Finite Group Databasebeta

← #27294792

Subgroups — #27294792

Order 13568.

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

normal subgroups only

This group has 488 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#20274798C211yesmaximal, characteristic, complemented
2#20273049C211yesmaximal, characteristic, complemented
3#20272718C214yesmaximal, complemented
4C53×#2619C211yesmaximal, characteristic
5#16810680—21no—
6#16810587—21no—
7#16810392—21no—
8#16810392C2212yes—
9#16810349C2212yes—
10#16810347—21no—
11C2×#127×C53C411yescharacteristic
12C2×#121×C53—21no—
13C2×#117×C53C2211yescharacteristic
14C53×#394—21no—
15C53×#394C2212yes—
16C53×#392—21no—
17C53×#392C412yes—
18C53×#390—21no—
19C53×#382C411yescharacteristic
20C2×#11719917—22no—
21C4⋊#88322—41no—
22C22⋊#88322—22no—
23C22⋊#88322—41no—
24C22⋊#88322—41no—
25#13133128—41no—
26C22×D4×C53—22no—
27C22×D4×C53—21no—
28C22×D4×C53—21no—
29C22×D4×C53C2×C412yes—
30C23×C4×C53D411yescharacteristic
31#127×C53—41no—
32#127×C53—41no—
33#121×C53—44no—
34#117×C53—41no—
35#117×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D411yescharacteristic
46#95×C53—21no—
47#95×C53C2×C412yes—
48C24×C53—22no—
49C24×C53C22⋊C411yescharacteristic
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—44no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53—82no—
58C2×D4×C53—82no—
59C2×D4×C53—82no—
60C2×D4×C53—82no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—42no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—44no—
78C22⋊C4×C53—82no—
79C22⋊C4×C53—82no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84#11719917—42no—
85#11719917—42no—
86C2×#88322—42no—
87C2×#88322—42no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53C2×C22⋊C411yescharacteristic
101D4×C53—82no—
102D4×C53—82no—
103D4×C53—84no—
104D4×C53—84no—
105D4×C53—88no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#9911yescharacteristic
125C2×C4×C53AGL(1,ℤ/8)12yes—
126#88322—82no—
127#88322—82no—
128#11806—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—42no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—81no—
136C22×C53—82no—
137C22×C53—82no—
138C22×C53—82no—
139C22×C53—82no—
140C22×C53—82no—
141C22×C53—82no—
142C22×C53—82no—
143C22×C53C2×#9911yescharacteristic
144C22×C53#41811yescharacteristic
145C22×C53#42111yescharacteristic
146C4×C53—22no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—82no—
154#2619D5311yescharacteristic
155#1822—531no—
156#1464—531no—
157#1345—534no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—81no—
161C2×C53—162no—
162C2×C53#151011yescharacteristic
163C2×C53#212211yescharacteristic
164C2×C53#213411yescharacteristic
165C2×#127Dic5311yescharacteristic
166C2×#121—21no—
167C2×#117C2×D5311yescharacteristic
168#432—1061no—
169#407—1061no—
170#394—21no—
171#394C2×D5312yes—
172#392—21no—
173#392Dic5312yes—
174#390—21no—
175#382Dic5311yescharacteristic
176#343—532no—
177#343—1061no—
178#324—532no—
179#323—1061no—
180C53#1180611yescharacteristic, complemented, Sylow
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C2×Dic5312yes—
185C23×C4C22⋊D5311yescharacteristic
186C2×M16—1062no—
187#127—41no—
188#127—41no—
189#121—44no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22⋊D5311yescharacteristic
194C2×C4⋊C4C22×D5311yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C2×Dic5311yescharacteristic
199C2×C22⋊C4C22⋊D5311yescharacteristic
200C2×C42C2×Dic5311yescharacteristic
201C2×C42C22⋊D5311yescharacteristic
202C4⋊C8—2121no—
203C22⋊C8—1062no—
204C22⋊C8—2121no—
205C22⋊C8—2121no—
206#97—2121no—
207#95—21no—
208#95C2×Dic5312yes—
209C24—22no—
210C24C22⋊Dic5311yescharacteristic
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4—82no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4C22⋊Dic5311yescharacteristic
230C22×C4C22×Dic5311yescharacteristic
231C22×C4C2×C22⋊D5311yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233M16—2122no—
234M16—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—44no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23C2×C22⋊Dic5311yescharacteristic
262D4—82no—
263D4—82no—
264D4—84no—
265D4—84no—
266D4—88no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4#1313313211yescharacteristic
285C2×C4#1313313211yescharacteristic
286C2×C4#1313324312yes—
287C8—4242no—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22C2×#1313313211yescharacteristic
304C22#1681062811yescharacteristic
305C22#1681064011yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—162no—
318C2#2027324411yescharacteristic
319C2#2027636511yescharacteristic
320C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
Fitting, prime radicalC53×#2619C2
socleC22×C53C2×#99
cosocleC2×C4⋊C4×C53C23
hypercenter#2619D53
hypocenterC53#11806
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #20274798C2semidirect
2. #20273049C2semidirect
3. #20272718C2semidirect
180. C53#11806semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202730492
#202727182
C53×#26192
#1180653