Finite Group Databasebeta

← #27274216

Subgroups — #27274216

Order 13568.

954 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 159 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#407×D53C211yesmaximal, characteristic
2#20277096C212yesmaximal
3#20272601C212yesmaximal, complemented
4#20272600C211yesmaximal, characteristic, complemented
5C53×#1297C211yesmaximal, characteristic, complemented
6C23×C4×D53C2211yescharacteristic
7C2×M16×D53C412yescomplemented
8C2×#13133174—22no—
9#16810587C2211yescharacteristic
10#16810586C412yescomplemented
11#16810585—22no—
12C22⋊C8×D53—22no—
13#16810485C414yescomplemented
14#16810482C414yescomplemented
15#16810481—22no—
16#16810478—22no—
17C4×#11719921—22no—
18C4×C22⋊Dic53C2212yes—
19C53×#407C2211yescharacteristic
20C4×C22⋊C4×C53C2212yes—
21C24×D53D411yescharacteristic
22C23×Dic53D411yescharacteristic
23C22×C4×D53—44no—
24C22×C4×D53C2×C411yescharacteristic
25C22×C4×D53C2×C411yescharacteristic
26C22×C4×D53C2×C412yes—
27C22×C4×D53C2×C412yes—
28C22×C4×D53D412yes—
29C22×C4×D53D412yes—
30C22×C4×D53Q811yescharacteristic
31C22×C4×D53Q811yescharacteristic
32C2×#11719917C2×C412yes—
33C2×#11719912—22no—
34M16×D53—44no—
35C2×C8×D53—22no—
36#13133174—48no—
37C2×#11719919—24no—
38C2×C22⋊Dic53—22no—
39C2×#11719921—24no—
40C2×#11719921—24no—
41C2×C4×Dic53—22no—
42C22⋊#88322—22no—
43#13133129—24no—
44#13133124—24no—
45#13133124C2×C414yes—
46C23×C4×C53C2311yescharacteristic
47C2×M16×C53C2×C412yes—
48C2×C22⋊C4×C53—22no—
49C2×C42×C53—22no—
50C22⋊C8×C53—22no—
51#95×C53C2×C414yes—
52C24×C53C2×D411yescharacteristic
53C23×D53—42no—
54C23×D53—44no—
55C23×D53C22⋊C412yes—
56C23×D53C22⋊C412yes—
57C23×D53C4⋊C411yescharacteristic
58C23×D53C4⋊C411yescharacteristic
59C22×C4×C53—24no—
60C22×C4×C53—42no—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C22×C412yes—
63C22×C4×C53C2×D412yes—
64C22×C4×C53C2×Q811yescharacteristic
65C22×Dic53—24no—
66C22×Dic53—42no—
67C22×Dic53C22⋊C412yes—
68C22×Dic53C22⋊C412yes—
69C22×Dic53C4⋊C411yescharacteristic
70C22×Dic53C4⋊C411yescharacteristic
71C2×C4×D53—28no—
72C2×C4×D53—24no—
73C2×C4×D53—44no—
74C2×C4×D53—42no—
75C2×C4×D53—42no—
76C2×C4×D53—44no—
77C2×C4×D53—44no—
78C2×C4×D53—42no—
79C2×C4×D53C4211yescharacteristic
80C2×C4×D53C4211yescharacteristic
81C2×C4×D53C4212yes—
82C2×C4×D53C22⋊C412yes—
83C2×C4×D53C22⋊C411yescharacteristic
84C2×C4×D53C22⋊C411yescharacteristic
85C2×C4×D53C4⋊C412yes—
86C2×C4×D53C4⋊C412yes—
87C2×C4×D53C4⋊C412yes—
88C2×C4×D53C4⋊C412yes—
89M16×C53—42no—
90C2×C8×C53—22no—
91C22⋊C4×C53—44no—
92C42×C53—44no—
93C22⋊Dic53—44no—
94#11719921—48no—
95#11719921—48no—
96#11719919—48no—
97C4×Dic53—44no—
98#11719917—42no—
99C2×#88322—22no—
100#11719912—44no—
101C8×D53—44no—
102C23×C53—24no—
103C23×C53—41no—
