Finite Group Databasebeta

← #27272379

Subgroups — #27272379

Order 13568.

646 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 121 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#391×D53C211yesmaximal, characteristic
2#20274694C212yesmaximal
3#20272541C211yesmaximal, characteristic, complemented
4#20272540C212yesmaximal, complemented
5C53×#1272C211yesmaximal, characteristic, complemented
6C22×Q8×D53C411yescharacteristic
7C2×C4⋊C4×D53C411yescharacteristic
8C2×C42×D53C2211yescharacteristic
9C2×#13133146—22no—
10C2×#13133129—22no—
11#16810530C411yescharacteristic
12#16810529C2211yescharacteristic
13#16810528C411yescharacteristic
14#16810527—22no—
15#95×D53—22no—
16#16810378C2212yes—
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106—21no—
20C22×Dic106C2×C411yescharacteristic
21C2×Q8×D53—22no—
22C2×Q8×D53—22no—
23C2×Q8×D53—42no—
24C2×Q8×D53—42no—
25C22×C4×D53—21no—
26C22×C4×D53—21no—
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D412yes—
29C22×#88322—22no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C4⋊C4×D53—42no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35C42×D53—42no—
36#13133146—44no—
37#13133129—44no—
38#13133124—22no—
39#13133124—22no—
40#13133124—22no—
41C22×Q8×C53C2×C411yescharacteristic
42C22×C8×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C42×C53C2311yescharacteristic
45#95×C53—22no—
46C23×D53C22⋊C411yescharacteristic
47C2×Q8×C53—21no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53M1612yes—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53C22×C411yescharacteristic
54C22×C4×C53C2×D412yes—
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C412yes—
59Q8×D53—48no—
60Q8×D53—48no—
61C2×C4×D53—22no—
62C2×C4×D53—21no—
63C2×C4×D53—21no—
64C2×C4×D53—22no—
65C2×C4×D53—24no—
66C2×C4×D53—42no—
67C2×C4×D53—42no—
68C2×C4×D53—42no—
69C2×C4×D53—42no—
70C2×C4×D53—44no—
71C2×C4×D53—42no—
72C2×C4×D53SD1614yes—
73C2×C4×D53Q1614yes—
74C2×Dic106—24no—
75C2×Dic106—21no—
76C2×Dic106—42no—
77C2×Dic106—42no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106M1612yes—
81C2×C8×C53—42no—
82C2×C8×C53—42no—
83C2×C8×C53—42no—
84C4⋊C4×C53—41no—
85C42×C53—41no—
86C4⋊Dic53—41no—
87#11719919—42no—
88C4×Dic53—24no—
89C4×Dic53—41no—
90C2×#88322—42no—
91C2×#88322—42no—
92C2×#88322—42no—
93C23×C53C2×C22⋊C411yescharacteristic
94C22×D53—21no—
95C22×D53—21no—
96C22×D53—22no—
97C22×D53#10012yes—
98C22×D53#10314yes—
99Q8×C53—42no—
100Q8×C53—42no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53C2×M1611yescharacteristic
111C2×C4×C53C2×M1611yescharacteristic
112C2×C4×C53C2×SD1612yes—
113C2×C4×C53C2×Q1612yes—
114C2×Dic53—21no—
115C2×Dic53—22no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—42no—
123C2×Dic53#9912yes—
124C2×Dic53#10314yes—
125C4×D53—24no—
126C4×D53—24no—
127C4×D53—44no—
128C4×D53—44no—
129C4×D53—48no—
130C4×D53—84no—
131Dic106—48no—
132Dic106—42no—
133Dic106—42no—
134C8×C53—82no—
135#88322—82no—
136#7643—531nomaximal, Sylow
137C22×C53#40711yescharacteristic
138C22×C53C2×#9911yescharacteristic
139C22×C53C2×#10011yescharacteristic
140C22×C53C2×#10312yes—
141C22×C53#42112yes—
142C2×D53—22no—
143C2×D53—24no—
144C2×D53—24no—
145C2×D53—22no—
146C2×D53#33314yes—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—22no—
150C4×C53—22no—
151C4×C53—42no—
152C4×C53—81no—
153Dic53—22no—
154Dic53—24no—
155Dic53—42no—
156Dic53—81no—
157C2×#391—531no—
158#1797—532no—
159#1272—531no—
