Finite Group Databasebeta

← #27272801

Subgroups — #27272801

Order 13568.

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

normal subgroups only

This group has 560 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#20275949C212yesmaximal, complemented
2#20275785C211yesmaximal, characteristic
3#20272538C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic, complemented
5C53×#2724C211yesmaximal, characteristic, complemented
6C22×#11719922—21no—
7C2×#13133214C2211yescharacteristic
8C2×#13133137—21no—
9C2×#13133137—22no—
10C2×#13133137—22no—
11#16810378C2211yescharacteristic
12#16810378C2212yes—
13C2×#123×C53—21no—
14C53×#398—21no—
15C53×#397—22no—
16C53×#390C2211yescharacteristic
17C53×#389C2212yes—
18C53×#385—22no—
19C22×D212—22no—
20C22×D212—21no—
21C2×#11719922—42no—
22C2×#11719922—42no—
23C2×#11719922—42no—
24C22×#88322—21no—
25C22×#88322—22no—
26#13133214—42no—
27#13133137—44no—
28#13133137—42no—
29#13133137—44no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32#123×C53—42no—
33#123×C53—42no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53D412yes—
37C2×C4⋊C4×C53D412yes—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2311yescharacteristic
43#95×C53—22no—
44#95×C53—21no—
45#95×C53—22no—
46C24×C53—21no—
47C23×D53—41no—
48C2×D4×C53—21no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53D812yes—
52C22×C4×C53—22no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—81no—
57C22×C4×C53—81no—
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D412yes—
60C2×D212—24no—
61C2×D212—22no—
62C2×D212—41no—
63C2×D212—41no—
64C2×D212—42no—
65C2×D212—42no—
66C2×D212—41no—
67C2×D212—42no—
68C2×D212—42no—
69C2×D212—41no—
70C2×D212—41no—
71C2×D212—42no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—81no—
79C22⋊C4×C53—44no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C42×C53—41no—
84#11719922—48no—
85C2×#88322—22no—
86C2×#88322—41no—
87C2×#88322—41no—
88C2×#88322—42no—
89C2×#88322—42no—
90C2×#88322—42no—
91C23×C53—22no—
92C23×C53—42no—
93C23×C53—41no—
94C23×C53—42no—
95C23×C53—41no—
96C23×C53#12011yescharacteristic
97C22×D53—42no—
98C22×D53—81no—
99C22×D53—82no—
100C22×D53—81no—
101C22×D53—82no—
102D4×C53—42no—
103D4×C53—81no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—84no—
117C2×C4×C53C2×D811yescharacteristic
118C2×C4×C53#13512yes—
119C2×C4×C53AGL(1,ℤ/8)12yes—
120C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
121D212—42no—
122D212—44no—
123D212—44no—
124D212—42no—
125D212—81no—
126D212—81no—
127D212—82no—
128D212—82no—
129#88322—42no—
130#88322—82no—
131#7736—531nomaximal, Sylow
132C22×C53—42no—
133C22×C53—44no—
134C22×C53—82no—
135C22×C53—81no—
136C22×C53—81no—
137C22×C53#44711yescharacteristic
138C22×C53#44912yes—
139C22×C53#45311yescharacteristic
140C22×C53#45412yes—
141C22×C53B(4,𝔽2)11yescharacteristic
142C2×D53—82no—
143C2×D53—84no—
144C2×D53—161no—
145C2×D53—162no—
146C2×D53—161no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—22no—
150C4×C53—22no—
151C4×C53—82no—
152C4×C53—81no—
153C4×C53—82no—
154#2724D5311yescharacteristic, complemented
155#2041—532no—
156#2011—531no—
157#1271—532no—
158#1270—531no—
159C2×C53—82no—
160C2×C53#159511yescharacteristic
161C2×C53#159712yes—
162C2×C53#162011yescharacteristic
163C2×C53#162512yes—
164C2×C53#165711yescharacteristic
165D53—162no—
166C22×D8—1061no—
167C2×#127—531no—
168C2×#123—21no—
169C2×#102—1061no—
170C2×#102—1062no—
171C2×#102—1062no—
172#398—21no—
173#397—22no—
174#390C2×D5311yescharacteristic
175#389C2×D5312yes—
176#385—22no—
177#336—531no—
178#336—532no—
179C53#773611yescharacteristic, complemented, Sylow
180C22×D4—1061no—
181C22×D4—1062no—
182C22×D4C22⋊D5311yescharacteristic
183C23×C4—21no—
184C2×D8—2122no—
185C2×D8—2122no—
186C2×D8—2122no—
187C22×C8—1061no—
188C22×C8—1062no—
189#127—2122no—
190#123—42no—
191#123—42no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22⋊D5312yes—
195C2×C4⋊C4C22⋊D5312yes—
196C2×C4⋊C4C22⋊D5311yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C42C22×D5311yescharacteristic
201#102—2124no—
202#102—2122no—
203#102—2124no—
204#95—22no—
205#95—21no—
206#95—22no—
207C24—21no—
208C24—2121no—
209C2×D4—21no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—1064no—
213C2×D4—1062no—
214C2×D4—2122no—
215C2×D4—2121no—
216C2×D4—2121no—
217C2×D4—2121no—
218C2×D4—2122no—
219C2×D4—2122no—
220C2×D4—2121no—
221C2×D4—2122no—
222C2×D4—2122no—
223C2×D4—2121no—
224C2×D4#1171992212yes—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4C2×C22⋊D5311yescharacteristic
232C22×C4C2×C22⋊D5312yes—
233D8—2128no—
234C2×C8—1062no—
235C2×C8—2122no—
236C2×C8—2122no—
237C2×C8—2122no—
238C2×C8—2121no—
239C2×C8—2121no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C42—41no—
252C23—22no—
253C23—42no—
254C23—41no—
255C23—42no—
256C23—41no—
257C23—4241no—
258C23—4242no—
259C23—4241no—
260C23—4242no—
261C23—2122no—
262C23#1313318611yescharacteristic
263D4—42no—
264D4—81no—
265D4—4241no—
266D4—4241no—
267D4—4242no—
268D4—4242no—
269D4—2122no—
270D4—2122no—
271D4—2124no—
272D4—2124no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—84no—
286C2×C4C2×#1171992211yescharacteristic
287C2×C4#1313323712yes—
288C2×C4#1313324212yes—
289C2×C4#1313324311yescharacteristic
290C8—4242no—
291C8—2122no—
292C22—44no—
293C22—42no—
294C22—81no—
295C22—81no—
296C22—82no—
297C22—8481no—
298C22—8481no—
299C22—8482no—
300C22—4242no—
301C22—4244no—
302C22#1681075711yescharacteristic
303C22#1681076812yes—
304C22#1681079111yescharacteristic
305C22#1681079612yes—
306C22#1681081011yescharacteristic
307C4—21no—
308C4—21no—
309C4—22no—
310C4—22no—
311C4—81no—
312C4—82no—
313C4—82no—
314C2—82no—
315C2—8482no—
316C2#2027363311yescharacteristic
317C2#2027364412yes—
318C2#2027377411yescharacteristic
319C2#2027380412yes—
320C2#2027401311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2724C2
socleC23×C53#120
prime radical#20275785C2
hypercenter#2724D53
hypocenterC53#7736
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
1. #20275949C2semidirect
3. #20272538C2semidirect
4. #20272536C2semidirect
5. C53×#2724C2semidirect
154. #2724D53semidirect
179. C53#7736semidirect

Maximal subgroups

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

subgroupindex
#202759492
#202757852
#202725382
#202725362
C53×#27242
#773653