Finite Group Databasebeta

← #27281869

Subgroups — #27281869

Order 13568.

716 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 115 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#20283744C211yesmaximal, characteristic
2#20278080C211yesmaximal, characteristic
3#20274649C212yesmaximal, complemented
4#20272582C212yesmaximal, complemented
5C53×#1799C211yesmaximal, characteristic
6C23×Dic106C2211yescharacteristic
7C2×#13133196—21no—
8#16810695—24no—
9#16810680—21no—
10C2×#13133140—24no—
11#16810587—21no—
12C2×C22⋊#88322C2212yes—
13#16810543C412yescomplemented
14#16810535C412yescomplemented
15#16810529—21no—
16#16810503—21no—
17#16810495—21no—
18#16810483C2211yescharacteristic
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C53×#380C2212yes—
22C53×#342—21no—
23C22×Dic106—22no—
24C22×Dic106—21no—
25C22×Dic106—22no—
26C22×Dic106—44no—
27C22×Dic106C2×C412yes—
28C23×Dic53—21no—
29C2×#11719917—22no—
30C22×#88322D414yes—
31#13133196—44no—
32C2×C4⋊Dic53—21no—
33C2×#11719919—22no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—21no—
36C4⋊#88322—22no—
37#13133140—48no—
38C22⋊#88322—22no—
39C22⋊#88322—22no—
40C22⋊#88322—22no—
41#13133128—22no—
42#13133124—22no—
43#13133124—22no—
44#13133124C2×C412yes—
45C22×D4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C23×C4×C53C2311yescharacteristic
48C4×D4×C53—22no—
49C4×D4×C53—22no—
50C2×C4⋊C4×C53—22no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53—22no—
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56#99×C53—42no—
57#95×C53—22no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C2×D411yescharacteristic
60C2×D4×C53—22no—
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—22no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D412yes—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53#4112yes—
76C22×Dic53—22no—
77C22×Dic53—22no—
78C22×Dic53—44no—
79C22×Dic53—42no—
80C2×Dic106—28no—
81C2×Dic106—24no—
82C2×Dic106—24no—
83C2×Dic106—44no—
84C2×Dic106—48no—
85C2×Dic106—44no—
86C2×Dic106—42no—
87C2×Dic106—41no—
88C2×Dic106—41no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—44no—
91C22⋊C4×C53—22no—
92C22⋊C4×C53—22no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—42no—
95C42×C53—21no—
96C42×C53—21no—
97C22⋊Dic53—44no—
98C22⋊Dic53—41no—
99C22⋊Dic53—41no—
100C4⋊Dic53—42no—
101#11719919—44no—
102#11719917—82no—
103C2×#88322—24no—
104C2×#88322—44no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53C2×C22⋊C411yescharacteristic
113C23×C53#12112yes—
114C23×C53AGL(1,ℤ/8)11yescharacteristic
115C23×C53#13712yes—
116C23×C53#13711yescharacteristic
117D4×C53—42no—
118D4×C53—42no—
119C2×C4×C53—22no—
120C2×C4×C53—24no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—81no—
132C2×C4×C53—82no—
133C2×C4×C53—82no—
134C2×C4×C53—82no—
135C2×C4×C53C4×D412yes—
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×Dic53—24no—
139C2×Dic53—48no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145Dic106—28no—
146Dic106—44no—
147Dic106—44no—
148Dic106—44no—
149#88322—44no—
150#9471—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—22no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—41no—
156C22×C53—42no—
157C22×C53—41no—
158C22×C53—42no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—81no—
163C22×C53#38611yescharacteristic
164C22×C53#41711yescharacteristic
165C22×C53#41911yescharacteristic
166C22×C53#46812yes—
167C22×C53#47012yes—
168C4×C53—22no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—162no—
174C4×C53B(4,𝔽2)12yes—
175Dic53—44no—
176Dic53—82no—
177#2973—531no—
178#2409—531no—
179#1799D5311yescharacteristic
