Finite Group Databasebeta

← #28307499

Subgroups — #28307499

Order 13568.

1042 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 160 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×#16811593C212yesmaximal
2C2×#16811503C214yesmaximal, complemented
3C2×#16810481C211yesmaximal, characteristic
4C22×#11719940C2214yes—
5#16811593—28no—
6C2×C22⋊#88324—22no—
7C2×C22⋊#88324—22no—
8#16811503—28no—
9C24×Dic53C2211yescharacteristic
10C22×#11719919C2212yes—
11C22×C22⋊Dic53C411yescharacteristic
12#16810481—22no—
13#16810481—22no—
14C2×#13133124C412yes—
15C22×C22⋊C4×C53C411yescharacteristic
16C2×#11719940—24no—
17C2×#11719940—24no—
18C2×#11719940—24no—
19C2×#11719940—24no—
20C2×#11719940—24no—
21C2×#11719940D418yes—
22C22×#88324—24no—
23C22⋊#88324—28no—
24C22⋊#88324—28no—
25C23×Dic53—24no—
26C23×Dic53—24no—
27C23×Dic53—24no—
28C23×Dic53C2×C411yescharacteristic
29C23×Dic53C2×C412yes—
30C23×Dic53D412yes—
31C23×Dic53D412yes—
32C23×Dic53C2311yescharacteristic
33C2×#11719919—24no—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919—24no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—42no—
40#13133124—24no—
41C25×C53C2×C411yescharacteristic
42C23×C4×C53C2×C412yes—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—42no—
46C24×C53—22no—
47C24×C53—22no—
48C24×C53—22no—
49C24×C53—24no—
50C24×C53—42no—
51C24×C53C22⋊C411yescharacteristic
52C24×C53C22⋊C411yescharacteristic
53C24×C53C22×C411yescharacteristic
54C22×C4×C53—22no—
55C22×C4×C53—24no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×Dic53—22no—
59C22×Dic53—24no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—24no—
63C22×Dic53—24no—
64C22×Dic53—24no—
65C22×Dic53—24no—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—24no—
70C22×Dic53—44no—
71C22×Dic53—44no—
72C22×Dic53—44no—
73C22×Dic53—44no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76C22×Dic53C22⋊C412yes—
77C22×Dic53C2×D414yes—
78C22×Dic53C2×D412yes—
79C22×Dic53#4114yes—
80C22×Dic53#4114yes—
81#11719940—28no—
82#11719940—28no—
83#11719940—44no—
84#11719940—44no—
85C2×#88324—28no—
86C2×#88324—44no—
87C2×#88324—44no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—42no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92#11719919—42no—
93#11719919—44no—
94#11719919—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—22no—
98C23×C53—28no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—22no—
106C23×C53—22no—
107C23×C53—48no—
108C23×C53—48no—
109C23×C53—44no—
110C23×C53—44no—
111C23×C53#10012yes—
112C23×C53#10112yes—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53#11712yes—
115C23×C53C4×D414yes—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—84no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—44no—
124C2×Dic53—44no—
125C2×Dic53—44no—
126C2×Dic53—44no—
127C2×Dic53—44no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—84no—
132C2×Dic53#12012yes—
133C2×Dic53#12114yes—
134C2×Dic53#12114yes—
135C2×Dic53#12314yes—
136C2×Dic53#12314yes—
137C2×Dic53#12312yes—
138C2×Dic53#12414yes—
139C2×Dic53#12614yes—
140#88324—48no—
141#11198—531nomaximal, Sylow
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—24no—
145C22×C53—24no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—24no—
149C22×C53—24no—
150C22×C53—416no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53#38111yescharacteristic
158C22×C53#38612yes—
159C22×C53#38814yes—
160C22×C53C2×#10011yescharacteristic
161C22×C53C2×#10111yescharacteristic
162C22×C53#41312yes—
163C4×C53—82no—
164Dic53—42no—
165Dic53—82no—
166Dic53#39414yes—
167Dic53#39714yes—
168C2×#407—532no—
169C2×#380—531no—
