Finite Group Databasebeta

← #20280450

Subgroups — #20280450

Order 6784.

824 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 143 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×#13133192C211yesmaximal, characteristic, complemented
2C2×#13133187C212yesmaximal, complemented
3C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810549C211yesmaximal, characteristic, complemented
5#16810541C212yesmaximal, complemented
6#16810538C212yesmaximal, complemented
7#16810533C211yesmaximal, characteristic, complemented
8#16810532C211yesmaximal, characteristic, complemented
9#16810501C212yesmaximal, complemented
10#16810482C211yesmaximal, characteristic, complemented
11C53×#397C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×D212C2212yes—
14C22×D212C2212yes—
15C22×D212C2211yescharacteristic, complemented
16C22×C22⋊D53C2212yes—
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19#13133192—24no—
20#13133187—28no—
21C2×#11719919C2212yescomplemented
22C2×C22⋊Dic53C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2212yes—
27C2×#11719921C2212yescomplemented
28C2×#11719921C2212yescomplemented
29C2×#11719921C2212yescomplemented
30C22⋊C4×D53—24no—
31#13133124C2212yescomplemented
32#13133124C2211yescharacteristic, complemented
33C2×C4⋊C4×C53C2211yescharacteristic, complemented
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2212yescomplemented
37#95×C53C2211yescharacteristic, complemented
38C24×C53C2311yescharacteristic
39C23×D53—22no—
40C23×D53—22no—
41C23×D53—42no—
42C23×D53—42no—
43C23×D53D414yescomplemented
44C23×D53C2312yes—
45C23×D53C2311yescharacteristic
46C23×D53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2312yes—
49C22×C4×C53C2312yes—
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—24no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53—42no—
54C22×Dic53C2312yes—
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2311yescharacteristic
57C2×D212—24no—
58C2×D212—24no—
59C2×D212—24no—
60C2×D212—24no—
61C2×D212—22no—
62C2×D212—42no—
63C2×D212—42no—
64C2×D212—42no—
65C2×C4×D53—22no—
66C2×C4×D53—42no—
67C2×C4×D53—42no—
68C2×C4×D53D414yescomplemented
69C4⋊C4×C53—22no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—22no—
72C22⋊Dic53—22no—
73#11719921—24no—
74#11719921—24no—
75#11719921—22no—
76#11719921—22no—
77#11719921—24no—
78#11719921—42no—
79#11719921—42no—
80#11719919—42no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53C2×D412yes—
86C23×C53C2411yescharacteristic
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53—28no—
94C22×D53—22no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—22no—
99C22×D53—22no—
100C22×D53—22no—
101C22×D53—48no—
102C22×D53—44no—
103C22×D53—44no—
104C22×D53—48no—
105C22×D53C2×D412yes—
106C22×D53C2×D412yes—
107C22×D53C2×D412yes—
108C22×D53#4112yes—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53C2×D412yes—
119C22⋊D53—48no—
120C2×Dic53—22no—
121C2×Dic53—21no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53C2×D412yes—
127D212—44no—
128D212—48no—
129D212—48no—
130C4×D53—44no—
131C4×D53—44no—
132C22×C53—22no—
133C22×C53—24no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—42no—
137C22×C53C22×D411yescharacteristic
138C22×C53C22×D411yescharacteristic
139C22×C53C2×#4111yescharacteristic
140C22×C53H(5,𝔽2)12yes—
141C22×C53H(5,𝔽2)12yes—
142C2×D53—22no—
143C2×D53—24no—
144C2×D53—22no—
145C2×D53—24no—
146C2×D53—22no—
147C2×D53—24no—
148C2×D53—416no—
149C2×D53—41no—
150C2×D53—41no—
151C2×D53—42no—
152C2×D53—42no—
153C2×D53—42no—
154C2×D53—42no—
155C2×D53—42no—
156C2×D53—42no—
157C2×D53—42no—
158C2×D53—42no—
159C2×D53#12114yes—
160C4×C53—22no—
161C4×C53—42no—
162C4×C53—42no—
163C4×C53—41no—
164Dic53—22no—
165Dic53—42no—
166Dic53—41no—
167#2617—531nomaximal, Sylow
168C2×C53—42no—
169C2×C53C2×#12111yescharacteristic
170C2×C53#53411yescharacteristic
171C2×C53#53412yes—
172C2×C53#53511yescharacteristic
173C2×C53#53812yes—
174D53—24no—
175D53—42no—
176D53—82no—
177C2×#121—531no—
