Finite Group Databasebeta

← #27282012

Subgroups — #27282012

Order 13568.

956 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277293C211yesmaximal, characteristic
2#20274789C212yesmaximal, complemented
3#20274630C211yesmaximal, characteristic
4#20272619C212yesmaximal, complemented
5C53×#2405C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7C2×#13133132—22no—
8#16810503—24no—
9#16810486—21no—
10#16810479C2212yes—
11#16810479C2212yes—
12#16810390—24no—
13C23×D4×C53C2211yescharacteristic
14C2×#123×C53C412yescomplemented
15C22×C22⋊C4×C53C2211yescharacteristic
16C53×#394—22no—
17C53×#386C412yescomplemented
18C53×#385—21no—
19C53×#379—21no—
20C53×#379C2211yescharacteristic
21C23×Dic53—22no—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—44no—
29C2×C22⋊Dic53D414yes—
30#13133132—44no—
31#13133124—24no—
32C25×C53D411yescharacteristic
33C25×C53C2311yescharacteristic
34C22×D4×C53—22no—
35C22×D4×C53—22no—
36C22×D4×C53—44no—
37C22×D4×C53C2×C412yes—
38C22×D4×C53D412yes—
39C23×C4×C53C2×C411yescharacteristic, complemented
40C23×C4×C53C2×C411yescharacteristic, complemented
41C23×C4×C53D411yescharacteristic
42#123×C53—44no—
43C2×C4⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53C2×C412yescomplemented
53#95×C53—22no—
54C24×C53—21no—
55C24×C53—24no—
56C24×C53—21no—
57C24×C53—42no—
58C24×C53—42no—
59C24×C53—42no—
60C24×C53—81no—
61C24×C53—81no—
62C24×C53C22⋊C412yes—
63C24×C53C22×C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C24×C53C2×D411yescharacteristic
66C24×C53C2×D412yes—
67C24×C53#4112yes—
68C2×D4×C53—24no—
69C2×D4×C53—24no—
70C2×D4×C53—24no—
71C2×D4×C53—24no—
72C2×D4×C53—42no—
73C2×D4×C53—42no—
74C2×D4×C53—44no—
75C2×D4×C53—44no—
76C2×D4×C53—48no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53—81no—
87C22×C4×C53C22⋊C412yes—
88C22×Dic53—24no—
89C22×Dic53—44no—
90C22×Dic53—42no—
91C22×Dic53—82no—
92C22×Dic53—82no—
93C4⋊C4×C53—44no—
94C22⋊C4×C53—42no—
95C22⋊C4×C53—44no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—42no—
100C22⋊C4×C53—82no—
101C22⋊C4×C53—81no—
102C22⋊Dic53—44no—
103C22⋊Dic53—44no—
104C22⋊Dic53—44no—
105C22⋊Dic53—44no—
106C22⋊Dic53—44no—
107C22⋊Dic53—82no—
108C22⋊Dic53—82no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—24no—
112C23×C53—22no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—22no—
118C23×C53—24no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—41no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—41no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—42no—
132C23×C53—42no—
133C23×C53—81no—
134C23×C53—81no—
135C23×C53—88no—
136C23×C53—88no—
137C23×C53#9914yes—
138C23×C53C2×C22⋊C411yescharacteristic
139C23×C53C4×D412yes—
140C23×C53#12011yescharacteristic
141C23×C53#12112yes—
142C23×C53#12311yescharacteristic
143D4×C53—48no—
144D4×C53—48no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×Dic53—44no—
160C2×Dic53—44no—
161C2×Dic53—44no—
162C2×Dic53—88no—
163C2×Dic53—82no—
164C2×Dic53—82no—
165C2×Dic53—82no—
166#9499—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—22no—
172C22×C53—21no—
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—21no—
176C22×C53—22no—
177C22×C53—24no—
178C22×C53—22no—
179C22×C53—22no—
180C22×C53—21no—
181C22×C53—44no—
182C22×C53—44no—
183C22×C53—44no—
184C22×C53—44no—
185C22×C53—44no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—84no—
189C22×C53—82no—
190C22×C53—82no—
191C22×C53—84no—
192C22×C53—84no—
193C22×C53#38611yescharacteristic
194C22×C53C2×#9912yes—
195C22×C53#45812yes—
196C4×C53—42no—
197C4×C53—82no—
198C4×C53—81no—
199C4×C53—81no—
200Dic53—84no—
201Dic53—162no—
202#2405D5311yescharacteristic
203#2308—531no—
204#1819—532no—
205#1781—531no—
206#1304—532no—
207C2×C53—22no—
208C2×C53—22no—
209C2×C53—42no—
210C2×C53—42no—
211C2×C53—41no—
212C2×C53—44no—
213C2×C53#189011yescharacteristic
