Finite Group Databasebeta

← #28306469

Subgroups — #28306469

Order 13568.

1678 conjugacy classes of subgroups; only the 244 normal subgroups are shown.

normal subgroups only

This group has 1678 conjugacy classes of subgroups — too many to draw. Only its 244 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×#16811595C212yesmaximal
2C2×#16811503C214yesmaximal, complemented
3C22×C4⋊C4×D53C211yesmaximal, characteristic
4C22×#11719940C2214yes—
5#16811595—24no—
6#16811595C418yescomplemented
7C2×#13133274—24no—
8#16811503—28no—
9C23×C4×D53C412yes—
10C23×C4×D53C2211yescharacteristic
11C22×C4⋊Dic53C411yescharacteristic
12C22×#11719919C2212yes—
13C2×C4⋊C4×D53—24no—
14C2×C4⋊C4×D53—28no—
15C22×C4⋊C4×C53C411yescharacteristic
16C2×#11719940—24no—
17C2×#11719940—28no—
18C2×#11719940—28no—
19C2×#11719940C2×C418yescomplemented
20C22×#88324—24no—
21#13133274—216no—
22C24×D53C2×C411yescharacteristic
23C23×Dic53C2×C412yes—
24C23×Dic53C2311yescharacteristic
25C22×C4×D53—24no—
26C22×C4×D53—24no—
27C22×C4×D53—24no—
28C22×C4×D53—24no—
29C22×C4×D53—24no—
30C22×C4×D53—48no—
31C22×C4×D53C2×C414yes—
32C22×C4×D53D412yes—
33C22×C4×D53D414yes—
34C22×C4×D53Q812yes—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53—24no—
37C2×#11719919—24no—
38C2×#11719919—24no—
39C2×#11719919D418yes—
40C4⋊C4×D53—232no—
41C23×C4×C53C2×C411yescharacteristic
42C23×C4×C53C2×C412yes—
43C2×C4⋊C4×C53—24no—
44C2×C4⋊C4×C53—22no—
45C24×C53C22×C411yescharacteristic
46C23×D53—22no—
47C23×D53—22no—
48C23×D53—24no—
49C23×D53—24no—
50C23×D53C4212yes—
51C23×D53C22⋊C412yes—
52C23×D53C4⋊C412yes—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—44no—
58C22×C4×C53C4212yes—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C22⋊C414yes—
61C22×C4×C53C4⋊C412yes—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—44no—
65C22×Dic53C22⋊C414yes—
66C22×Dic53C22×C412yes—
67C22×Dic53C2×D411yescharacteristic
68C22×Dic53C2×D412yes—
69C22×Dic53C2×D414yes—
70C22×Dic53C2×Q811yescharacteristic
71C22×Dic53#4114yes—
72C2×C4×D53—216no—
73C2×C4×D53—28no—
74C2×C4×D53—28no—
75C2×C4×D53—28no—
76C2×C4×D53—28no—
77C2×C4×D53—216no—
78C2×C4×D53—416no—
79C2×C4×D53—48no—
80C2×C4×D53—416no—
81C2×C4×D53—48no—
82C2×C4×D53C22⋊C418yes—
83C2×C4×D53C4⋊C418yes—
84#11719940—216no—
85#11719940—48no—
86C2×#88324—28no—
87C2×#88324—44no—
88C2×#88324—44no—
89C4⋊C4×C53—28no—
90C4⋊Dic53—28no—
91#11719919—216no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53C2×C4211yescharacteristic
95C23×C53C2×C22⋊C412yes—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53C2×C4⋊C411yescharacteristic
98C23×C53#11712yes—
99C22×D53—28no—
100C22×D53—216no—
101C22×D53—28no—
102C22×D53—24no—
103C22×D53—22no—
104C22×D53—24no—
105C22×D53—22no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—28no—
109C22×D53—28no—
110C22×D53#9514yes—
111C22×D53#10014yes—
112C22×D53#10114yes—
113C2×C4×C53—24no—
114C2×C4×C53—22no—
115C2×C4×C53—24no—
116C2×C4×C53—22no—
117C2×C4×C53—24no—
118C2×C4×C53—44no—
119C2×C4×C53—44no—
120C2×C4×C53—44no—
121C2×C4×C53#9514yes—
122C2×Dic53—24no—
123C2×Dic53—44no—
124C2×Dic53—44no—
125C2×Dic53—44no—
126C2×Dic53C2×C22⋊C412yes—
127C2×Dic53C2×C4⋊C412yes—
128C2×Dic53C4×D418yes—
129C2×Dic53#12014yes—
130C2×Dic53#12114yes—
131C2×Dic53C22⋊Q814yes—
132C2×Dic53#12314yes—
133C4×D53—216no—
134C4×D53—216no—
135C4×D53—432no—
136#88324—48no—
137#10727—531nomaximal, Sylow
138C22×C53—24no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53—22no—
143C22×C53C2×#9511yescharacteristic
144C22×C53C4×C22⋊C412yes—
145C22×C53#37911yescharacteristic
146C22×C53#38012yes—
147C22×C53#38111yescharacteristic
148C22×C53C2×#10012yes—
149C22×C53C2×#10112yes—
150C2×D53—216no—
151C2×D53—216no—
152C2×D53—24no—
153C2×D53—24no—
154C2×D53—24no—
155C2×D53—24no—
156C2×D53—24no—
157C2×D53—24no—
158C2×D53#34318yescomplemented