170C2×#343—534no—
171C2×C53—24no—
172C2×C53—42no—
173C2×C53—42no—
174C2×C53#144011yescharacteristic
175C2×C53#182111yescharacteristic
176C2×C53#182211yescharacteristic
177C2×C53#190012yes—
178C2×C53#190112yes—
179C24×C4—531no—
180C22×M16—534no—
181C22×C4⋊C4—532no—
182C22×C22⋊C4—531no—
183C22×C22⋊C4#615711yescharacteristic
184#407—1068no—
185C2×C22⋊C8—1062no—
186C2×C22⋊C8—1062no—
187#380—1062no—
188#380—1062no—
189C2×#95—532no—
190#343—1068no—
191C53#1119811yescharacteristic, complemented, Sylow
192C25C2×#615711yescharacteristic
193C23×C4—531no—
194C23×C4—532no—
195C23×C4—532no—
196C23×C4—531no—
197C23×C4—532no—
198C23×C4—1064no—
199C23×C4—1064no—
200C23×C4—1064no—
201C23×C4C2×#615712yes—
202C2×M16—538no—
203C2×M16—1064no—
204C2×M16—1064no—
205C2×M16—1064no—
206C2×M16—1064no—
207C2×M16—1064no—
208C22×C8—1064no—
209C2×C4⋊C4—1064no—
210C2×C4⋊C4—1064no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1062no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—2122no—
219C22⋊C8—1068no—
220C22⋊C8—1068no—
221#95—1064no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—24no—
226C24—42no—
227C24C22⋊#615711yescharacteristic
228C24C22⋊#615711yescharacteristic
229C24C22×#615711yescharacteristic
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—532no—
235C22×C4—534no—
236C22×C4—534no—
237C22×C4—534no—
238C22×C4—532no—
239C22×C4—1062no—
240C22×C4—1062no—
241C22×C4—1064no—
242C22×C4—1064no—
243C22×C4—1064no—
244C22×C4—1064no—
245C22×C4—1062no—
246C22×C4—1062no—
247C22×C4—1062no—
248C22×C4—1064no—
249C22×C4—1064no—
250C22×C4—1062no—
251C22×C4—2124no—
252C22×C4—2124no—
253C22×C4—2124no—
254C22×C4—2124no—
255C22×C4—2122no—
256C22×C4—2122no—
257M16—1068no—
258M16—1068no—
259M16—2124no—
260M16—2124no—
261C2×C8—1068no—
262C2×C8—2124no—
263C2×C8—2124no—
264C4⋊C4—2122no—
265C4⋊C4—2124no—
266C4⋊C4—2122no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—2122no—
270C22⋊C4—2122no—
271C23—21no—
272C23—21no—
273C23—22no—
274C23—28no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—22no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—48no—
284C23—48no—
285C23—44no—
286C23—44no—
287C23#1313327712yes—
288C23#1313327812yes—
289C23C2×C22⋊#615711yescharacteristic
290C23#1313329612yes—
291C23D4×#615714yes—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—84no—
296C2×C4—534no—
297C2×C4—532no—
298C2×C4—534no—
299C2×C4—532no—
300C2×C4—534no—
301C2×C4—534no—
302C2×C4—534no—
303C2×C4—534no—
304C2×C4—1062no—
305C2×C4—1062no—
306C2×C4—1062no—
307C2×C4—4244no—
308C2×C4—2124no—
309C2×C4—2124no—
310C2×C4—2124no—
311C2×C4—2124no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4—2122no—
315C2×C4—2124no—
316C8—2128no—
317C22—22no—
318C22—22no—
319C22—24no—
320C22—24no—
321C22—22no—
322C22—22no—
323C22—24no—
324C22—24no—
325C22—416no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—42no—
331C22—42no—
332C22#1681156611yescharacteristic
333C22#1681157112yes—
334C22#1681157314yes—
335C22C2×#1313327711yescharacteristic
336C22C2×#1313327811yescharacteristic
337C22#1681161012yes—
338C4—82no—
339C4—534no—
340C4—534no—
341C4—4242no—
342C4—2122no—
343C2—24no—
344C2—42no—
345C2—42no—
346C2#2028989911yescharacteristic
347C2#2029061411yescharacteristic
348C2#2029061711yescharacteristic
349C2#2029074012yes—
350C2#2029074212yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53#11198
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16811503C2semidirect
191. C53#11198semidirect

Maximal subgroups

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

subgroupindex
C2×#168115932
C2×#168115032
C2×#168104812
#1119853