Finite Group Databasebeta

← #20279184

Subgroups — #20279184

Order 6784.

708 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133201C211yesmaximal, characteristic, complemented
2C2×#13133200C211yesmaximal, characteristic, complemented
3C2×#13133198C211yesmaximal, characteristic, complemented
4C2×#13133189C211yesmaximal, characteristic
5C22×#11719919C211yesmaximal, characteristic, complemented
6#16810559C212yesmaximal, complemented
7#16810558C212yesmaximal, complemented
8#16810523C212yesmaximal, complemented
9#16810492C212yesmaximal, complemented
10C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
11C2×#123×C53C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15C23×Dic53C2211yescharacteristic
16#13133201—24no—
17#13133200—24no—
18#13133198—24no—
19#13133189—24no—
20C2×C4⋊Dic53C2212yes—
21C2×#11719919—24no—
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C2×#11719919C2212yes—
26C2×#11719919C2211yescharacteristic
27C2×C22⋊Dic53C2212yes—
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×#11719921C2212yes—
31C2×#11719921C2212yes—
32C2×#11719921C2211yescharacteristic
33C2×C4×Dic53C2212yes—
34#13133124C2212yes—
35C22×D4×C53C2211yescharacteristic
36C23×C4×C53C2211yescharacteristic
37#123×C53—24no—
38C2×C4⋊C4×C53C2212yes—
39C2×C22⋊C4×C53C2212yes—
40C2×C22⋊C4×C53C2211yescharacteristic
41C24×C53C2311yescharacteristic
42C24×C53C2311yescharacteristic
43C23×D53C2311yescharacteristic
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53C2312yes—
51C22×C4×C53C2311yescharacteristic
52C22×C4×C53C2312yes—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—22no—
56C2×C22⋊D53—22no—
57C2×C22⋊D53—24no—
58C22×Dic53—24no—
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53D414yes—
63C22×Dic53C2312yes—
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2312yes—
67C22×Dic53C2311yescharacteristic
68C4⋊C4×C53—24no—
69C22⋊C4×C53—24no—
70C22⋊C4×C53D414yescomplemented
71C22⋊Dic53—22no—
72C22⋊Dic53—24no—
73C22⋊Dic53D414yescomplemented
74#11719921—24no—
75#11719921—22no—
76#11719921—24no—
77C4⋊Dic53—24no—
78#11719919—24no—
79#11719919—24no—
80#11719919—22no—
81#11719919—24no—
82#11719919—24no—
83C4×Dic53—24no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53C2×D412yes—
92C23×C53C2×D412yes—
93C23×C53C2×D412yes—
94C23×C53C2411yescharacteristic
95C22×D53—21no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—22no—
99C22×D53—22no—
100D4×C53—44no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—44no—
110C2×C4×C53C2×D414yes—
111C22⋊D53—44no—
112C22⋊D53—44no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—28no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—44no—
124C2×Dic53C2×D412yes—
125C2×Dic53C2×D414yes—
126C2×Dic53C2×D412yes—
127C2×Dic53#4114yes—
128C22×C53—22no—
129C22×C53—22no—
130C22×C53—24no—
131C22×C53—24no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53#12714yes—
138C22×C53C22×D411yescharacteristic
139C22×C53C22×D411yescharacteristic
140C22×C53C22×D411yescharacteristic
141C22×C53C2×#4112yes—
142C22×C532−1+412yes—
143C2×D53—41no—
144C2×D53—41no—
145C2×D53—41no—
146C2×D53—42no—
147C2×D53—42no—
148C4×C53—24no—
149C4×C53—42no—
150C4×C53—41no—
151Dic53—22no—
152Dic53—24no—
153Dic53—42no—
154Dic53—41no—
155Dic53#12314yescomplemented
156#2501—531nomaximal, Sylow
157C2×C53—24no—
158C2×C53—81no—
159C2×C53C2×#12311yescharacteristic
160C2×C53C2×#12711yescharacteristic
161C2×C53#53611yescharacteristic
162C2×C53#54712yes—
163C2×C53#54712yes—
