Finite Group Databasebeta

← #28303367

Subgroups — #28303367

Order 13568.

514 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 114 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#20290571C211yesmaximal, characteristic
2#20290571C211yesmaximal, characteristic
3#20289593C212yesmaximal, complemented
4#20289593C212yesmaximal, complemented
5#20277117C211yesmaximal, characteristic
6C2×#13133284C2212yes—
7C2×#13133284C2212yes—
8C2×C22⋊#88324—21no—
9C2×C22⋊#88324—21no—
10#16811489—21no—
11#16811489—21no—
12C2×#13133173—21no—
13C22×C4×Dic53C2211yescharacteristic
14#16810492C2211yescharacteristic
15#16810492C2211yescharacteristic
16#16810487—21no—
17#16810483C411yescharacteristic
18#16810481—21no—
19C4⋊C4×Dic53C411yescharacteristic
20C4⋊C4×Dic53C411yescharacteristic
21#16810474—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—24no—
26#13133284—24no—
27C22⋊#88324—42no—
28C22⋊#88324—42no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133173—42no—
32#13133173—42no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×C22⋊Dic53—21no—
40C2×C4×Dic53—24no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C23×C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53—21no—
54#95×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53D812yes—
74C22×Dic53SD1612yes—
75C22×Dic53C2×D411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C22⋊C4×C53—42no—
86C22⋊Dic53—42no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89#11719919—41no—
90#11719919—41no—
91#11719919—41no—
92#11719919—41no—
93#11719919—42no—
94#11719919—41no—
95#11719919—41no—
96C4×Dic53—22no—
97C4×Dic53—22no—
98C4×Dic53—22no—
99C4×Dic53—42no—
100C4×Dic53—42no—
101C4×Dic53#4114yes—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53#10212yes—
109C23×C53C2×C22⋊C411yescharacteristic
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—44no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#11712yes—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—24no—
126C2×Dic53—24no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53#10212yes—
139C2×Dic53#12312yes—
140C2×Dic53#12312yes—
141C2×Dic53C2×D811yescharacteristic
142C2×Dic53C2×SD1611yescharacteristic
143C2×Dic53#13512yes—
144C2×Dic53AGL(1,ℤ/8)11yescharacteristic
145C2×Dic53#13712yes—
146C2×Dic53#13711yescharacteristic
147#88324—82no—
148#88324—82no—
149#8726—531nomaximal, Sylow
150C22×C53—22no—
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53#38111yescharacteristic
158C22×C53#41311yescharacteristic
159C22×C53#41311yescharacteristic
160C22×C53C2×#10211yescharacteristic
161C22×C53#41611yescharacteristic
162C22×C53#41711yescharacteristic
163C22×C53#41911yescharacteristic
164C4×C53—22no—
165C4×C53—41no—
166C4×C53—81no—
167C4×C53—81no—
168Dic53—22no—
169Dic53—41no—
170Dic53—81no—
171Dic53—81no—
172Dic53#48012yes—
173Dic53#48212yes—
174Dic53#48312yes—
175Dic53#48512yes—
176#2285—531no—
177#1797—531no—
178#1797—531no—
179#1296—532no—
180#1296—532no—
181C2×C53—42no—
182C2×C53#154011yescharacteristic
183C2×C53#154711yescharacteristic
184C2×C53#155411yescharacteristic
185C2×C53#155911yescharacteristic
186C2×C53#182211yescharacteristic
187C2×C53#182311yescharacteristic
188C2×C53#182511yescharacteristic
189C2×#117—1061no—
190C22×C42—531no—
191C2×C4⋊C8—532no—
192C2×C4⋊C8—532no—
193C2×C22⋊C8—1061no—
194C2×C22⋊C8—1061no—
195#384—531no—
196#384—531no—
197#382—1061no—
198#380—531no—
199#380—1061no—
200#380#615711yescharacteristic
201C4×C4⋊C4—531no—
202C4×C4⋊C4—531no—
203C4×C4⋊C4—1061no—
204#336—1061no—
205#336—1061no—
206C53#872611yescharacteristic, complemented, Sylow
207C23×C4—531no—
208C23×C4—531no—
209C23×C4C2×#615711yescharacteristic
210C22×C8—1062no—
211C22×C8—1062no—
212#117—2122no—
213#117—2122no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4C2×#615711yescharacteristic
221C2×C4⋊C4C2×#615711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—1061no—
224C2×C42—531no—
225C2×C42—531no—
226C2×C42—531no—
227C2×C42—531no—
228C2×C42—1064no—
229C2×C42—1061no—
230C2×C42—1061no—
231C4⋊C8—1064no—
232C4⋊C8—1064no—
233C22⋊C8—2122no—
234C22⋊C8—2122no—
235#95—21no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239C24C22⋊#615711yescharacteristic
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—532no—
248C22×C4—532no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—1062no—
252C22×C4—1062no—
253C22×C4—1061no—
254C22×C4—1062no—
255C22×C4—1062no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4C22⋊#615711yescharacteristic
260C22×C4C22×#615711yescharacteristic
261C2×C8—2122no—
262C2×C8—2122no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2122no—
266C2×C8—2122no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C4⋊C4—2122no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C22⋊C4—42no—
279C22⋊C4—2122no—
280C42—534no—
281C42—1062no—
282C42—1062no—
283C42—1062no—
284C42—2122no—
285C42—2122no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23#1313327912yes—
293C23C2×C22⋊#615711yescharacteristic
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—532no—
307C2×C4—532no—
308C2×C4—531no—
309C2×C4—531no—
310C2×C4—532no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—532no—
314C2×C4—532no—
315C2×C4—1064no—
316C2×C4—1064no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4#1313329612yes—
331C8—4242no—
332C8—4242no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#1681156611yescharacteristic
341C22#1681161011yescharacteristic
342C22#1681161011yescharacteristic
343C22C2×#1313327911yescharacteristic
344C22#1681161311yescharacteristic
345C22#1681161411yescharacteristic
346C22#1681161611yescharacteristic
347C4—22no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—532no—
352C4—532no—
353C4—532no—
354C4—532no—
355C4—1062no—
356C4—4241no—
357C4—4241no—
358C4—2121no—
359C2—42no—
360C2#2029021311yescharacteristic
361C2#2029022711yescharacteristic
362C2#2029024111yescharacteristic
363C2#2029025111yescharacteristic
364C2#2029061711yescharacteristic
365C2#2029061811yescharacteristic
366C2#2029062011yescharacteristic
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#8726
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. #20289593C2semidirect
4. #20289593C2semidirect
206. C53#8726semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202895932
#202771172
#872653