Finite Group Databasebeta

← #27281897

Subgroups — #27281897

Order 13568.

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

normal subgroups only

This group has 818 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#20277072C211yesmaximal, characteristic, complemented
2#20274659C212yesmaximal
3#20273045C211yesmaximal, characteristic, complemented
4#20272567C212yesmaximal, complemented
5C53×#2406C211yesmaximal, characteristic
6C2×#13133176—21no—
7C2×#13133144—24no—
8#16810587C2212yes—
9C4⋊C4×Dic53—24no—
10C4×C22⋊Dic53—21no—
11C4×C22⋊Dic53C2212yes—
12#16810349—22no—
13C22×#41×C53C2211yescharacteristic
14C2×#123×C53—21no—
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#395C412yescomplemented
17C53×#394C412yescomplemented
18C53×#386—21no—
19C53×#385—22no—
20C53×#381C2211yescharacteristic
21C2×#11719917—22no—
22#13133176—28no—
23C2×C4⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—22no—
26C2×C4×Dic53—22no—
27C2×C4×Dic53D414yes—
28#13133144—48no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124—22no—
33C2×#41×C53—22no—
34C2×#41×C53—22no—
35C2×#41×C53—44no—
36C22×Q8×C53C2×C411yescharacteristic
37C22×D4×C53—21no—
38C22×D4×C53C2×C411yescharacteristic
39C23×C4×C53—21no—
40C23×C4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C23×C4×C53C2311yescharacteristic
43#123×C53—44no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53—22no—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53C2×C412yes—
48C2×C4⋊C4×C53D412yes—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53—22no—
52#95×C53—22no—
53#95×C53—21no—
54#95×C53C2×C412yes—
55C24×C53—21no—
56C24×C53C2×D411yescharacteristic
57#41×C53—28no—
58#41×C53—44no—
59#41×C53—44no—
60#41×C53—44no—
61C2×Q8×C53—22no—
62C2×Q8×C53—42no—
63C2×D4×C53—24no—
64C2×D4×C53—22no—
65C2×D4×C53—42no—
66C2×D4×C53—42no—
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—24no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—22no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53C22⋊C412yes—
82C22×C4×C53C22⋊C412yes—
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D412yes—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4112yes—
87C22×Dic53—24no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—44no—
91C22⋊C4×C53—44no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—44no—
94C22⋊Dic53—28no—
95C4⋊Dic53—28no—
96C4×Dic53—28no—
97C4×Dic53—44no—
98C4×Dic53—82no—
99#11719917—44no—
100C2×#88322—42no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—42no—
107C23×C53—41no—
108C23×C53—42no—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53#12011yescharacteristic
111C23×C53#12311yescharacteristic
112Q8×C53—24no—
113Q8×C53—42no—
114D4×C53—24no—
115D4×C53—42no—
116D4×C53—44no—
117D4×C53—44no—
118C2×C4×C53—24no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—24no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—48no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—81no—
138C2×C4×C53#10414yescomplemented
139C2×C4×C53C4×D412yes—
140C2×C4×C53#12112yes—
141C2×Dic53—28no—
142C2×Dic53—44no—
143C2×Dic53—44no—
144#88322—82no—
145#9476—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—44no—
154C22×C53—44no—
155C22×C53—42no—
156C22×C53—82no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53#38611yescharacteristic
160C22×C53C2×#10412yes—
161C22×C53B(4,𝔽2)12yes—
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—24no—
165C4×C53—42no—
166C4×C53—42no—
167C4×C53—82no—
168Dic53—48no—
