Finite Group Databasebeta

← #27465216

Subgroups — #27465216

Order 13568.

1124 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 231 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
1C2×#16810483C212yesmaximal
2#20276138C214yesmaximal, complemented
3#20274905C214yesmaximal, complemented
4#20274802C214yesmaximal, complemented
5C53×#1779C211yesmaximal, characteristic
6C22×C4⋊Dic53C2212yes—
7C22×C4⋊Dic53C2212yes—
8C22×C22⋊Dic53—22no—
9C2×#13133151C2214yescomplemented
10#16810483—22no—
11#16810483—22no—
12#16810483C2214yes—
13#16810483C2214yes—
14C2×#13133124—22no—
15#16810386C2214yescomplemented
16#16810386C2218yescomplemented
17C24×C4×C53C2211yescharacteristic
18C2×C22⋊C8×C53C2214yes—
19C2×C22⋊C8×C53C2212yes—
20C23×Dic53—22no—
21C23×Dic53—22no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53—28no—
25C2×C4⋊Dic53—44no—
26C2×C4⋊Dic53C2314yes—
27C2×C4⋊Dic53C2314yes—
28C2×C22⋊Dic53—24no—
29C2×C22⋊Dic53—24no—
30C2×C22⋊Dic53—48no—
31#13133151—44no—
32#13133124—24no—
33#13133124—24no—
34C25×C53D411yescharacteristic
35C23×C4×C53—28no—
36C23×C4×C53—42no—
37C23×C4×C53D412yes—
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C23×C4×C53C2312yes—
41C22×C8×C53C2314yes—
42C22⋊C8×C53—28no—
43C22⋊C8×C53—42no—
44C24×C53—24no—
45C24×C53—24no—
46C24×C53—41no—
47C24×C53—41no—
48C24×C53D814yes—
49C24×C53C2×D411yescharacteristic
50C24×C53C2×D412yes—
51C22×C4×C53—24no—
52C22×C4×C53—24no—
53C22×C4×C53—24no—
54C22×C4×C53—24no—
55C22×C4×C53—24no—
56C22×C4×C53—24no—
57C22×C4×C53—22no—
58C22×C4×C53—41no—
59C22×C4×C53—42no—
60C22×C4×C53—416no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D412yes—
63C22×C4×C53#4118yes—
64C22×C4×C53C2411yescharacteristic
65C22×Dic53—24no—
66C22×Dic53—24no—
67C22×Dic53—44no—
68C22×Dic53—44no—
69C22×Dic53—44no—
70C22×Dic53—44no—
71C22×Dic53—44no—
72C22×Dic53—44no—
73C2×C8×C53—28no—
74C2×C8×C53—24no—
75C22⋊Dic53—48no—
76C22⋊Dic53—48no—
77C4⋊Dic53—28no—
78C4⋊Dic53—28no—
79C4⋊Dic53—44no—
80C4⋊Dic53—44no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—21no—
87C23×C53—22no—
88C23×C53—21no—
89C23×C53—216no—
90C23×C53—44no—
91C23×C53—44no—
92C23×C53—48no—
93C23×C53—44no—
94C23×C53#12314yes—
95C23×C53C2×D812yes—
96C23×C53C2×D814yes—
97C23×C53#13714yes—
98C23×C53C22×D411yescharacteristic
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—216no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—44no—
105C2×C4×C53—48no—
106C2×C4×C53—44no—
107C2×C4×C53#12314yes—
108C2×C4×C53C2×#4114yes—
109C2×C4×C53H(5,𝔽2)12yes—
110C2×Dic53—44no—
111C2×Dic53—44no—
112C2×Dic53—44no—
113C2×Dic53—44no—
114C2×Dic53—44no—
115C2×Dic53—44no—
116C2×Dic53—88no—
117C2×Dic53—88no—
118C8×C53—44no—
119#27600—531nomaximal, Sylow
120C22×C53—24no—
121C22×C53—24no—
122C22×C53—24no—
123C22×C53—24no—
124C22×C53—24no—
125C22×C53—24no—
126C22×C53—24no—
127C22×C53—22no—
128C22×C53—416no—
129C22×C53—42no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53#48018yes—
133C22×C53C2×#12312yes—
134C22×C53#53411yescharacteristic
135C22×C53C22×D811yescharacteristic
136C22×C53C2×#13712yes—
137C22×C53#57511yescharacteristic
138C4×C53—24no—
139C4×C53—41no—
140C4×C53#55114yes—
141Dic53—84no—
142Dic53—84no—
143C2×#380—532no—
144#2075—534no—
145#1852—534no—
146#1823—534no—
147#1779D5311yescharacteristic
148C2×C53—28no—