159C4×C53—24no—
160C4×C53—44no—
161Dic53—44no—
162Dic53#38514yes—
163Dic53#38614yes—
164C23×C4⋊C4—531no—
165C2×#407—532no—
166C2×#343—534no—
167C2×C53—22no—
168C2×C53—22no—
169C2×C53C22⋊#9511yescharacteristic
170C2×C53C2×#34312yes—
171C2×C53#178311yescharacteristic
172C2×C53#178411yescharacteristic
173C2×C53#182112yes—
174D53—28no—
175C24×C4—531no—
176C24×C4—532no—
177C22×M16—534no—
178C22×C4⋊C4—532no—
179C22×C4⋊C4—531no—
180C22×C4⋊C4—1068no—
181C22×C4⋊C4—1064no—
182C22×C4⋊C4#615711yescharacteristic
183#407—538no—
184#407—1064no—
185C2×C22⋊C8—1064no—
186#343—1068no—
187C53#1072711yescharacteristic, complemented, Sylow
188C25—531no—
189C23×C4—531no—
190C23×C4—534no—
191C23×C4—532no—
192C23×C4—532no—
193C23×C4—534no—
194C23×C4—532no—
195C23×C4—1064no—
196C23×C4—1064no—
197C23×C4—1064no—
198C23×C4—1064no—
199C23×C4—1064no—
200C23×C4—2128no—
201C23×C4C2×#615712yes—
202C23×C4C2×#615711yescharacteristic
203C2×M16—538no—
204C2×M16—1064no—
205C2×M16—1068no—
206C2×M16—1068no—
207C22×C8—1064no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—538no—
211C2×C4⋊C4—10632no—
212C2×C4⋊C4—1064no—
213C2×C4⋊C4—1064no—
214C2×C4⋊C4—1064no—
215C2×C4⋊C4—1062no—
216C22⋊C8—10616no—
217C24—532no—
218C24—532no—
219C24—532no—
220C24—1064no—
221C24—1064no—
222C24—1062no—
223C24—1062no—
224C24C22×#615711yescharacteristic
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—44no—
230C22×C4—532no—
231C22×C4—534no—
232C22×C4—532no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—534no—
236C22×C4—538no—
237C22×C4—538no—
238C22×C4—534no—
239C22×C4—1068no—
240C22×C4—10616no—
241C22×C4—1068no—
242C22×C4—1068no—
243C22×C4—1068no—
244C22×C4—10616no—
245C22×C4—1062no—
246C22×C4—1062no—
247C22×C4—2128no—
248C22×C4—21216no—
249C22×C4—2128no—
250C22×C4—21216no—
251C22×C4—2124no—
252C22×C4C4×#615712yes—
253C22×C4C4⋊#615712yes—
254C22×C4C22⋊#615712yes—
255C22×C4C22⋊#615714yes—
256M16—10616no—
257M16—2128no—
258C2×C8—1068no—
259C2×C8—2124no—
260C2×C8—2124no—
261C4⋊C4—28no—
262C4⋊C4—10616no—
263C4⋊C4—1068no—
264C23—22no—
265C23—22no—
266C23—534no—
267C23—534no—
268C23—534no—
269C23—1068no—
270C23—1068no—
271C23—1068no—
272C23—10616no—
273C23—1068no—
274C23—1064no—
275C23—1062no—
276C23—1064no—
277C23—1062no—
278C23—1062no—
279C23—1062no—
280C23C2×C4×#615711yescharacteristic
281C23C2×C22⋊#615712yes—
282C23C2×C22⋊#615711yescharacteristic
283C23C2×C4⋊#615711yescharacteristic
284C23#1313329612yes—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—532no—
294C2×C4—532no—
295C2×C4—538no—
296C2×C4—534no—
297C2×C4—534no—
298C2×C4—534no—
299C2×C4—534no—
300C2×C4—10616no—
301C2×C4—10616no—
302C2×C4—1064no—
303C2×C4—21232no—
304C2×C4—2124no—
305C2×C4—2124no—
306C2×C4—2124no—
307C2×C4(C2×#6157)⋊C414yes—
308C8—2128no—
309C22—24no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—538no—
315C22—10616no—
316C22—10616no—
317C22—1064no—
318C22—1064no—
319C22—1064no—
320C22—1064no—
321C22—1064no—
322C22—1064no—
323C22C2×(C2×#6157)⋊C411yescharacteristic
324C22C4×C22⋊#615712yes—
325C22#1681156411yescharacteristic
326C22#1681156512yes—
327C22#1681156611yescharacteristic
328C22D53⋊#10012yes—
329C22D53⋊#10112yes—
330C4—24no—
331C4—44no—
332C4—534no—
333C4—534no—
334C4—2124no—
335C2—22no—
336C2—22no—
337C2—1068no—
338C2#2028988911yescharacteristic
339C2#2029041812yes—
340C2#2029054811yescharacteristic
341C2#2029055011yescharacteristic
342C2#2029061512yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#10727
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
2. C2×#16811503C2semidirect
6. #16811595C4semidirect
19. C2×#11719940C2×C4semidirect
158. C2×D53#343semidirect
187. C53#10727semidirect

Maximal subgroups

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

subgroupindex
C2×#168115952
C2×#168115032
C22×C4⋊C4×D532
#1072753