169#2406D5311yescharacteristic
170#2274—531no—
171#1789—532no—
172#1463—531no—
173#1284—532no—
174C2×C53—22no—
175C2×C53—22no—
176C2×C53—41no—
177C2×C53—82no—
178C2×C53#188911yescharacteristic
179C2×C53#189611yescharacteristic
180C2×C53#191211yescharacteristic
181C22×#41C2×D5311yescharacteristic
182C2×#123—21no—
183C2×#117—1061no—
184C22×C4⋊C4C2×D5311yescharacteristic
185C2×#104—1064no—
186#407—532no—
187#395Dic5312yescomplemented
188#394Dic5312yescomplemented
189#386—21no—
190#385—22no—
191#381C2×D5311yescharacteristic
192C4×C4⋊C4—1064no—
193C4×C22⋊C4—532no—
194C4×C22⋊C4—1061no—
195#324—1062no—
196C53#947611yescharacteristic, complemented, Sylow
197C2×#41—22no—
198C2×#41—22no—
199C2×#41—44no—
200C22×Q8C2×Dic5311yescharacteristic
201C22×D4—21no—
202C22×D4C2×Dic5311yescharacteristic
203C23×C4—21no—
204C23×C4C22⋊D5311yescharacteristic
205C23×C4C22⋊D5311yescharacteristic
206C23×C4C22×D5311yescharacteristic
207C2×M16—1062no—
208#123—44no—
209#117—1068no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—1062no—
214C2×C4⋊C4C2×Dic5312yes—
215C2×C4⋊C4C22⋊D5312yes—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—1062no—
220C2×C42—534no—
221C2×C42—1062no—
222C2×C42—1062no—
223#104—2128no—
224C22⋊C8—1062no—
225C22⋊C8—1062no—
226#95—21no—
227#95—22no—
228#95—1062no—
229#95—1062no—
230#95C2×Dic5312yes—
231C24—21no—
232C24C2×C22⋊D5311yescharacteristic
233#41—28no—
234#41—44no—
235#41—44no—
236#41—44no—
237C2×Q8—22no—
238C2×Q8—42no—
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—1064no—
258C22×C4C22⋊Dic5312yes—
259C22×C4C22⋊Dic5312yes—
260C22×C4D4×D5312yes—
261C22×C4#1171994712yes—
262C22×C4C22×Dic5311yescharacteristic
263C22×C4C2×C22⋊D5311yescharacteristic
264M16—2124no—
265C2×C8—2122no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—44no—
269C4⋊C4—1068no—
270C22⋊C4—44no—
271C22⋊C4—42no—
272C22⋊C4—44no—
273C22⋊C4—1068no—
274C42—1068no—
275C42—4242no—
276C42—2124no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—41no—
284C23—42no—
285C23C2×C22⋊Dic5311yescharacteristic
286C23#1313318511yescharacteristic
287C23#1313319811yescharacteristic
288Q8—24no—
289Q8—42no—
290D4—24no—
291D4—42no—
292D4—44no—
293D4—44no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—48no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—1068no—
315C2×C4—2124no—
316C2×C4—2124no—
317C2×C4#1313314414yescomplemented
318C2×C4D4×Dic5312yes—
319C2×C4#1313318912yes—
320C8—4242no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22—44no—
330C22—42no—
331C22—82no—
332C22—81no—
333C22—81no—
334C22#1681050311yescharacteristic
335C22C2×#1313314412yes—
336C22#1681080912yes—
337C4—22no—
338C4—22no—
339C4—24no—
340C4—42no—
341C4—42no—
342C4—82no—
343C4—2128no—
344C2—22no—
345C2—22no—
346C2—41no—
347C2—82no—
348C2#2027512411yescharacteristic
349C2#2027515911yescharacteristic
350C2#2027525311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2406C2
socleC22×C53#386
prime radicalC22×#41×C53C22
cosocleC23×C4×C53C23
hypercenter#2406D53
hypocenterC53#9476
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. #20277072C2semidirect
3. #20273045C2semidirect
4. #20272567C2semidirect
16. C53×#395C4semidirect
17. C53×#394C4semidirect
138. C2×C4×C53#104semidirect
187. #395Dic53semidirect
188. #394Dic53semidirect
196. C53#9476semidirect
317. C2×C4#13133144semidirect

Maximal subgroups

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

subgroupindex
#202770722
#202746592
#202730452
#202725672
C53×#24062
#947653