Finite Group Databasebeta

← #27293755

Subgroups — #27293755

Order 13568.

358 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 73 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#20279928C211yesmaximal, characteristic
2#20273057C211yesmaximal, characteristic
3#20273037C211yesmaximal, characteristic
4#20272659C211yesmaximal, characteristic
5#20272658C211yesmaximal, characteristic
6#20272658C211yesmaximal, characteristic
7C53×#1324C211yesmaximal, characteristic
8C2×#13133216C411yescharacteristic
9C2×#13133182C2211yescharacteristic
10#16810677—21no—
11#16810673—21no—
12#16810564—21no—
13#16810564—21no—
14#16810529C411yescharacteristic
15#16810528C411yescharacteristic
16#16810527—21no—
17#16810527—21no—
18#16810488C2211yescharacteristic
19C4×#11719919C411yescharacteristic
20#16810349C2211yescharacteristic
21#16810349C2211yescharacteristic
22#16810348—21no—
23#16810348—21no—
24C53×#382C2211yescharacteristic
25C53×#324C2211yescharacteristic
26C53×#324C2211yescharacteristic
27C22×Dic106—21no—
28C22×Dic106C2×C411yescharacteristic
29#13133216—41no—
30#13133216—41no—
31#13133182—41no—
32#13133182—41no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919D411yescharacteristic
37C2×#11719919D411yescharacteristic
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42#13133146—41no—
43#13133146—41no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46#13133127—41no—
47C8×Dic53—41no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C4⋊C4×C53C2311yescharacteristic
57C2×C42×C53C2×C411yescharacteristic
58C22⋊C8×C53—21no—
59C22⋊C8×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C411yescharacteristic
73C2×Dic106—21no—
74C2×Dic106—21no—
75C2×Dic106—22no—
76C2×Dic106—41no—
77C2×Dic106—44no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C42×C53—21no—
85C42×C53—21no—
86C4⋊Dic53—41no—
87#11719919—41no—
88#11719919—41no—
89#11719919—42no—
90#11719919—42no—
91C4×Dic53—21no—
92C4×Dic53—21no—
93C4×Dic53—21no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C2×#88322—41no—
100C2×#88322—41no—
101C23×C53C2×C22⋊C411yescharacteristic
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53#13111yescharacteristic
114C2×C4×C53#13111yescharacteristic
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—41no—
123C2×Dic53—42no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53#9911yescharacteristic
129C2×Dic53#9911yescharacteristic
130C2×Dic53#10112yes—
131Dic106—42no—
132Dic106—42no—
133Dic106—82no—
134Dic106—82no—
135C8×C53—81no—
136C8×C53—81no—
137#88322—81no—
138#88322—81no—
139#11573—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53#40811yescharacteristic
143C22×C53C2×#9911yescharacteristic
144C22×C53C2×#10111yescharacteristic
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53—81no—
150C4×C53—81no—
151Dic53—21no—
152Dic53—22no—
153Dic53—41no—
154Dic53—41no—
155Dic53—81no—
156Dic53—81no—
157Dic53#35512yes—
158#2568—531no—
159#1467—531no—
160#1461—531no—
161#1324—531no—
162#1324—531no—
163#1324—531no—
164#1324D5311yescharacteristic
165C2×C53—21no—
166C2×C53—21no—
167C2×C53#147211yescharacteristic
168C2×C53#212911yescharacteristic
169C2×C53C2×#35511yescharacteristic
170C2×C4⋊Q8—531no—
171C2×C4×Q8—531no—
172#432—1061no—
173#431—1061no—
174#400—1061no—
175#400—1061no—
176#391—531no—
177#391—531no—
178#391—1061no—
179#391—1061no—
180#382—531no—
181#382C2×D5311yescharacteristic
182C4×C4⋊C4—531no—
183#324—531no—
184#324—531no—
185#324—1061no—
186#324—1061no—
187#324C2×D5311yescharacteristic
188#324C2×D5311yescharacteristic
189C53#1157311yescharacteristic, complemented, Sylow
190C22×Q8—531no—
191C22×Q8—1061no—
192C4⋊Q8—2121no—
193C4⋊Q8—2121no—
194C4×Q8—2121no—
195C4×Q8—2121no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4C4×D5311yescharacteristic
202C2×C4⋊C4C22×D5311yescharacteristic
203C2×C42—531no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42C4×D5311yescharacteristic
208C4⋊C8—2121no—
209C4⋊C8—2121no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—1061no—
213C22⋊C8—1061no—
214#97—2121no—
215C4×C8—2121no—
216#95—21no—
217#95—21no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225C2×Q8—1062no—
226C2×Q8—1061no—
227C2×Q8—1061no—
228C2×Q8—2121no—
229C2×Q8—2121no—
230C2×Q8—2121no—
231C2×Q8—2124no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4C2×C4×D5311yescharacteristic
241C22×C4D4×D5311yescharacteristic
242C22×C4D4×D5311yescharacteristic
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—2121no—
246C2×C8—2121no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—2121no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C4⋊C4—2122no—
253C4⋊C4—2122no—
254C42—21no—
255C42—21no—
256C42—1061no—
257C42—1061no—
258C42—1061no—
259C42—1061no—
260C42—1061no—
261C42—1061no—
262C42—1061no—
263C42—1061no—
264C23C22⋊C4×D5311yescharacteristic
265Q8—4242no—
266Q8—4242no—
267Q8—2122no—
268Q8—2122no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—531no—
281C2×C4—531no—
282C2×C4—532no—
283C2×C4—1062no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2122no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4#1313322311yescharacteristic
297C2×C4#1313322511yescharacteristic
298C8—81no—
299C8—81no—
300C8—4241no—
301C8—4241no—
302C22—21no—
303C22—21no—
304C22#1681058811yescharacteristic
305C22#99×D5311yescharacteristic
306C22#101×D5311yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—532no—
314C4—1061no—
315C4—1062no—
316C4—4241no—
317C4—4241no—
318C4—2121no—
319C4—2121no—
320C2—21no—
321C2—21no—
322C2#2027307711yescharacteristic
323C2#2027638811yescharacteristic
324C2#355×D5311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1324C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1324D53
hypocenterC53#11573
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
189. C53#11573semidirect

Maximal subgroups

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

subgroupindex
#202799282
#202730572
#202730372
#202726592
#202726582
C53×#13242
#1157353