214C2×C53#189811yescharacteristic
215C2×C53#191311yescharacteristic
216C23×D4C2×D5311yescharacteristic
217C2×#123Dic5312yescomplemented
218C22×C22⋊C4—1061no—
219C22×C22⋊C4C2×D5311yescharacteristic
220C2×#99—1062no—
221#394—22no—
222#386—1064no—
223#386Dic5312yescomplemented
224#385—21no—
225#381—1061no—
226#379—21no—
227#379—532no—
228#379—532no—
229#379C2×D5311yescharacteristic
230#342—1064no—
231C53#949911yescharacteristic, complemented, Sylow
232C25C22⋊D5311yescharacteristic
233C25C22×D5311yescharacteristic
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—44no—
237C22×D4C2×Dic5312yes—
238C22×D4C22⋊D5312yes—
239C23×C4—1062no—
240C23×C4C2×Dic5311yescharacteristic, complemented
241C23×C4C2×Dic5311yescharacteristic, complemented
242C23×C4C22⋊D5311yescharacteristic
243#123—44no—
244C2×C4⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—534no—
254C2×C22⋊C4—1062no—
255C2×C22⋊C4—1062no—
256C2×C22⋊C4—1062no—
257C2×C22⋊C4—1062no—
258C2×C22⋊C4—1062no—
259C2×C22⋊C4—1062no—
260C2×C22⋊C4—2124no—
261C2×C22⋊C4C2×Dic5312yescomplemented
262#99—2124no—
263#95—22no—
264#95—1064no—
265C24—21no—
266C24—24no—
267C24—21no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—81no—
272C24—81no—
273C24C22⋊Dic5312yes—
274C24D4×D5312yes—
275C24#1171994712yes—
276C24C22×Dic5311yescharacteristic
277C24C2×C22⋊D5311yescharacteristic
278C24C2×C22⋊D5311yescharacteristic
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—48no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—1064no—
299C22×C4—4242no—
300C22×C4—4242no—
301C22×C4—2122no—
302C22×C4—2124no—
303C22×C4C22⋊Dic5312yes—
304C4⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—41no—
308C22⋊C4—42no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—81no—
312C22⋊C4—82no—
313C22⋊C4—4242no—
314C22⋊C4—4242no—
315C22⋊C4—2124no—
316C22⋊C4—2124no—
317C22⋊C4—2124no—
318C22⋊C4—2124no—
319C22⋊C4—2124no—
320C23—22no—
321C23—22no—
322C23—24no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—24no—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23—88no—
347C23—88no—
348C23#1313313214yes—
349C23C2×C22⋊Dic5311yescharacteristic
350C23D4×Dic5312yes—
351C23#1313318511yescharacteristic
352C23#1313318912yes—
353C23#1313319811yescharacteristic
354D4—48no—
355D4—48no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—4242no—
371C2×C4—4242no—
372C2×C4—4242no—
373C2×C4—4248no—
374C2×C4—2124no—
375C2×C4—2124no—
376C2×C4—2124no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—21no—
386C22—22no—
387C22—21no—
388C22—22no—
389C22—22no—
390C22—24no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—44no—
396C22—42no—
397C22—42no—
398C22—82no—
399C22—82no—
400C22—84no—
401C22—84no—
402C22—84no—
403C22#1681050311yescharacteristic
404C22C2×#1313313212yes—
405C22#1681081212yes—
406C4—42no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4—8482no—
411C4—4244no—
412C2—22no—
413C2—22no—
414C2—42no—
415C2—41no—
416C2—42no—
417C2—44no—
418C2#2027512911yescharacteristic
419C2#2027516911yescharacteristic
420C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC53×#2405C2
socleC22×C53#386
prime radicalC23×D4×C53C22
cosocleC25×C53C23
hypercenter#2405D53
hypocenterC53#9499
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #2405

Complements

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

NH ≅ G/Ntype
2. #20274789C2semidirect
4. #20272619C2semidirect
14. C2×#123×C53C4semidirect
17. C53×#386C4semidirect
39. C23×C4×C53C2×C4semidirect
40. C23×C4×C53C2×C4semidirect
52. C2×C22⋊C4×C53C2×C4semidirect
217. C2×#123Dic53semidirect
223. #386Dic53semidirect
231. C53#9499semidirect
240. C23×C4C2×Dic53semidirect
241. C23×C4C2×Dic53semidirect
261. C2×C22⋊C4C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202772932
#202747892
#202746302
#202726192
C53×#24052
#949953