Finite Group Databasebeta

← #28307920

Subgroups — #28307920

Order 13568.

594 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 90 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
1D53⋊#324C212yesmaximal, complemented
2#20289646C212yesmaximal, complemented
3#20289643C212yesmaximal, complemented
4#20277305C211yesmaximal, characteristic
5D53⋊(C22⋊C8)—22no—
6#16811469—22no—
7#16811468C2214yes—
8C22×#11719947C411yescharacteristic
9C2×#13133174C411yescharacteristic
10C2×C4⋊C4×D53C2211yescharacteristic
11#16810529C2211yescharacteristic
12#16810528C411yescharacteristic
13#16810526C2211yescharacteristic
14#16810499—22no—
15#16810481—22no—
16C53×#390C411yescharacteristic
17C2×D53⋊C8—42no—
18#13133274—42no—
19C22⋊#88324—24no—
20C2×#11719947—22no—
21C2×#11719947—42no—
22C2×#11719947—42no—
23C2×#11719947C814yescomplemented
24C22×Dic106C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C23×Dic53—21no—
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D412yes—
29#13133174—42no—
30C2×C4⋊Dic53C2311yescharacteristic
31C2×#11719919—21no—
32C2×#11719919D412yes—
33C4⋊C4×D53—42no—
34C2×C22⋊Dic53—22no—
35C2×#11719921—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37#13133124—22no—
38#13133124—22no—
39C22×D4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2×C411yescharacteristic
43C24×C53—21no—
44C23×D53C22⋊C411yescharacteristic
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53C2×C812yescomplemented
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C412yes—
51C22×C4×C53C22×C411yescharacteristic
52C2×C22⋊D53—24no—
53C2×C22⋊D53—42no—
54C2×C22⋊D53—42no—
55C22×Dic53—21no—
56C22×Dic53—24no—
57C22×Dic53—21no—
58C22×Dic53—42no—
59C22×Dic53—42no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C2×D412yes—
62#11719947—48no—
63#11719947—48no—
64C2×C4×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—82no—
69C2×C4×D53C2×C812yes—
70C2×C4×D53M1612yes—
71C2×Dic106—21no—
72C2×Dic106—41no—
73C2×Dic106—41no—
74C2×Dic106C2×C812yescomplemented
75C2×#88324—44no—
76D53⋊C8—84no—
77C4⋊C4×C53—41no—
78C22⋊C4×C53—82no—
79C42×C53—41no—
80C22⋊Dic53—82no—
81#11719921—42no—
82C4⋊Dic53—41no—
83#11719919—42no—
84#11719919—42no—
85C4×Dic53—41no—
86C23×C53—22no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—41no—
93C22×D53—41no—
94C22×D53C22⋊C812yes—
95C22×D53#9912yes—
96C22×D53#10012yes—
97D4×C53—42no—
98D4×C53—42no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53C22×C811yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C22⋊D53—48no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—48no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—42no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53C22⋊C812yes—
121C4×D53—24no—
122C4×D53—24no—
123C4×D53—44no—
124C4×D53—82no—
125Dic106—42no—
126Dic106—42no—
127#88324—84no—
128#11432—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—44no—
132C22×C53—42no—
133C22×C53—82no—
134C22×C53—81no—
135C22×C53—81no—
136C22×C53C2×C22⋊C811yescharacteristic
137C22×C53C2×#9911yescharacteristic
138C22×C53C2×#10011yescharacteristic
139C2×D53—22no—
140C2×D53—22no—
141C2×D53—42no—
142C2×D53—42no—
143C2×D53#32314yes—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—42no—
147C4×C53—81no—
148Dic53—22no—
149Dic53—42no—
150Dic53—42no—
151Dic53—81no—
152#2311—531no—
153C2×#324—532no—
154#1319—532no—
