Finite Group Databasebeta

← #28306466

Subgroups — #28306466

Order 13568.

1510 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 1510 conjugacy classes of subgroups — too many to draw. Only its 228 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×#16811566C212yesmaximal
2C2×#16811501C214yesmaximal
3C22×#13133175C211yesmaximal, characteristic
4C22×C22⋊#6157C2214yes—
5#16811566—24no—
6#16811566C418yescomplemented
7C2×(C2×#6157)⋊C4—24no—
8#16811501—28no—
9C23×C4×D53C2211yescharacteristic
10C2×#13133175—24no—
11C2×#13133175—28no—
12C22×C4⋊Dic53C411yescharacteristic, complemented
13C22×#11719921C2212yes—
14C22×C4×Dic53C412yescomplemented
15C22×C4⋊C4×C53C411yescharacteristic, complemented
16C23×#6157—24no—
17C2×C22⋊#6157—28no—
18C2×C22⋊#6157—28no—
19C2×C22⋊#6157—24no—
20C2×C22⋊#6157C2×C418yes—
21(C2×#6157)⋊C4—216no—
22C24×D53C2311yescharacteristic
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C412yes—
25C22×C4×D53—24no—
26C22×C4×D53—24no—
27C22×C4×D53C2×C414yes—
28C22×C4×D53D412yes—
29C22×C4×D53D414yes—
30C22×C4×D53Q812yes—
31#13133175—232no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53—24no—
34C2×#11719921—24no—
35C2×#11719921—24no—
36C2×#11719921D418yes—
37C2×C4×Dic53—24no—
38C2×C4×Dic53—24no—
39C2×C4×Dic53—24no—
40C23×C4×C53C2×C411yescharacteristic
41C23×C4×C53C2×C412yes—
42C2×C4⋊C4×C53—24no—
43C2×C4⋊C4×C53—22no—
44C24×C53C22×C411yescharacteristic
45C23×D53—42no—
46C23×D53—42no—
47C23×D53C22×C412yes—
48C23×D53C2×D412yes—
49C23×D53C2×D411yescharacteristic
50C23×D53C2×D414yes—
51C23×D53C2×Q811yescharacteristic
52C23×D53#4114yes—
53C22×#6157—28no—
54C22×#6157—48no—
55C22×#6157—48no—
56C22×#6157—44no—
57C22×#6157—44no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—44no—
63C22×C4×C53C4212yes—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C414yes—
66C22×C4×C53C4⋊C412yes—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—44no—
72C22×Dic53C4212yes—
73C22×Dic53C22⋊C412yes—
74C22×Dic53C22⋊C414yes—
75C22×Dic53C4⋊C412yes—
76C2×C4×D53—216no—
77C2×C4×D53—28no—
78C2×C4×D53—28no—
79C2×C4×D53—28no—
80C2×C4×D53—28no—
81C2×C4×D53C22⋊C418yes—
82C2×C4×D53C4⋊C418yes—
83C22⋊#6157—216no—
84C22⋊#6157—48no—
85C4⋊C4×C53—28no—
86#11719921—216no—
87C4⋊Dic53—28no—
88C4×Dic53—216no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53C2×C4211yescharacteristic
92C23×C53C2×C22⋊C412yes—
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53C2×C4⋊C411yescharacteristic
95C23×C53#11712yes—
96C22×D53—24no—
97C22×D53—24no—
98C22×D53—44no—
99C22×D53—42no—
100C22×D53—44no—
101C22×D53—44no—
102C22×D53—42no—
103C22×D53—48no—
104C22×D53C2×C22⋊C412yes—
105C22×D53C2×C4⋊C412yes—
106C22×D53C4×D418yes—
107C22×D53#12014yes—
108C22×D53#12114yes—
109C22×D53C22⋊Q814yes—
110C22×D53#12314yes—
111C2×#6157—416no—
112C2×#6157—48no—
113C2×#6157—88no—
114C2×#6157—84no—
115C2×#6157—84no—
116C2×C4×C53—24no—
117C2×C4×C53—22no—
118C2×C4×C53—24no—
119C2×C4×C53—22no—
120C2×C4×C53—24no—
121C2×C4×C53—44no—
122C2×C4×C53—44no—
123C2×C4×C53—44no—
124C2×C4×C53#9514yes—
125C2×Dic53—24no—
126C2×Dic53—22no—
127C2×Dic53—24no—
128C2×Dic53—22no—
129C2×Dic53—24no—
130C2×Dic53—44no—
131C2×Dic53—44no—
132C2×Dic53—44no—
133C2×Dic53#9514yes—
134C4×D53—216no—
135C4×D53—216no—
136#10726—531nomaximal, Sylow
137C22×C53—24no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53C2×#9511yescharacteristic
143C22×C53C4×C22⋊C412yes—
144C22×C53#37911yescharacteristic
145C22×C53#38012yes—
146C22×C53#38111yescharacteristic
147C22×C53#41012yes—
148C22×C53#41012yes—
149C2×D53—28no—
150C2×D53—24no—
151C2×D53—24no—
152C2×D53—24no—
153C2×D53—24no—
154C2×D53—48no—
155C2×D53—44no—
156C2×D53—44no—
157C2×D53#38514yes—
158C2×D53#38614yes—
159#6157—88no—
160C4×C53—24no—
161C4×C53—44no—
162Dic53—24no—
163Dic53—44no—
164C22×#117—531no—
165C2×#381—532no—
