Finite Group Databasebeta

← #27280141

Subgroups — #27280141

Order 13568.

560 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 135 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#20277333C211yesmaximal, characteristic
2#20273287C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272582C212yesmaximal
5C53×#2627C211yesmaximal, characteristic
6C2×#13133176—21no—
7C2×C22⋊#88322C2212yes—
8#16810522—24no—
9#16810483—21no—
10#16810483C2212yes—
11#16810360—24no—
12#16810349—22no—
13C2×C22⋊Q8×C53C2211yescharacteristic
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#400C412yescomplemented
16C53×#398—22no—
17C53×#395C412yescomplemented
18C53×#385—21no—
19C53×#381—21no—
20C53×#380C2211yescharacteristic
21C22×#88322—22no—
22#13133176—42no—
23#13133176—42no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53—22no—
27C2×C22⋊Dic53—22no—
28C2×C4×Dic53—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124D414yes—
33C22×Q8×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37C22⋊Q8×C53—22no—
38C22⋊Q8×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53C2×C412yes—
46C2×C4⋊C4×C53D412yes—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49#95×C53—22no—
50#95×C53—22no—
51#95×C53—22no—
52#95×C53C2×C412yes—
53C24×C53C2×D411yescharacteristic
54C2×Q8×C53—22no—
55C2×Q8×C53—81no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—21no—
65C22×C4×C53—81no—
66C22×C4×C53—81no—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53SD1612yes—
70C22×C4×C53Q1612yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4112yes—
75C22×Dic53—24no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—81no—
81C4⋊C4×C53—82no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C4⋊Dic53—44no—
89C4⋊Dic53—82no—
90C4⋊Dic53—82no—
91C4×Dic53—44no—
92C2×#88322—42no—
93C2×#88322—82no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53#12011yescharacteristic
98C23×C53#12311yescharacteristic
99C23×C53C2×SD1611yescharacteristic
100C23×C53C2×Q1611yescharacteristic
101C23×C53AGL(1,ℤ/8)11yescharacteristic
102C23×C53#13711yescharacteristic
103Q8×C53—82no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—81no—
125C2×C4×C53#10314yes—
126C2×C4×C53C4×D412yes—
127C2×C4×C53#12112yes—
128C2×Dic53—44no—
129C2×Dic53—44no—
130C2×Dic53—44no—
131#88322—82no—
132#9200—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53#38611yescharacteristic
144C22×C53C2×#10311yescharacteristic
145C22×C53#41711yescharacteristic
146C22×C53#45011yescharacteristic
147C22×C53#45111yescharacteristic
148C22×C53#48311yescharacteristic
149C22×C53#48411yescharacteristic
150C4×C53—24no—
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—42no—
154C4×C53—82no—
155C4×C53—81no—
156C4×C53—81no—
157Dic53—84no—
158#2627D5311yescharacteristic
159#2316—531no—
160#1520—532no—
161C2×#324—531no—
162#1289—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53#187211yescharacteristic
170C2×C53#191211yescharacteristic
171C2×C53#191311yescharacteristic
172C2×C22⋊Q8C2×D5311yescharacteristic
173C2×#117—1061no—
174C22×C4⋊C4C2×D5311yescharacteristic
175C2×C22⋊C8—532no—
176#400Dic5312yescomplemented
177#398—22no—
178#395Dic5312yescomplemented
179#389—1064no—
180#385—21no—
181#381—21no—
182#380—532no—
183#380—1061no—
184#380C2×D5311yescharacteristic
185#328—1064no—
186#324—1062no—
187C53#920011yescharacteristic, complemented, Sylow
188C22×Q8C2×Dic5311yescharacteristic
189C23×C4C22⋊D5311yescharacteristic
190C23×C4C22⋊D5311yescharacteristic
191C23×C4C22×D5311yescharacteristic
192C22×C8—1062no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195#117—2122no—
196#117—2122no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—1062no—
203C2×C4⋊C4—1062no—
204C2×C4⋊C4—1062no—
205C2×C4⋊C4C2×Dic5311yescharacteristic
206C2×C4⋊C4C2×Dic5312yes—
207C2×C4⋊C4C22⋊D5312yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—1062no—
211C2×C42—1062no—
212C22⋊C8—1062no—
213C22⋊C8—1062no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—534no—
218#95—1062no—
219#95C2×Dic5312yes—
220C24C2×C22⋊D5311yescharacteristic
221C2×Q8—22no—
222C2×Q8—81no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—1064no—
235C22×C4#1171992412yes—
236C22×C4#1171992512yes—
237C22×C4C22⋊Dic5312yes—
238C22×C4C22⋊Dic5312yes—
239C22×C4D4×D5312yes—
240C22×C4#1171994712yes—
241C22×C4C22×Dic5311yescharacteristic
242C22×C4C2×C22⋊D5311yescharacteristic
243C2×C8—4242no—
244C2×C8—2122no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—81no—
250C4⋊C4—82no—
251C4⋊C4—4242no—
252C4⋊C4—4242no—
253C4⋊C4—2124no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—2122no—
259C22⋊C4—2122no—
260C42—2124no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊Dic5311yescharacteristic
264C23#1313318511yescharacteristic
265C23#1313319811yescharacteristic
266C23C2×#1171992411yescharacteristic
267C23C2×#1171992511yescharacteristic
268C23#1313324211yescharacteristic
269C23#1313324711yescharacteristic
270Q8—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—2124no—
293C2×C4—2124no—
294C2×C4—2124no—
295C2×C4#1313314114yes—
296C2×C4D4×Dic5312yes—
297C2×C4#1313318912yes—
298C8—4242no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#1681050311yescharacteristic
310C22C2×#1313314111yescharacteristic
311C22#1681062411yescharacteristic
312C22#1681077311yescharacteristic
313C22#1681077811yescharacteristic
314C22#1681095111yescharacteristic
315C22#1681095611yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4—81no—
323C4—4244no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#2027502311yescharacteristic
331C2#2027525311yescharacteristic
332C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2627C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2627D53
hypocenterC53#9200
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
15. C53×#400C4semidirect
17. C53×#395C4semidirect
176. #400Dic53semidirect
178. #395Dic53semidirect
187. C53#9200semidirect

Maximal subgroups

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

subgroupindex
#202773332
#202732872
C2×#168103492
#202725822
C53×#26272
#920053