Finite Group Databasebeta

← #28302886

Subgroups — #28302886

Order 13568.

1102 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 116 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#20290614C212yesmaximal, complemented
2#20289963C212yesmaximal, complemented
3#20289960C212yesmaximal
4C22×#13133196C211yesmaximal, characteristic
5#16811593C2214yes—
6#16811503—24no—
7#16811468—24no—
8#16811466—24no—
9C23×Dic106C411yescharacteristic
10C2×#13133196—24no—
11C2×#13133196—24no—
12C2×#13133196—24no—
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C411yescharacteristic
15C22×#11719919C2211yescharacteristic
16C22×C22⋊Dic53C411yescharacteristic
17C22×C22⋊Dic53C2211yescharacteristic
18C24×C4×C53C411yescharacteristic
19C2×#11719940—24no—
20C22⋊#88324—24no—
21C22⋊#88324—24no—
22C22×Dic106—22no—
23C22×Dic106—22no—
24C22×Dic106—22no—
25C22×Dic106—44no—
26C23×Dic53C2×C411yescharacteristic
27C23×Dic53C2×C411yescharacteristic
28C23×Dic53C2×C411yescharacteristic
29C23×Dic53C2311yescharacteristic
30#13133196—232no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53D414yes—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919—22no—
37C2×#11719919—22no—
38C2×#11719919—22no—
39C2×#11719919D414yes—
40C2×C22⋊Dic53—22no—
41C2×C22⋊Dic53—22no—
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—22no—
45C2×C22⋊Dic53D414yes—
46C25×C53C2×C411yescharacteristic
47C23×C4×C53—22no—
48C23×C4×C53—22no—
49C23×C4×C53—22no—
50C23×C4×C53—44no—
51C23×C4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53C24×C53—24no—
54C24×C53—21no—
55C24×C53—21no—
56C24×C53—42no—
57C24×C53—42no—
58C24×C53C22⋊C412yes—
59C24×C53C22×C411yescharacteristic
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—24no—
63C22×C4×C53—28no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—48no—
67C22×C4×C53—48no—
68C22×C4×C53—44no—
69C22×C4×C53—44no—
70C22×C4×C53—42no—
71C22×C4×C53C22⋊C412yes—
72C22×C4×C53C22⋊C412yes—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—42no—
80C22×Dic53—42no—
81C22×Dic53—42no—
82C22×Dic53—42no—
83C22×Dic53C22⋊C412yes—
84C22×Dic53C22⋊C412yes—
85C22×Dic53C22⋊C412yes—
86C22×Dic53C2×D412yes—
87C22×Dic53C2×D412yes—
88C22×Dic53C2×D412yes—
89C2×Dic106—28no—
90C2×Dic106—48no—
91C2×Dic106—48no—
92C2×Dic106—44no—
93C2×Dic106—44no—
94#11719940—48no—
95C2×#88324—44no—
96C22⋊Dic53—28no—
97C22⋊Dic53—28no—
98C4⋊Dic53—28no—
99#11719919—28no—
100#11719919—28no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—24no—
105C23×C53—24no—
106C23×C53—22no—
107C23×C53—22no—
108C23×C53—42no—
109C23×C53—42no—
110C23×C53—42no—
111C23×C53—416no—
112C23×C53—42no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53#10114yes—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53C2×C22⋊C411yescharacteristic
118C23×C53C2×C22⋊C411yescharacteristic
119C2×C4×C53—24no—
120C2×C4×C53—24no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—416no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53#10014yes—
129C2×Dic53—24no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53#12014yes—
145Dic106—416no—
146#88324—84no—
147#8555—531nomaximal, Sylow
148C22×C53—28no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—24no—
156C22×C53—44no—
157C22×C53—48no—
158C22×C53—48no—
159C22×C53—44no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53#37911yescharacteristic
163C22×C53#41012yes—
164C22×C53C2×#10012yes—
165C22×C53C2×#10112yes—
166C4×C53—24no—
167C4×C53—42no—
168Dic53—24no—
169Dic53—42no—
170Dic53—42no—
171Dic53—42no—
172C22×C22⋊Q8—531no—
173#1821—532no—
174#1463—532no—
175#1462—532no—
