Finite Group Databasebeta

← #28303368

Subgroups — #28303368

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#20290572C211yesmaximal, characteristic
2#20290572C211yesmaximal, characteristic
3#20289593C212yesmaximal, complemented
4#20289593C212yesmaximal, complemented
5#20277119C211yesmaximal, characteristic
6C2×#13133284C2212yes—
7C2×#13133284C2212yes—
8C2×#13133274—21no—
9C2×#13133274—21no—
10#16811489—21no—
11#16811489—21no—
12C2×#13133174—21no—
13C2×C42×D53C2211yescharacteristic
14#16810492C2211yescharacteristic
15#16810492C2211yescharacteristic
16#16810487—21no—
17#16810482—21no—
18#16810480C411yescharacteristic
19#16810480C411yescharacteristic
20C4×#11719919—21no—
21C4×C4⋊Dic53C411yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—24no—
26#13133284—24no—
27#13133274—42no—
28#13133274—42no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32#13133174—44no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45C42×D53—24no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C42×C53—21no—
52C2×C42×C53C2×C411yescharacteristic
53#95×C53—21no—
54C23×D53C22⋊C411yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C4×D53SD1612yes—
75C2×C4×D53Q1612yes—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C42×C53—22no—
85C42×C53—42no—
86#11719921—42no—
87#11719921—42no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90#11719919—42no—
91#11719919—41no—
92#11719919—42no—
93#11719919—42no—
94#11719919—41no—
95C4×Dic53—22no—
96C4×Dic53—22no—
97C4×Dic53—42no—
98C4×Dic53#4114yes—
99C23×C53C2×C22⋊C411yescharacteristic
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53#10312yes—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
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#10312yes—
121C2×C4×C53#11712yes—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53#12312yes—
132C2×Dic53#12312yes—
133C2×Dic53C2×SD1611yescharacteristic
134C2×Dic53C2×Q1611yescharacteristic
135C2×Dic53#13512yes—
136C2×Dic53AGL(1,ℤ/8)12yes—
137C2×Dic53AGL(1,ℤ/8)11yescharacteristic
138C2×Dic53#13711yescharacteristic
139C4×D53—24no—
140C4×D53—24no—
141C4×D53—44no—
142C4×D53—42no—
143C4×D53—42no—
144#88324—82no—
145#88324—82no—
146#8727—531nomaximal, Sylow
147C22×C53#38111yescharacteristic
148C22×C53#41311yescharacteristic
149C22×C53#41311yescharacteristic
150C22×C53C2×#10311yescharacteristic
151C22×C53#41611yescharacteristic
152C22×C53#41711yescharacteristic
153C22×C53#41811yescharacteristic
154C2×D53—22no—
155C2×D53—22no—
156C2×D53—22no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—41no—
164C4×C53—81no—
165C4×C53—81no—
166Dic53—41no—
167Dic53—81no—
168Dic53—81no—
169Dic53#48112yes—
170Dic53#48212yes—
171Dic53#48412yes—
172Dic53#48512yes—
173#2285—531no—
174#1797—531no—
175#1797—531no—
176#1296—532no—
177#1296—532no—
178C2×C53#154211yescharacteristic
179C2×C53#155011yescharacteristic
180C2×C53#155311yescharacteristic
181C2×C53#155711yescharacteristic
182C2×C53#182211yescharacteristic
183C2×C53#182411yescharacteristic
184C2×C53#182511yescharacteristic
185D53—42no—
186C2×#117—1061no—
187C22×C42—531no—
188C2×C4⋊C8—532no—
189C2×C4⋊C8—532no—
190C2×C22⋊C8—1061no—
191C2×C22⋊C8—1061no—
192#384—531no—
193#384—531no—
194#382—1061no—
195#380—531no—
196#380—531no—
197#380—1061no—
198C4×C4⋊C4—531no—
199C4×C4⋊C4—1061no—
200C4×C4⋊C4#615711yescharacteristic
201#336—1061no—
202#336—1061no—
203C53#872711yescharacteristic, complemented, Sylow
204C23×C4—531no—
205C23×C4—531no—
206C23×C4—531no—
207C22×C8—1062no—
208C22×C8—1062no—
209#117—2124no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4C2×#615711yescharacteristic
217C2×C4⋊C4C2×#615711yescharacteristic
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C42—21no—
221C2×C42—531no—
222C2×C42—531no—
223C2×C42—531no—
224C2×C42—1064no—
225C2×C42—1061no—
226C2×C42C2×#615711yescharacteristic
227C4⋊C8—1064no—
228C4⋊C8—1064no—
229C22⋊C8—2122no—
230C22⋊C8—2122no—
231#95—21no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235C24—531no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—532no—
241C22×C4—532no—
242C22×C4—531no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4—1061no—
246C22×C4—1062no—
247C22×C4—1062no—
248C22×C4—1061no—
249C22×C4—1062no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—2121no—
253C22×C4—2121no—
254C22×C4C22⋊#615711yescharacteristic
255C22×C4C22⋊#615711yescharacteristic
256C22×C4C22×#615711yescharacteristic
257C2×C8—2122no—
258C2×C8—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C2×C8—2122no—
262C2×C8—2122no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2122no—
268C4⋊C4—2122no—
269C4⋊C4—2122no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C22⋊C4—2122no—
273C22⋊C4—2122no—
274C42—22no—
275C42—42no—
276C42—534no—
277C42—1062no—
278C42—1062no—
279C42—2122no—
280C23—532no—
281C23—1061no—
282C23—1061no—
283C23—1061no—
284C23—1061no—
285C23—1061no—
286C23—1061no—
287C23C2×C22⋊#615711yescharacteristic
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—532no—
302C2×C4—532no—
303C2×C4—531no—
304C2×C4—531no—
305C2×C4—532no—
306C2×C4—531no—
307C2×C4—531no—
308C2×C4—532no—
309C2×C4—1064no—
310C2×C4—1064no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—2124no—
315C2×C4—2122no—
316C2×C4—2122no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4#1313328012yes—
324C2×C4#1313329612yes—
325C8—4242no—
326C8—4242no—
327C22—1062no—
328C22—1062no—
329C22—1062no—
330C22—2121no—
331C22—2121no—
332C22—2121no—
333C22—2121no—
334C22#1681156611yescharacteristic
335C22#1681160911yescharacteristic
336C22#1681160911yescharacteristic
337C22C2×#1313328011yescharacteristic
338C22#1681161311yescharacteristic
339C22#1681161411yescharacteristic
340C22#1681161511yescharacteristic
341C4—22no—
342C4—22no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—532no—
347C4—532no—
348C4—532no—
349C4—532no—
350C4—4241no—
351C4—4241no—
352C4—2121no—
353C2—2122no—
354C2#2029021611yescharacteristic
355C2#2029023211yescharacteristic
356C2#2029023811yescharacteristic
357C2#2029024611yescharacteristic
358C2#2029061611yescharacteristic
359C2#2029061911yescharacteristic
360C2#2029062011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#8727
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289593C2semidirect
4. #20289593C2semidirect
203. C53#8727semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202895932
#202771192
#872753