Finite Group Databasebeta

← #27282067

Subgroups — #27282067

Order 13568.

522 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283756C211yesmaximal, characteristic
2#20274795C212yesmaximal, complemented
3#20273053C211yesmaximal, characteristic
4#20272621C212yesmaximal, complemented
5C53×#2487C211yesmaximal, characteristic
6#16810701—24no—
7#16810676—21no—
8#16810587—21no—
9#16810587C2212yes—
10#16810587C2212yes—
11#16810392—24no—
12#16810349—21no—
13#16810347—21no—
14C2×C4×D4×C53C2211yescharacteristic
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#400—21no—
17C53×#399—21no—
18C53×#389—21no—
19C53×#386C412yes—
20C53×#385C412yes—
21C53×#380—21no—
22C53×#380C2211yescharacteristic
23C2×#11719917—22no—
24C2×#11719917—22no—
25C2×#11719917D414yes—
26C4⋊#88322—22no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C4×#88322—22no—
33C22×D4×C53C2×C411yescharacteristic
34C23×C4×C53C2×C412yes—
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37C4×D4×C53—22no—
38C4×D4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53C2×C412yes—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51#95×C53—22no—
52#95×C53—22no—
53#95×C53—21no—
54C24×C53C22⋊C411yescharacteristic
55C24×C53C22×C411yescharacteristic
56C2×D4×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—24no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53—82no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C2×D412yes—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4112yes—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—82no—
80C22⋊C4×C53—22no—
81C22⋊C4×C53—22no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—82no—
84C42×C53—21no—
85C42×C53—21no—
86#11719917—44no—
87#11719917—44no—
88#11719917—82no—
89C2×#88322—24no—
90C2×#88322—42no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53C4×D412yes—
102D4×C53—44no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—24no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
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—81no—
121C2×C4×C53—84no—
122C2×C4×C53#10112yes—
123C2×C4×C53#12011yescharacteristic
124C2×C4×C53#12112yes—
125C2×C4×C53#12311yescharacteristic
126#88322—44no—
127#88322—82no—
128#9510—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—81no—
141C22×C53#38611yescharacteristic
142C22×C53C2×#10111yescharacteristic
143C22×C53#41311yescharacteristic
144C4×C53—22no—
145C4×C53—21no—
146C4×C53—22no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53—82no—
150C4×C53—82no—
151C4×C53#45812yes—
152#2975—531no—
153#2487D5311yescharacteristic
154#1821—532no—
155#1465—531no—
156#1305—532no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—42no—
160C2×C53—81no—
161C2×C53#189011yescharacteristic
162C2×C53#190111yescharacteristic
163C2×C53#191411yescharacteristic
164C2×C4×D4C2×D5311yescharacteristic
165C22×C4⋊C4C2×D5311yescharacteristic
166#436—1064no—
167#431—1061no—
168#407—532no—
169#407—532no—
170#407—1061no—
171#400—21no—
172#399—21no—
173#389—21no—
174#386Dic5312yes—
175#385Dic5312yes—
176#380—21no—
177#380C2×D5311yescharacteristic
178#343—1064no—
179#324—1061no—
180#323—1061no—
181C53#951011yescharacteristic, complemented, Sylow
182C22×D4C2×Dic5311yescharacteristic
183C23×C4C2×Dic5312yes—
184C23×C4C22⋊D5311yescharacteristic
185C23×C4C22×D5311yescharacteristic
186C2×M16—534no—
187C2×M16—1062no—
188C2×M16—1062no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D5311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×Dic5311yescharacteristic
200C2×C22⋊C4C2×Dic5312yes—
201C2×C22⋊C4C22⋊D5311yescharacteristic
202C2×C42C22⋊D5311yescharacteristic
203C4⋊C8—1062no—
204C22⋊C8—1062no—
205C22⋊C8—1062no—
206C22⋊C8—1062no—
207C22⋊C8—1062no—
208C22⋊C8—1062no—
209C4×C8—1062no—
210#95—21no—
211#95—22no—
212#95—22no—
213C24C22⋊Dic5311yescharacteristic
214C24C22×Dic5311yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4C22⋊Dic5311yescharacteristic
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4D4×D5312yes—
230C22×C4#1171994712yes—
231C22×C4C2×C22⋊D5311yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233M16—4242no—
234M16—2124no—
235M16—2124no—
236C2×C8—1064no—
237C2×C8—2122no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—82no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23C2×C22⋊Dic5311yescharacteristic
260C23D4×Dic5312yes—
261D4—44no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4#1313313612yes—
282C2×C4#1313318511yescharacteristic
283C2×C4#1313318912yes—
284C2×C4#1313319811yescharacteristic
285C8—4242no—
286C8—2124no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1681050311yescharacteristic
300C22C2×#1313313611yescharacteristic
301C22#1681060711yescharacteristic
302C4—22no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4#1681081212yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2027512911yescharacteristic
315C2#2027518411yescharacteristic
316C2#2027526311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2487C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#2487D53
hypocenterC53#9510
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #20274795C2semidirect
4. #20272621C2semidirect
181. C53#9510semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202747952
#202730532
#202726212
C53×#24872
#951053