Finite Group Databasebeta

← #28307724

Subgroups — #28307724

Order 13568.

1426 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1426 conjugacy classes of subgroups — too many to draw. Only its 176 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×#16811501C212yesmaximal
2C2×#16811501C212yesmaximal
3#20289889C212yesmaximal
4C22×#13133195C211yesmaximal, characteristic
5C22×C22⋊#6157C2214yes—
6C2×(C2×#6157)⋊C4—24no—
7#16811501—24no—
8#16811501—24no—
9C23×Dic106C411yescharacteristic, complemented
10C23×C4×D53C2211yescharacteristic
11C2×#13133195—24no—
12C2×#13133195—24no—
13C2×#13133195—24no—
14C22×#11719919C411yescharacteristic, complemented
15C22×#11719919C411yescharacteristic, complemented
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2211yescharacteristic
18C22×C4⋊C4×C53C411yescharacteristic, complemented
19C23×#6157—24no—
20C2×C22⋊#6157—24no—
21C2×C22⋊#6157—24no—
22C2×C22⋊#6157—24no—
23C2×C22⋊#6157—24no—
24C2×C22⋊#6157—24no—
25C2×C22⋊#6157Q818yescomplemented
26(C2×#6157)⋊C4—48no—
27C24×D53C2311yescharacteristic
28C22×Dic106—22no—
29C22×Dic106—22no—
30C22×Dic106—22no—
31C22×Dic106—44no—
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2×C411yescharacteristic
35C22×C4×D53—22no—
36C22×C4×D53—22no—
37C22×C4×D53—44no—
38C22×C4×D53D414yes—
39#13133195—232no—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719919—22no—
43C2×#11719919—22no—
44C2×#11719919—22no—
45C2×#11719919—22no—
46C2×#11719921—22no—
47C2×#11719921—22no—
48C2×#11719921—22no—
49C2×#11719921—22no—
50C2×#11719921D414yes—
51C2×#11719921D414yes—
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54C23×C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—22no—
58C24×C53C22×C411yescharacteristic
59C23×D53—42no—
60C23×D53—42no—
61C23×D53C2×D412yes—
62C23×D53C2×D412yes—
63C23×D53C2×D412yes—
64C23×D53C2×Q814yes—
65C23×D53#4114yes—
66C22×#6157—28no—
67C22×#6157—48no—
68C22×#6157—48no—
69C22×#6157—44no—
70C22×#6157—44no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53C22⋊C412yes—
81C22×C4×C53C22⋊C412yes—
82C22×C4×C53C22⋊C412yes—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—24no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—42no—
91C22×Dic53—42no—
92C22×Dic53C22⋊C412yes—
93C22×Dic53C22⋊C412yes—
94C22×Dic53C22⋊C412yes—
95C2×C4×D53—28no—
96C2×C4×D53—28no—
97C2×C4×D53—44no—
98C2×C4×D53—48no—
99C2×C4×D53—48no—
100C2×Dic106—28no—
101C2×Dic106—48no—
102C2×Dic106—48no—
103C2×Dic106—44no—
104C2×Dic106—44no—
105C22⋊#6157—28no—
106C22⋊#6157—28no—
107C22⋊#6157—44no—
108C22⋊#6157—44no—
109C4⋊C4×C53—28no—
110#11719921—28no—
111#11719921—28no—
112#11719919—28no—
113#11719919—28no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53C2×C22⋊C411yescharacteristic
118C23×C53C2×C22⋊C411yescharacteristic
119C23×C53C4×Q814yes—
120C22×D53—24no—
121C22×D53—24no—
122C22×D53—42no—
123C22×D53—42no—
124C22×D53—42no—
125C22×D53—42no—
126C22×D53—48no—
127C22×D53—48no—
128C22×D53#12014yes—
129C22×D53C22⋊Q814yes—
130C22×D53C22⋊Q814yes—
131C22×D53C22⋊Q814yes—
132C22×D53#12412yes—
133C22×D53#12412yes—
134C22×D53#12412yes—
135C22×D53C4⋊Q812yes—
136C22×D53C4⋊Q812yes—
137C22×D53C4⋊Q812yes—
138C2×#6157—416no—
139C2×#6157—48no—
140C2×#6157—88no—
141C2×#6157—84no—
142C2×#6157—84no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—24no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×Dic53—24no—
156C2×Dic53—24no—
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53#9914yes—
168C4×D53—416no—
169Dic106—416no—
170#11381—531nomaximal, Sylow
171C22×C53—28no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53#37911yescharacteristic
177C22×C53#39112yes—
178C22×C53#39112yes—
179C22×C53#39112yes—
180C22×C53C2×#9912yes—
181C22×C53#41012yes—
182C2×D53—28no—
183C2×D53—24no—
184C2×D53—24no—
185C2×D53—24no—
186C2×D53—24no—
187C2×D53—44no—
188C2×D53—44no—
189C2×D53—42no—
190C2×D53—42no—
191C2×D53—42no—
192C2×D53—42no—
193C2×D53#39314yes—
194C2×D53#39514yes—
195#6157—88no—
196C4×C53—42no—
197C4×C53—42no—
198C4×C53—42no—
199Dic53—24no—
200Dic53—42no—
201Dic53—42no—
202C22×C22⋊Q8—531no—
203C2×#342—532no—
204C2×#342—532no—
205C22⋊#95—532no—
206C2×C53—22no—
207C2×C53—22no—
208C2×C53#144611yescharacteristic
