Finite Group Databasebeta

← #28302874

Subgroups — #28302874

Order 13568.

1026 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1026 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#20290549C212yesmaximal
2#20289965C212yesmaximal
3#20289963C212yesmaximal
4C22×#13133216C211yesmaximal, characteristic
5C2×#13133278—24no—
6#16811593C2214yes—
7#16811469—24no—
8#16811468—24no—
9C23×Dic106C411yescharacteristic
10C23×Dic106C2211yescharacteristic
11C2×#13133216—24no—
12C2×#13133216—24no—
13C2×#13133216—24no—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919C411yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×C4×Dic53C411yescharacteristic
18C22×C4⋊C4×C53C411yescharacteristic
19C2×#11719940—24no—
20#13133278—48no—
21C22⋊#88324—24no—
22C22⋊#88324—24no—
23C22×Dic106—22no—
24C22×Dic106—22no—
25C22×Dic106—22no—
26C22×Dic106—22no—
27C22×Dic106—22no—
28C22×Dic106—44no—
29C22×Dic106—44no—
30C22×Dic106D414yes—
31C23×Dic53C2×C411yescharacteristic
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2311yescharacteristic
35#13133216—232no—
36C2×C4⋊Dic53—22no—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53D414yes—
39C2×#11719919—22no—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719919—22no—
43C2×#11719919—22no—
44C2×#11719919D414yes—
45C2×C4×Dic53—22no—
46C2×C4×Dic53—22no—
47C2×C4×Dic53—22no—
48C23×C4×C53C2×C411yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53—22no—
52C2×C4⋊C4×C53—22no—
53C2×C4⋊C4×C53—22no—
54C24×C53C22×C411yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—24no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C412yes—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—24no—
72C22×Dic53—24no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—42no—
76C22×Dic53—42no—
77C22×Dic53C22⋊C412yes—
78C22×Dic53C22⋊C412yes—
79C22×Dic53C22⋊C412yes—
80C22×Dic53C2×D412yes—
81C22×Dic53C2×D412yes—
82C22×Dic53C2×D412yes—
83C2×Dic106—28no—
84C2×Dic106—28no—
85C2×Dic106—28no—
86C2×Dic106—44no—
87C2×Dic106—48no—
88C2×Dic106—48no—
89C2×Dic106—48no—
90C2×Dic106—48no—
91C2×Dic106—44no—
92C2×Dic106—44no—
93#11719940—48no—
94C2×#88324—44no—
95C4⋊C4×C53—28no—
96C4⋊Dic53—28no—
97#11719919—28no—
98#11719919—28no—
99C4×Dic53—28no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53—24no—
107C2×C4×C53—24no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
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#10114yescomplemented
119C2×Dic53—24no—
120C2×Dic53—24no—
121C2×Dic53—24no—
122C2×Dic53—24no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—24no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53#9914yes—
135C2×Dic53#10114yescomplemented
136C2×Dic53#12014yes—
137Dic106—416no—
138Dic106—416no—
139#88324—84no—
140#8551—531nomaximal, Sylow
141C22×C53—28no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53#37911yescharacteristic
149C22×C53C2×#9912yes—
150C22×C53C2×#10112yes—
151C22×C53C2×#10112yes—
152C4×C53—24no—
153C4×C53—42no—
154C4×C53—42no—
155Dic53—24no—
156Dic53—24no—
157Dic53—24no—
158Dic53—42no—
159C22×C4⋊Q8—531no—
160#1784—532no—
161#1463—532no—
162#1463—532no—
163C2×C53—22no—
164C2×C53—22no—
165C2×C53—22no—
166C2×C53#178111yescharacteristic
167C2×C53#178411yescharacteristic
