Finite Group Databasebeta

← #27276276

Subgroups — #27276276

Order 13568.

1002 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 139 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#20274633C211yesmaximal, characteristic, complemented
2#20274627C211yesmaximal, characteristic, complemented
3#20273271C214yesmaximal, complemented
4C53×#3487C211yesmaximal, characteristic
5C2×#13133132—22no—
6C2×C22⋊#88322—22no—
7C2×C22⋊#88322C2212yes—
8#16810486C2212yes—
9#16810360—24no—
10#16810357—24no—
11C2×2−1+4×C53C411yescharacteristic, complemented
12C22×#41×C53C2211yescharacteristic
13C53×#550—24no—
14C53×#548—24no—
15C53×#535—22no—
16C53×#535C412yescomplemented
17C53×#525—22no—
18C53×#525C2212yes—
19C53×#519C411yescharacteristic, complemented
20C22×#88322—22no—
21C2×C22⋊Dic53—22no—
22#13133132—44no—
23C22⋊#88322—24no—
24C22⋊#88322—44no—
25#13133124—24no—
262−1+4×C53—44no—
27C2×#41×C53—22no—
28C2×#41×C53—22no—
29C2×#41×C53—24no—
30C2×#41×C53—44no—
31C2×#41×C53C2×C412yes—
32C22×Q8×C53—22no—
33C22×Q8×C53C2×C411yescharacteristic
34C22×D4×C53C2×C411yescharacteristic
35C22×D4×C53D412yes—
36C23×C4×C53D412yes—
37C23×C4×C53C2311yescharacteristic
38#127×C53—24no—
39#127×C53—22no—
40#124×C53—24no—
41#124×C53—22no—
42#123×C53—22no—
43C22⋊Q8×C53—24no—
44C22⋊Q8×C53D414yes—
45#121×C53—24no—
46#121×C53—28no—
47#121×C53—24no—
48#121×C53D414yes—
49#120×C53—24no—
50#120×C53—22no—
51C4×Q8×C53—24no—
52C4×D4×C53—28no—
53C4×D4×C53—24no—
54#117×C53—22no—
55#117×C53C2×C412yes—
56C24×C53C22⋊C412yes—
57C24×C53C2×D411yescharacteristic
58#41×C53—44no—
59#41×C53—44no—
60#41×C53—416no—
61#41×C53—48no—
62#41×C53—44no—
63C2×Q8×C53—22no—
64C2×Q8×C53—24no—
65C2×Q8×C53—22no—
66C2×Q8×C53—48no—
67C2×Q8×C53C22⋊C412yes—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—24no—
71C2×D4×C53—28no—
72C2×D4×C53—24no—
73C2×D4×C53—24no—
74C2×D4×C53—22no—
75C2×D4×C53—42no—
76C2×D4×C53—42no—
77C2×D4×C53C22⋊C412yes—
78C22×C4×C53—24no—
79C22×C4×C53—24no—
80C22×C4×C53—22no—
81C22×C4×C53—24no—
82C22×C4×C53—42no—
83C22×C4×C53—42no—
84C22×C4×C53C22⋊C412yes—
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D414yes—
88C22×Dic53—42no—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53C22⋊C414yes—
92C22⋊C4×C53—28no—
93C22⋊C4×C53—22no—
94C22⋊C4×C53—24no—
95C22⋊C4×C53—24no—
96C42×C53—24no—
97C42×C53—22no—
98C22⋊Dic53—44no—
99C2×#88322—44no—
100C2×#88322—42no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—42no—
107C23×C53—44no—
108C23×C53C2×C22⋊C411yescharacteristic
109C23×C53#12012yes—
110Q8×C53—28no—
111Q8×C53—44no—
112Q8×C53—42no—
113D4×C53—48no—
114D4×C53—42no—
115D4×C53—44no—
116D4×C53—44no—
117D4×C53—44no—
118D4×C53—84no—
119D4×C53—88no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—24no—
126C2×C4×C53—44no—
127C2×C4×C53—48no—
128C2×C4×C53—48no—
129C2×C4×C53—44no—
130C2×C4×C53#9914yes—
131C2×C4×C53C2×C22⋊C412yes—
132C2×C4×C53#12012yes—
133C2×Dic53—84no—
134C2×Dic53—82no—
135#88322—82no—
136#8464—531nomaximal, Sylow
137C22×C53—22no—
138C22×C53—24no—
139C22×C53—22no—
140C22×C53—44no—
141C22×C53—44no—
142C22×C53—44no—
143C22×C53—42no—
144C22×C53—42no—
145C22×C53—82no—
146C22×C53—82no—
147C22×C53#37911yescharacteristic
148C22×C53C2×#9912yes—
149C4×C53—22no—
150C4×C53—24no—
151C4×C53—22no—
152C4×C53—44no—
153C4×C53—42no—
154Dic53—162no—
155#3487D5311yescharacteristic
156#1782—531no—
157#1780—531no—
158#1516—534no—
