Finite Group Databasebeta

← #27293355

Subgroups — #27293355

Order 13568.

460 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277380C211yesmaximal, characteristic
2#20273036C211yesmaximal, characteristic
3#20272662C212yesmaximal, complemented
4#20272644C212yesmaximal, complemented
5C53×#1459C211yesmaximal, characteristic
6C2×#13133198C411yescharacteristic
7C22×#11719919C411yescharacteristic
8#16810676—21no—
9#16810518—21no—
10#16810517—21no—
11#16810507—21no—
12#16810503C411yescharacteristic
13#16810496—21no—
14#16810495C2211yescharacteristic
15#16810484C2211yescharacteristic
16C4×C22⋊Dic53C411yescharacteristic
17#16810347—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25#13133198—22no—
26#13133198C814yescomplemented
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—41no—
32C2×#11719919—41no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53C2×C411yescharacteristic
36C2×C22⋊Dic53C2×C411yescharacteristic
37C2×C4×Dic53—21no—
38C4⋊#88322—41no—
39C22⋊#88322—22no—
40C4×#88322—41no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×D4×C53C2×C411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C22×C8×C53—41no—
50C4×D4×C53—41no—
51C4×D4×C53—41no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53C2311yescharacteristic
55C2×C42×C53D411yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C22×C411yescharacteristic
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53C2×C812yes—
63C2×D4×C53M1612yes—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—81no—
69C22×C4×C53—81no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—81no—
84C22×Dic53C2×C812yescomplemented
85C2×C8×C53—42no—
86C2×C8×C53—82no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53—21no—
91C42×C53—21no—
92C22⋊Dic53—24no—
93C22⋊Dic53—21no—
94C22⋊Dic53—41no—
95C22⋊Dic53—81no—
96C22⋊Dic53C2×C812yescomplemented
97#11719919—24no—
98#11719919—42no—
99#11719919—41no—
100#11719919—41no—
101#11719919—81no—
102C4×Dic53—81no—
103C2×#88322—42no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53C22⋊C812yes—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C22×C811yescharacteristic
115C23×C53C2×M1611yescharacteristic
116D4×C53—42no—
117D4×C53—42no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53C22⋊C812yes—
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53#9911yescharacteristic
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—41no—
137C2×Dic53—42no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—42no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—82no—
149C8×C53—82no—
150#88322—82no—
151#11474—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—22no—
157C22×C53—42no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—42no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—81no—
166C22×C53C2×C22⋊C811yescharacteristic
167C22×C53C2×#9911yescharacteristic
168C22×C53#41311yescharacteristic
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—22no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175Dic53—42no—
176Dic53—42no—
177Dic53—81no—
178Dic53—81no—
179#2322—531no—
180#1460—531no—
181#1459D5311yescharacteristic
182#1326—532no—
183#1317—532no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—41no—
187C2×C53—41no—
188C2×C53—42no—
189C2×C53#145911yescharacteristic
190C2×C53#212111yescharacteristic
191C2×C53#212311yescharacteristic
192C2×#123—531no—
193C2×C4×D4C2×D5311yescharacteristic
194C22×C4⋊C4—531no—
195#431—1061no—
196C2×C22⋊C8—21no—
197#389—1061no—
198#388—1061no—
199#387—1061no—
200#386—531no—
201#385—531no—
202#385—1061no—
203#381—531no—
204C4×C22⋊C4—531no—
205#324—21no—
206#323—532no—
207#323—1061no—
208#323C2×D5312yes—
209C53#1147411yescharacteristic, complemented, Sylow
210C22×D4C4×D5311yescharacteristic
211C23×C4—531no—
212C23×C4—531no—
213C23×C4C4×D5311yescharacteristic
214C23×C4C22⋊D5311yescharacteristic
215C22×C8—41no—
216#123—534no—
217#123—1062no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—2121no—
225C2×C4⋊C4—2121no—
226C2×C4⋊C4C22⋊D5311yescharacteristic
227C2×C22⋊C4—531no—
228C2×C22⋊C4—531no—
229C2×C22⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C22⋊C4D21211yescharacteristic
232C2×C22⋊C4C22×D5311yescharacteristic
233C2×C42—1061no—
234C2×C42D21211yescharacteristic
235C4⋊C8—2121no—
236C22⋊C8—22no—
237C22⋊C8—41no—
238C22⋊C8—1062no—
239C4×C8—2121no—
240#95—1061no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245C24#1171992111yescharacteristic
246C24C2×C4×D5311yescharacteristic
247C2×D4—41no—
248C2×D4—41no—
249C2×D4C8×D5312yes—
250C2×D4#1171991212yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—532no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—4241no—
264C22×C4—2121no—
265C22×C4—2121no—
266C22×C4—2121no—
267C22×C4#1171992111yescharacteristic
268C22×C4#1171992111yescharacteristic
269C22×C4#1171992111yescharacteristic
270C22×C4C2×D21211yescharacteristic
271C22×C4C2×C22⋊D5311yescharacteristic
272C2×C8—42no—
273C2×C8—82no—
274C2×C8—2122no—
275C4⋊C4—41no—
276C4⋊C4—1064no—
277C4⋊C4—4241no—
278C4⋊C4—2122no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—532no—
284C22⋊C4—1064no—
285C22⋊C4—1061no—
286C22⋊C4—4241no—
287C22⋊C4—2121no—
288C42—21no—
289C42—21no—
290C42—4241no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23#1313312912yes—
300C23C2×#1171992111yescharacteristic
301C23C2×C8×D5311yescharacteristic
302C23C2×#1171991211yescharacteristic
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—4242no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2122no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2122no—
333C2×C4#1313312912yes—
334C2×C4#1313313111yescharacteristic
335C2×C4#1313313111yescharacteristic
336C8—82no—
337C8—4242no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—81no—
352C22C2×#1313312911yescharacteristic
353C22C2×#1313313111yescharacteristic
354C22#1681060511yescharacteristic
355C4—22no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—4241no—
362C4—4241no—
363C4—2122no—
364C4—2122no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—41no—
369C2—42no—
370C2#2027303111yescharacteristic
371C2#2027635911yescharacteristic
372C2#2027637011yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC23×C53C22×C8
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1459C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1459D53
hypocenterC53#11474
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20272662C2semidirect
4. #20272644C2semidirect
26. #13133198C8semidirect
84. C22×Dic53C2×C8semidirect
96. C22⋊Dic53C2×C8semidirect
209. C53#11474semidirect

Maximal subgroups

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

subgroupindex
#202773802
#202730362
#202726622
#202726442
C53×#14592
#1147453