Finite Group Databasebeta

← #27281917

Subgroups — #27281917

Order 13568.

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

normal subgroups only

This group has 726 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#20277318C211yesmaximal, characteristic, complemented
2#20274659C212yesmaximal
3#20273045C211yesmaximal, characteristic, complemented
4#20272567C212yesmaximal, complemented
5C53×#2406C211yesmaximal, characteristic
6C2×#13133144—24no—
7#16810587C2212yes—
8#16810509—24no—
9#16810486—21no—
10#16810483—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—
22C2×C4⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53D414yes—
25#13133144—48no—
26C22⋊#88322—22no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—24no—
30#13133124—22no—
31C2×#41×C53—22no—
32C2×#41×C53—22no—
33C2×#41×C53—44no—
34C22×Q8×C53C2×C411yescharacteristic
35C22×D4×C53—21no—
36C22×D4×C53C2×C411yescharacteristic
37C23×C4×C53—21no—
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41#123×C53—44no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C4⋊C4×C53D412yes—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—22no—
50#95×C53—22no—
51#95×C53—21no—
52#95×C53C2×C412yes—
53C24×C53—21no—
54C24×C53C2×D411yescharacteristic
55#41×C53—28no—
56#41×C53—44no—
57#41×C53—44no—
58#41×C53—44no—
59C2×Q8×C53—22no—
60C2×Q8×C53—42no—
61C2×D4×C53—24no—
62C2×D4×C53—22no—
63C2×D4×C53—42no—
64C2×D4×C53—42no—
65C2×D4×C53—41no—
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—22no—
73C22×C4×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53C22⋊C412yes—
80C22×C4×C53C22⋊C412yes—
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4112yes—
85C22×Dic53—24no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—44no—
89C22⋊C4×C53—44no—
90C22⋊C4×C53—44no—
91C22⋊C4×C53—42no—
92C22⋊Dic53—44no—
93C4⋊Dic53—82no—
94C4×Dic53—44no—
95#11719917—44no—
96C2×#88322—42no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—42no—
103C23×C53—41no—
104C23×C53—42no—
105C23×C53C2×C22⋊C411yescharacteristic
106C23×C53#12011yescharacteristic
107C23×C53#12311yescharacteristic
108Q8×C53—24no—
109Q8×C53—42no—
110D4×C53—24no—
111D4×C53—42no—
112D4×C53—44no—
113D4×C53—44no—
114C2×C4×C53—24no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—48no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—81no—
134C2×C4×C53C4×D412yes—
135C2×C4×C53#12112yes—
136C2×Dic53—44no—
137C2×Dic53—44no—
138C2×Dic53—44no—
139#88322—82no—
140#9480—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—44no—
149C22×C53—44no—
150C22×C53—42no—
151C22×C53—82no—
152C22×C53—81no—
153C22×C53—81no—
154C22×C53#38611yescharacteristic
155C22×C53#42112yes—
156C22×C53B(4,𝔽2)12yes—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—42no—
160C4×C53—42no—
161C4×C53—42no—
162C4×C53—82no—
163Dic53—84no—
164#2406D5311yescharacteristic
165#2313—531no—
166#1789—532no—
167#1463—531no—
168#1284—532no—
169C2×C53—22no—
170C2×C53—22no—
171C2×C53—41no—
172C2×C53—82no—
173C2×C53#188911yescharacteristic
174C2×C53#189711yescharacteristic
175C2×C53#191211yescharacteristic
176C22×#41C2×D5311yescharacteristic
177C2×#123—21no—
178C22×C4⋊C4C2×D5311yescharacteristic
179C2×#104—1064no—
180#407—532no—
181#395Dic5312yescomplemented
182#394Dic5312yescomplemented
183#387—1064no—
184#386—21no—
185#385—22no—
186#381—1061no—
187#381C2×D5311yescharacteristic
188#380—1061no—
189C4×C22⋊C4—532no—
190#324—1062no—
191C53#948011yescharacteristic, complemented, Sylow
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—44no—
195C22×Q8C2×Dic5311yescharacteristic
196C22×D4—21no—
197C22×D4C2×Dic5311yescharacteristic
198C23×C4—21no—
199C23×C4C22⋊D5311yescharacteristic
200C23×C4C22⋊D5311yescharacteristic
201C23×C4C22×D5311yescharacteristic
202C2×M16—1062no—
203#123—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—1062no—
208C2×C4⋊C4C2×Dic5312yes—
209C2×C4⋊C4C22⋊D5312yes—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1062no—
214C2×C42—534no—
215#104—2128no—
216C22⋊C8—1062no—
217C22⋊C8—1062no—
218#95—21no—
219#95—22no—
220#95—1062no—
221#95—1064no—
222#95—1062no—
223#95C2×Dic5312yes—
224C24—21no—
225C24C2×C22⋊D5311yescharacteristic
226#41—28no—
227#41—44no—
228#41—44no—
229#41—44no—
230C2×Q8—22no—
231C2×Q8—42no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—1064no—
251C22×C4C22⋊Dic5312yes—
252C22×C4C22⋊Dic5312yes—
253C22×C4D4×D5312yes—
254C22×C4#1171994712yes—
255C22×C4C22×Dic5311yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257M16—2124no—
258C2×C8—2122no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—4242no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C22⋊C4—2124no—
267C42—2124no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—41no—
275C23—42no—
276C23C2×C22⋊Dic5311yescharacteristic
277C23#1313318511yescharacteristic
278C23#1313319811yescharacteristic
279Q8—24no—
280Q8—42no—
281D4—24no—
282D4—42no—
283D4—44no—
284D4—44no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—48no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—81no—
305C2×C4—2124no—
306C2×C4—2124no—
307C2×C4—2124no—
308C2×C4D4×Dic5312yes—
309C2×C4#1313318912yes—
310C8—4242no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—44no—
320C22—42no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22#1681050311yescharacteristic
325C22#1681064012yes—
326C22#1681080912yes—
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—82no—
333C4—4244no—
334C2—22no—
335C2—22no—
336C2—41no—
337C2—82no—
338C2#2027512411yescharacteristic
339C2#2027516411yescharacteristic
340C2#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#9480
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. #20277318C2semidirect
3. #20273045C2semidirect
4. #20272567C2semidirect
16. C53×#395C4semidirect
17. C53×#394C4semidirect
181. #395Dic53semidirect
182. #394Dic53semidirect
191. C53#9480semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202746592
#202730452
#202725672
C53×#24062
#948053