Finite Group Databasebeta

← #27273249

Subgroups — #27273249

Order 13568.

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

normal subgroups only

This group has 350 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#20282668C211yesmaximal, characteristic
2#20276187C211yesmaximal, characteristic
3#20276124C211yesmaximal, characteristic
4#20272540C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#2099C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×#13133140—21no—
10#16810562—21no—
11#16810562—21no—
12#16810544—21no—
13#16810529C2211yescharacteristic
14#16810522—21no—
15#16810521C2211yescharacteristic
16#16810510—21no—
17#16810509—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C53×#389C2211yescharacteristic
23C53×#384C2211yescharacteristic
24C53×#340—21no—
25C53×#340—21no—
26C53×#336C2211yescharacteristic
27C22×Dic106D411yescharacteristic
28C22×#88322—21no—
29C22×#88322—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53D411yescharacteristic
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×#11719919—21no—
36C2×C4×Dic53—21no—
37#13133150—42no—
38#13133140—42no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C22×C8×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×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×C53—41no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C2×Dic106—21no—
67C2×Dic106—21no—
68C2×Dic106—41no—
69C2×Dic106—41no—
70C2×C8×C53—41no—
71C2×C8×C53—41no—
72C2×C8×C53—41no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53Q1612yes—
83C42×C53—41no—
84C4⋊Dic53—21no—
85C4⋊Dic53—21no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C4⋊Dic53—81no—
89C4⋊Dic53—81no—
90#11719919—41no—
91#11719919—41no—
92C4×Dic53—81no—
93C2×#88322—22no—
94C2×#88322—41no—
95C2×#88322—41no—
96C2×#88322—41no—
97C2×#88322—41no—
98C2×#88322—41no—
99C23×C53#12011yescharacteristic
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—81no—
113C2×C4×C53—81no—
114C2×C4×C53—81no—
115C2×C4×C53C2×Q1611yescharacteristic
116C2×C4×C53#13511yescharacteristic
117C2×C4×C53#13511yescharacteristic
118C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C53#13711yescharacteristic
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133Dic106—42no—
134Dic106—81no—
135C8×C53—81no—
136#88322—42no—
137#88322—81no—
138#7806—531nomaximal, Sylow
139C22×C53#44911yescharacteristic
140C22×C53#44911yescharacteristic
141C22×C53#45111yescharacteristic
142C22×C53#45411yescharacteristic
143C22×C53#45511yescharacteristic
144C22×C53#45611yescharacteristic
145C22×C53#45811yescharacteristic
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—42no—
153C4×C53—81no—
154C4×C53—161no—
155Dic53—81no—
156Dic53—81no—
157Dic53—81no—
158Dic53—81no—
159#2805—531no—
160#2099D5311yescharacteristic
161#2083—531no—
162#2072—531no—
163#1272—531no—
164#1271—531no—
165#1271—531no—
166C2×C53#160911yescharacteristic
167C2×C53#161211yescharacteristic
168C2×C53#161711yescharacteristic
169C2×C53#162811yescharacteristic
170C2×C53#164411yescharacteristic
171C2×C53#164811yescharacteristic
172C2×C53#166411yescharacteristic
173C2×#107—1061no—
174C2×#103—1061no—
175#400—1061no—
176#400—1061no—
177#395—1061no—
178#391—531no—
179#389—531no—
180#389—1061no—
181#389C2×D5311yescharacteristic
182#387—1061no—
183#387—1061no—
184#384C2×D5311yescharacteristic
185#340—21no—
186#340—21no—
187#340—1061no—
188#340—1061no—
189#336—531no—
190#336—531no—
191#336C2×D5311yescharacteristic
192C53#780611yescharacteristic, complemented, Sylow
193C22×Q8—531no—
194C22×C8—21no—
195C22×C8—1061no—
196C22×C8—1061no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4C22⋊D5311yescharacteristic
208C2×C4⋊C4C22⋊D5311yescharacteristic
209C2×C42—1061no—
210C2×C42C22×D5311yescharacteristic
211#107—2122no—
212#103—2122no—
213#95—21no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217#95—1061no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222C2×Q8—1061no—
223C2×Q8—1061no—
224C2×Q8—2121no—
225C2×Q8—2121no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—1061no—
230C22×C4—1061no—
231C22×C4—1061no—
232C22×C4—1061no—
233C22×C4D4×D5311yescharacteristic
234C22×C4D4×D5311yescharacteristic
235C22×C4C2×C22⋊D5311yescharacteristic
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—1062no—
240C2×C8—2121no—
241C2×C8—2121no—
242C2×C8—2121no—
243C2×C8—2121no—
244C2×C8—2121no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—1061no—
255C4⋊C4—1061no—
256C4⋊C4—4241no—
257C4⋊C4—4241no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4#1171992512yes—
263C42—41no—
264C42—4241no—
265C23#1313318511yescharacteristic
266Q8—4241no—
267Q8—2122no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4C2×#1171992511yescharacteristic
296C2×C4#1313323611yescharacteristic
297C2×C4#1313323811yescharacteristic
298C2×C4#1313324111yescharacteristic
299C2×C4#1313324411yescharacteristic
300C2×C4#1313324711yescharacteristic
301C8—81no—
302C8—4241no—
303C8—2122no—
304C22#1681076511yescharacteristic
305C22#1681076711yescharacteristic
306C22#1681077811yescharacteristic
307C22#1681079511yescharacteristic
308C22#1681079911yescharacteristic
309C22#1681080411yescharacteristic
310C22#1681081211yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—42no—
318C4—81no—
319C4—161no—
320C4—4241no—
321C4—4241no—
322C4—4241no—
323C4—4241no—
324C2#2027370911yescharacteristic
325C2#2027372811yescharacteristic
326C2#2027375611yescharacteristic
327C2#2027382211yescharacteristic
328C2#2027392811yescharacteristic
329C2#2027395611yescharacteristic
330C2#2027404611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2099C2
socle, prime radicalC23×C53#120
hypercenter#2099D53
hypocenterC53#7806
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
192. C53#7806semidirect

Maximal subgroups

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

subgroupindex
#202826682
#202761872
#202761242
#202725402
#202725382
#202725372
C53×#20992
#780653