Finite Group Databasebeta

← #28303980

Subgroups — #28303980

Order 13568.

558 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 102 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#20289543C212yesmaximal, complemented
5#124×Dic53C211yesmaximal, 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#16810489C411yescharacteristic
18C22⋊C4×Dic53—22no—
19C42×Dic53C2211yescharacteristic
20C2×#124×C53C411yescharacteristic
21C2×#11719940—24no—
22C4⋊#88324—22no—
23C4⋊#88324D414yes—
24C22⋊#88324—24no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27C4×#88324—22no—
28C4×#88324—22no—
29C23×Dic53D412yes—
30Q8×Dic53—22no—
31Q8×Dic53—22no—
32D4×Dic53—22no—
33D4×Dic53—22no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53D412yes—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2311yescharacteristic
44#13133124—22no—
45#13133124—22no—
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—21no—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—21no—
68C22×Dic53—44no—
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—44no—
80C4⋊Dic53—22no—
81C4⋊Dic53—22no—
82C4⋊Dic53—42no—
83C4×Dic53—22no—
84C4×Dic53—42no—
85C4×Dic53—42no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C4×Dic53C2×D412yes—
89C4×Dic53#4112yes—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53C2×C22⋊C411yescharacteristic
96Q8×C53—24no—
97Q8×C53—42no—
98D4×C53—42no—
99D4×C53—42no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53C4×D412yes—
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C2×C4×C53#13112yes—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—44no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—41no—
120C2×Dic53—82no—
121C2×Dic53#12011yescharacteristic
122C2×Dic53#12112yes—
123C2×Dic53#12311yescharacteristic
124#88324—44no—
125#88324—82no—
126#8961—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—44no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—82no—
134C22×C53#38611yescharacteristic
135C22×C53#40711yescharacteristic
136C22×C53#40811yescharacteristic
137C4×C53—42no—
138C4×C53—42no—
139C4×C53#43512yes—
140C4×C53#43612yes—
141Dic53—22no—
142Dic53—21no—
143Dic53—42no—
144Dic53—42no—
145Dic53—81no—
146Dic53#45512yes—
147#2975—531no—
148C4×#124—531no—
149#1479—532no—
150#1460—531no—
151#1276—532no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53—82no—
155C2×C53#187011yescharacteristic
156C2×C53#188411yescharacteristic
157C2×C53#188511yescharacteristic
158C2×#124#615711yescharacteristic
159C2×C4×Q8—531no—
160C2×C4×D4—531no—
161#436—1064no—
162#431—532no—
163#431—532no—
164#407—1062no—
165#391—531no—
166#390—531no—
167#388—1062no—
168#382—531no—
169C4×C22⋊C4—1062no—
170C43—531no—
171Q8⋊C8—1064no—
172#323—1062no—
173C53#896111yescharacteristic, complemented, Sylow
174C22×Q8C2×#615711yescharacteristic
175C22×D4C2×#615711yescharacteristic
176C23×C4—532no—
177C2×M16—1064no—
178#124—22no—
179#124—22no—
180C4×Q8—1062no—
181C4×Q8—1062no—
182C4×D4—1062no—
183C4×D4—1062no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
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×C42—1061no—
194C2×C42—1061no—
195C2×C42C2×#615711yescharacteristic
196C4⋊C8—534no—
197C4⋊C8—1062no—
198C22⋊C8—1062no—
199C22⋊C8—1064no—
200C22⋊C8—1062no—
201C4×C8—1062no—
202C4×C8—1062no—
203#95—1062no—
204#95—1062no—
205C24C22⋊#615712yes—
206C2×Q8—21no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8#1171994012yescomplemented
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—532no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4—2124no—
224C22×C4C22×#615711yescharacteristic
225M16—4244no—
226C2×C8—1064no—
227C2×C8—2122no—
228C4⋊C4—1062no—
229C4⋊C4—1062no—
230C4⋊C4—2122no—
231C22⋊C4—28no—
232C22⋊C4—1064no—
233C22⋊C4—2124no—
234C42—21no—
235C42—532no—
236C42—532no—
237C42—1062no—
238C42—2121no—
239C42—2121no—
240C42—2122no—
241C42—2122no—
242C42#1171994012yescomplemented
243C23—22no—
244C23—22no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23C2×C22⋊#615711yescharacteristic
249Q8—24no—
250Q8—42no—
251D4—42no—
252D4—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—531no—
260C2×C4—532no—
261C2×C4—531no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—4242no—
266C2×C4—2124no—
267C2×C4—2121no—
268C2×C4—2122no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2122no—
273C2×C4D4×#615712yes—
274C2×C4C2×#1171994011yescharacteristic
275C2×C4C2×#1171994011yescharacteristic
276C2×C4#1313330412yes—
277C8—4242no—
278C8—2124no—
279C22—21no—
280C22—21no—
281C22—44no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—82no—
286C22#1681157111yescharacteristic
287C22#1681159311yescharacteristic
288C22#1681159611yescharacteristic
289C4—42no—
290C4—42no—
291C4—532no—
292C4—1062no—
293C4—1061no—
294C4—4241no—
295C4—2122no—
296C4—2122no—
297C4#1681165412yes—
298C4#1681165812yes—
299C2—21no—
300C2—21no—
301C2—82no—
302C2#2029069211yescharacteristic
303C2#2029071611yescharacteristic
304C2#2029072011yescharacteristic
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#8961
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. #20289543C2semidirect
56. C2×Q8×C53M16semidirect
77. C42×C53M16semidirect
173. C53#8961semidirect
209. C2×Q8#11719940semidirect
242. 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
#202895432
#124×Dic532
#896153