Finite Group Databasebeta

← #28308359

Subgroups — #28308359

Order 13568.

414 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289963C211yesmaximal, characteristic
2C2×#16811468C211yesmaximal, characteristic
3#20289658C212yesmaximal, complemented
4#20289645C212yesmaximal, complemented
5#20279194C211yesmaximal, characteristic
6#16811593—21no—
7C2×C22⋊#88324—21no—
8#16811468—21no—
9#16811468—21no—
10#16811468C2212yes—
11#16811468C2212yes—
12C22×C4⋊Dic53C2211yescharacteristic
13#16810553—21no—
14#16810553—21no—
15#16810530C411yescharacteristic
16#16810530C2211yescharacteristic
17#16810495—21no—
18#16810495—21no—
19#16810492C411yescharacteristic
20#16810492C2211yescharacteristic
21C4×C22⋊Dic53C411yescharacteristic
22C2×C22⋊Q8×C53C411yescharacteristic
23C2×#11719940—41no—
24C22×#88324—41no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27C22⋊#88324—41no—
28C22⋊#88324—41no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31C2×C4⋊Dic53—41no—
32C2×C4⋊Dic53—41no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53C2311yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×C22⋊Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×Q8×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C22⋊Q8×C53—41no—
52C22⋊Q8×C53—41no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—21no—
56C24×C53C22⋊C411yescharacteristic
57C2×Q8×C53—22no—
58C2×Q8×C53—41no—
59C2×Q8×C53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—81no—
75C22×Dic53—81no—
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C2×D411yescharacteristic
79C22×Dic53C2×D411yescharacteristic
80#11719940—82no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—82no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊Dic53—24no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93#11719919—81no—
94#11719919—81no—
95#11719919—81no—
96#11719919—81no—
97C4×Dic53—22no—
98C4×Dic53—41no—
99C23×C53—22no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53#9911yescharacteristic
104C23×C53#9911yescharacteristic
105C23×C53C2×C22⋊C411yescharacteristic
106Q8×C53—82no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53#13112yes—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—81no—
138C2×Dic53—81no—
139#88324—82no—
140#88324—82no—
141#11555—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53#35612yes—
152C22×C53#40811yescharacteristic
153C22×C53C2×#9911yescharacteristic
154C22×C53#41311yescharacteristic
155C4×C53—22no—
156C4×C53—41no—
157C4×C53—81no—
158C4×C53—81no—
159Dic53—42no—
160Dic53—42no—
161Dic53—41no—
162Dic53—41no—
163Dic53—41no—
164Dic53—41no—
165#2502—531no—
166#1463—531no—
167C2×#324—531no—
168#1325—532no—
169#1319—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—22no—
173C2×C53—41no—
174C2×C53#146811yescharacteristic
175C2×C53#212711yescharacteristic
176C2×C53C2×#35611yescharacteristic
177C2×C22⋊Q8#615711yescharacteristic
178C22×C4⋊C4—531no—
179#407—1061no—
180C2×C22⋊C8—1061no—
181#398—1061no—
182#398—1061no—
183#391—531no—
184#391—531no—
185#385—1061no—
186#385—1061no—
187#384—531no—
188#384—531no—
189C4×C22⋊C4—531no—
190#324—532no—
191#324—532no—
192#324—1061no—
193#324—1061no—
194C53#1155511yescharacteristic, complemented, Sylow
195C22×Q8C2×#615711yescharacteristic
196C23×C4—531no—
197C23×C4—531no—
198C23×C4C2×#615711yescharacteristic
199C2×M16—2121no—
200C22×C8—2121no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—2121no—
213C2×C4⋊C4—2121no—
214C2×C4⋊C4C2×#615711yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—1061no—
217C2×C42—531no—
218C2×C42—531no—
219C22⋊C8—1062no—
220C22⋊C8—1062no—
221C22⋊C8—2121no—
222C22⋊C8—2121no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228C24C22⋊#615711yescharacteristic
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—4241no—
249C22×C4—4241no—
250C22×C4C22⋊#615711yescharacteristic
251C22×C4C22×#615711yescharacteristic
252M16—4242no—
253C2×C8—4242no—
254C2×C8—2122no—
255C2×C8—2122no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—4241no—
259C4⋊C4—4241no—
260C4⋊C4—4241no—
261C4⋊C4—4241no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—1064no—
269C42—1062no—
270C42—2121no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23#1313327511yescharacteristic
276C23#1313327511yescharacteristic
277C23C2×C22⋊#615711yescharacteristic
278Q8—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—4241no—
299C2×C4—4241no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2122no—
303C2×C4—2122no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4#1313330412yes—
311C8—4242no—
312C8—4242no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—42no—
321C22—42no—
322C22#1681153312yes—
323C22#1681159611yescharacteristic
324C22C2×#1313327511yescharacteristic
325C22#1681161011yescharacteristic
326C4—22no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—2122no—
331C4—2122no—
332C4—2121no—
333C4—2121no—
334C4—2121no—
335C4—2121no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—41no—
340C2#2028998011yescharacteristic
341C2#2029099011yescharacteristic
342C2C2×#1681153311yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C22⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C22⋊Q8#6157
hypocenterC53#11555
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289658C2semidirect
4. #20289645C2semidirect
194. C53#11555semidirect

Maximal subgroups

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

subgroupindex
#202899632
C2×#168114682
#202896582
#202896452
#202791942
#1155553