Finite Group Databasebeta

← #28302233

Subgroups — #28302233

Order 13568.

874 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 124 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#20290605C212yesmaximal, complemented
2C2×#16811468C212yesmaximal, complemented
3C2×#16811467C212yesmaximal
4C22×#13133212C211yesmaximal, characteristic
5C2×C22⋊#88324C2214yes—
6#16811489—24no—
7#16811468—24no—
8#16811467—24no—
9C2×#13133212—24no—
10C2×#13133212—24no—
11C2×#13133212—24no—
12C22×C4⋊Dic53C2211yescharacteristic
13C22×#11719919C411yescharacteristic
14C22×#11719921C411yescharacteristic
15C22×#11719921C411yescharacteristic
16C22×#11719921C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C22×C4⋊C4×C53C411yescharacteristic
19C22×#88324—24no—
20C22⋊#88324—24no—
21C22⋊#88324—24no—
22C24×D53C2×C411yescharacteristic
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2311yescharacteristic
26#13133212—232no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53D414yes—
30C2×#11719919—22no—
31C2×#11719919—22no—
32C2×#11719919—22no—
33C2×#11719921—22no—
34C2×#11719921—22no—
35C2×#11719921—22no—
36C2×#11719921—22no—
37C2×#11719921—22no—
38C2×#11719921—22no—
39C2×#11719921—22no—
40C2×#11719921—22no—
41C2×#11719921D414yes—
42C2×C4×Dic53—22no—
43C2×C4×Dic53—22no—
44C2×C4×Dic53D414yes—
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53—22no—
51C24×C53C22×C411yescharacteristic
52C23×D53—21no—
53C23×D53—21no—
54C23×D53—24no—
55C23×D53—42no—
56C23×D53—42no—
57C23×D53C22⋊C412yes—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—24no—
65C22×C4×C53—24no—
66C22×C4×C53—42no—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53C22⋊C412yes—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76C22×Dic53—42no—
77C22×Dic53C22⋊C412yes—
78C22×Dic53C22⋊C412yes—
79C22×Dic53C2×D412yes—
80C22×Dic53C2×D412yes—
81C22×Dic53C2×D412yes—
82C2×#88324—48no—
83C2×#88324—44no—
84C4⋊C4×C53—28no—
85#11719921—28no—
86#11719921—28no—
87#11719921—28no—
88C4⋊Dic53—28no—
89#11719919—28no—
90C4×Dic53—28no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53C2×C22⋊C411yescharacteristic
95C23×C53C2×C22⋊C411yescharacteristic
96C23×C53#13114yes—
97C22×D53—22no—
98C22×D53—24no—
99C22×D53—42no—
100C22×D53—42no—
101C22×D53—42no—
102C22×D53—42no—
103C22×D53—42no—
104C22×D53—42no—
105C22×D53—42no—
106C22×D53—42no—
107C22×D53—88no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—24no—
112C2×C4×C53—24no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×Dic53—24no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—24no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53#12014yes—
132#88324—84no—
133#8309—531nomaximal, Sylow
134C22×C53—28no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53#37911yescharacteristic
140C22×C53#40812yes—
141C22×C53#40812yes—
142C22×C53#40812yes—
143C22×C53#41012yes—
144C22×C53#41312yes—
145C2×D53—42no—
146C2×D53—42no—
147C2×D53—42no—
148C2×D53—84no—
149C2×D53—84no—
150C2×D53—84no—
151C4×C53—42no—
152C4×C53—42no—
153C4×C53—42no—
154Dic53—24no—
155Dic53—42no—
156Dic53—42no—
157C22×#126—531no—
158#1817—532no—
159C2×#324—532no—
160C2×#323—532no—
161C2×C53—22no—
162C2×C53—22no—
163C2×C53#147314yes—
