Finite Group Databasebeta

← #20278107

Subgroups — #20278107

Order 6784.

814 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13133185C211yesmaximal, characteristic
2C22×#11719921C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810549C212yesmaximal, complemented
5#16810539C212yesmaximal, complemented
6#16810503C211yesmaximal, characteristic, complemented
7#16810499C212yesmaximal, complemented
8#16810498C211yesmaximal, characteristic, complemented
9#16810486C211yesmaximal, characteristic, complemented
10#16810480C212yesmaximal, complemented
11C2×#123×C53C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C23×Dic53C2211yescharacteristic
16C22×C4×D53C2212yes—
17#13133185—24no—
18C2×C4⋊Dic53C2212yescomplemented
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2212yescomplemented
22C2×#11719921—24no—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C22⋊C4×D53—24no—
29#13133124C2212yescomplemented
30#13133124C2212yescomplemented
31#13133124C2212yescomplemented
32#13133124C2212yescomplemented
33C22×D4×C53C2211yescharacteristic
34C23×C4×C53C2211yescharacteristic
35#123×C53—24no—
36C2×C4⋊C4×C53C2212yescomplemented
37C2×C22⋊C4×C53C2211yescharacteristic
38C2×C22⋊C4×C53C2212yescomplemented
39C24×C53C2311yescharacteristic
40C24×C53C2311yescharacteristic
41C23×D53—22no—
42C23×D53—22no—
43C23×D53—24no—
44C23×D53—42no—
45C23×D53D414yes—
46C23×D53C2311yescharacteristic
47C23×D53C2311yescharacteristic
48C2×D4×C53—22no—
49C2×D4×C53—22no—
50C2×D4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53C2312yes—
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2312yes—
57C2×C22⋊D53—22no—
58C2×C22⋊D53—22no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—42no—
61C22×Dic53—22no—
62C22×Dic53—42no—
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2312yes—
65C22×Dic53C2312yes—
66C22×Dic53C2311yescharacteristic
67C2×C4×D53—24no—
68C2×C4×D53—24no—
69C2×C4×D53—42no—
70C4⋊C4×C53—24no—
71C22⋊C4×C53—24no—
72C22⋊C4×C53D414yescomplemented
73C22⋊Dic53—22no—
74C22⋊Dic53—22no—
75C22⋊Dic53—42no—
76#11719921—24no—
77#11719921—24no—
78#11719921—22no—
79#11719921—22no—
80#11719921—24no—
81C4⋊Dic53—42no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—22no—
87C23×C53—21no—
88C23×C53—22no—
89C23×C53C2×D412yes—
90C23×C53C2×D412yes—
91C23×C53#4112yes—
92C23×C53C2411yescharacteristic
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—22no—
96C22×D53—28no—
97C22×D53—21no—
98C22×D53—22no—
99C22×D53—21no—
100C22×D53—22no—
101C22×D53—28no—
102C22×D53—44no—
103C22×D53—48no—
104C22×D53—48no—
105C22×D53C2×D412yes—
106C22×D53C2×D412yes—
107C22×D53#4114yes—
108D4×C53—44no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—44no—
118C2×C4×C53C2×D414yes—
119C22⋊D53—44no—
120C22⋊D53—44no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—44no—
132C4×D53—48no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—24no—
136C22×C53—24no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53#12114yes—
143C22×C53C22×D411yescharacteristic
144C22×C53C22×D411yescharacteristic
145C22×C53C2×#4112yes—
146C22×C53C2×#4111yescharacteristic
147C22×C53H(5,𝔽2)12yes—
148C2×D53—22no—
149C2×D53—22no—
150C2×D53—24no—
151C2×D53—24no—
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—22no—
155C2×D53—416no—
156C2×D53—42no—
157C2×D53—42no—
158C2×D53#12314yescomplemented
159C4×C53—24no—
160C4×C53—42no—
161C4×C53—41no—
162Dic53—41no—
163Dic53—42no—
164Dic53—41no—
165Dic53—42no—
166#2412—531nomaximal, Sylow
167C2×C53—24no—
168C2×C53—81no—
169C2×C53C2×#12111yescharacteristic
170C2×C53C2×#12311yescharacteristic
171C2×C53#53411yescharacteristic
172C2×C53#54612yes—
173C2×C53#55112yes—
174D53—24no—
175D53—42no—
176C2×#123D5311yescharacteristic, complemented
177C2×#120—531no—
178C22×C22⋊C4—531no—
