Finite Group Databasebeta

← #27299263

Subgroups — #27299263

Order 13568.

564 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 123 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#20277427C211yesmaximal, characteristic
2C2×#16810417C212yesmaximal
3C4×#13133132C211yesmaximal, characteristic
4#20272839C212yesmaximal
5C4×#99×C53C211yesmaximal, characteristic
6C2×#13133204C2211yescharacteristic
7C2×C4×Dic106C411yescharacteristic, complemented
8C2×#13133132C2212yes—
9#16810528C411yescharacteristic, complemented
10#16810526C2211yescharacteristic
11#16810511—21no—
12#16810511—21no—
13#16810496—21no—
14#16810496—21no—
15#16810492C411yescharacteristic, complemented
16C4×C22⋊Dic53—21no—
17#16810469C411yescharacteristic, complemented
18#16810417C418yescomplemented
19#16810390—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#99×C53C2212yes—
22C4×C22⋊C4×C53—21no—
23C53×#342—21no—
24C22×Dic106C2×C411yescharacteristic
25#13133204—22no—
26#13133204C2×C414yes—
27C4×Dic106—41no—
28C4×Dic106—41no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—22no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53—41no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42#13133132—22no—
43#13133132C2×C414yes—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—41no—
49C22×D4×C53C2311yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C4×D4×C53—41no—
53C4×D4×C53—41no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53—41no—
59C2×C42×C53C2×C411yescharacteristic
60#99×C53—22no—
61#99×C53C2×C414yes—
62#95×C53—41no—
63C24×C53C2×D411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53C22×C412yes—
68C2×D4×C53#4112yes—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—42no—
74C22×C4×C53—81no—
75C22×C4×C53—81no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—42no—
86C2×Dic106—21no—
87C2×Dic106—41no—
88C2×Dic106—41no—
89C2×Dic106C4212yescomplemented
90C4⋊C4×C53—41no—
91C22⋊C4×C53—24no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—42no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—82no—
98C22⋊Dic53—24no—
99C22⋊Dic53—24no—
100C22⋊Dic53—24no—
101C22⋊Dic53—42no—
102C22⋊Dic53—81no—
103C22⋊Dic53—81no—
104C4⋊Dic53—41no—
105#11719919—41no—
106#11719919—41no—
107#11719919—81no—
108#11719919—81no—
109C4×Dic53—21no—
110C4×Dic53—42no—
111C4×Dic53—41no—
112C4×Dic53—82no—
113C4×Dic53C4212yescomplemented
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53C2×C22⋊C411yescharacteristic
123C23×C53C4×D412yes—
124C23×C53C4×D412yes—
125D4×C53—42no—
126D4×C53—42no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—44no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53—82no—
142C2×C4×C53—82no—
143C2×C4×C53#9911yescharacteristic
144C2×C4×C53#9911yescharacteristic
145C2×C4×C53C2×C4211yescharacteristic
146C2×C4×C53#11711yescharacteristic
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—44no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—82no—
161C2×Dic53—82no—
162Dic106—42no—
163Dic106—42no—
164#12627—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53C4×C22⋊C411yescharacteristic
178C22×C53C2×#9911yescharacteristic
179C22×C53#41011yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—84no—
188Dic53—42no—
189Dic53—42no—
190Dic53—81no—
191Dic53—84no—
192Dic53—81no—
193#2326—531no—
194C2×#354—532no—
195C4×#99—531no—
196C4×#99D5311yescharacteristic
197#1393—532no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—42no—
201C2×C53—42no—
202C2×C53C4×#9911yescharacteristic
203C2×C53#212711yescharacteristic
204C2×C53#212911yescharacteristic
205C2×#124—531no—
206C2×C4×Q8—531no—
207C2×C4×D4C2×D5311yescharacteristic
208C2×#99—532no—
209C2×#99C2×D5312yes—
210#391—531no—
211#390—531no—
212#388—1061no—
213#388—1061no—
214#385—1061no—
215#385—1061no—
216#384—531no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—1061no—
219#376—531no—
220#354—538no—
221#342—21no—
222#342—1061no—
223C53#1262711yescharacteristic, complemented, Sylow
224C22×Q8—531no—
225C22×D4C22×D5311yescharacteristic
226C23×C4D21211yescharacteristic
227C23×C4C22⋊D5311yescharacteristic
228#124—534no—
229#124—1062no—
230C4×Q8—2121no—
231C4×Q8—2121no—
232C4×D4—41no—
233C4×D4—41no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4C4×D5311yescharacteristic
239C2×C22⋊C4—22no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1062no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4D21211yescharacteristic
246C2×C22⋊C4C22⋊D5311yescharacteristic
247C2×C42—41no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—1061no—
251C2×C42—2121no—
252C2×C42C4×D5311yescharacteristic
253#99—22no—
254#99—534no—
255#99—1062no—
256#99C4×D5314yes—
257#95—41no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262#95—2121no—
263C24C2×D21211yescharacteristic
264C24C2×C22⋊D5311yescharacteristic
265C2×Q8—532no—
266C2×Q8—1061no—
267C2×Q8—2121no—
268C2×Q8—2121no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4C2×C4×D5312yes—
272C2×D4#1171994512yes—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—2122no—
285C22×C4#1171992111yescharacteristic
286C22×C4#1171992111yescharacteristic
287C22×C4#1171992111yescharacteristic
288C22×C4#1171992111yescharacteristic
289C22×C4C2×C4×D5311yescharacteristic
290C4⋊C4—41no—
291C4⋊C4—4241no—
292C4⋊C4—4241no—
293C4⋊C4—2121no—
294C4⋊C4—2121no—
295C4⋊C4—2121no—
296C22⋊C4—24no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—1064no—
301C22⋊C4—1064no—
302C22⋊C4—1064no—
303C22⋊C4—4241no—
304C22⋊C4—4241no—
305C22⋊C4—2122no—
306C42—21no—
307C42—21no—
308C42—82no—
309C42—532no—
310C42—1061no—
311C42—4242no—
312C42—2121no—
313C42—2122no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23C2×#1171992111yescharacteristic
323C23C4×C22⋊D5312yes—
324C23C4×D21212yes—
325Q8—2122no—
326Q8—2122no—
327D4—42no—
328D4—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—44no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—4242no—
348C2×C4—4242no—
349C2×C4—2124no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4#1313313111yescharacteristic
361C2×C4#1313313111yescharacteristic
362C2×C4C42×D5311yescharacteristic
363C2×C4#1313317411yescharacteristic
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—81no—
376C22C4×#1171992111yescharacteristic
377C22C2×#1313313111yescharacteristic
378C22#1681059611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—84no—
387C4—4241no—
388C4—4244no—
389C4—4241no—
390C4—2122no—
391C4—2122no—
392C2—21no—
393C2—21no—
394C2—42no—
395C2—42no—
396C2C4×#1313313111yescharacteristic
397C2#2027638211yescharacteristic
398C2#2027638911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC4×#99D53
hypocenterC53#12627
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
7. C2×C4×Dic106C4semidirect
9. #16810528C4semidirect
15. #16810492C4semidirect
17. #16810469C4semidirect
18. #16810417C4semidirect
89. C2×Dic106C42semidirect
113. C4×Dic53C42semidirect
223. C53#12627semidirect

Maximal subgroups

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

subgroupindex
#202774272
C2×#168104172
C4×#131331322
#202728392
C4×#99×C532
#1262753