159C2×C53—22no—
160C2×C53—42no—
161C2×C53—82no—
162C2×C53—82no—
163C2×C53#178111yescharacteristic
164C2×C53#179412yes—
165C2×2−1+4Dic5311yescharacteristic, complemented
166C22×#41C2×D5311yescharacteristic
167#550—24no—
168#548—24no—
169#535—22no—
170#535Dic5312yescomplemented
171#525—22no—
172#525C2×D5312yes—
173#519Dic5311yescharacteristic, complemented
174C2×#99—1062no—
175C2×C22⋊C8—532no—
176C2×C22⋊C8—1062no—
177#381—532no—
178#328—1064no—
179#327—1064no—
180C53#846411yescharacteristic, complemented, Sylow
1812−1+4—44no—
182C2×#41—22no—
183C2×#41—24no—
184C2×#41—22no—
185C2×#41—44no—
186C2×#41C2×Dic5312yes—
187C22×Q8—22no—
188C22×Q8C2×Dic5311yescharacteristic
189C22×D4C2×Dic5311yescharacteristic
190C22×D4C22⋊D5312yes—
191C23×C4C22⋊D5312yes—
192C23×C4C22×D5311yescharacteristic
193C22×C8—1062no—
194#127—24no—
195#127—22no—
196#124—24no—
197#124—22no—
198#123—22no—
199C22⋊Q8—24no—
200C22⋊Q8C22⋊D5314yes—
201#121—24no—
202#121—28no—
203#121—24no—
204#121C22⋊D5314yes—
205#120—22no—
206#120—24no—
207C4×Q8—24no—
208C4×D4—28no—
209C4×D4—24no—
210#117—22no—
211#117C2×Dic5312yes—
212C2×C22⋊C4—1062no—
213#99—2124no—
214C22⋊C8—1064no—
215C22⋊C8—2124no—
216#95—1064no—
217C24C22⋊Dic5312yes—
218C24C2×C22⋊D5311yescharacteristic
219#41—44no—
220#41—44no—
221#41—416no—
222#41—48no—
223#41—44no—
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—22no—
227C2×Q8—48no—
228C2×Q8C22⋊Dic5312yes—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—28no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4C22⋊Dic5312yes—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—2122no—
246C22×C4C22⋊Dic5312yes—
247C22×C4C22×Dic5311yescharacteristic
248C22×C4C2×C22⋊D5311yescharacteristic
249C22×C4C2×C22⋊D5314yes—
250C2×C8—2124no—
251C2×C8—2122no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4C22⋊Dic5314yes—
255C22⋊C4—28no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—2124no—
260C42—24no—
261C42—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—44no—
269C23C2×C22⋊Dic5311yescharacteristic
270C23#1313318612yes—
271Q8—28no—
272Q8—42no—
273Q8—44no—
274D4—48no—
275D4—42no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—84no—
280D4—88no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—4242no—
292C2×C4—4244no—
293C2×C4#1313313214yes—
294C2×C4C2×C22⋊Dic5312yes—
295C2×C4#1313318612yes—
296C8—4242no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—44no—
304C22—44no—
305C22—82no—
306C22—82no—
307C22#1681047911yescharacteristic
308C22C2×#1313313212yes—
309C4—22no—
310C4—24no—
311C4—22no—
312C4—42no—
313C4—44no—
314C4—8482no—
315C2—22no—
316C2—42no—
317C2—82no—
318C2—82no—
319C2#2027463011yescharacteristic
320C2#2027467912yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3487C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenter#3487D53
hypocenterC53#8464
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3487

Complements

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

NH ≅ G/Ntype
1. #20274633C2semidirect
2. #20274627C2semidirect
3. #20273271C2semidirect
11. C2×2−1+4×C53C4semidirect
16. C53×#535C4semidirect
19. C53×#519C4semidirect
165. C2×2−1+4Dic53semidirect
170. #535Dic53semidirect
173. #519Dic53semidirect
180. C53#8464semidirect

Maximal subgroups

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

subgroupindex
#202746332
#202746272
#202732712
C53×#34872
#846453