Finite Group Databasebeta

← #27293714

Subgroups — #27293714

Order 13568.

616 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279661C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic, complemented
3#20272646C214yesmaximal, complemented
4C53×#1465C211yesmaximal, characteristic, complemented
5C2×#13133214C411yescharacteristic
6C2×#13133213C412yes—
7C2×#13133191—21no—
8C2×#13133191—21no—
9C2×C22⋊#88322—21no—
10#16810532—21no—
11#16810532—21no—
12#16810524C2212yes—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349C2212yes—
15#16810347—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324C2212yes—
19C53×#323—21no—
20C22×D212—21no—
21C22×D212—21no—
22C22×D212C2×C412yes—
23C22×C22⋊D53—22no—
24C22×C22⋊D53—22no—
25C22×#88322—41no—
26#13133214—41no—
27#13133214—41no—
28#13133213—44no—
29#13133191—44no—
30#13133191—44no—
31C2×C4⋊Dic53—21no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×C4×Dic53C2×C412yes—
35C22⋊#88322—22no—
36C22⋊#88322—41no—
37#13133124—21no—
38C22×D4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C2×M16×C53—41no—
42C4×D4×C53—41no—
43C4×D4×C53—41no—
44C2×C4⋊C4×C53C2311yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48C22⋊C8×C53—22no—
49C22⋊C8×C53—41no—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C22⋊C411yescharacteristic
52C23×D53—22no—
53C2×D4×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53—41no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C2×C22⋊D53—44no—
68C2×C22⋊D53—44no—
69C2×C22⋊D53—44no—
70C2×C22⋊D53—44no—
71C2×C22⋊D53—82no—
72C2×C22⋊D53—82no—
73C22×Dic53—22no—
74C2×D212—24no—
75C2×D212—42no—
76C2×D212—42no—
77C2×D212—42no—
78C2×D212—42no—
79C2×D212—42no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—82no—
83M16×C53—82no—
84C2×C8×C53—42no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C42×C53—21no—
89C42×C53—21no—
90#11719921—44no—
91#11719921—44no—
92C4⋊Dic53—24no—
93C4×Dic53—42no—
94C2×#88322—42no—
95C2×#88322—82no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53#10012yes—
106C23×C53C2×C22⋊C411yescharacteristic
107C22×D53—42no—
108C22×D53—42no—
109C22×D53—42no—
110C22×D53—42no—
111C22×D53—42no—
112C22×D53—42no—
113C22×D53—42no—
114D4×C53—44no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#9911yescharacteristic
130C2×C4×C53#13112yes—
131C22⋊D53—88no—
132C22⋊D53—88no—
133C2×Dic53—24no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136D212—84no—
137D212—82no—
138D212—84no—
139D212—82no—
140C8×C53—82no—
141#88322—82no—
142#11563—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53#40811yescharacteristic
156C22×C53C2×#9911yescharacteristic
157C22×C53C2×#10011yescharacteristic
158C2×D53—82no—
159C2×D53—82no—
160C2×D53—82no—
161C2×D53—82no—
162C2×D53—82no—
163C2×D53—82no—
164C2×D53—82no—
165C4×C53—21no—
166C4×C53—22no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53#35312yes—
171Dic53—44no—
172#2541—531no—
173#1465D5311yescharacteristic, complemented
174#1459—531no—
175#1318—534no—
176C2×C53—21no—
177C2×C53—21no—
178C2×C53—42no—
179C2×C53—81no—
180C2×C53#147111yescharacteristic
181C2×C53C2×#35311yescharacteristic
182C2×C53#213411yescharacteristic
183D53—162no—
184C2×#127—532no—
185C2×#127—531no—
186C2×#121—1061no—
187C2×#121—1061no—
188C2×C4×D4C2×D5311yescharacteristic
189#407—21no—
190C2×C22⋊C8—1061no—
191#392—1061no—
192#392—1061no—
193#390—532no—
194C4×C4⋊C4—531no—
195#324—532no—
196#324C2×D5312yes—
197#323—21no—
198#323—1061no—
199C53#1156311yescharacteristic, complemented, Sylow
200C22×D4—532no—
201C22×D4—1062no—
202C22×D4—1062no—
203C22×D4—1061no—
204C22×D4—1061no—
205C22×D4C4×D5311yescharacteristic
206C23×C4D21211yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C2×M16—41no—
209C22×C8—2121no—
210#127—2121no—
211#127—2121no—
212#127—2124no—
213#121—2124no—
214#121—2124no—
215C4×D4—41no—
216C4×D4—41no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4C22×D5311yescharacteristic
219C2×C22⋊C4—1062no—
220C2×C22⋊C4—1062no—
221C2×C22⋊C4D21211yescharacteristic
222C2×C22⋊C4C22⋊D5311yescharacteristic
223C2×C42—532no—
224C2×C42C4×D5311yescharacteristic
225C22⋊C8—22no—
226C22⋊C8—41no—
227C22⋊C8—1062no—
228C22⋊C8—2121no—
229#95—1061no—
230C24—1062no—
231C24#1171992111yescharacteristic
232C24#1171992111yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—41no—
236C2×D4—1064no—
237C2×D4—4242no—
238C2×D4—4242no—
239C2×D4—4242no—
240C2×D4—2122no—
241C2×D4—2122no—
242C2×D4—2122no—
243C2×D4—2122no—
244C2×D4—2122no—
245C2×D4—2124no—
246C2×D4—2124no—
247C2×D4—2124no—
248C2×D4—2124no—
249C2×D4—2121no—
250C2×D4—2121no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—1062no—
258C22×C4#1171992111yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4C2×C4×D5311yescharacteristic
261C22×C4C2×D21211yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263M16—82no—
264C2×C8—42no—
265C2×C8—4242no—
266C2×C8—2122no—
267C4⋊C4—41no—
268C4⋊C4—1064no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—2124no—
272C22⋊C4—2124no—
273C42—21no—
274C42—21no—
275C42—2122no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—2122no—
286C23—2122no—
287C23—2122no—
288C23—2122no—
289C23—2122no—
290C23—2122no—
291C23—2122no—
292C23#1313313312yes—
293C23C2×#1171992111yescharacteristic
294D4—44no—
295D4—4244no—
296D4—4242no—
297D4—4244no—
298D4—4248no—
299D4—4248no—
300D4—4242no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—1064no—
315C2×C4—2122no—
316C2×C4—2122no—
317C2×C4#1313313111yescharacteristic
318C2×C4#1313313111yescharacteristic
319C2×C4#1313322512yes—
320C8—82no—
321C8—4242no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—4242no—
335C22—4242no—
336C22—4242no—
337C22—4242no—
338C22—4242no—
339C22—4242no—
340C22—4242no—
341C22#1681059111yescharacteristic
342C22C2×#1313313111yescharacteristic
343C22C2×#1313313311yescharacteristic
344C4—21no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—2124no—
350C4#1681041512yes—
351C2—21no—
352C2—21no—
353C2—42no—
354C2—81no—
355C2—8482no—
356C2#2027307611yescharacteristic
357C2C2×#1681041511yescharacteristic
358C2#2027640611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1465C2
socleC22×C53C2×#99
prime radical#20279661C2
cosocleC2×C4⋊C4×C53C23
hypercenter#1465D53
hypocenterC53#11563
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #20273033C2semidirect
3. #20272646C2semidirect
4. C53×#1465C2semidirect
173. #1465D53semidirect
199. C53#11563semidirect

Maximal subgroups

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

subgroupindex
#202796612
#202730332
#202726462
C53×#14652
#1156353