Finite Group Databasebeta

← #28303354

Subgroups — #28303354

Order 13568.

482 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290441C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3#20289591C212yesmaximal, complemented
4#20289591C212yesmaximal, complemented
5#20277110C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×#13133272C2212yes—
9C2×#13133272C2212yes—
10#16811489—21no—
11#16811489—21no—
12C22×C4×Dic53C2211yescharacteristic
13#16810490—21no—
14#16810488C2211yescharacteristic
15#16810488C2211yescharacteristic
16#16810484—21no—
17#16810483C411yescharacteristic
18C4⋊C4×Dic53C411yescharacteristic
19C4⋊C4×Dic53C411yescharacteristic
20C22⋊C4×Dic53—21no—
21#16810468—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324—24no—
27#13133272—24no—
28#13133272—24no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53—21no—
34C2×C4×Dic53—24no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53C2311yescharacteristic
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C23×C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—21no—
52#95×C53—21no—
53C24×C53C22⋊C411yescharacteristic
54C22×C4×C53—22no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73C2×#88324—24no—
74C2×#88324—24no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C22⋊Dic53—42no—
83C4⋊Dic53—41no—
84C4⋊Dic53—41no—
85C4×Dic53—22no—
86C4×Dic53—22no—
87C4×Dic53—22no—
88C4×Dic53—42no—
89C4×Dic53—42no—
90C4×Dic53#4114yes—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53#10412yes—
98C23×C53C2×C22⋊C411yescharacteristic
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—44no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53#11712yes—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—44no—
122C2×Dic53—44no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53#10412yes—
130C2×Dic53#12312yes—
131C2×Dic53#12312yes—
132#88324—44no—
133#88324—44no—
134#8720—531nomaximal, Sylow
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53#38111yescharacteristic
143C22×C53#41311yescharacteristic
144C22×C53#41311yescharacteristic
145C22×C53C2×#10411yescharacteristic
146C22×C53C2×#10411yescharacteristic
147C22×C53#42111yescharacteristic
148C22×C53#42111yescharacteristic
149C4×C53—22no—
150C4×C53—41no—
151C4×C53—81no—
152C4×C53—81no—
153Dic53—22no—
154Dic53—22no—
155Dic53—22no—
156Dic53—41no—
157Dic53—81no—
158Dic53—81no—
159#2283—531no—
160#1749—531no—
161#1749—531no—
162#1295—532no—
163#1295—532no—
164C2×C53—42no—
165C2×C53#152411yescharacteristic
166C2×C53#152911yescharacteristic
167C2×C53#153211yescharacteristic
168C2×C53#153511yescharacteristic
169C2×C53#182211yescharacteristic
170C2×C53#182611yescharacteristic
171C2×C53#182611yescharacteristic
172C22×C42—531no—
173C2×C22⋊C8—1061no—
174C2×C22⋊C8—1061no—
175C2×#97—532no—
176C2×#97—532no—
177#383—1061no—
178#382—531no—
179#382—531no—
180#381—1061no—
181#380—531no—
182#380#615711yescharacteristic
183C4×C4⋊C4—531no—
184C4×C4⋊C4—531no—
185C4×C22⋊C4—1061no—
186#376—1061no—
187#336—1061no—
188#336—1061no—
189C53#872011yescharacteristic, complemented, Sylow
190C23×C4—531no—
191C23×C4—531no—
192C23×C4C2×#615711yescharacteristic
193C22×C8—1062no—
194C22×C8—1062no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4C2×#615711yescharacteristic
198C2×C4⋊C4C2×#615711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—1061no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42—531no—
205C2×C42—1064no—
206C2×C42—1061no—
207C2×C42—1061no—
208C22⋊C8—1064no—
209C22⋊C8—1064no—
210#97—1064no—
211#97—1064no—
212#95—21no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217#95—1061no—
218#95—1061no—
219#95—1061no—
220C24C22⋊#615711yescharacteristic
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—1062no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4—2121no—
237C22×C4—2121no—
238C22×C4C22⋊#615711yescharacteristic
239C22×C4C22×#615711yescharacteristic
240C2×C8—1064no—
241C2×C8—1064no—
242C2×C8—2122no—
243C2×C8—2122no—
244C2×C8—2122no—
245C2×C8—2122no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—2121no—
249C4⋊C4—2121no—
250C22⋊C4—42no—
251C22⋊C4—2122no—
252C42—534no—
253C42—1062no—
254C42—1062no—
255C42—1062no—
256C42—2122no—
257C42—2122no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23#1313328212yes—
265C23C2×C22⋊#615711yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—532no—
279C2×C4—532no—
280C2×C4—532no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—2124no—
291C2×C4—2124no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4#1313329612yes—
299C8—2124no—
300C8—2124no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22#1681156611yescharacteristic
309C22#1681161011yescharacteristic
310C22#1681161011yescharacteristic
311C22D53⋊#10411yescharacteristic
312C22C2×#1313328211yescharacteristic
313C22#1681162011yescharacteristic
314C22#1681162111yescharacteristic
315C4—22no—
316C4—41no—
317C4—81no—
318C4—81no—
319C4—1062no—
320C4—1062no—
321C4—1062no—
322C4—4241no—
323C4—4241no—
324C4—2121no—
325C2—42no—
326C2#2029016811yescharacteristic
327C2#2029018111yescharacteristic
328C2#2029019311yescharacteristic
329C2#2029020011yescharacteristic
330C2#2029061711yescharacteristic
331C2#2029062211yescharacteristic
332C2#2029062311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#8720
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289591C2semidirect
4. #20289591C2semidirect
189. C53#8720semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202895912
#202771102
#872053