Finite Group Databasebeta

← #27275455

Subgroups — #27275455

Order 13568.

874 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C22×#13133212C211yesmaximal, characteristic
2#20274781C212yesmaximal, complemented
3C2×#16810349C211yesmaximal, characteristic, complemented
4C2×#16810346C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic, complemented
6C2×#13133212—24no—
7C2×#13133212—24no—
8C2×#13133212—22no—
9C2×#13133212—22no—
10C22×C4⋊Dic53C411yescharacteristic
11C22×#11719919C411yescharacteristic
12C22×#11719921C411yescharacteristic
13C22×#11719921C411yescharacteristic
14C22×#11719921C2211yescharacteristic
15C22×C4×Dic53C2211yescharacteristic
16C2×C22⋊#88322C2212yes—
17#16810377—24no—
18#16810349—22no—
19#16810346—24no—
20C22×C4⋊C4×C53C2211yescharacteristic
21C2×C22⋊C8×C53C2212yes—
22C53×#324—22no—
23C24×D53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C22×#88322—22no—
28#13133212—216no—
29#13133212—216no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—22no—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719921—22no—
37C2×#11719921—22no—
38C2×#11719921—22no—
39C2×#11719921—22no—
40C2×#11719921—22no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×#11719921—22no—
44C2×#11719921—22no—
45C2×#11719921D414yes—
46C2×C4×Dic53—21no—
47C2×C4×Dic53—22no—
48C2×C4×Dic53—21no—
49C2×C4×Dic53D414yes—
50C22⋊#88322—22no—
51C22⋊#88322—22no—
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54C23×C4×C53C2311yescharacteristic
55C22×C8×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—22no—
58C2×C4⋊C4×C53D412yes—
59C2×C4⋊C4×C53D412yes—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62C24×C53C22×C411yescharacteristic
63C23×D53—21no—
64C23×D53—21no—
65C23×D53—22no—
66C23×D53—22no—
67C23×D53—42no—
68C23×D53—42no—
69C23×D53C22⋊C412yes—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53C22⋊C412yes—
79C22×C4×C53C22⋊C412yes—
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—22no—
91C22×Dic53—22no—
92C22×Dic53—42no—
93C22×Dic53—42no—
94C22×Dic53C22⋊C412yes—
95C22×Dic53C22⋊C412yes—
96C22×Dic53C22⋊C412yes—
97C2×C8×C53—44no—
98C2×C8×C53—42no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53—24no—
101#11719921—28no—
102#11719921—28no—
103#11719921—24no—
104#11719921—24no—
105C4⋊Dic53—28no—
106#11719919—28no—
107C4×Dic53—24no—
108C4×Dic53—24no—
109C2×#88322—44no—
110C2×#88322—42no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53#13112yes—
117C23×C53#13112yes—
118C22×D53—22no—
119C22×D53—24no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—42no—
126C22×D53—42no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—84no—
130C22×D53—84no—
131C2×C4×C53—24no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53#12012yes—
145C2×C4×C53#12012yes—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—24no—
150C2×Dic53—24no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C8×C53—82no—
160#88322—82no—
161#8309—531nomaximal, Sylow
162C22×C53—24no—
163C22×C53—24no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53#37911yescharacteristic
169C22×C53#40812yes—
170C22×C53#40812yes—
171C22×C53#40811yescharacteristic
172C22×C53#40811yescharacteristic
173C22×C53#41012yes—
174C22×C53#41312yes—
175C2×D53—42no—
176C2×D53—42no—
177C2×D53—42no—
178C2×D53—82no—
179C2×D53—82no—
180C2×D53—84no—
181C2×D53—84no—
182C4×C53—22no—
183C4×C53—22no—
184C4×C53—42no—
185C4×C53—42no—
186Dic53—42no—
187Dic53—42no—
188Dic53—42no—
189C22×#126—531no—
190#1817—532no—
191C2×#324—531no—
192C2×#324D5311yescharacteristic, complemented
193C2×#323—532no—
194C2×C53—22no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53#147314yes—
198C2×C53#178011yescharacteristic
199C2×C53#178211yescharacteristic
200C2×C53#178511yescharacteristic
