Finite Group Databasebeta

← #27299629

Subgroups — #27299629

Order 13568.

1002 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1002 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#20276384C214yesmaximal, complemented
2#20274633C212yesmaximal, complemented
3C53×#3487C211yesmaximal, characteristic
4C2×#13133132—22no—
5C2×#13133132C2214yes—
6#16810486—22no—
7#16810418—24no—
8#16810416—24no—
9C2×2−1+4×C53C411yescharacteristic, complemented
10C22×#41×C53C2211yescharacteristic
11C53×#550—24no—
12C53×#548—24no—
13C53×#535—22no—
14C53×#535C2212yes—
15C53×#525—22no—
16C53×#525C412yescomplemented
17C53×#519C411yescharacteristic, complemented
18C2×C22⋊Dic53—24no—
19#13133132—28no—
20#13133132—44no—
21#13133124—44no—
222−1+4×C53—44no—
23C2×#41×C53—22no—
24C2×#41×C53—24no—
25C2×#41×C53—22no—
26C2×#41×C53—44no—
27C2×#41×C53C2×C412yes—
28C22×Q8×C53—22no—
29C22×Q8×C53C2×C411yescharacteristic
30C22×D4×C53D412yes—
31C22×D4×C53C2311yescharacteristic
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53D412yes—
34#127×C53—24no—
35#127×C53D414yes—
36#124×C53—24no—
37#124×C53D414yes—
38#123×C53—22no—
39C22⋊Q8×C53—24no—
40C22⋊Q8×C53—22no—
41#121×C53—24no—
42#121×C53—28no—
43#121×C53—22no—
44#121×C53—24no—
45#120×C53—22no—
46#120×C53—24no—
47C4×Q8×C53—24no—
48C4×D4×C53—28no—
49C4×D4×C53—24no—
50#117×C53—22no—
51#117×C53C2×C412yes—
52C24×C53C22⋊C411yescharacteristic
53C24×C53C2×D412yes—
54#41×C53—44no—
55#41×C53—44no—
56#41×C53—416no—
57#41×C53—48no—
58#41×C53—44no—
59C2×Q8×C53—22no—
60C2×Q8×C53—24no—
61C2×Q8×C53—22no—
62C2×Q8×C53—48no—
63C2×Q8×C53C22⋊C412yes—
64C2×D4×C53—22no—
65C2×D4×C53—22no—
66C2×D4×C53—24no—
67C2×D4×C53—24no—
68C2×D4×C53—28no—
69C2×D4×C53—24no—
70C2×D4×C53—42no—
71C2×D4×C53—42no—
72C2×D4×C53C22⋊C412yes—
73C2×D4×C53C2×D414yes—
74C22×C4×C53—24no—
75C22×C4×C53—24no—
76C22×C4×C53—24no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C412yes—
83C22×C4×C53C22×C411yescharacteristic
84C22×Dic53—44no—
85C4⋊C4×C53—24no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—22no—
88C22⋊C4×C53—24no—
89C22⋊C4×C53—24no—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—28no—
92C42×C53—24no—
93C42×C53C22⋊C414yes—
94C22⋊Dic53—44no—
95C22⋊Dic53—44no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—21no—
100C23×C53—22no—
101C23×C53—82no—
102C23×C53—81no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53#12014yes—
105Q8×C53—28no—
106Q8×C53—44no—
107Q8×C53—42no—
108D4×C53—44no—
109D4×C53—48no—
110D4×C53—42no—
111D4×C53—44no—
112D4×C53—44no—
113D4×C53—88no—
114D4×C53—84no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—24no—
121C2×C4×C53—22no—
122C2×C4×C53—48no—
123C2×C4×C53—44no—
124C2×C4×C53—48no—
125C2×C4×C53—44no—
126C2×C4×C53#9914yes—
127C2×C4×C53C2×C22⋊C412yes—
128C2×Dic53—84no—
129C2×Dic53—88no—
130#12708—531nomaximal, Sylow
131C22×C53—24no—
132C22×C53—41no—
133C22×C53—44no—
134C22×C53—44no—
135C22×C53—44no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—41no—
139C22×C53—82no—
140C22×C53—82no—
141C22×C53#37911yescharacteristic
142C22×C53C2×#9912yes—
143C4×C53—24no—
144C4×C53—22no—
145C4×C53—44no—
146C4×C53—41no—
147C4×C53—42no—
148Dic53—164no—
149#3487D5311yescharacteristic
150#2127—534no—
151#1782—532no—
152C2×C53—21no—
153C2×C53—22no—
