Finite Group Databasebeta

← #27273250

Subgroups — #27273250

Order 13568.

350 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 87 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#20276273C211yesmaximal, characteristic
2#20276186C211yesmaximal, characteristic
3#20276123C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272538C211yesmaximal, characteristic
7C53×#2805C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×#13133141—21no—
10#16810493C2211yescharacteristic
11#16810385—21no—
12#16810385—21no—
13#16810385—21no—
14#16810385—21no—
15#16810378C2211yescharacteristic
16#16810378C2211yescharacteristic
17#16810378C2211yescharacteristic
18C53×#400—21no—
19C53×#400—21no—
20C53×#395—21no—
21C53×#391C2211yescharacteristic
22C53×#389—21no—
23C53×#389C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#387—21no—
26C53×#387—21no—
27C22×#88322—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33#13133150—42no—
34#13133141—42no—
35C22×Q8×C53D411yescharacteristic
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C42×C53—21no—
46C2×C42×C53C2311yescharacteristic
47#95×C53—21no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56C2×Q8×C53—21no—
57C2×Q8×C53—21no—
58C2×Q8×C53—41no—
59C2×Q8×C53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—41no—
70C4⋊C4×C53—21no—
71C4⋊C4×C53—21no—
72C4⋊C4×C53—21no—
73C4⋊C4×C53—41no—
74C4⋊C4×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—81no—
81C4⋊C4×C53—81no—
82C4⋊C4×C53Q1612yes—
83C42×C53—41no—
84C42×C53—81no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90C2×#88322—22no—
91C2×#88322—41no—
92C2×#88322—41no—
93C2×#88322—41no—
94C2×#88322—41no—
95C2×#88322—41no—
96C2×#88322—41no—
97C2×#88322—41no—
98C2×#88322—41no—
99C23×C53#12011yescharacteristic
100Q8×C53—42no—
101Q8×C53—81no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
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—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53C2×Q1611yescharacteristic
127C2×C4×C53#13511yescharacteristic
128C2×C4×C53#13511yescharacteristic
129C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C53#13711yescharacteristic
132C2×Dic53—81no—
133C2×Dic53—81no—
134C2×Dic53—81no—
135#88322—42no—
136#88322—81no—
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—81no—
155C4×C53—81no—
156C4×C53—81no—
157C4×C53—81no—
158Dic53—161no—
159#2805D5311yescharacteristic
160#2099—531no—
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—21no—
176#400—21no—
177#395—21no—
178#391C2×D5311yescharacteristic
179#389—21no—
180#389C2×D5311yescharacteristic
181#389C2×D5311yescharacteristic
182#387—21no—
183#387—21no—
184#384—531no—
185#340—1061no—
186#340—1061no—
187#340—1061no—
188#340—1061no—
189#336—531no—
190#336—531no—
191#336—531no—
192C53#780611yescharacteristic, complemented, Sylow
193C22×Q8C22⋊D5311yescharacteristic
194C22×C8—1061no—
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—1061no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4C22⋊D5311yescharacteristic
205C2×C4⋊C4C22⋊D5311yescharacteristic
206C2×C4⋊C4C22⋊D5311yescharacteristic
207C2×C4⋊C4C22⋊D5311yescharacteristic
208C2×C4⋊C4C22⋊D5311yescharacteristic
209C2×C42—21no—
210C2×C42C22×D5311yescharacteristic
211#107—2122no—
212#103—2122no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—41no—
225C2×Q8—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—2121no—
233C22×C4C2×C22⋊D5311yescharacteristic
234C22×C4C2×C22⋊D5311yescharacteristic
235C22×C4C2×C22⋊D5311yescharacteristic
236C2×C8—1062no—
237C2×C8—2121no—
238C2×C8—2121no—
239C2×C8—2121no—
240C2×C8—2121no—
241C2×C8—2121no—
242C2×C8—2121no—
243C2×C8—2121no—
244C2×C8—2121no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—2122no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4#1171992512yes—
263C42—41no—
264C42—81no—
265C23#1313318611yescharacteristic
266Q8—42no—
267Q8—81no—
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—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—4241no—
293C2×C4—4241no—
294C2×C4—4241no—
295C2×C4C2×#1171992511yescharacteristic
296C2×C4#1313323711yescharacteristic
297C2×C4#1313323711yescharacteristic
298C2×C4#1313324211yescharacteristic
299C2×C4#1313324211yescharacteristic
300C2×C4#1313324811yescharacteristic
301C8—4241no—
302C8—4241no—
303C8—2122no—
304C22#1681076811yescharacteristic
305C22#1681076811yescharacteristic
306C22#1681077911yescharacteristic
307C22#1681079611yescharacteristic
308C22#1681080111yescharacteristic
309C22#1681080611yescharacteristic
310C22#1681081311yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—8481no—
324C2#2027371211yescharacteristic
325C2#2027372911yescharacteristic
326C2#2027375711yescharacteristic
327C2#2027382511yescharacteristic
328C2#2027392911yescharacteristic
329C2#2027395711yescharacteristic
330C2#2027404911yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2805C2
socle, prime radicalC23×C53#120
hypercenter#2805D53
hypocenterC53#7806
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2805

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
#202762732
#202761862
#202761232
#202725412
#202725382
C53×#28052
#780653