Finite Group Databasebeta

← #27273231

Subgroups — #27273231

Order 13568.

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

normal subgroups only

This group has 338 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#20282706C211yesmaximal, characteristic
2#20276274C211yesmaximal, characteristic
3#20276193C211yesmaximal, characteristic
4#20276186C211yesmaximal, characteristic
5#20272539C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#1271C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×#13133149—21no—
10#16810563—21no—
11#16810562—21no—
12#16810561—21no—
13#16810521C2211yescharacteristic
14#16810521C2211yescharacteristic
15#16810510—21no—
16#16810509—21no—
17#16810504—21no—
18#16810493C2211yescharacteristic
19#16810386—21no—
20#16810386—21no—
21#16810386—21no—
22#16810385—21no—
23#16810378C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×#88322—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53D411yescharacteristic
32C2×C4⋊Dic53D411yescharacteristic
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C4×Dic53—21no—
38#13133150—42no—
39#13133149—42no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×C8×C53—21no—
50C22×C8×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C42×C53C2311yescharacteristic
55#95×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53—41no—
66C2×C8×C53—22no—
67C2×C8×C53—41no—
68C2×C8×C53—41no—
69C2×C8×C53—41no—
70C2×C8×C53—41no—
71C2×C8×C53—41no—
72C4⋊C4×C53—21no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C42×C53—41no—
78C4⋊Dic53—21no—
79C4⋊Dic53—21no—
80C4⋊Dic53—21no—
81C4⋊Dic53—21no—
82C4⋊Dic53—41no—
83C4⋊Dic53—41no—
84C4⋊Dic53—42no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89#11719919—41no—
90#11719919—41no—
91#11719919—41no—
92#11719919—41no—
93C4×Dic53—81no—
94C2×#88322—22no—
95C2×#88322—41no—
96C2×#88322—41no—
97C23×C53#12011yescharacteristic
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53#13511yescharacteristic
111C2×C4×C53#13511yescharacteristic
112C2×C4×C53#13511yescharacteristic
113C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
114C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C53#13711yescharacteristic
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—81no—
129C2×Dic53—81no—
130C2×Dic53—81no—
131C8×C53—42no—
132C8×C53—81no—
133#88322—42no—
134#7803—531nomaximal, Sylow
135C22×C53#44911yescharacteristic
136C22×C53#44911yescharacteristic
137C22×C53#45211yescharacteristic
138C22×C53#45411yescharacteristic
139C22×C53#45611yescharacteristic
140C22×C53#45611yescharacteristic
141C22×C53#45811yescharacteristic
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—81no—
149C4×C53—81no—
150Dic53—81no—
151Dic53—81no—
152Dic53—81no—
153Dic53—81no—
154Dic53—161no—
155#2810—531no—
156#2099—531no—
157#2084—531no—
158#2083—531no—
159#1271—531no—
160#1271—531no—
161#1271D5311yescharacteristic
162C2×C53#160911yescharacteristic
163C2×C53#160911yescharacteristic
164C2×C53#161511yescharacteristic
165C2×C53#162811yescharacteristic
166C2×C53#165011yescharacteristic
167C2×C53#165211yescharacteristic
168C2×C53#166311yescharacteristic
169C2×#107—1061no—
170C2×#106—1061no—
171#400—1061no—
172#400—1061no—
173#400—1061no—
174#389—531no—
175#389—531no—
176#389C2×D5311yescharacteristic
177#387—1061no—
178#387—1061no—
179#387—1061no—
180#384—531no—
181#340—1061no—
182#340—1061no—
183#340—1061no—
184#340—1061no—
185#336—531no—
186#336C2×D5311yescharacteristic
187#336C2×D5311yescharacteristic
188C53#780311yescharacteristic, complemented, Sylow
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—1061no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4D21211yescharacteristic
203C2×C4⋊C4D21211yescharacteristic
204C2×C42—1061no—
205C2×C42C22×D5311yescharacteristic
206#107—2122no—
207#106—2122no—
208#95—21no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217#95—1061no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4—1061no—
224C22×C4—2121no—
225C22×C4C2×D21211yescharacteristic
226C22×C4D4×D5311yescharacteristic
227C22×C4D4×D5311yescharacteristic
228C2×C8—22no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—1062no—
235C2×C8—2121no—
236C2×C8—2121no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—1061no—
243C4⋊C4—1061no—
244C4⋊C4—1061no—
245C4⋊C4—1061no—
246C4⋊C4—2122no—
247C4⋊C4—2121no—
248C4⋊C4—2121no—
249C4⋊C4—2121no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C4⋊C4—2121no—
253C4⋊C4—2121no—
254C4⋊C4—2121no—
255C4⋊C4—2121no—
256C4⋊C4—2121no—
257C42—41no—
258C42—4241no—
259C23#1313318711yescharacteristic
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—4241no—
273C2×C4—4241no—
274C2×C4—4241no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4#1313323811yescharacteristic
288C2×C4#1313323811yescharacteristic
289C2×C4#1313323911yescharacteristic
290C2×C4#1313324111yescharacteristic
291C2×C4#1313324511yescharacteristic
292C2×C4#1313324911yescharacteristic
293C8—42no—
294C8—81no—
295C8—2122no—
296C22#1681076911yescharacteristic
297C22#1681077011yescharacteristic
298C22#1681078811yescharacteristic
299C22#1681079711yescharacteristic
300C22#1681080711yescharacteristic
301C22#1681080811yescharacteristic
302C22#1681081411yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—81no—
310C4—81no—
311C4—8481no—
312C4—4241no—
313C4—4241no—
314C4—4241no—
315C4—4241no—
316C2#2027371311yescharacteristic
317C2#2027371411yescharacteristic
318C2#2027374911yescharacteristic
319C2#2027382711yescharacteristic
320C2#2027397211yescharacteristic
321C2#2027398711yescharacteristic
322C2#2027404511yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socle, prime radicalC23×C53#120
hypercenter#1271D53
hypocenterC53#7803
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
188. C53#7803semidirect

Maximal subgroups

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

subgroupindex
#202827062
#202762742
#202761932
#202761862
#202725392
#202725372
C53×#12712
#780353