Finite Group Databasebeta

← #27294619

Subgroups — #27294619

Order 13568.

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

normal subgroups only

This group has 420 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#20280413C211yesmaximal, characteristic
2#20274454C211yesmaximal, characteristic
3#20272743C212yesmaximal, complemented
4#20272722C212yesmaximal, complemented
5C53×#1464C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×#13133209C411yescharacteristic
8#16810562C2211yescharacteristic
9#16810559C411yescharacteristic
10#16810553C2211yescharacteristic
11#16810548C411yescharacteristic
12#16810518—21no—
13#16810517—21no—
14#16810507—21no—
15#16810496—21no—
16#16810469C411yescharacteristic
17#16810392—21no—
18#16810392C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324C2212yes—
22C53×#323—21no—
23C2×#11719917—24no—
24C2×#11719917—22no—
25C2×#11719917—41no—
26C2×#11719917—42no—
27C22×#88322—41no—
28#13133209—41no—
29#13133209—41no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C4×Dic53—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×C53D411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C4⋊C8×C53—41no—
56C22⋊C8×C53—22no—
57#97×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C2×C8×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C22⋊Dic53—42no—
84C22⋊Dic53—42no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87#11719919—42no—
88#11719919—42no—
89#11719919—41no—
90#11719919—41no—
91#11719919—41no—
92#11719919—41no—
93C4×Dic53—42no—
94#11719917—24no—
95#11719917—44no—
96#11719917—42no—
97C2×#88322—44no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53#10412yes—
104C23×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—21no—
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—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C53#13711yescharacteristic
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C8×C53—82no—
138#88322—44no—
139#11767—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—22no—
143C22×C53—42no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—81no—
147C22×C53C2×#9911yescharacteristic
148C22×C53#41711yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158Dic53—42no—
159Dic53—81no—
160Dic53—81no—
161Dic53—81no—
162#2611—531no—
163#1742—531no—
164#1464D5311yescharacteristic
165#1353—532no—
166#1346—532no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53—42no—
170C2×C53#150411yescharacteristic
171C2×C53#212311yescharacteristic
172C2×C53#213211yescharacteristic
173C22×M16—1061no—
174C2×#125—531no—
175C2×#117C2×D5311yescharacteristic
176#432—21no—
177#400—531no—
178#399—531no—
179#398—531no—
180#397—531no—
181#389—1061no—
182#388—1061no—
183#387—1061no—
184#385—1061no—
185#376—531no—
186#343—532no—
187#343—1061no—
188#324C2×D5312yes—
189#323—21no—
190C53#1176711yescharacteristic, complemented, Sylow
191C23×C4D21211yescharacteristic
192C2×M16—1064no—
193C2×M16—1062no—
194C2×M16—2121no—
195C2×M16—2122no—
196C22×C8—2121no—
197#125—2121no—
198#125—2121no—
199#117—41no—
200#117—41no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4D21211yescharacteristic
208C2×C4⋊C4C22×D5311yescharacteristic
209C2×C22⋊C4—1061no—
210C2×C22⋊C4—1061no—
211C2×C22⋊C4C4×D5311yescharacteristic
212C2×C22⋊C4C22⋊D5311yescharacteristic
213C2×C42—1061no—
214C2×C42C4×D5311yescharacteristic
215C2×C42C22⋊D5311yescharacteristic
216C4⋊C8—41no—
217C22⋊C8—22no—
218#97—41no—
219#95—531no—
220#95—531no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226C24#1171992111yescharacteristic
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—1061no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4#1171992111yescharacteristic
237C22×C4#1171992111yescharacteristic
238C22×C4#1171992111yescharacteristic
239C22×C4C2×C4×D5311yescharacteristic
240C22×C4C2×D21211yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—1064no—
243M16—2124no—
244M16—2122no—
245C2×C8—42no—
246C2×C8—2124no—
247C2×C8—2122no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C4⋊C4—2122no—
253C4⋊C4—2122no—
254C4⋊C4—2121no—
255C4⋊C4—2121no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—2122no—
261C22⋊C4—2122no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—2122no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#1313314312yes—
272C23C2×#1171992111yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—1062no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4#1313313111yescharacteristic
302C2×C4#1313313111yescharacteristic
303C2×C4#1313324511yescharacteristic
304C2×C4#1313325011yescharacteristic
305C8—82no—
306C8—2124no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#1313313111yescharacteristic
315C22#1681062511yescharacteristic
316C22C2×#1313314311yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—4241no—
326C4—4241no—
327C4—4241no—
328C4—2122no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#2027322011yescharacteristic
333C2#2027636811yescharacteristic
334C2#2027639811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1464C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1464D53
hypocenterC53#11767
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
3. #20272743C2semidirect
4. #20272722C2semidirect
190. C53#11767semidirect

Maximal subgroups

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

subgroupindex
#202804132
#202744542
#202727432
#202727222
C53×#14642
#1176753