166C2×#342—534no—
167C2×C53—22no—
168C2×C53—22no—
169C2×C53C22⋊#9511yescharacteristic
170C2×C53#174012yes—
171C2×C53#178211yescharacteristic
172C2×C53#178211yescharacteristic
173C2×C53#182012yes—
174D53—24no—
175D53—24no—
176C24×C4—531no—
177C2×#117—1068no—
178C2×#117—1064no—
179C22×C4⋊C4—531no—
180C22×C4⋊C4#615711yescharacteristic, complemented
181C22×C22⋊C4—532no—
182C22×C22⋊C4—534no—
183C22×C42—532no—
184#381—538no—
185#381—1064no—
186C2×#95—1064no—
187#342—1068no—
188C53#1072611yescharacteristic, complemented, Sylow
189C25—531no—
190C23×C4—531no—
191C23×C4—534no—
192C23×C4—532no—
193C23×C4—532no—
194C23×C4—534no—
195C23×C4—532no—
196C23×C4—1064no—
197C23×C4—1064no—
198C23×C4—1064no—
199C23×C4C2×#615712yes—
200C23×C4C2×#615711yescharacteristic
201#117—10632no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—1064no—
205C2×C4⋊C4—1062no—
206C2×C22⋊C4—538no—
207C2×C22⋊C4—538no—
208C2×C22⋊C4—1064no—
209C2×C22⋊C4—1064no—
210C2×C22⋊C4—1068no—
211C2×C22⋊C4—1068no—
212C2×C22⋊C4—1064no—
213C2×C42—1064no—
214C2×C42—1064no—
215C2×C42—1064no—
216#95—10616no—
217C24—532no—
218C24—532no—
219C24—534no—
220C24—531no—
221C24—531no—
222C24—534no—
223C24—2122no—
224C24—2122no—
225C24C22×#615711yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—44no—
231C22×C4—532no—
232C22×C4—532no—
233C22×C4—532no—
234C22×C4—538no—
235C22×C4—538no—
236C22×C4—534no—
237C22×C4—1068no—
238C22×C4—10616no—
239C22×C4—1068no—
240C22×C4—1068no—
241C22×C4—1068no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4—1062no—
246C22×C4—1068no—
247C22×C4—2124no—
248C22×C4—2124no—
249C22×C4—2128no—
250C22×C4—2128no—
251C22×C4—2124no—
252C22×C4C4×#615712yes—
253C22×C4C4⋊#615712yes—
254C22×C4C22⋊#615712yes—
255C22×C4C22⋊#615714yes—
256C4⋊C4—28no—
257C4⋊C4—1068no—
258C22⋊C4—10616no—
259C22⋊C4—10616no—
260C22⋊C4—2128no—
261C42—10616no—
262C23—22no—
263C23—22no—
264C23—532no—
265C23—532no—
266C23—534no—
267C23—534no—
268C23—538no—
269C23—534no—
270C23—534no—
271C23—1064no—
272C23—1064no—
273C23—2128no—
274C23—2124no—
275C23—2122no—
276C23—2122no—
277C23—2124no—
278C23—2124no—
279C23C2×C4×#615711yescharacteristic
280C23C2×C22⋊#615712yes—
281C23C2×C22⋊#615711yescharacteristic
282C23C2×C4⋊#615711yescharacteristic
283C23#1313329612yes—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—534no—
293C2×C4—10616no—
294C2×C4—10616no—
295C2×C4—1064no—
296C2×C4—1062no—
297C2×C4—1064no—
298C2×C4—1062no—
299C2×C4—1064no—
300C2×C4—4248no—
301C2×C4—4244no—
302C2×C4—4244no—
303C2×C4—2128no—
304C2×C4—21216no—
305C2×C4—2124no—
306C2×C4—2124no—
307C2×C4—2124no—
308C2×C4(C2×#6157)⋊C414yes—
309C22—24no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—534no—
315C22—534no—
316C22—1068no—
317C22—1064no—
318C22—1064no—
319C22—1064no—
320C22—1064no—
321C22—2128no—
322C22—2124no—
323C22—2124no—
324C22C2×(C2×#6157)⋊C411yescharacteristic
325C22C4×C22⋊#615712yes—
326C22#1681156411yescharacteristic
327C22#1681156512yes—
328C22#1681156611yescharacteristic
329C22#1681160312yes—
330C22#1681160412yes—
331C4—24no—
332C4—44no—
333C4—1064no—
334C4—4248no—
335C4—2124no—
336C2—22no—
337C2—22no—
338C2—1064no—
339C2—1064no—
340C2#2028988911yescharacteristic
341C2#2029041512yes—
342C2#2029054511yescharacteristic
343C2#2029054611yescharacteristic
344C2#2029061312yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radical, cosocleC24×D53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#10726
LCSG ▷ C23×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
6. #16811566C4semidirect
12. C22×C4⋊Dic53C4semidirect
14. C22×C4×Dic53C4semidirect
15. C22×C4⋊C4×C53C4semidirect
180. C22×C4⋊C4#6157semidirect
188. C53#10726semidirect

Maximal subgroups

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

subgroupindex
C2×#168115662
C2×#168115012
C22×#131331752
#1072653