176C2×C53—22no—
177C2×C53—22no—
178C2×C53—22no—
179C2×C53—44no—
180C2×C53#178211yescharacteristic
181C2×C53#178311yescharacteristic
182C2×C53#178411yescharacteristic
183C23×Q8—531no—
184C24×C4#615711yescharacteristic
185C2×C22⋊Q8—1064no—
186C2×C22⋊Q8—1064no—
187C2×C22⋊Q8—1064no—
188C22×C4⋊C4—531no—
189C22×C4⋊C4—531no—
190C22×C4⋊C4—531no—
191C22×C22⋊C4—531no—
192C22×C22⋊C4—531no—
193#407—534no—
194#343—1064no—
195#324—1064no—
196#323—1064no—
197C53#855511yescharacteristic, complemented, Sylow
198C25C2×#615711yescharacteristic
199C22×Q8—1062no—
200C22×Q8—1062no—
201C22×Q8—1062no—
202C22×Q8—2124no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—44no—
207C23×C4—531no—
208C23×C4—531no—
209C23×C4—531no—
210C23×C4—531no—
211C23×C4C2×#615711yescharacteristic
212C23×C4C2×#615711yescharacteristic
213C2×M16—1064no—
214C22⋊Q8—10632no—
215C2×C4⋊C4—534no—
216C2×C4⋊C4—534no—
217C2×C4⋊C4—1062no—
218C2×C4⋊C4—1062no—
219C2×C4⋊C4—1062no—
220C2×C4⋊C4—1062no—
221C2×C4⋊C4—1062no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1062no—
224C2×C22⋊C4—534no—
225C2×C22⋊C4—1062no—
226C2×C22⋊C4—1062no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4—1062no—
229C2×C22⋊C4—1062no—
230C22⋊C8—1064no—
231C22⋊C8—1064no—
232C24—24no—
233C24—21no—
234C24—21no—
235C24—42no—
236C24—42no—
237C24C22⋊#615712yes—
238C24C22×#615711yescharacteristic
239C2×Q8—1068no—
240C2×Q8—2128no—
241C2×Q8—2128no—
242C2×Q8—2124no—
243C2×Q8—2124no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—532no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—532no—
259C22×C4—532no—
260C22×C4—532no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—2122no—
268C22×C4—2122no—
269C22×C4—2122no—
270C22×C4—2122no—
271C22×C4C22⋊#615712yes—
272C22×C4C22⋊#615712yes—
273M16—2128no—
274C2×C8—2124no—
275C4⋊C4—1068no—
276C4⋊C4—1068no—
277C4⋊C4—1068no—
278C22⋊C4—1068no—
279C22⋊C4—1068no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—24no—
284C23—24no—
285C23—22no—
286C23—22no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—416no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23#1313327814yes—
295C23C2×C22⋊#615711yescharacteristic
296C23C2×C22⋊#615711yescharacteristic
297C23C2×C22⋊#615711yescharacteristic
298Q8—21216no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—416no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—534no—
309C2×C4—1064no—
310C2×C4—1062no—
311C2×C4—1062no—
312C2×C4—1062no—
313C2×C4—2122no—
314C2×C4—2122no—
315C2×C4—2122no—
316C2×C4—2122no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4#1313327714yes—
325C8—4244no—
326C22—28no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—24no—
334C22—44no—
335C22—48no—
336C22—48no—
337C22—44no—
338C22—42no—
339C22—42no—
340C22#1681156411yescharacteristic
341C22#1681160212yes—
342C22C2×#1313327712yes—
343C22C2×#1313327812yes—
344C4—24no—
345C4—42no—
346C4—1064no—
347C4—2122no—
348C4—2122no—
349C4—2122no—
350C2—22no—
351C2—22no—
352C2—22no—
353C2—44no—
354C2#2029054411yescharacteristic
355C2#2029054711yescharacteristic
356C2#2029054911yescharacteristic
subgroupG/H
centerC22#16811564
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC24×C4×C53C4
socleC22×C53#379
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenterC24×C4#6157
hypocenterC53#8555
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #20290614C2semidirect
2. #20289963C2semidirect
197. C53#8555semidirect

Maximal subgroups

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

subgroupindex
#202906142
#202899632
#202899602
C22×#131331962
#855553