168C2×C53#178411yescharacteristic
169C23×Q8—531no—
170C23×Q8—531no—
171C2×C4⋊Q8—1064no—
172C2×C4⋊Q8—1064no—
173C2×C4⋊Q8—1064no—
174C22×C4⋊C4—531no—
175C22×C4⋊C4—531no—
176C22×C4⋊C4—531no—
177C22×C4⋊C4#615711yescharacteristic
178C22×C42—531no—
179C2×#101—1064no—
180#407—534no—
181#324—1064no—
182#324—1064no—
183C53#855111yescharacteristic, complemented, Sylow
184C22×Q8—534no—
185C22×Q8—1062no—
186C22×Q8—1062no—
187C22×Q8—1062no—
188C22×Q8—1062no—
189C22×Q8—1062no—
190C22×Q8—2124no—
191C22×Q8—2124no—
192C23×C4—531no—
193C23×C4—531no—
194C23×C4—531no—
195C23×C4—531no—
196C23×C4C2×#615711yescharacteristic
197C23×C4C2×#615711yescharacteristic
198C23×C4C2×#615711yescharacteristic
199C2×M16—1064no—
200C4⋊Q8—10632no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—534no—
205C2×C4⋊C4—534no—
206C2×C4⋊C4—1062no—
207C2×C4⋊C4—1062no—
208C2×C4⋊C4—1062no—
209C2×C4⋊C4—1062no—
210C2×C4⋊C4—1062no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1062no—
213C2×C42—1062no—
214C2×C42—1062no—
215C2×C42—1062no—
216#101—2128no—
217C22⋊C8—1064no—
218C22⋊C8—1064no—
219C24C22×#615711yescharacteristic
220C2×Q8—1068no—
221C2×Q8—1068no—
222C2×Q8—1068no—
223C2×Q8—2124no—
224C2×Q8—2124no—
225C2×Q8—2128no—
226C2×Q8—2128no—
227C2×Q8—2128no—
228C2×Q8—2124no—
229C2×Q8—2128no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—532no—
240C22×C4—532no—
241C22×C4—532no—
242C22×C4—532no—
243C22×C4—532no—
244C22×C4—532no—
245C22×C4—1064no—
246C22×C4—1064no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—2122no—
254C22×C4—2122no—
255C22×C4C22⋊#615712yes—
256C22×C4C22⋊#615712yes—
257C22×C4C22⋊#615712yes—
258M16—2128no—
259C2×C8—2124no—
260C4⋊C4—28no—
261C4⋊C4—1068no—
262C4⋊C4—1068no—
263C4⋊C4—1068no—
264C42—1068no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23C2×C22⋊#615711yescharacteristic
269C23C2×C22⋊#615711yescharacteristic
270C23C2×C22⋊#615711yescharacteristic
271Q8—21216no—
272Q8—21216no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—534no—
286C2×C4—534no—
287C2×C4—534no—
288C2×C4—1064no—
289C2×C4—1064no—
290C2×C4—1064no—
291C2×C4—1064no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1064no—
295C2×C4—1062no—
296C2×C4—1062no—
297C2×C4—1062no—
298C2×C4—2122no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4—2122no—
303C2×C4#1313327814yescomplemented
304C8—4244no—
305C22—28no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22#1681156411yescharacteristic
313C22D53⋊#9912yes—
314C22C2×#1313327812yes—
315C22D53⋊#10112yes—
316C4—24no—
317C4—42no—
318C4—42no—
319C4—1064no—
320C4—1064no—
321C4—1064no—
322C4—2122no—
323C2—22no—
324C2—22no—
325C2—22no—
326C2#2029054211yescharacteristic
327C2#2029054911yescharacteristic
328C2#2029055011yescharacteristic
subgroupG/H
centerC22#16811564
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53#379
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8551
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
118. C2×C4×C53#101semidirect
135. C2×Dic53#101semidirect
183. C53#8551semidirect
303. C2×C4#13133278semidirect

Maximal subgroups

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

subgroupindex
#202905492
#202899652
#202899632
C22×#131332162
#855153