209C2×C53#178111yescharacteristic
210C2×C53#178211yescharacteristic
211C2×C53#199614yes—
212D53—24no—
213D53—24no—
214C23×Q8—531no—
215C24×C4—531no—
216C2×C22⋊Q8—1064no—
217C2×C22⋊Q8—1064no—
218C2×C22⋊Q8—1064no—
219C22×C4⋊C4—531no—
220C22×C4⋊C4—531no—
221C22×C4⋊C4#615711yescharacteristic, complemented
222C22×C22⋊C4—531no—
223C22×C22⋊C4—531no—
224C22×C22⋊C4—534no—
225C2×#95—1064no—
226#342—1064no—
227#342—1064no—
228C53#1138111yescharacteristic, complemented, Sylow
229C25—531no—
230C22×Q8—1062no—
231C22×Q8—1062no—
232C22×Q8—1062no—
233C22×Q8—2124no—
234C23×C4—531no—
235C23×C4—531no—
236C23×C4—531no—
237C23×C4—534no—
238C23×C4—1062no—
239C23×C4—1062no—
240C23×C4—1064no—
241C23×C4—2124no—
242C23×C4C2×#615711yescharacteristic
243C23×C4C2×#615711yescharacteristic
244C23×C4C2×#615711yescharacteristic
245C22⋊Q8—10632no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—1062no—
250C2×C4⋊C4—1062no—
251C2×C4⋊C4—1062no—
252C2×C4⋊C4—1062no—
253C2×C4⋊C4—1062no—
254C2×C4⋊C4—1062no—
255C2×C22⋊C4—534no—
256C2×C22⋊C4—534no—
257C2×C22⋊C4—538no—
258C2×C22⋊C4—1062no—
259C2×C22⋊C4—1062no—
260C2×C22⋊C4—1062no—
261C2×C22⋊C4—1062no—
262C2×C22⋊C4—1064no—
263C2×C22⋊C4—1064no—
264C2×C22⋊C4—1064no—
265C2×C22⋊C4—1064no—
266C2×C22⋊C4—1064no—
267#95—2128no—
268C24—532no—
269C24—532no—
270C24—532no—
271C24—534no—
272C24—534no—
273C24—2122no—
274C24—2122no—
275C24C22×#615711yescharacteristic
276C2×Q8—1068no—
277C2×Q8—2124no—
278C2×Q8—2128no—
279C2×Q8—2124no—
280C2×Q8—2128no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—532no—
291C22×C4—532no—
292C22×C4—532no—
293C22×C4—1068no—
294C22×C4—1068no—
295C22×C4—1064no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4—1061no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4—1068no—
303C22×C4—2124no—
304C22×C4—2128no—
305C22×C4—2128no—
306C22×C4—2124no—
307C22×C4—2124no—
308C22×C4—2128no—
309C22×C4—2128no—
310C22×C4—2122no—
311C22×C4—2122no—
312C22×C4C22⋊#615712yes—
313C22×C4C22⋊#615712yes—
314C22×C4C22⋊#615712yes—
315C4⋊C4—28no—
316C4⋊C4—1068no—
317C4⋊C4—1068no—
318C22⋊C4—1068no—
319C22⋊C4—1068no—
320C22⋊C4—1068no—
321C22⋊C4—1068no—
322C22⋊C4—2124no—
323C22⋊C4—2124no—
324C23—22no—
325C23—22no—
326C23—534no—
327C23—534no—
328C23—532no—
329C23—532no—
330C23—534no—
331C23—534no—
332C23—532no—
333C23—532no—
334C23—532no—
335C23—532no—
336C23—1064no—
337C23—1064no—
338C23—2122no—
339C23—2122no—
340C23—2128no—
341C23—2128no—
342C23—2122no—
343C23—2122no—
344C23C2×C22⋊#615711yescharacteristic
345C23C2×C22⋊#615711yescharacteristic
346C23C2×C22⋊#615711yescharacteristic
347C23Q8×#615714yes—
348Q8—21216no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—534no—
362C2×C4—1064no—
363C2×C4—1064no—
364C2×C4—1062no—
365C2×C4—1062no—
366C2×C4—1062no—
367C2×C4—1062no—
368C2×C4—4248no—
369C2×C4—4244no—
370C2×C4—4244no—
371C2×C4—21216no—
372C2×C4—2128no—
373C2×C4—21216no—
374C2×C4—2122no—
375C2×C4—2122no—
376C2×C4—2122no—
377C2×C4—2122no—
378C2×C4—2122no—
379C2×C4—2122no—
380C22—28no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—534no—
386C22—534no—
387C22—1064no—
388C22—1068no—
389C22—1064no—
390C22—1064no—
391C22—1064no—
392C22—2122no—
393C22—2122no—
394C22—2124no—
395C22—2124no—
396C22—2122no—
397C22—2122no—
398C22#1681156411yescharacteristic
399C22(C2×#6157)⋊Q812yes—
400C22(C2×#6157)⋊Q812yes—
401C22(C2×#6157)⋊Q812yes—
402C22D53⋊#9912yes—
403C22#1681160212yes—
404C4—42no—
405C4—42no—
406C4—42no—
407C4—1064no—
408C4—4248no—
409C4—2122no—
410C4—2122no—
411C2—22no—
412C2—22no—
413C2—1064no—
414C2—1064no—
415C2#2028990511yescharacteristic
416C2#2029054211yescharacteristic
417C2#2029054411yescharacteristic
418C2#2029090614yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radical, cosocleC24×D53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11381
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
9. C23×Dic106C4semidirect
14. C22×#11719919C4semidirect
15. C22×#11719919C4semidirect
18. C22×C4⋊C4×C53C4semidirect
25. C2×C22⋊#6157Q8semidirect
221. C22×C4⋊C4#6157semidirect
228. C53#11381semidirect

Maximal subgroups

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

subgroupindex
C2×#168115012
#202898892
C22×#131331952
#1138153