Finite Group Databasebeta

← #28307728

Subgroups — #28307728

Order 13568.

926 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 926 conjugacy classes of subgroups — too many to draw. Only its 172 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×#16811503C212yesmaximal
2C2×#16811503C212yesmaximal
3#20290378C212yesmaximal
4C22×#13133207C211yesmaximal, characteristic
5C22×#11719940C2214yes—
6#16811503—24no—
7#16811503—24no—
8#16811489—24no—
9C2×#13133207—24no—
10C2×#13133207—24no—
11C2×#13133207—24no—
12C22×C4⋊Dic53C411yescharacteristic
13C22×#11719919C411yescharacteristic
14C22×#11719919C411yescharacteristic
15C22×#11719919C2211yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C22×C4⋊C4×C53C411yescharacteristic
19C2×#11719940—24no—
20C2×#11719940—24no—
21C2×#11719940—24no—
22C2×#11719940—24no—
23C2×#11719940—24no—
24C2×#11719940Q818yes—
25C22×#88324—24no—
26C23×Dic53C2×C411yescharacteristic
27C23×Dic53C2×C411yescharacteristic
28C23×Dic53C2×C411yescharacteristic
29C23×Dic53C2311yescharacteristic
30#13133207—232no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53—22no—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919—22no—
37C2×#11719919—22no—
38C2×#11719919—22no—
39C2×#11719919—22no—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719919—22no—
43C2×#11719919—22no—
44C2×#11719919D414yes—
45C2×#11719919D414yes—
46C2×C4×Dic53—22no—
47C2×C4×Dic53—22no—
48C2×C4×Dic53D414yes—
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53—22no—
53C2×C4⋊C4×C53—22no—
54C2×C4⋊C4×C53—22no—
55C24×C53C22×C411yescharacteristic
56C22×C4×C53—21no—
57C22×C4×C53—21no—
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×C53—42no—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22⋊C412yes—
68C22×Dic53—21no—
69C22×Dic53—21no—
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×Dic53C22⋊C412yes—
80C22×Dic53C2×D412yes—
81C22×Dic53C2×D412yes—
82C22×Dic53C2×D412yes—
83C22×Dic53C2×Q814yes—
84C22×Dic53#4114yes—
85#11719940—28no—
86#11719940—28no—
87#11719940—44no—
88#11719940—44no—
89C2×#88324—28no—
90C2×#88324—44no—
91C2×#88324—44no—
92C4⋊C4×C53—28no—
93C4⋊Dic53—28no—
94#11719919—28no—
95#11719919—28no—
96#11719919—28no—
97#11719919—28no—
98C4×Dic53—28no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53C2×C22⋊C411yescharacteristic
102C23×C53C2×C22⋊C411yescharacteristic
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53C4×Q814yes—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—24no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—24no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53#12014yes—
130C2×Dic53C22⋊Q814yes—
131C2×Dic53C22⋊Q814yes—
132C2×Dic53C22⋊Q814yes—
133C2×Dic53#12412yes—
134C2×Dic53#12412yes—
135C2×Dic53#12412yes—
136C2×Dic53C4⋊Q812yes—
137C2×Dic53C4⋊Q812yes—
138C2×Dic53C4⋊Q812yes—
139#88324—48no—
140#11383—531nomaximal, Sylow
141C22×C53—28no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53#37911yescharacteristic
147C22×C53#39112yes—
148C22×C53#39112yes—
149C22×C53#39112yes—
150C22×C53#41312yes—
151C22×C53#41312yes—
152C4×C53—42no—
153C4×C53—42no—
154C4×C53—42no—
155Dic53—42no—
156Dic53—42no—
157Dic53—42no—
158Dic53#39314yes—
159Dic53#39514yes—
160C22×#125—531no—
161C2×#343—532no—
162C2×#343—532no—
163#1728—532no—
164C2×C53—22no—
165C2×C53—22no—
166C2×C53#144611yescharacteristic
167C2×C53#178511yescharacteristic
168C2×C53#178511yescharacteristic
169C2×C53#199814yes—
170C22×M16—534no—
171C2×#125—1064no—
172C2×#125—1064no—
173C2×#125—1064no—
174C22×C4⋊C4—531no—
175C22×C4⋊C4—531no—
176C22×C4⋊C4—531no—
177C22×C4⋊C4—531no—
178C22×C4⋊C4—531no—
179C22×C4⋊C4#615711yescharacteristic
180C22×C42—531no—
181#343—1064no—
182#343—1064no—
183#336—1064no—
184C53#1138311yescharacteristic, complemented, Sylow
185C23×C4—531no—
186C23×C4—531no—
187C23×C4—531no—
188C23×C4—531no—
189C23×C4C2×#615711yescharacteristic
190C23×C4C2×#615711yescharacteristic
191C23×C4C2×#615711yescharacteristic
192C2×M16—538no—
193C2×M16—1064no—
194C2×M16—1064no—
195C2×M16—1064no—
196C2×M16—1064no—
197C2×M16—1064no—
198C22×C8—1064no—
199#125—10632no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—534no—
204C2×C4⋊C4—534no—
205C2×C4⋊C4—1062no—
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×C4⋊C4—1062no—
214C2×C4⋊C4—1062no—
215C2×C4⋊C4—1062no—
216C2×C4⋊C4—1062no—
217C2×C4⋊C4—1062no—
218C2×C42—534no—
219C2×C42—1062no—
220C2×C42—1062no—
221C24C22×#615711yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—532no—
232C22×C4—532no—
233C22×C4—532no—
234C22×C4—534no—
235C22×C4—534no—
236C22×C4—532no—
237C22×C4—532no—
238C22×C4—532no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—2122no—
246C22×C4—2122no—
247C22×C4—2122no—
248C22×C4C22⋊#615712yes—
249C22×C4C22⋊#615712yes—
250C22×C4C22⋊#615712yes—
251M16—1068no—
252M16—1068no—
253M16—2124no—
254M16—2124no—
255C2×C8—1068no—
256C2×C8—2124no—
257C2×C8—2124no—
258C4⋊C4—28no—
259C4⋊C4—1068no—
260C4⋊C4—1068no—
261C4⋊C4—1068no—
262C4⋊C4—1068no—
263C4⋊C4—1068no—
264C42—1068no—
265C23—22no—
266C23—22no—
267C23C2×C22⋊#615711yescharacteristic
268C23C2×C22⋊#615711yescharacteristic
269C23C2×C22⋊#615711yescharacteristic
270C23Q8×#615714yes—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—534no—
284C2×C4—534no—
285C2×C4—534no—
286C2×C4—534no—
287C2×C4—532no—
288C2×C4—532no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—532no—
292C2×C4—532no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—1064no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4—2122no—
303C2×C4—2122no—
304C2×C4—2122no—
305C8—2128no—
306C22—28no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22#1681156411yescharacteristic
312C22(C2×#6157)⋊Q812yes—
313C22(C2×#6157)⋊Q812yes—
314C22(C2×#6157)⋊Q812yes—
315C22#1681160912yes—
316C22#1681161012yes—
317C4—42no—
318C4—42no—
319C4—42no—
320C4—534no—
321C4—534no—
322C4—2122no—
323C4—2122no—
324C4—2122no—
325C2—22no—
326C2—22no—
327C2#2028990511yescharacteristic
328C2#2029055111yescharacteristic
329C2#2029055211yescharacteristic
330C2#2029090914yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11383
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
184. C53#11383semidirect

Maximal subgroups

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

subgroupindex
C2×#168115032
#202903782
C22×#131332072
#1138353