180#1787—532no—
181#1289—532no—
182C2×C53—21no—
183C2×C53—22no—
184C2×C53—21no—
185C2×C53—22no—
186C2×C53—81no—
187C2×C53#188911yescharacteristic
188C2×C53#189411yescharacteristic
189C2×C53#190911yescharacteristic
190C23×Q8—531no—
191C2×C22⋊Q8—1061no—
192C2×C4×D4C2×D5311yescharacteristic
193#435—1064no—
194#432—1061no—
195C2×#103—1064no—
196C2×#99—21no—
197#407—1061no—
198C2×C22⋊C8—532no—
199#395—532no—
200#393—532no—
201#391—1061no—
202#386—1061no—
203#385—1061no—
204#380—531no—
205#380C2×D5312yes—
206#342—21no—
207C53#947111yescharacteristic, complemented, Sylow
208C22×Q8—532no—
209C22×Q8—1061no—
210C22×Q8—1062no—
211C22×Q8—1062no—
212C22×Q8—2124no—
213C22×D4C22⋊D5311yescharacteristic
214C23×C4—1061no—
215C23×C4D21211yescharacteristic
216C23×C4C22×D5311yescharacteristic
217C2×M16—1062no—
218C22×C8—534no—
219C22⋊Q8—2124no—
220C4×D4—22no—
221C4×D4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1062no—
225C2×C4⋊C4C4×D5311yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—1061no—
228C2×C22⋊C4—1062no—
229C2×C22⋊C4C4×D5311yescharacteristic
230C2×C22⋊C4C22⋊D5311yescharacteristic
231C2×C42D21211yescharacteristic
232C4⋊C8—1062no—
233#103—2128no—
234#99—42no—
235C22⋊C8—1062no—
236C22⋊C8—1062no—
237C22⋊C8—1062no—
238#97—1062no—
239#95—22no—
240#95—532no—
241#95—1062no—
242#95—1062no—
243C24#1171992111yescharacteristic
244C24C2×C22⋊D5311yescharacteristic
245C2×Q8—1064no—
246C2×Q8—1068no—
247C2×Q8—1064no—
248C2×Q8—2124no—
249C2×Q8—2122no—
250C2×Q8—2124no—
251C2×Q8—2128no—
252C2×Q8—2121no—
253C2×Q8—2121no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—22no—
257C2×D4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—1062no—
264C22×C4—1062no—
265C22×C4—2124no—
266C22×C4—2122no—
267C22×C4#1171992111yescharacteristic
268C22×C4#1171992111yescharacteristic
269C22×C4#1171992111yescharacteristic
270C22×C4C2×C4×D5311yescharacteristic
271C22×C4C2×D21211yescharacteristic
272C22×C4D4×D5312yes—
273C22×C4#1171994712yes—
274M16—4242no—
275C2×C8—1064no—
276C2×C8—2124no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C4⋊C4—2122no—
280C4⋊C4—2124no—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—2124no—
286C22⋊C4—2121no—
287C22⋊C4—2121no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×#1171992111yescharacteristic
298C23#1313318912yes—
299C23#1313324311yescharacteristic
300C23#1313324712yes—
301C23#1313324811yescharacteristic
302Q8—1068no—
303Q8—2124no—
304Q8—2124no—
305Q8—2124no—
306D4—42no—
307D4—42no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—1064no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4—2128no—
330C2×C4—2122no—
331C2×C4#1313317712yes—
332C2×C4#1313318711yescharacteristic
333C2×C4#1313320111yescharacteristic
334C8—2124no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#1681050011yescharacteristic
348C22#1681062211yescharacteristic
349C22#1681063111yescharacteristic
350C22#1681086712yes—
351C22#1681087912yes—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—162no—
358C4—4242no—
359C4—2124no—
360C4#1681081112yes—
361C2—22no—
362C2—21no—
363C2—21no—
364C2—22no—
365C2—81no—
366C2#2027512111yescharacteristic
367C2#2027514611yescharacteristic
368C2#2027523511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1799C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#1799D53
hypocenterC53#9471
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20274649C2semidirect
4. #20272582C2semidirect
13. #16810543C4semidirect
14. #16810535C4semidirect
207. C53#9471semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202780802
#202746492
#202725822
C53×#17992
#947153