Finite Group Databasebeta

← #27281762

Subgroups — #27281762

Order 13568.

490 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 107 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#20273342C212yesmaximal, complemented
3#20273056C211yesmaximal, characteristic
4#20272571C212yesmaximal, complemented
5C53×#2566C211yesmaximal, characteristic
6#16810695—24no—
7#16810680C2212yes—
8#16810680C2212yes—
9#16810587—21no—
10C2×C22⋊#88322—21no—
11#16810365—24no—
12#16810347—21no—
13#16810347—21no—
14C2×#125×C53C2211yescharacteristic
15C2×C4×D4×C53C2211yescharacteristic
16C53×#396—21no—
17C53×#396—21no—
18C53×#390C411yescharacteristic
19C53×#390C411yescharacteristic
20C53×#385—21no—
21C53×#385—21no—
22C53×#384C411yescharacteristic
23C53×#382C411yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C2×#11719917—22no—
26C22×#88322—22no—
27C4⋊#88322—22no—
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133128—22no—
34#13133128D414yes—
35C22×D4×C53C2×C411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38#125×C53—22no—
39#125×C53—22no—
40C4×D4×C53—22no—
41C4×D4×C53—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53C2×C411yescharacteristic
58C2×C42×C53C2311yescharacteristic
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62C24×C53C22⋊C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C2×D4×C53—22no—
65C2×D4×C53—22no—
66C2×D4×C53—41no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—81no—
90C4⋊C4×C53—81no—
91C22⋊C4×C53—22no—
92C22⋊C4×C53—22no—
93C22⋊C4×C53—42no—
94C22⋊C4×C53—42no—
95C22⋊C4×C53—81no—
96C22⋊C4×C53—81no—
97C42×C53—41no—
98C42×C53—41no—
99C42×C53C2×D412yes—
100C42×C53#4112yes—
101#11719917—82no—
102C2×#88322—24no—
103C2×#88322—42no—
104C2×#88322—82no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53#10112yes—
115C23×C53C2×C22⋊C411yescharacteristic
116D4×C53—44no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53C4×D412yes—
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12112yes—
144C2×C4×C53#12311yescharacteristic
145#88322—44no—
146#88322—82no—
147#9449—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—22no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53#38611yescharacteristic
161C22×C53C2×#10111yescharacteristic
162C22×C53#41311yescharacteristic
163C4×C53—22no—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—42no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—81no—
171C4×C53—81no—
172C4×C53—81no—
173#2973—531no—
174#2566D5311yescharacteristic
175#1534—532no—
176#1466—531no—
177#1285—532no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53—42no—
181C2×C53—81no—
182C2×C53#188411yescharacteristic
183C2×C53#190111yescharacteristic
184C2×C53#191111yescharacteristic
185C2×#125C2×D5311yescharacteristic
186C2×C4×D4C2×D5311yescharacteristic
187#435—1064no—
188#432—532no—
189#432—532no—
190#407—1061no—
191C2×C22⋊C8—1061no—
192#396—21no—
193#396—21no—
194#390Dic5311yescharacteristic
195#390Dic5311yescharacteristic
196#385—21no—
197#385—21no—
198#384Dic5311yescharacteristic
199#382Dic5311yescharacteristic
200C4×C4⋊C4C2×D5311yescharacteristic
201#330—1064no—
202#323—1061no—
203#323—1061no—
204C53#944911yescharacteristic, complemented, Sylow
205C22×D4C2×Dic5311yescharacteristic
206C23×C4C22⋊D5311yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C2×M16—1062no—
209C22×C8—1062no—
210#125—22no—
211#125—22no—
212C4×D4—22no—
213C4×D4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C2×Dic5311yescharacteristic
220C2×C4⋊C4C2×Dic5311yescharacteristic
221C2×C4⋊C4C2×Dic5311yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4C22⋊D5311yescharacteristic
227C2×C22⋊C4C22⋊D5311yescharacteristic
228C2×C42C2×Dic5311yescharacteristic
229C2×C42C2×Dic5311yescharacteristic
230C2×C42C22×D5311yescharacteristic
231C4⋊C8—1062no—
232C4⋊C8—1062no—
233C22⋊C8—1062no—
234C22⋊C8—1062no—
235C22⋊C8—1062no—
236C22⋊C8—1062no—
237#97—534no—
238#97—1062no—
239#95—21no—
240#95—21no—
241#95—21no—
242C24C22⋊Dic5311yescharacteristic
243C24C22⋊Dic5311yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4C22⋊Dic5311yescharacteristic
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22×Dic5311yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262M16—4242no—
263C2×C8—1064no—
264C2×C8—4242no—
265C2×C8—2122no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—81no—
280C22⋊C4—81no—
281C42—41no—
282C42—41no—
283C42D4×D5312yes—
284C42#1171994712yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1313313612yes—
295C23C2×C22⋊Dic5311yescharacteristic
296D4—44no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4D4×Dic5312yes—
322C2×C4#1313318511yescharacteristic
323C2×C4#1313318912yes—
324C2×C4#1313319811yescharacteristic
325C8—4242no—
326C8—2124no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1681050311yescharacteristic
340C22C2×#1313313611yescharacteristic
341C22#1681060711yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#2027509911yescharacteristic
357C2#2027518411yescharacteristic
358C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2566C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2566D53
hypocenterC53#9449
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2566

Complements

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

NH ≅ G/Ntype
2. #20273342C2semidirect
4. #20272571C2semidirect
204. C53#9449semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202733422
#202730562
#202725712
C53×#25662
#944953