179C22×C22⋊C4—531no—
180#397—532no—
181#394—532no—
182#386—531no—
183#386—532no—
184#385—531no—
185#381—531no—
186#380—532no—
187C53#241211yescharacteristic, complemented, Sylow
188C25—531no—
189C22×D4—531no—
190C22×D4—531no—
191C22×D4C2×D5311yescharacteristic
192C23×C4—531no—
193C23×C4—532no—
194C23×C4C2×D5311yescharacteristic
195#123—24no—
196#120—1064no—
197C2×C4⋊C4—532no—
198C2×C4⋊C4C2×D5312yescomplemented
199C2×C22⋊C4—531no—
200C2×C22⋊C4—531no—
201C2×C22⋊C4—532no—
202C2×C22⋊C4—532no—
203C2×C22⋊C4—532no—
204C2×C22⋊C4—531no—
205C2×C22⋊C4—531no—
206C2×C22⋊C4—532no—
207C2×C22⋊C4—1064no—
208C2×C22⋊C4—1064no—
209C2×C22⋊C4C2×D5311yescharacteristic
210C2×C22⋊C4C2×D5312yescomplemented
211#95—532no—
212#95—532no—
213#95—532no—
214#95—532no—
215C24—531no—
216C24—531no—
217C24—534no—
218C24—1064no—
219C24—1062no—
220C24—1062no—
221C24—2122no—
222C24C22×D5311yescharacteristic
223C24C22×D5311yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—1062no—
228C2×D4—1062no—
229C2×D4—1064no—
230C2×D4—2122no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—531no—
235C22×C4—532no—
236C22×C4—532no—
237C22×C4—531no—
238C22×C4—1064no—
239C22×C4—1062no—
240C22×C4—1064no—
241C22×C4—2122no—
242C22×C4—2122no—
243C22×C4C22×D5312yes—
244C22×C4C22×D5311yescharacteristic
245C22×C4C22×D5312yes—
246C4⋊C4—24no—
247C4⋊C4—2122no—
248C22⋊C4—24no—
249C22⋊C4—1064no—
250C22⋊C4—1064no—
251C22⋊C4—1062no—
252C22⋊C4—1062no—
253C22⋊C4—1062no—
254C22⋊C4—1062no—
255C22⋊C4—1064no—
256C22⋊C4—2122no—
257C22⋊C4C22⋊D5314yescomplemented
258C23—22no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—22no—
265C23—532no—
266C23—534no—
267C23—532no—
268C23—1068no—
269C23—1061no—
270C23—1061no—
271C23—1062no—
272C23—1068no—
273C23—1061no—
274C23—1062no—
275C23—1061no—
276C23—1062no—
277C23—2124no—
278C23—2128no—
279C23—2128no—
280C23D4×D5312yes—
281C23#1171994712yes—
282C23C2×C22⋊D5312yes—
283C23C23×D5311yescharacteristic
284D4—44no—
285D4—2124no—
286D4—2124no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—1062no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1062no—
300C2×C4—1062no—
301C2×C4—1062no—
302C2×C4—1062no—
303C2×C4—1062no—
304C2×C4—1062no—
305C2×C4—1062no—
306C2×C4—2128no—
307C2×C4—2124no—
308C2×C4C2×C22⋊D5314yes—
309C22—24no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—534no—
319C22—1064no—
320C22—1062no—
321C22—1062no—
322C22—1064no—
323C22—1062no—
324C22—1062no—
325C22—1062no—
326C22—21216no—
327C22—2122no—
328C22—2122no—
329C22#1313318814yes—
330C22C2×D4×D5311yescharacteristic
331C22C22×C22⋊D5311yescharacteristic
332C22#41×D5312yes—
333C22C2×#1171994711yescharacteristic
334C22#1313326312yes—
335C4—24no—
336C4—42no—
337C4—41no—
338C4—2121no—
339C4—2122no—
340C4—2121no—
341C4—2122no—
342C2—24no—
343C2—81no—
344C2—1064no—
345C2—2122no—
346C2C2×#1313318811yescharacteristic
347C2#123×D5311yescharacteristic
348C2#1681118411yescharacteristic
349C2#1681126112yes—
350C2#1681129612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2412
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C22×#11719921C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. #16810549C2semidirect
5. #16810539C2semidirect
6. #16810503C2semidirect
7. #16810499C2semidirect
8. #16810498C2semidirect
9. #16810486C2semidirect
10. #16810480C2semidirect
11. C2×#123×C53C2semidirect
18. C2×C4⋊Dic53C22semidirect
21. C2×C22⋊Dic53C22semidirect
29. #13133124C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
32. #13133124C22semidirect
36. C2×C4⋊C4×C53C22semidirect
38. C2×C22⋊C4×C53C22semidirect
72. C22⋊C4×C53D4semidirect
158. C2×D53#123semidirect
176. C2×#123D53semidirect
187. C53#2412semidirect
198. C2×C4⋊C4C2×D53semidirect
210. C2×C22⋊C4C2×D53semidirect
257. C22⋊C4C22⋊D53semidirect