Finite Group Databasebeta

← #27293252

Subgroups — #27293252

Order 13568.

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

normal subgroups only

This group has 454 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#20277339C211yesmaximal, characteristic
2C2×#16810349C211yesmaximal, characteristic
3#20272660C212yesmaximal, complemented
4#20272658C212yesmaximal, complemented
5C2×C53×#324C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7C2×#13133176C411yescharacteristic
8C2×C22⋊#88322—21no—
9#16810519—21no—
10#16810519—21no—
11#16810496—21no—
12#16810496—21no—
13#16810492C411yescharacteristic
14#16810492C2211yescharacteristic
15#16810488C411yescharacteristic
16#16810488C2211yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×#88322—41no—
24#13133176—22no—
25#13133176—41no—
26#13133176—41no—
27#13133176C814yescomplemented
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40C22⋊#88322—22no—
41C22⋊#88322—41no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C22×C8×C53—41no—
50C2×C4⋊C4×C53—41no—
51C2×C4⋊C4×C53—41no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53C2311yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—81no—
67C22×C4×C53—81no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C2×C812yes—
72C22×C4×C53M1612yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×C8×C53—42no—
81C2×C8×C53—82no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C22⋊Dic53—24no—
87C22⋊Dic53—41no—
88C22⋊Dic53—41no—
89C4⋊Dic53—24no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92#11719919—81no—
93#11719919—81no—
94#11719919—81no—
95#11719919—81no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—42no—
99C4×Dic53C2×C812yescomplemented
100C4×Dic53C2×C812yescomplemented
101C2×#88322—42no—
102C2×#88322—82no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53C22⋊C812yes—
106C23×C53#9911yescharacteristic
107C23×C53#9911yescharacteristic
108C23×C53#10112yes—
109C23×C53C2×C22⋊C411yescharacteristic
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53C22⋊C812yes—
131C2×C4×C53C22×C811yescharacteristic
132C2×C4×C53C2×M1611yescharacteristic
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—42no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C8×C53—82no—
145#88322—82no—
146#11448—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53#32412yes—
156C22×C53#32412yes—
157C22×C53C2×C22⋊C811yescharacteristic
158C22×C53C2×#9911yescharacteristic
159C22×C53C2×#10111yescharacteristic
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167Dic53—42no—
168Dic53—42no—
169Dic53—81no—
170Dic53—81no—
171#2317—531no—
172C2×#324—531no—
173C2×#324D5311yescharacteristic
174#1325—532no—
175#1324—532no—
176C2×C53—21no—
177C2×C53—22no—
178C2×C53—22no—
179C2×C53—21no—
180C2×C53C2×#32411yescharacteristic
181C2×C53#212711yescharacteristic
182C2×C53#213311yescharacteristic
183C2×#117—531no—
184C2×#117—531no—
185C22×C4⋊C4C2×D5311yescharacteristic
186C2×C22⋊C8—21no—
187C2×C22⋊C8—1061no—
188#389—1061no—
189#389—1061no—
190#385—1061no—
191#385—1061no—
192#384—531no—
193#384—531no—
194#382—531no—
195#382—531no—
196#324—21no—
197#324—532no—
198#324—1061no—
199#324C2×D5312yes—
200C53#1144811yescharacteristic, complemented, Sylow
201C23×C4C4×D5311yescharacteristic
202C23×C4D21211yescharacteristic
203C23×C4C22⋊D5311yescharacteristic
204C22×C8—41no—
205C22×C8—2121no—
206#117—534no—
207#117—1062no—
208#117—2121no—
209#117—2121no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4C4×D5311yescharacteristic
220C2×C4⋊C4D21211yescharacteristic
221C2×C4⋊C4C22⋊D5311yescharacteristic
222C2×C4⋊C4C22×D5311yescharacteristic
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C42—531no—
226C2×C42—531no—
227C2×C42—1061no—
228C22⋊C8—22no—
229C22⋊C8—41no—
230C22⋊C8—1062no—
231C22⋊C8—2121no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235#95—1061no—
236C24#1171992111yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4C8×D5312yes—
251C22×C4#1171991212yes—
252C22×C4#1171992111yescharacteristic
253C22×C4#1171992111yescharacteristic
254C22×C4#1171992111yescharacteristic
255C22×C4C2×C4×D5311yescharacteristic
256C22×C4C2×D21211yescharacteristic
257C22×C4C2×C22⋊D5311yescharacteristic
258C2×C8—42no—
259C2×C8—82no—
260C2×C8—4242no—
261C2×C8—2122no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—1064no—
267C4⋊C4—4241no—
268C4⋊C4—4241no—
269C4⋊C4—4241no—
270C4⋊C4—4241no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C22⋊C4—1064no—
274C22⋊C4—2121no—
275C22⋊C4—2121no—
276C42—532no—
277C42—532no—
278C42—1061no—
279C42—1061no—
280C42—2122no—
281C23—41no—
282C23—41no—
283C23#1313312912yes—
284C23#1313313111yescharacteristic
285C23#1313313111yescharacteristic
286C23#1313313512yes—
287C23C2×#1171992111yescharacteristic
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—1062no—
309C2×C4—1062no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2122no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4#1313312912yes—
320C2×C4C2×C8×D5311yescharacteristic
321C2×C4C2×#1171991211yescharacteristic
322C8—82no—
323C8—4242no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—42no—
331C22—42no—
332C22#1681034812yes—
333C22#1681034812yes—
334C22C2×#1313312911yescharacteristic
335C22C2×#1313313111yescharacteristic
336C22C2×#1313313511yescharacteristic
337C4—22no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—4241no—
345C4—4241no—
346C4—2122no—
347C4—2122no—
348C2—21no—
349C2—21no—
350C2—22no—
351C2—22no—
352C2C2×#1681034811yescharacteristic
353C2#2027638211yescharacteristic
354C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#324C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#324D53
hypocenterC53#11448
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #20272660C2semidirect
4. #20272658C2semidirect
27. #13133176C8semidirect
99. C4×Dic53C2×C8semidirect
100. C4×Dic53C2×C8semidirect
200. C53#11448semidirect

Maximal subgroups

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

subgroupindex
#202773392
C2×#168103492
#202726602
#202726582
C2×C53×#3242
#1144853