154C2×C53—44no—
155C2×C53—82no—
156C2×C53—81no—
157C2×C53#178111yescharacteristic
158C2×2−1+4Dic5311yescharacteristic, complemented
159C22×#41C2×D5311yescharacteristic
160#550—24no—
161#548—24no—
162#535—22no—
163#535C2×D5312yes—
164#525—22no—
165#525Dic5312yescomplemented
166#519Dic5311yescharacteristic, complemented
167C2×#99—534no—
168C2×#99—1062no—
169#381—1062no—
170#354—1064no—
171#353—1064no—
172C53#1270811yescharacteristic, complemented, Sylow
1732−1+4—44no—
174C2×#41—24no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—44no—
178C2×#41C2×Dic5312yes—
179C22×Q8—22no—
180C22×Q8C2×Dic5311yescharacteristic
181C22×D4C22⋊D5312yes—
182C22×D4C22×D5311yescharacteristic
183C23×C4C2×Dic5311yescharacteristic
184C23×C4C22⋊D5312yes—
185#127—24no—
186#127C22⋊D5314yes—
187#124—24no—
188#124C22⋊D5314yes—
189#123—22no—
190C22⋊Q8—24no—
191C22⋊Q8—22no—
192#121—24no—
193#121—28no—
194#121—24no—
195#121—22no—
196#120—24no—
197#120—22no—
198C4×Q8—24no—
199C4×D4—28no—
200C4×D4—24no—
201#117—22no—
202#117C2×Dic5312yes—
203C2×C22⋊C4—1064no—
204#99—1068no—
205#99—2124no—
206#95—2124no—
207C24C22⋊Dic5311yescharacteristic
208C24C2×C22⋊D5312yes—
209#41—44no—
210#41—44no—
211#41—416no—
212#41—48no—
213#41—44no—
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—22no—
217C2×Q8—48no—
218C2×Q8C22⋊Dic5312yes—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—28no—
224C2×D4—24no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4C22⋊Dic5312yes—
228C2×D4C2×C22⋊D5314yes—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—2124no—
237C22×C4C22⋊Dic5311yescharacteristic
238C22×C4C22⋊Dic5312yes—
239C22×C4C22×Dic5311yescharacteristic
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—22no—
246C22⋊C4—28no—
247C22⋊C4—2124no—
248C22⋊C4—2124no—
249C42—24no—
250C42C22⋊Dic5314yes—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—82no—
257C23—81no—
258C23C2×C22⋊Dic5311yescharacteristic
259C23#1313318614yes—
260Q8—28no—
261Q8—42no—
262Q8—44no—
263D4—44no—
264D4—48no—
265D4—42no—
266D4—44no—
267D4—44no—
268D4—88no—
269D4—84no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—4244no—
282C2×C4—4248no—
283C2×C4#1313313214yes—
284C2×C4C2×C22⋊Dic5312yes—
285C22—24no—
286C22—41no—
287C22—44no—
288C22—44no—
289C22—44no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—82no—
294C22—82no—
295C22#1681047911yescharacteristic
296C22C2×#1313313212yes—
297C4—24no—
298C4—22no—
299C4—44no—
300C4—41no—
301C4—42no—
302C4—8484no—
303C2—21no—
304C2—22no—
305C2—44no—
306C2—81no—
307C2—82no—
308C2#2027463011yescharacteristic
subgroupG/H
centerC2#20274630
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC53×#3487C2
socleC2×C53#1781
cosocleC22×D4×C53C23
hypercenter#3487D53
hypocenterC53#12708
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #3487

Complements

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

NH ≅ G/Ntype
1. #20276384C2semidirect
2. #20274633C2semidirect
9. C2×2−1+4×C53C4semidirect
16. C53×#525C4semidirect
17. C53×#519C4semidirect
158. C2×2−1+4Dic53semidirect
165. #525Dic53semidirect
166. #519Dic53semidirect
172. C53#12708semidirect

Maximal subgroups

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

subgroupindex
#202763842
#202746332
C53×#34872
#1270853