149C2×C53—42no—
150C2×C53#248311yescharacteristic
151C2×C53C2×#48012yes—
152C2×C53#309512yes—
153C2×C53#318611yescharacteristic
154C2×C53#319511yescharacteristic
155C24×C4C2×D5311yescharacteristic
156C22×C4⋊C4—532no—
157C22×C4⋊C4—532no—
158C22×C22⋊C4—1062no—
159C2×#107—534no—
160C2×C22⋊C8C2×D5314yes—
161C2×C22⋊C8C2×D5312yes—
162#380—534no—
163#380—534no—
164#380—1062no—
165#380—1062no—
166C2×#95—1062no—
167#340—534no—
168#340—538no—
169C53#2760011yescharacteristic, complemented, Sylow
170C25D21211yescharacteristic
171C23×C4—28no—
172C23×C4—42no—
173C23×C4—1062no—
174C23×C4—1062no—
175C23×C4D21212yes—
176C23×C4D21211yescharacteristic
177C23×C4C22×D5311yescharacteristic
178C23×C4C22×D5312yes—
179C22×C8C22×D5314yes—
180C2×C4⋊C4—534no—
181C2×C4⋊C4—534no—
182C2×C4⋊C4—1068no—
183C2×C4⋊C4—1062no—
184C2×C4⋊C4—1062no—
185C2×C4⋊C4—2124no—
186C2×C22⋊C4—1064no—
187C2×C22⋊C4—1064no—
188C2×C22⋊C4—2128no—
189#107—2124no—
190C22⋊C8—28no—
191C22⋊C8—42no—
192#95—1064no—
193#95—1064no—
194C24—24no—
195C24—24no—
196C24—41no—
197C24—41no—
198C24D42414yes—
199C24C2×D21211yescharacteristic
200C24C2×D21212yes—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—416no—
211C22×C4—1064no—
212C22×C4—1064no—
213C22×C4—2124no—
214C22×C4—2124no—
215C22×C4—2124no—
216C22×C4—2124no—
217C22×C4—2124no—
218C22×C4—2124no—
219C22×C4C2×D21211yescharacteristic
220C22×C4C2×D21212yes—
221C22×C4#1171994718yes—
222C22×C4C23×D5311yescharacteristic
223C2×C8—28no—
224C2×C8—24no—
225C4⋊C4—1068no—
226C4⋊C4—1068no—
227C4⋊C4—2124no—
228C4⋊C4—2124no—
229C22⋊C4—2128no—
230C22⋊C4—2128no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—216no—
240C23—44no—
241C23—44no—
242C23—44no—
243C23—48no—
244C23#1313320114yes—
245C23C2×D42412yes—
246C23C2×D42414yes—
247C23#1313325014yes—
248C23C22×D21211yescharacteristic
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—216no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—44no—
255C2×C4—48no—
256C2×C4—44no—
257C2×C4—4248no—
258C2×C4—4248no—
259C2×C4—2124no—
260C2×C4—2124no—
261C2×C4—2124no—
262C2×C4—2124no—
263C2×C4—2124no—
264C2×C4—2124no—
265C2×C4#1313320114yes—
266C2×C4C2×#1171994714yes—
267C2×C4#1313326212yes—
268C8—44no—
269C22—24no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—22no—
276C22—24no—
277C22—416no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22#1681093718yes—
282C22C2×#1313320112yes—
283C22#1681118511yescharacteristic
284C22C22×D42411yescharacteristic
285C22C2×#1313325012yes—
286C22#1681142311yescharacteristic
287C4—24no—
288C4—41no—
289C4—4244no—
290C4—4244no—
291C4#1681130414yes—
292C2—28no—
293C2—42no—
294C2#2027899611yescharacteristic
295C2C2×#1681093712yes—
296C2#2028466412yes—
297C2#2028546611yescharacteristic
298C2#2028553911yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1779C2
socleC23×C53C22×D4
prime radicalC25×C53D4
hypercenter#1779D53
hypocenterC53#27600
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. #20276138C2semidirect
3. #20274905C2semidirect
4. #20274802C2semidirect
9. C2×#13133151C22semidirect
15. #16810386C22semidirect
16. #16810386C22semidirect
169. C53#27600semidirect

Maximal subgroups

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

subgroupindex
C2×#168104832
#202761382
#202749052
#202748022
C53×#17792
#2760053