178C2×#120—532no—
179C22×C22⋊C4—531no—
180#397—531no—
181#397D5311yescharacteristic, complemented
182#394—532no—
183#394—532no—
184#392—531no—
185#392—531no—
186#386—532no—
187#380—531no—
188C53#261711yescharacteristic, complemented, Sylow
189C25—531no—
190C22×D4—532no—
191C22×D4—532no—
192C22×D4—532no—
193C22×D4—531no—
194C23×C4—531no—
195C23×C4—531no—
196#121—1064no—
197#120—1068no—
198C2×C4⋊C4—532no—
199C2×C4⋊C4C2×D5311yescharacteristic, complemented
200C2×C22⋊C4—531no—
201C2×C22⋊C4—531no—
202C2×C22⋊C4—531no—
203C2×C22⋊C4—532no—
204C2×C22⋊C4—532no—
205C2×C22⋊C4—532no—
206C2×C22⋊C4—532no—
207C2×C22⋊C4—532no—
208C2×C22⋊C4—1064no—
209C2×C22⋊C4C2×D5311yescharacteristic
210C2×C22⋊C4C2×D5312yes—
211#95—531no—
212#95—532no—
213#95C2×D5311yescharacteristic, complemented
214#95C2×D5312yescomplemented
215C24—531no—
216C24—531no—
217C24—534no—
218C24—532no—
219C24—1062no—
220C24—1062no—
221C24—2122no—
222C24—2122no—
223C24C22×D5311yescharacteristic
224C2×D4—1064no—
225C2×D4—1064no—
226C2×D4—1064no—
227C2×D4—1064no—
228C2×D4—1064no—
229C2×D4—1062no—
230C2×D4—2122no—
231C2×D4—2122no—
232C2×D4—2122no—
233C2×D4—2122no—
234C2×D4—2122no—
235C22×C4—531no—
236C22×C4—534no—
237C22×C4—531no—
238C22×C4—532no—
239C22×C4—1062no—
240C22×C4—2122no—
241C22×C4—2122no—
242C22×C4C22×D5312yes—
243C22×C4C22×D5311yescharacteristic
244C22×C4C22×D5312yes—
245C22×C4C22×D5311yescharacteristic
246C4⋊C4—22no—
247C4⋊C4—2122no—
248C22⋊C4—24no—
249C22⋊C4—22no—
250C22⋊C4—1062no—
251C22⋊C4—1064no—
252C22⋊C4—1064no—
253C22⋊C4—1064no—
254C22⋊C4—1062no—
255C22⋊C4—1062no—
256C22⋊C4—2122no—
257C22⋊C4—2122no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—22no—
262C23—532no—
263C23—532no—
264C23—532no—
265C23—532no—
266C23—1068no—
267C23—1062no—
268C23—1062no—
269C23—1061no—
270C23—1061no—
271C23—1062no—
272C23—1062no—
273C23—1062no—
274C23—1062no—
275C23—1062no—
276C23—1062no—
277C23—1062no—
278C23—1062no—
279C23—1062no—
280C23—2128no—
281C23—2124no—
282C23—2124no—
283C23—2128no—
284C23D4×D5312yes—
285C23C23×D5311yescharacteristic
286D4—2124no—
287D4—2128no—
288D4—2128no—
289D4—2128no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—532no—
300C2×C4—1062no—
301C2×C4—1061no—
302C2×C4—1062no—
303C2×C4—1062no—
304C2×C4—1062no—
305C2×C4—1062no—
306C2×C4—2124no—
307C2×C4—2124no—
308C2×C4D4×D5312yes—
309C22—22no—
310C22—24no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—534no—
315C22—1062no—
316C22—1064no—
317C22—1062no—
318C22—1064no—
319C22—1062no—
320C22—1064no—
321C22—21216no—
322C22—2122no—
323C22—2122no—
324C22—2122no—
325C22—2122no—
326C22—2122no—
327C22—2122no—
328C22—2122no—
329C22—2121no—
330C22—2121no—
331C22—2122no—
332C22C2×D4×D5311yescharacteristic
333C22C2×D4×D5311yescharacteristic
334C22#41×D5311yescharacteristic
335C22#1313326312yes—
336C22#1313326312yes—
337C4—22no—
338C4—42no—
339C4—42no—
340C4—41no—
341C4—1062no—
342C4—2122no—
343C4—2121no—
344C2—42no—
345C2—1064no—
346C2—4242no—
347C2—2122no—
348C2#121×D5311yescharacteristic
349C2#1681118411yescharacteristic
350C2#1681118412yes—
351C2#1681119111yescharacteristic
352C2#1681121212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2617
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#13133192C2semidirect
2. C2×#13133187C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. #16810549C2semidirect
5. #16810541C2semidirect
6. #16810538C2semidirect
7. #16810533C2semidirect
8. #16810532C2semidirect
9. #16810501C2semidirect
10. #16810482C2semidirect
11. C53×#397C2semidirect
15. C22×D212C22semidirect
21. C2×#11719919C22semidirect
27. C2×#11719921C22semidirect
28. C2×#11719921C22semidirect
29. C2×#11719921C22semidirect
31. #13133124C22semidirect
32. #13133124C22semidirect
33. C2×C4⋊C4×C53C22semidirect
36. #95×C53C22semidirect
37. #95×C53C22semidirect
43. C23×D53D4semidirect
68. C2×C4×D53D4semidirect
181. #397D53semidirect
188. C53#2617semidirect
199. C2×C4⋊C4C2×D53semidirect
213. #95C2×D53semidirect
214. #95C2×D53semidirect