164C2×C53#178011yescharacteristic
165C2×C53#178211yescharacteristic
166C2×C53#178511yescharacteristic
167D53—82no—
168C2×#126—1064no—
169C2×#126—1064no—
170C2×#126—1064no—
171C22×C4⋊C4—531no—
172C22×C4⋊C4—531no—
173C22×C4⋊C4#615711yescharacteristic
174C22×C22⋊C4—531no—
175C22×C22⋊C4—531no—
176C22×C22⋊C4—531no—
177C22×C42—531no—
178C2×C22⋊C8—534no—
179#336—1064no—
180#324—1064no—
181#323—1064no—
182C53#830911yescharacteristic, complemented, Sylow
183C25—531no—
184C23×C4—531no—
185C23×C4—531no—
186C23×C4—531no—
187C23×C4C2×#615711yescharacteristic
188C23×C4C2×#615711yescharacteristic
189C23×C4C2×#615711yescharacteristic
190C22×C8—1064no—
191#126—10632no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—534no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4—1062no—
198C2×C4⋊C4—1062no—
199C2×C4⋊C4—1062no—
200C2×C4⋊C4—1062no—
201C2×C22⋊C4—534no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4—1062no—
204C2×C22⋊C4—1062no—
205C2×C22⋊C4—1062no—
206C2×C22⋊C4—1062no—
207C2×C22⋊C4—1062no—
208C2×C22⋊C4—1062no—
209C2×C22⋊C4—1062no—
210C2×C42—534no—
211C2×C42—1062no—
212C2×C42—1062no—
213C22⋊C8—1064no—
214C22⋊C8—1064no—
215C24—532no—
216C24—1064no—
217C24—1061no—
218C24—1061no—
219C24—2122no—
220C24—2122no—
221C24C22×#615711yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—42no—
231C22×C4—532no—
232C22×C4—532no—
233C22×C4—532no—
234C22×C4—532no—
235C22×C4—532no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—2122no—
241C22×C4—2122no—
242C22×C4—2122no—
243C22×C4C22⋊#615712yes—
244C22×C4C22⋊#615712yes—
245C22×C4C22⋊#615712yes—
246C2×C8—2124no—
247C2×C8—2128no—
248C4⋊C4—28no—
249C4⋊C4—1068no—
250C4⋊C4—1068no—
251C22⋊C4—1068no—
252C22⋊C4—1068no—
253C22⋊C4—1068no—
254C42—1068no—
255C23—22no—
256C23—22no—
257C23—1064no—
258C23—1062no—
259C23—4248no—
260C23—2122no—
261C23—2122no—
262C23—2122no—
263C23—2122no—
264C23—2122no—
265C23—2122no—
266C23—2122no—
267C23—2122no—
268C23C2×C22⋊#615711yescharacteristic
269C23C2×C22⋊#615711yescharacteristic
270C23C2×C22⋊#615711yescharacteristic
271C23#1313330414yes—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—534no—
285C2×C4—1064no—
286C2×C4—1062no—
287C2×C4—1062no—
288C2×C4—1064no—
289C2×C4—2122no—
290C2×C4—2122no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4—2122no—
295C2×C4—2122no—
296C8—4244no—
297C22—28no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—4244no—
303C22—4244no—
304C22—4244no—
305C22—2122no—
306C22—2122no—
307C22—2122no—
308C22#1681156411yescharacteristic
309C22#1681159612yes—
310C22#1681159612yes—
311C22#1681159712yes—
312C22#1681160412yes—
313C22#1681160912yes—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—1064no—
318C4—2122no—
319C4—2122no—
320C2—22no—
321C2—22no—
322C2—4242no—
323C2#2028999914yes—
324C2#2029053911yescharacteristic
325C2#2029054611yescharacteristic
326C2#2029055111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8309
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
1. #20290605C2semidirect
2. C2×#16811468C2semidirect
182. C53#8309semidirect

Maximal subgroups

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

subgroupindex
#202906052
C2×#168114682
C2×#168114672
C22×#131332122
#830953