Finite Group Databasebeta

← #27279096

Subgroups — #27279096

Order 13568.

836 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16810535C211yesmaximal, characteristic
2#20274649C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272593C212yesmaximal, complemented
5C53×#2312C211yesmaximal, characteristic
6C23×Dic106C2211yescharacteristic
7C22×C22⋊Dic53—21no—
8C2×#13133140—24no—
9C2×C22⋊#88322C2212yes—
10#16810535—22no—
11#16810535C414yescomplemented
12C2×#13133124—21no—
13C2×#13133124C2211yescharacteristic
14#16810347—22no—
15C22×C22⋊C4×C53C2211yescharacteristic
16C53×#388—24no—
17C53×#380C2212yes—
18C53×#379—21no—
19C4×C22⋊C4×C53—21no—
20C22×Dic106—22no—
21C22×Dic106—22no—
22C22×Dic106—21no—
23C22×Dic106—44no—
24C22×Dic106C2×C412yes—
25C23×Dic53—21no—
26C23×Dic53—21no—
27C22×#88322—22no—
28C2×C22⋊Dic53—24no—
29C2×C22⋊Dic53—44no—
30#13133140—48no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133124—24no—
34#13133124—21no—
35#13133124C2×C412yes—
36C25×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39C2×C4⋊C4×C53D414yes—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53C2×C412yes—
46C2×C22⋊C4×C53D412yes—
47C2×C42×C53—22no—
48#95×C53—22no—
49#95×C53—22no—
50C24×C53—21no—
51C24×C53—22no—
52C24×C53—21no—
53C24×C53—22no—
54C24×C53—42no—
55C24×C53—42no—
56C24×C53C22⋊C412yes—
57C24×C53C2×D411yescharacteristic
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—24no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D412yes—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4112yes—
68C22×Dic53—24no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—42no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76C22×Dic53—42no—
77C2×Dic106—28no—
78C2×Dic106—24no—
79C2×Dic106—44no—
80C2×Dic106—48no—
81C2×Dic106—44no—
82C2×Dic106—44no—
83C2×Dic106—42no—
84C4⋊C4×C53—44no—
85C22⋊C4×C53—24no—
86C22⋊C4×C53—24no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—82no—
90C22⋊C4×C53—82no—
91C42×C53—82no—
92C22⋊Dic53—48no—
93C2×#88322—44no—
94C2×#88322—42no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—48no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53—81no—
108C23×C53—81no—
109C23×C53—84no—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53#12011yescharacteristic
112C23×C53#12311yescharacteristic
113C23×C53#13512yes—
114C23×C53#13712yes—
115C2×C4×C53—24no—
116C2×C4×C53—22no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—44no—
123C2×C4×C53—44no—
124C2×C4×C53—44no—
125C2×C4×C53—81no—
126C2×C4×C53C4×D412yes—
127C2×C4×C53#12112yes—
128C2×Dic53—24no—
129C2×Dic53—42no—
130C2×Dic53—44no—
131C2×Dic53—42no—
132C2×Dic53—44no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—84no—
137Dic106—28no—
138Dic106—48no—
139Dic106—44no—
140#88322—82no—
141#9016—531nomaximal, Sylow
142C22×C53—24no—
143C22×C53—24no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—44no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—82no—
153C22×C53—84no—
154C22×C53—84no—
155C22×C53#38611yescharacteristic
156C22×C53#41611yescharacteristic
157C22×C53#41911yescharacteristic
158C22×C53#45212yes—
159C22×C53#45512yes—
160C22×C53B(4,𝔽2)12yes—
161C22×C53#48512yes—
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—42no—
165C4×C53—84no—
166Dic53—44no—
167Dic53—82no—
168C2×#393—531no—
169#2312D5311yescharacteristic
170#1787—532no—
171C2×#323—531no—
