×C4×Q8—531no—
160#446—1064no—
161#432—532no—
162#432—532no—
163#432—1061no—
164#431—1061no—
165#400—1061no—
166#400—1061no—
167#391—531no—
168#391—1061no—
169#391—1061no—
170#391#615711yescharacteristic
171#382—531no—
172#382—531no—
173C4×C4⋊C4—531no—
174#332—1064no—
175#324—1061no—
176#324—1061no—
177C53#945611yescharacteristic, complemented, Sylow
178C22×Q8—1061no—
179C22×Q8C2×#615711yescharacteristic
180C4⋊Q8—1062no—
181C4⋊Q8—1062no—
182C4×Q8—1062no—
183C4×Q8—1062no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—531no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4C2×#615711yescharacteristic
191C2×C42—531no—
192C2×C42—531no—
193C2×C42—531no—
194C2×C42—531no—
195C2×C42C2×#615711yescharacteristic
196C4⋊C8—534no—
197C4⋊C8—1062no—
198C4⋊C8—1062no—
199C4⋊C8—1062no—
200C22⋊C8—1062no—
201C22⋊C8—1062no—
202#97—1062no—
203#97—1062no—
204#97—1062no—
205C4×C8—1062no—
206#95—21no—
207#95—21no—
208#95—1061no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—2121no—
219C2×Q8—2121no—
220C2×Q8—2124no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4C22⋊#615711yescharacteristic
230C22×C4C22⋊#615711yescharacteristic
231C22×C4C22×#615711yescharacteristic
232C2×C8—1064no—
233C2×C8—2122no—
234C4⋊C4—22no—
235C4⋊C4—1062no—
236C4⋊C4—1062no—
237C4⋊C4—1062no—
238C4⋊C4—1062no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C42—41no—
242C42—532no—
243C42—532no—
244C42—1061no—
245C42—1061no—
246C42—1061no—
247C42—1061no—
248C42—2121no—
249C23C2×C22⋊#615711yescharacteristic
250Q8—44no—
251Q8—4242no—
252Q8—4242no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—531no—
266C2×C4—532no—
267C2×C4—531no—
268C2×C4—1062no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2122no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4#1313327812yes—
280C2×C4D4×#615712yes—
281C8—4242no—
282C8—2124no—
283C22—21no—
284C22—21no—
285C22#1681157111yescharacteristic
286C22C2×#1313327811yescharacteristic
287C22#1681161011yescharacteristic
288C4—22no—
289C4—22no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—532no—
294C4—1061no—
295C4—1062no—
296C4—4241no—
297C4—4241no—
298C4—2121no—
299C4—2121no—
300C2—21no—
301C2—21no—
302C2#2029072311yescharacteristic
303C2#2029074211yescharacteristic
304C2#2029075711yescharacteristic
subgroupG/H
centerC22#16811571
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#9456
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
177. C53#9456semidirect

Maximal subgroups

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

subgroupindex
#202916112
#202902442
#202899782
#202895662
#202799322
#945653