160#1272—532no—
161#1272D5311yescharacteristic, complemented
162C2×C53#146211yescharacteristic
163C2×C53#148712yes—
164C2×C53#152012yes—
165C2×C53#153311yescharacteristic
166C2×C53C2×#33311yescharacteristic
167D53—24no—
168C23×Q8—531no—
169C22×C4⋊C4—531no—
170C22×C42—531no—
171C2×C4⋊C8—1062no—
172C2×C22⋊C8—1062no—
173#391—531no—
174#391—531no—
175#391—531no—
176#391—1062no—
177#391C2×D5311yescharacteristic
178C2×#95—1062no—
179#336—532no—
180#336C2×D5312yes—
181C53#764311yescharacteristic, complemented, Sylow
182C22×Q8—531no—
183C22×Q8—1061no—
184C22×Q8—1062no—
185C22×Q8—1062no—
186C22×Q8—2122no—
187C22×Q8—2122no—
188C22×Q8C4×D5311yescharacteristic
189C23×C4—532no—
190C23×C4—531no—
191C23×C4—1061no—
192C23×C4—1061no—
193C22×C8—22no—
194C22×C8—1062no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—2122no—
198C2×C4⋊C4C4×D5311yescharacteristic
199C2×C42—532no—
200C2×C42—531no—
201C2×C42—2122no—
202C2×C42C22×D5311yescharacteristic
203C4⋊C8—2124no—
204C22⋊C8—2124no—
205#95—22no—
206#95—1062no—
207#95—1062no—
208#95—1062no—
209C24—531no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—532no—
214C2×Q8—1064no—
215C2×Q8—1061no—
216C2×Q8—2121no—
217C2×Q8—2121no—
218C2×Q8—2122no—
219C2×Q8—2122no—
220C2×Q8—2128no—
221C2×Q8—2128no—
222C2×Q8#1171991212yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—532no—
226C22×C4—531no—
227C22×C4—534no—
228C22×C4—534no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4—1064no—
232C22×C4—1061no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4—2122no—
237C22×C4—2122no—
238C22×C4—2122no—
239C22×C4—2124no—
240C22×C4—2122no—
241C22×C4—2122no—
242C22×C4C2×C4×D5311yescharacteristic
243C22×C4D4×D5312yes—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—2122no—
248C2×C8—2122no—
249C2×C8—2122no—
250C4⋊C4—41no—
251C4⋊C4—2121no—
252C4⋊C4—2122no—
253C42—41no—
254C42—1064no—
255C42—2121no—
256C23—532no—
257C23—534no—
258C23—1061no—
259C23—1061no—
260C23—1062no—
261C23C22⋊C4×D5311yescharacteristic
262Q8—42no—
263Q8—42no—
264Q8—2122no—
265Q8—2122no—
266Q8—2128no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—532no—
277C2×C4—534no—
278C2×C4—1064no—
279C2×C4—1064no—
280C2×C4—1061no—
281C2×C4—1062no—
282C2×C4—1061no—
283C2×C4—1062no—
284C2×C4—1062no—
285C2×C4—4244no—
286C2×C4—2124no—
287C2×C4—2124no—
288C2×C4—2128no—
289C2×C4—2121no—
290C2×C4—2122no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4M16×D5311yescharacteristic
294C2×C4C2×#1171991211yescharacteristic
295C2×C4SD16×D5312yes—
296C2×C4Q16×D5312yes—
297C8—82no—
298C8—4242no—
299C22—534no—
300C22—1064no—
301C22—1062no—
302C22—1062no—
303C22—1064no—
304C22#1681058511yescharacteristic
305C22#99×D5311yescharacteristic
306C22#100×D5311yescharacteristic
307C22#103×D5312yes—
308C22#1681063812yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—81no—
315C4—1064no—
316C4—1062no—
317C4—4241no—
318C4—2122no—
319C2—1064no—
320C2#2027304011yescharacteristic
321C2#2027314212yes—
322C2#2027328512yes—
323C2#2027333711yescharacteristic
324C2#333×D5311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7643
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
3. #20272541C2semidirect
4. #20272540C2semidirect
5. C53×#1272C2semidirect
161. #1272D53semidirect
181. C53#7643semidirect

Maximal subgroups

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

subgroupindex
#391×D532
#202746942
#202725412
#202725402
C53×#12722
#764353