172#1294—532no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—22no—
176C2×C53—82no—
177C2×C53#160614yes—
178C2×C53#187311yescharacteristic
179C2×C53#188511yescharacteristic
180C2×C53#188911yescharacteristic
181C23×Q8—531no—
182C22×C22⋊C4—1061no—
183C22×C22⋊C4C2×D5311yescharacteristic
184C2×#103—1064no—
185C2×C22⋊C8—532no—
186#393—534no—
187#393—1062no—
188#388—24no—
189#380C2×D5312yes—
190#379—21no—
191C4×C22⋊C4—21no—
192C2×#95—531no—
193C2×#95—1061no—
194#323—1062no—
195C53#901611yescharacteristic, complemented, Sylow
196C25C22⋊D5311yescharacteristic
197C22×Q8—532no—
198C22×Q8—1061no—
199C22×Q8—1062no—
200C22×Q8—1062no—
201C22×Q8—2124no—
202C23×C4—1061no—
203C23×C4—1061no—
204C23×C4C22⋊D5311yescharacteristic
205C23×C4C22×D5311yescharacteristic
206C22×C8—1062no—
207C2×C4⋊C4C22⋊D5314yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1064no—
214C2×C22⋊C4—2124no—
215C2×C22⋊C4C4×D5312yes—
216C2×C22⋊C4D21212yes—
217C2×C42—22no—
218#103—2128no—
219C22⋊C8—1062no—
220C22⋊C8—1062no—
221#95—22no—
222#95—22no—
223#95—532no—
224#95—1061no—
225#95—1064no—
226C24—21no—
227C24—22no—
228C24—21no—
229C24—22no—
230C24—42no—
231C24—42no—
232C24#1171992112yes—
233C24C2×C22⋊D5311yescharacteristic
234C2×Q8—1068no—
235C2×Q8—1064no—
236C2×Q8—2124no—
237C2×Q8—2124no—
238C2×Q8—2124no—
239C2×Q8—2128no—
240C2×Q8—2122no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—1064no—
247C22×C4—1062no—
248C22×C4—1062no—
249C22×C4—2122no—
250C22×C4—2122no—
251C22×C4—2122no—
252C22×C4—2122no—
253C22×C4—2121no—
254C22×C4—2121no—
255C22×C4#1171992112yes—
256C22×C4C2×C4×D5311yescharacteristic
257C22×C4C2×D21211yescharacteristic
258C22×C4#1171994512yes—
259C22×C4C2×C22⋊D5312yes—
260C2×C8—2124no—
261C2×C8—2122no—
262C4⋊C4—44no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C22⋊C4—2128no—
270C42—82no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—81no—
284C23—81no—
285C23—84no—
286C23C2×#1171992111yescharacteristic
287C23#1313318611yescharacteristic
288C23#1313320011yescharacteristic
289C23#1313323712yes—
290C23#1313324712yes—
291Q8—1068no—
292Q8—2128no—
293Q8—2124no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—81no—
305C2×C4—1064no—
306C2×C4—4244no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4—2124no—
312C2×C4—2122no—
313C2×C4—2124no—
314C2×C4C4×C22⋊D5312yes—
315C2×C4#1313319112yes—
316C8—4242no—
317C22—24no—
318C22—24no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—84no—
330C22#1681050211yescharacteristic
331C22#1681061711yescharacteristic
332C22#1681063111yescharacteristic
333C22#1681078512yes—
334C22#1681080112yes—
335C22#1681081012yes—
336C22#1681096412yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—84no—
341C4—4242no—
342C4—2124no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#2027369514yes—
348C2#2027502711yescharacteristic
349C2#2027510311yescharacteristic
350C2#2027512311yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2312C2
socleC23×C53#120
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#2312D53
hypocenterC53#9016
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #20274649C2semidirect
4. #20272593C2semidirect
11. #16810535C4semidirect
195. C53#9016semidirect

Maximal subgroups

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

subgroupindex
C2×#168105352
#202746492
C2×#168103472
#202725932
C53×#23122
#901653