Finite Group Databasebeta

← #27294889

Subgroups — #27294889

Order 13568.

418 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280583C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20272740C212yesmaximal, complemented
4#20272716C212yesmaximal, complemented
5C53×#1465C211yesmaximal, characteristic
6C2×#13133209C411yescharacteristic, complemented
7C2×#13133132—21no—
8#16810559C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810554—21no—
11#16810548C411yescharacteristic, complemented
12#16810548C2211yescharacteristic
13#16810537—21no—
14#16810496—21no—
15#16810496—21no—
16#16810491C411yescharacteristic, complemented
17#16810483—21no—
18#16810390C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23#13133209—41no—
24#13133209—41no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—22no—
37#13133132—42no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—41no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×D4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C2×M16×C53—41no—
49C4×D4×C53—41no—
50C4×D4×C53—41no—
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C2×C42×C53C2×C411yescharacteristic
55C22⋊C8×C53—22no—
56C22⋊C8×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—81no—
68C22×C4×C53—81no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—42no—
79M16×C53—82no—
80C2×C8×C53—42no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C42×C53—21no—
85C42×C53—21no—
86C22⋊Dic53—42no—
87C22⋊Dic53—41no—
88C22⋊Dic53—41no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C22⋊Dic53—81no—
92C22⋊Dic53—81no—
93C4⋊Dic53—41no—
94C4⋊Dic53—41no—
95C4⋊Dic53—82no—
96#11719919—42no—
97#11719919—41no—
98#11719919—41no—
99#11719919—81no—
100#11719919—81no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53AGL(1,ℤ/8)11yescharacteristic
112C23×C53#13711yescharacteristic
113D4×C53—42no—
114D4×C53—42no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53#9911yescharacteristic
130C2×C4×C53#9911yescharacteristic
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—82no—
144C2×Dic53—82no—
145C2×Dic53—82no—
146C8×C53—82no—
147#11827—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53C2×#9911yescharacteristic
163C22×C53#41711yescharacteristic
164C22×C53#42111yescharacteristic
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171Dic53—81no—
172Dic53—81no—
173Dic53—81no—
174Dic53—81no—
175Dic53—162no—
176#2633—531no—
177#1799—531no—
178#1465D5311yescharacteristic
179#1352—532no—
180#1344—532no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—41no—
184C2×C53—41no—
185C2×C53—81no—
186C2×C53#151311yescharacteristic
187C2×C53#212311yescharacteristic
188C2×C53#212711yescharacteristic
189C2×#125—531no—
190C2×C4×D4C2×D5311yescharacteristic
191C2×#99—1061no—
192#407—21no—
193#399—531no—
194#399—531no—
195#398—1061no—
196#397—531no—
197#397—531no—
198#394—1061no—
199#385—1061no—
200#385—1061no—
201#383—531no—
202#380—1061no—
203#342—532no—
204#324—21no—
205#323C2×D5312yes—
206C53#1182711yescharacteristic, complemented, Sylow
207C22×D4D21211yescharacteristic
208C23×C4D21211yescharacteristic
209C23×C4C22⋊D5311yescharacteristic
210C2×M16—41no—
211#125—2121no—
212#125—2121no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—2121no—
222C2×C4⋊C4C22⋊D5311yescharacteristic
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1061no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4C4×D5311yescharacteristic
229C2×C22⋊C4C22×D5311yescharacteristic
230C2×C42C4×D5311yescharacteristic
231#99—2122no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234#95—531no—
235#95—531no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—2121no—
241C24#1171992111yescharacteristic
242C24C2×D21211yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—2122no—
258C22×C4#1171992111yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4#1171992111yescharacteristic
261C22×C4C2×C4×D5311yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263M16—82no—
264C2×C8—42no—
265C4⋊C4—41no—
266C4⋊C4—4242no—
267C4⋊C4—4241no—
268C4⋊C4—4241no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4—2122no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—4241no—
277C22⋊C4—4241no—
278C22⋊C4—2122no—
279C22⋊C4—2122no—
280C22⋊C4—2122no—
281C22⋊C4—2121no—
282C22⋊C4—2121no—
283C42—21no—
284C42—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23C2×#1171992111yescharacteristic
295C23#1313324511yescharacteristic
296C23#1313325011yescharacteristic
297D4—42no—
298D4—42no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—4242no—
314C2×C4—4242no—
315C2×C4—4242no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4#1313313111yescharacteristic
329C2×C4#1313313111yescharacteristic
330C8—82no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22C2×#1313313111yescharacteristic
346C22#1681062511yescharacteristic
347C22#1681063911yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—8482no—
355C4—4241no—
356C4—4241no—
357C4—4241no—
358C4—4241no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2#2027325911yescharacteristic
365C2#2027637011yescharacteristic
366C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1465C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1465D53
hypocenterC53#11827
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20272740C2semidirect
4. #20272716C2semidirect
6. C2×#13133209C4semidirect
8. #16810559C4semidirect
11. #16810548C4semidirect
16. #16810491C4semidirect
206. C53#11827semidirect

Maximal subgroups

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

subgroupindex
#202805832
#202747032
#202727402
#202727162
C53×#14652
#1182753