164D53—81no—
165C2×#123—531no—
166C2×#123—531no—
167C2×#123—531no—
168C2×#123D5311yescharacteristic, complemented
169C2×#121—531no—
170C22×C4⋊C4—531no—
171#399—532no—
172#399—532no—
173#390—532no—
174#384—532no—
175C4×C22⋊C4—531no—
176C53#250111yescharacteristic, complemented, Sylow
177C22×D4—531no—
178C22×D4—531no—
179C22×D4C2×D5311yescharacteristic
180C23×C4—531no—
181C23×C4—531no—
182C23×C4C2×D5311yescharacteristic
183#123—24no—
184#123—1064no—
185#123—1064no—
186#123—1064no—
187#121—1064no—
188C2×C4⋊C4—532no—
189C2×C4⋊C4—532no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—532no—
192C2×C4⋊C4—532no—
193C2×C4⋊C4—532no—
194C2×C4⋊C4—1064no—
195C2×C4⋊C4C2×D5312yes—
196C2×C22⋊C4—531no—
197C2×C22⋊C4—531no—
198C2×C22⋊C4—532no—
199C2×C22⋊C4—531no—
200C2×C22⋊C4—532no—
201C2×C22⋊C4—532no—
202C2×C22⋊C4C2×D5312yes—
203C2×C22⋊C4C2×D5311yescharacteristic
204C2×C42—532no—
205#95—532no—
206C24—531no—
207C24C22×D5311yescharacteristic
208C24C22×D5311yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—1062no—
213C2×D4—1064no—
214C2×D4—1062no—
215C2×D4—1062no—
216C2×D4—1062no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—531no—
221C22×C4—534no—
222C22×C4—531no—
223C22×C4—532no—
224C22×C4—532no—
225C22×C4—531no—
226C22×C4—1064no—
227C22×C4—1062no—
228C22×C4—1062no—
229C22×C4—1062no—
230C22×C4C22×D5312yes—
231C22×C4C22×D5311yescharacteristic
232C22×C4C22×D5312yes—
233C4⋊C4—24no—
234C4⋊C4—1064no—
235C4⋊C4—1064no—
236C4⋊C4—1062no—
237C4⋊C4—1064no—
238C4⋊C4—1064no—
239C4⋊C4—1064no—
240C22⋊C4—24no—
241C22⋊C4—534no—
242C22⋊C4—1062no—
243C22⋊C4—1064no—
244C22⋊C4—1062no—
245C22⋊C4—1064no—
246C22⋊C4—1064no—
247C22⋊C4C22⋊D5314yescomplemented
248C42—1064no—
249C23—22no—
250C23—22no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—1061no—
257C23—1061no—
258C23—1061no—
259C23—1062no—
260C23—1062no—
261C23D4×D5312yes—
262C23D4×D5312yes—
263C23C2×C22⋊D5312yes—
264C23C23×D5311yescharacteristic
265D4—44no—
266D4—2124no—
267D4—2124no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—44no—
277C2×C4—532no—
278C2×C4—534no—
279C2×C4—534no—
280C2×C4—532no—
281C2×C4—1068no—
282C2×C4—1062no—
283C2×C4—1062no—
284C2×C4—1062no—
285C2×C4—1062no—
286C2×C4—1061no—
287C2×C4—1062no—
288C2×C4—1061no—
289C2×C4—1062no—
290C2×C4—1061no—
291C2×C4—2124no—
292C2×C4C2×C22⋊D5314yes—
293C22—24no—
294C22—24no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—2122no—
303C22—2122no—
304C22—2121no—
305C22—2121no—
306C22—2121no—
307C22#1313321314yes—
308C22C2×D4×D5311yescharacteristic
309C22C2×D4×D5311yescharacteristic
310C22C22×C22⋊D5311yescharacteristic
311C22#41×D5312yes—
312C22#1313326412yes—
313C4—24no—
314C4—42no—
315C4—41no—
316C4—534no—
317C4—1064no—
318C4—1062no—
319C4—2122no—
320C4—2121no—
321C2—24no—
322C2—81no—
323C2—4241no—
324C2#123×D5311yescharacteristic
325C2C2×#1313321311yescharacteristic
326C2#1681119511yescharacteristic
327C2#1681127112yes—
328C2#1681127212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2501
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#13133201C2semidirect
2. C2×#13133200C2semidirect
3. C2×#13133198C2semidirect
5. C22×#11719919C2semidirect
6. #16810559C2semidirect
7. #16810558C2semidirect
8. #16810523C2semidirect
9. #16810492C2semidirect
10. C22⋊C4×Dic53C2semidirect
11. C2×#123×C53C2semidirect
70. C22⋊C4×C53D4semidirect
73. C22⋊Dic53D4semidirect
155. Dic53#123semidirect
168. C2×#123D53semidirect
176. C53#2501semidirect
247. C22⋊C4C22⋊D53semidirect