155#1318—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—82no—
159C2×C53C2×#32311yescharacteristic
160C2×C53#212711yescharacteristic
161C2×C53#213311yescharacteristic
162D53—24no—
163C22×#41—531no—
164C2×#117—531no—
165C22×C4⋊C4—531no—
166C2×C22⋊C8—1062no—
167#391—531no—
168#391—531no—
169#390—531no—
170#390#615711yescharacteristic
171#386—1062no—
172#380—1062no—
173#324—534no—
174#324—1062no—
175C53#1143211yescharacteristic, complemented, Sylow
176C2×#41—534no—
177C2×#41—1062no—
178C2×#41—2122no—
179C2×#41—2122no—
180C22×Q8—531no—
181C22×D4—1061no—
182C22×D4C2×#615711yescharacteristic
183C23×C4—531no—
184C23×C4—532no—
185C23×C4—1061no—
186C22×C8—2122no—
187#117—2122no—
188C2×C4⋊C4—532no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—2122no—
192C2×C4⋊C4C2×#615711yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1061no—
196C2×C42—531no—
197C2×C42C2×#615711yescharacteristic
198C22⋊C8—1064no—
199C22⋊C8—2122no—
200#95—1062no—
201#95—1062no—
202C24—21no—
203C24—531no—
204#41—2128no—
205#41—2128no—
206C2×Q8—532no—
207C2×Q8—1061no—
208C2×Q8—2121no—
209C2×Q8—2121no—
210C2×D4—21no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—1064no—
214C2×D4—2122no—
215C2×D4—2122no—
216C2×D4D53⋊C812yescomplemented
217C22×C4—21no—
218C22×C4—531no—
219C22×C4—532no—
220C22×C4—532no—
221C22×C4—532no—
222C22×C4—1062no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—1062no—
226C22×C4—1064no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4—4242no—
230C22×C4—2122no—
231C22×C4—2122no—
232C22×C4C22⋊#615712yes—
233C22×C4C22×#615711yescharacteristic
234C2×C8—4244no—
235C2×C8—2124no—
236C4⋊C4—41no—
237C4⋊C4—2122no—
238C4⋊C4—2121no—
239C4⋊C4—2122no—
240C22⋊C4—82no—
241C22⋊C4—4242no—
242C22⋊C4—2122no—
243C42—41no—
244C42—2121no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—532no—
251C23—532no—
252C23—532no—
253C23—2121no—
254C23—2121no—
255C23C2×C22⋊#615711yescharacteristic
256Q8—2122no—
257Q8—2122no—
258D4—42no—
259D4—42no—
260D4—2128no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—532no—
269C2×C4—1064no—
270C2×C4—1064no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—4242no—
276C2×C4—2124no—
277C2×C4—2128no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2122no—
284C2×C4C2×D53⋊C811yescharacteristic
285C2×C4D53⋊M1611yescharacteristic
286C8—4244no—
287C22—21no—
288C22—21no—
289C22—44no—
290C22—42no—
291C22—82no—
292C22—81no—
293C22—81no—
294C22—534no—
295C22—1062no—
296C22—1062no—
297C22—2122no—
298C22—2122no—
299C22D53⋊(C22⋊C8)11yescharacteristic
300C22D53⋊#9911yescharacteristic
301C22D53⋊#10011yescharacteristic
302C4—21no—
303C4—21no—
304C4—42no—
305C4—81no—
306C4—1062no—
307C4—4241no—
308C4—2122no—
309C4—2122no—
310C2—21no—
311C2—21no—
312C2—82no—
313C2—1064no—
314C2D53⋊#32311yescharacteristic
315C2#2029099111yescharacteristic
316C2#2029101111yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC22×C53C2×#99
prime radicalC22×#11719947C4
cosocleC2×C4⋊Dic53C23
hypercenter#390#6157
hypocenterC53#11432
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. D53⋊#324C2semidirect
2. #20289646C2semidirect
3. #20289643C2semidirect
23. C2×#11719947C8semidirect
48. C2×D4×C53C2×C8semidirect
74. C2×Dic106C2×C8semidirect
175. C53#11432semidirect
216. C2×D4D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
D53⋊#3242
#202896462
#202896432
#202773052
#1143253