Finite Group Databasebeta

← #27294564

Subgroups — #27294564

Order 13568.

444 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 83 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#20280292C211yesmaximal, characteristic
2#20274448C211yesmaximal, characteristic
3#20272737C212yesmaximal
4#20272716C212yesmaximal
5C53×#1464C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic, complemented
7C2×#13133176—21no—
8#16810553C411yescharacteristic, complemented
9#16810548C2211yescharacteristic
10#16810543C411yescharacteristic, complemented
11#16810535C2211yescharacteristic
12#16810511—21no—
13#16810507—21no—
14#16810504—21no—
15#16810496—21no—
16#16810491C411yescharacteristic, complemented
17#16810390—21no—
18#16810390C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C4×Dic106—41no—
25C4×Dic106—41no—
26#13133176—24no—
27#13133176—42no—
28C2×C4⋊Dic53—41no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47#117×C53—41no—
48#117×C53—41no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C4⋊C8×C53—41no—
56C22⋊C8×C53—22no—
57#97×C53—41no—
58C24×C53C2×D411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—42no—
75C2×Dic106—41no—
76C2×Dic106—42no—
77C2×C8×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C22⋊Dic53—24no—
87C22⋊Dic53—42no—
88C22⋊Dic53—41no—
89C22⋊Dic53—42no—
90C22⋊Dic53—41no—
91C4⋊Dic53—44no—
92C4⋊Dic53—41no—
93#11719919—42no—
94#11719919—41no—
95#11719919—41no—
96#11719919—42no—
97C4×Dic53—42no—
98C4×Dic53—44no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53AGL(1,ℤ/8)11yescharacteristic
105C23×C53#13711yescharacteristic
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#10412yes—
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—44no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—82no—
139C2×Dic53—82no—
140Dic106—42no—
141Dic106—42no—
142C8×C53—82no—
143#11756—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—81no—
153C22×C53C2×#9911yescharacteristic
154C22×C53#41711yescharacteristic
155C22×C53C2×#10411yescharacteristic
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163Dic53—42no—
164Dic53—81no—
165Dic53—81no—
166Dic53—84no—
167Dic53—81no—
168#2599—531no—
169#1740—531no—
170#1464D5311yescharacteristic
171#1351—532no—
172#1344—532no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—41no—
176C2×C53—41no—
177C2×C53#150411yescharacteristic
178C2×C53#212011yescharacteristic
179C2×C53#212911yescharacteristic
180C2×C4×Q8—531no—
181C2×#117—1061no—
182C2×#117C2×D5311yescharacteristic
183#432—21no—
184#398—531no—
185#397—531no—
186#395—531no—
187#393—531no—
188#388—1061no—
189#387—1061no—
190#387—1061no—
191#385—1061no—
192#383—531no—
193#342—532no—
194#342—1061no—
195#324—21no—
196#323C2×D5312yes—
197C53#1175611yescharacteristic, complemented, Sylow
198C22×Q8—531no—
199C23×C4D21211yescharacteristic
200C4×Q8—2121no—
201C4×Q8—2121no—
202#117—41no—
203#117—41no—
204#117—1064no—
205#117—2122no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—2121no—
211C2×C4⋊C4C4×D5311yescharacteristic
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4—1062no—
214C2×C22⋊C4—1061no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4D21211yescharacteristic
217C2×C22⋊C4C22×D5311yescharacteristic
218C2×C42—1061no—
219C2×C42—2121no—
220C2×C42C4×D5311yescharacteristic
221C2×C42C22⋊D5311yescharacteristic
222C4⋊C8—41no—
223C22⋊C8—22no—
224#97—41no—
225#95—531no—
226#95—531no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233C24C2×D21211yescharacteristic
234C2×Q8—2121no—
235C2×Q8—2122no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—2122no—
246C22×C4#1171992111yescharacteristic
247C22×C4#1171992111yescharacteristic
248C22×C4#1171992111yescharacteristic
249C22×C4#1171992111yescharacteristic
250C22×C4C2×C4×D5311yescharacteristic
251C22×C4C2×C22⋊D5311yescharacteristic
252C2×C8—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2121no—
256C4⋊C4—2124no—
257C4⋊C4—2122no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2122no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—1064no—
264C22⋊C4—2122no—
265C22⋊C4—2122no—
266C22⋊C4—2121no—
267C22⋊C4—2121no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—2122no—
273C42—2124no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23C2×#1171992111yescharacteristic
279C23#1313324511yescharacteristic
280C23#1313325011yescharacteristic
281Q8—2122no—
282Q8—2122no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—1062no—
300C2×C4—4242no—
301C2×C4—4242no—
302C2×C4—2124no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4#1313313111yescharacteristic
315C2×C4#1313313111yescharacteristic
316C2×C4#1313314312yes—
317C8—82no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#1313313111yescharacteristic
328C22#1681062511yescharacteristic
329C22C2×#1313314311yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—4241no—
338C4—4241no—
339C4—4244no—
340C4—4241no—
341C4—2122no—
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2#2027322011yescharacteristic
347C2#2027635711yescharacteristic
348C2#2027638911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1464C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1464D53
hypocenterC53#11756
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic106C4semidirect
8. #16810553C4semidirect
10. #16810543C4semidirect
16. #16810491C4semidirect
197. C53#11756semidirect

Maximal subgroups

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

subgroupindex
#202802922
#202744482
#202727372
#202727162
C53×#14642
#1175653