104C23×C53C2×C22⋊C412yes—
105C23×C53C2×C4⋊C411yescharacteristic
106C22×D53—24no—
107C22×D53—24no—
108C22×D53—24no—
109C22×D53—22no—
110C22×D53—42no—
111C22×D53—41no—
112C22×D53—42no—
113C22×D53—42no—
114C22×D53—42no—
115C22×D53—42no—
116C22×D53—416no—
117C22×D53#9511yescharacteristic
118C22×D53#9512yes—
119C22×D53#9511yescharacteristic
120C2×C4×C53—22no—
121C2×C4×C53—24no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—44no—
125C2×C4×C53—44no—
126C2×C4×C53—44no—
127C2×C4×C53C2×C4211yescharacteristic
128C2×C4×C53C2×C22⋊C411yescharacteristic
129C2×C4×C53C2×C4⋊C412yes—
130C2×Dic53—24no—
131C2×Dic53—22no—
132C2×Dic53—42no—
133C2×Dic53—41no—
134C2×Dic53—44no—
135C2×Dic53—44no—
136C2×Dic53—44no—
137C2×Dic53#9511yescharacteristic
138C2×Dic53#9512yes—
139C2×Dic53#9511yescharacteristic
140C4×D53—22no—
141C4×D53—24no—
142C4×D53—24no—
143C4×D53—22no—
144C4×D53—22no—
145C4×D53—24no—
146C4×D53—42no—
147C4×D53—42no—
148C4×D53—42no—
149C4×D53#9914yes—
150C8×C53—42no—
151#88322—42no—
152#8049—531nomaximal, Sylow
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—21no—
157C22×C53—44no—
158C22×C53C2×#9511yescharacteristic
159C22×C53#42114yes—
160C2×D53—22no—
161C2×D53—22no—
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—22no—
165C2×D53—21no—
166C2×D53—48no—
167C2×D53—44no—
168C2×D53—48no—
169C2×D53—44no—
170C2×D53#34212yes—
171C4×C53—21no—
172C4×C53—22no—
173C4×C53—84no—
174C4×C53C2×#9912yes—
175Dic53—21no—
176Dic53—22no—
177Dic53—84no—
178Dic53#34212yes—
179C2×#407—531no—
180#2280—532no—
181#1297—531no—
182#1297—532no—
183#1297D5311yescharacteristic, complemented
184C2×C53—24no—
185C2×C53—41no—
186C2×C53#173312yes—
187C2×C53C2×#34211yescharacteristic
188D53—22no—
189D53—42no—
190D53—41no—
191C24×C4—531no—
192C22×M16—532no—
193C2×#117—1062no—
194#407—531no—
195#407—532no—
196#407—1062no—
197#407C2×D5311yescharacteristic
198C2×C22⋊C8—1062no—
199#381—534no—
200#380—534no—
201#380—1062no—
202#379—1062no—
203C4×C22⋊C4—532no—
204C4×C22⋊C4—1062no—
205C4×C22⋊C4C2×D5312yes—
206C53#804911yescharacteristic, complemented, Sylow
207C25—531no—
208C23×C4—531no—
209C23×C4—531no—
210C23×C4—532no—
211C23×C4—532no—
212C23×C4—532no—
213C23×C4—531no—
214C23×C4—532no—
215C23×C4—531no—
216C23×C4—531no—
217C23×C4—2124no—
218C23×C4C22×D5311yescharacteristic
219C2×M16—532no—
220C2×M16—1062no—
221C2×M16—2124no—
222C2×M16C4×D5312yes—
223C22×C8—1062no—
224#117—2128no—
225C2×C4⋊C4—1064no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—1064no—
228C2×C22⋊C4—1064no—
229C2×C22⋊C4—1062no—
230C2×C42—22no—
231C2×C42—1062no—
232C22⋊C8—22no—
233C22⋊C8—1064no—
234C22⋊C8—1062no—
235#95—534no—
236#95—1064no—
237#95C4×D5314yes—
238C24—532no—
239C24—532no—
240C24—531no—
241C24—531no—
