Finite Group Databasebeta

← #27273256

Subgroups — #27273256

Order 13568.

366 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 81 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#20282670C211yesmaximal, characteristic
2#20276192C211yesmaximal, characteristic
3#20276187C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic
5#20272539C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#2096C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×#13133148—21no—
10#16810565—21no—
11#16810563—21no—
12#16810542—21no—
13#16810521C2211yescharacteristic
14#16810521C2211yescharacteristic
15#16810520—21no—
16#16810506—21no—
17#16810504—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C2×#106×C53—21no—
23C2×#103×C53—21no—
24C53×#391C2211yescharacteristic
25C53×#384C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×#88322—21no—
28C22×#88322—21no—
29C2×C4⋊Dic53D411yescharacteristic
30C2×C4⋊Dic53D411yescharacteristic
31C2×C4⋊Dic53D411yescharacteristic
32C2×C4⋊Dic53D411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C4×Dic53—21no—
38#13133150—42no—
39#13133148—42no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C22×Q8×C53D411yescharacteristic
48C22×C8×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C42×C53C2311yescharacteristic
54#106×C53—42no—
55#103×C53—42no—
56#95×C53—21no—
57#95×C53—21no—
58C2×Q8×C53—21no—
59C2×Q8×C53—21no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C2×C8×C53—22no—
73C2×C8×C53—41no—
74C2×C8×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C42×C53—41no—
83C4⋊Dic53—21no—
84C4⋊Dic53—21no—
85C4⋊Dic53—21no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C4⋊Dic53SD1612yes—
89#11719919—41no—
90#11719919—41no—
91#11719919—41no—
92#11719919—41no—
93#11719919—81no—
94#11719919—81no—
95C4×Dic53—81no—
96C2×#88322—22no—
97C2×#88322—22no—
98C2×#88322—41no—
99C2×#88322—41no—
100C2×#88322—41no—
101C2×#88322—41no—
102C23×C53#12011yescharacteristic
103Q8×C53—42no—
104Q8×C53—81no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—81no—
118C2×C4×C53—81no—
119C2×C4×C53—81no—
120C2×C4×C53C2×SD1611yescharacteristic
121C2×C4×C53#13511yescharacteristic
122C2×C4×C53#13511yescharacteristic
123C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C53#13711yescharacteristic
125C2×C4×C53#13711yescharacteristic
126C2×Dic53—22no—
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—
137C2×Dic53—41no—
138C8×C53—42no—
139#88322—42no—
140#88322—42no—
141#7807—531nomaximal, Sylow
142C22×C53#44811yescharacteristic
143C22×C53#44911yescharacteristic
144C22×C53#45211yescharacteristic
145C22×C53#45311yescharacteristic
146C22×C53#45611yescharacteristic
147C22×C53#45611yescharacteristic
148C22×C53#45811yescharacteristic
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—81no—
156C4×C53—81no—
157C4×C53—161no—
158Dic53—42no—
159Dic53—81no—
160Dic53—81no—
161Dic53—81no—
162#2805—531no—
163#2096D5311yescharacteristic
164#2084—531no—
165#2083—531no—
166#1272—531no—
167#1271—531no—
168#1271—531no—
169C2×C53#160911yescharacteristic
170C2×C53#161111yescharacteristic
171C2×C53#161811yescharacteristic
172C2×C53#162211yescharacteristic
173C2×C53#165011yescharacteristic
174C2×C53#165311yescharacteristic
175C2×C53#166011yescharacteristic
176C2×#107—1061no—
177C2×#106—21no—
178C2×#106—1061no—
179C2×#103—21no—
180#400—1061no—
181#400—1061no—
182#395—1061no—
183#391C2×D5311yescharacteristic
184#389—531no—
185#389—531no—
186#389—1061no—
187#387—1061no—
188#387—1061no—
189#384C2×D5311yescharacteristic
190#340—1061no—
191#340—1061no—
192#336—531no—
193#336—531no—
194#336C2×D5311yescharacteristic
195C53#780711yescharacteristic, complemented, Sylow
196C22×Q8C22⋊D5311yescharacteristic
197C22×C8—21no—
198C22×C8—1061no—
199C22×C8—1061no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4C22⋊D5311yescharacteristic
212C2×C42—1061no—
213C2×C42C22×D5311yescharacteristic
214#107—2122no—
215#106—42no—
216#106—2122no—
217#103—42no—
218#95—21no—
219#95—21no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—41no—
230C2×Q8—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4D4×D5311yescharacteristic
239C22×C4D4×D5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—1062no—
245C2×C8—1062no—
246C2×C8—2121no—
247C2×C8—2121no—
248C2×C8—2121no—
249C2×C8—2121no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—532no—
258C4⋊C4—1061no—
259C4⋊C4—1061no—
260C4⋊C4—1061no—
261C4⋊C4—4241no—
262C4⋊C4—4241no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C42—41no—
270C42—4241no—
271C23#1313318511yescharacteristic
272Q8—42no—
273Q8—81no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
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×C4SD16×D5311yescharacteristic
302C2×C4#1313323711yescharacteristic
303C2×C4#1313323811yescharacteristic
304C2×C4#1313324411yescharacteristic
305C2×C4#1313324611yescharacteristic
306C2×C4#1313324811yescharacteristic
307C8—42no—
308C8—2122no—
309C8—2122no—
310C22#1681076011yescharacteristic
311C22#1681076711yescharacteristic
312C22#1681078311yescharacteristic
313C22#1681079011yescharacteristic
314C22#1681080411yescharacteristic
315C22#1681080411yescharacteristic
316C22#1681081211yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—81no—
324C4—81no—
325C4—161no—
326C4—4241no—
327C4—4241no—
328C4—4241no—
329C4—2122no—
330C2#2027371111yescharacteristic
331C2#2027372111yescharacteristic
332C2#2027376111yescharacteristic
333C2#2027378411yescharacteristic
334C2#2027397011yescharacteristic
335C2#2027399111yescharacteristic
336C2#2027402411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2096C2
socle, prime radicalC23×C53#120
hypercenter#2096D53
hypocenterC53#7807
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
195. C53#7807semidirect

Maximal subgroups

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

subgroupindex
#202826702
#202761922
#202761872
#202725412
#202725392
#202725372
C53×#20962
#780753