201D53—82no—
202C2×#126—1064no—
203C2×#126—1064no—
204C2×#126—1062no—
205C2×#126—1062no—
206C22×C4⋊C4—531no—
207C22×C4⋊C4—531no—
208C22×C4⋊C4C2×D5311yescharacteristic
209C22×C22⋊C4—531no—
210C22×C22⋊C4—531no—
211C22×C22⋊C4—531no—
212C22×C42—531no—
213C2×C22⋊C8—532no—
214C2×C22⋊C8C2×D5312yes—
215#336—1064no—
216#324—22no—
217#324—1062no—
218#323—1064no—
219C53#830911yescharacteristic, complemented, Sylow
220C25—531no—
221C23×C4—531no—
222C23×C4—531no—
223C23×C4—531no—
224C23×C4C4×D5311yescharacteristic
225C23×C4C4×D5311yescharacteristic
226C23×C4C22×D5311yescharacteristic
227C22×C8—22no—
228C22×C8—1062no—
229#126—10616no—
230#126—10616no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—1062no—
234C2×C4⋊C4—1062no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1062no—
237C2×C4⋊C4—1062no—
238C2×C4⋊C4—1062no—
239C2×C4⋊C4D21212yes—
240C2×C4⋊C4C22⋊D5312yes—
241C2×C22⋊C4—534no—
242C2×C22⋊C4—1062no—
243C2×C22⋊C4—1062no—
244C2×C22⋊C4—1062no—
245C2×C22⋊C4—1062no—
246C2×C22⋊C4—1061no—
247C2×C22⋊C4—1062no—
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1062no—
250C2×C22⋊C4—1062no—
251C2×C42—534no—
252C2×C42—1061no—
253C2×C42—1062no—
254C2×C42—1061no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—1062no—
258C22⋊C8—1062no—
259C24—532no—
260C24—1062no—
261C24—1062no—
262C24—1061no—
263C24—1061no—
264C24—2122no—
265C24—2122no—
266C24C2×C4×D5311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—532no—
276C22×C4—532no—
277C22×C4—532no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1062no—
285C22×C4—1062no—
286C22×C4—2122no—
287C22×C4—2122no—
288C22×C4#1171992112yes—
289C22×C4#1171992112yes—
290C22×C4C2×D21211yescharacteristic
291C22×C4D4×D5312yes—
292C22×C4D4×D5312yes—
293C22×C4C2×C22⋊D5311yescharacteristic
294C2×C8—44no—
295C2×C8—42no—
296C2×C8—2124no—
297C2×C8—2122no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—1068no—
301C4⋊C4—1068no—
302C22⋊C4—1068no—
303C22⋊C4—1068no—
304C22⋊C4—1064no—
305C22⋊C4—1064no—
306C42—1064no—
307C42—1064no—
308C23—22no—
309C23—22no—
310C23—1064no—
311C23—1062no—
312C23—4244no—
313C23—4244no—
314C23—2122no—
315C23—2122no—
316C23—2122no—
317C23—2122no—
318C23—2122no—
319C23—2121no—
320C23—2121no—
321C23—2122no—
322C23—2122no—
323C23C22⋊C4×D5311yescharacteristic
324C23C22⋊C4×D5311yescharacteristic
325C23C2×#1171992111yescharacteristic
326C23#1313322312yes—
327C23#1313322512yes—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—1064no—
342C2×C4—1064no—
343C2×C4—1062no—
344C2×C4—1062no—
345C2×C4—1062no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4—2122no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4#1313318512yes—
355C2×C4#1313318712yes—
356C8—82no—
357C8—4242no—
358C22—24no—
359C22—24no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—4244no—
365C22—4242no—
366C22—4242no—
367C22—4244no—
368C22—2122no—
369C22—2122no—
370C22—2122no—
371C22#1681047811yescharacteristic
372C22#1681058812yes—
373C22#1681058812yes—
374C22#1681058911yescharacteristic
375C22#1681059111yescharacteristic
376C22#1681059512yes—
377C22#1681060612yes—
378C4—22no—
379C4—22no—
380C4—42no—
381C4—42no—
382C4—2122no—
383C4—2122no—
384C4—2122no—
385C2—22no—
386C2—21no—
387C2—21no—
388C2—4242no—
389C2#2027308314yes—
390C2#2027462611yescharacteristic
391C2#2027463111yescharacteristic
392C2#2027464211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#8309
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20274781C2semidirect
3. C2×#16810349C2semidirect
5. C2×C53×#324C2semidirect
192. C2×#324D53semidirect
219. C53#8309semidirect

Maximal subgroups

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

subgroupindex
C22×#131332122
#202747812
C2×#168103492
C2×#168103462
C2×C53×#3242
#830953