242C24—2124no—
243C24—2122no—
244C24D4×D5311yescharacteristic
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—531no—
248C22×C4—532no—
249C22×C4—531no—
250C22×C4—532no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—532no—
255C22×C4—531no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—532no—
259C22×C4—532no—
260C22×C4—532no—
261C22×C4—1068no—
262C22×C4—1064no—
263C22×C4—1064no—
264C22×C4—2124no—
265C22×C4—2122no—
266C22×C4—2124no—
267C22×C4—2122no—
268C22×C4—2124no—
269C22×C4—2122no—
270C22×C4—2122no—
271C22×C4C2×C4×D5311yescharacteristic
272C22×C4C2×C4×D5312yes—
273C22×C4D4×D5312yes—
274C22×C4Q8×D5311yescharacteristic
275M16—42no—
276M16—2124no—
277M16—2122no—
278C2×C8—22no—
279C2×C8—1062no—
280C2×C8—2124no—
281C4⋊C4—2128no—
282C22⋊C4—44no—
283C22⋊C4—2128no—
284C22⋊C4—2128no—
285C22⋊C4—2124no—
286C42—44no—
287C42—2124no—
288C23—24no—
289C23—41no—
290C23—531no—
291C23—531no—
292C23—532no—
293C23—1064no—
294C23—1062no—
295C23—1064no—
296C23—1064no—
297C23—2122no—
298C23—2121no—
299C23—2122no—
300C23—2122no—
301C23—2122no—
302C23—2122no—
303C23—21216no—
304C23C22⋊C4×D5312yes—
305C23C4⋊C4×D5311yescharacteristic
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—531no—
314C2×C4—534no—
315C2×C4—532no—
316C2×C4—531no—
317C2×C4—1062no—
318C2×C4—1064no—
319C2×C4—1062no—
320C2×C4—1064no—
321C2×C4—1064no—
322C2×C4—1062no—
323C2×C4—1062no—
324C2×C4—1064no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4—2121no—
330C2×C4—2124no—
331C2×C4—2124no—
332C2×C4—2124no—
333C2×C4C42×D5311yescharacteristic
334C2×C4C22⋊C4×D5311yescharacteristic
335C2×C4C4⋊C4×D5312yes—
336C8—42no—
337C8—2122no—
338C22—22no—
339C22—22no—
340C22—21no—
341C22—22no—
342C22—44no—
343C22—532no—
344C22—1062no—
345C22—1061no—
346C22—1062no—
347C22—1062no—
348C22—1062no—
349C22—1062no—
350C22—2128no—
351C22—2128no—
352C22—2124no—
353C22—2124no—
354C22#95×D5311yescharacteristic
355C22#1681063814yes—
356C4—21no—
357C4—22no—
358C4—84no—
359C4—532no—
360C4—1062no—
361C4—1061no—
362C4—4244no—
363C4#99×D5312yes—
364C2—24no—
365C2—41no—
366C2—1062no—
367C2—2122no—
368C2—2121no—
369C2#2027442312yes—
370C2#342×D5311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1297C2
socleC22×C53C2×#95
prime radicalC24×D53D4
cosocleC23×C4×C53C23
hypercenter#1297D53
hypocenterC53#8049
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
3. #20272601C2semidirect
4. #20272600C2semidirect
5. C53×#1297C2semidirect
7. C2×M16×D53C4semidirect
10. #16810586C4semidirect
13. #16810485C4semidirect
14. #16810482C4semidirect
183. #1297D53semidirect
206. C53#8049semidirect

Maximal subgroups

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

subgroupindex
#407×D532
#202770962
#202726012
#202726002
C53×#12972
#804953