Finite Group Databasebeta

← #20280455

Subgroups — #20280455

Order 6784.

824 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133189C211yesmaximal, characteristic, complemented
2C2×#13133187C211yesmaximal, characteristic, complemented
3C2×#13133185C211yesmaximal, characteristic, complemented
4C22×#11719921C211yesmaximal, characteristic, complemented
5#16810552C212yesmaximal, complemented
6#16810541C212yesmaximal, complemented
7#16810538C212yesmaximal, complemented
8#16810532C212yesmaximal, complemented
9#16810502C211yesmaximal, characteristic, complemented
10#16810481C211yesmaximal, characteristic, complemented
11C53×#386C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×D212C2212yes—
14C22×C22⋊D53C2211yescharacteristic
15C22×C22⋊D53C2212yes—
16C22×C22⋊D53C2211yescharacteristic, complemented
17C23×Dic53C2211yescharacteristic
18#13133189—24no—
19#13133187—24no—
20#13133185—24no—
21C2×#11719919C2211yescharacteristic, complemented
22C2×#11719919C2212yescomplemented
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic, complemented
25C2×#11719921—24no—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×#11719921C2212yes—
30C2×#11719921C2211yescharacteristic, complemented
31C2×#11719921C2212yescomplemented
32#13133124C2212yescomplemented
33#13133124C2212yescomplemented
34C22×D4×C53C2211yescharacteristic
35C23×C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2212yescomplemented
38#95×C53C2212yescomplemented
39C24×C53C2311yescharacteristic
40C24×C53C2311yescharacteristic
41C23×D53—22no—
42C23×D53—22no—
43C23×D53—42no—
44C23×D53—42no—
45C23×D53D414yescomplemented
46C23×D53C2311yescharacteristic
47C23×D53C2312yes—
48C2×D4×C53—22no—
49C2×D4×C53—22no—
50C2×D4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—42no—
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2312yes—
55C22×C4×C53C2312yes—
56C2×C22⋊D53—22no—
57C2×C22⋊D53—22no—
58C2×C22⋊D53—22no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—22no—
61C2×C22⋊D53—42no—
62C2×C22⋊D53—42no—
63C2×C22⋊D53—42no—
64C22×Dic53—42no—
65C22×Dic53D414yescomplemented
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2312yes—
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C2×D212—24no—
71C2×D212—42no—
72C2×D212—42no—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—42no—
75C22⋊Dic53—22no—
76C22⋊Dic53—22no—
77#11719921—24no—
78#11719921—24no—
79#11719921—24no—
80#11719921—24no—
81#11719921—22no—
82#11719921—42no—
83#11719919—22no—
84#11719919—42no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53C2×D412yes—
93C23×C53C2×D412yes—
94C23×C53#4112yes—
95C23×C53C2411yescharacteristic
96C22×D53—28no—
97C22×D53—22no—
98C22×D53—22no—
99C22×D53—22no—
100C22×D53—22no—
101C22×D53—22no—
102C22×D53—22no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—48no—
109C22×D53—44no—
110C22×D53—44no—
111C22×D53—48no—
112C22×D53C2×D414yes—
113D4×C53—44no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—44no—
124C22⋊D53—44no—
125C22⋊D53—48no—
126C22⋊D53—44no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—22no—
132C2×Dic53—21no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—44no—
136C2×Dic53C2×D412yes—
137C2×Dic53C2×D412yes—
138D212—48no—
139C22×C53—22no—
140C22×C53—24no—
141C22×C53—22no—
142C22×C53—24no—
143C22×C53—41no—
144C22×C53—42no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—42no—
148C22×C53#12114yes—
149C22×C53C22×D411yescharacteristic
150C22×C53C22×D411yescharacteristic
151C22×C53C2×#4111yescharacteristic
152C22×C53H(5,𝔽2)12yes—
153C22×C53H(5,𝔽2)12yes—
154C2×D53—24no—
155C2×D53—24no—
156C2×D53—24no—
157C2×D53—416no—
158C2×D53—42no—
159C2×D53—42no—
160C2×D53—42no—
161C2×D53—42no—
162C2×D53—41no—
163C2×D53—42no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—42no—
167C4×C53—41no—
168C4×C53—42no—
169C4×C53—42no—
170Dic53—22no—
171Dic53—22no—
172Dic53—41no—
173Dic53—42no—
174#2617—531nomaximal, Sylow
175C2×C53—24no—
176C2×C53—81no—
177C2×C53C2×#12111yescharacteristic
178C2×C53#53412yes—
179C2×C53#53411yescharacteristic
180C2×C53#53511yescharacteristic
181C2×C53#53812yes—
182D53—44no—
183D53—81no—
184C2×#121—531no—
185C2×#120—531no—
186C2×#120—531no—
187C22×C22⋊C4—531no—
188#397—532no—
189#394—532no—
190#394—532no—
191#392—532no—
192#386—531no—
193#386D5311yescharacteristic, complemented
194#380—531no—
195C53#261711yescharacteristic, complemented, Sylow
196C25—531no—
197C22×D4—531no—
198C22×D4—532no—
199C22×D4—531no—
200C22×D4—532no—
201C22×D4C2×D5311yescharacteristic
202C23×C4—531no—
203C23×C4C2×D5311yescharacteristic
204#121—1064no—
205#120—1064no—
206#120—1064no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—532no—
209C2×C22⋊C4—531no—
210C2×C22⋊C4—531no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4—532no—
213C2×C22⋊C4—532no—
214C2×C22⋊C4—532no—
215C2×C22⋊C4—531no—
216C2×C22⋊C4—532no—
217C2×C22⋊C4—1064no—
218C2×C22⋊C4C2×D5311yescharacteristic
219C2×C22⋊C4C2×D5312yescomplemented
220#95—532no—
221#95—532no—
222#95C2×D5312yescomplemented
223C24—532no—
224C24—534no—
225C24—531no—
226C24—1062no—
227C24—1062no—
228C24—2122no—
229C24—2122no—
230C24C22×D5311yescharacteristic
231C24C22×D5311yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—1064no—
236C2×D4—1062no—
237C2×D4—1064no—
238C2×D4—1062no—
239C2×D4—1062no—
240C2×D4—1062no—
241C2×D4—2122no—
242C2×D4—2122no—
243C2×D4—2122no—
244C2×D4—2122no—
245C2×D4—2122no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—534no—
251C22×C4—531no—
252C22×C4—532no—
253C22×C4—2122no—
254C22×C4C22×D5312yes—
255C22×C4C22×D5312yes—
256C22×C4C22×D5311yescharacteristic
257C4⋊C4—1062no—
258C4⋊C4—2122no—
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—1064no—
262C22⋊C4—1064no—
263C22⋊C4—1062no—
264C22⋊C4—1064no—
265C22⋊C4—1064no—
266C22⋊C4—1062no—
267C22⋊C4—1062no—
268C22⋊C4—2122no—
269C23—22no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—534no—
277C23—1068no—
278C23—1062no—
279C23—1062no—
280C23—1062no—
281C23—1062no—
282C23—1062no—
283C23—1062no—
284C23—1061no—
285C23—1061no—
286C23—1061no—
287C23—1062no—
288C23—1062no—
289C23—2128no—
290C23—2124no—
291C23—2124no—
292C23—2128no—
293C23#1171994512yes—
294C23D4×D5312yes—
295C23D4×D5312yes—
296C23C23×D5311yescharacteristic
297D4—44no—
298D4—2128no—
299D4—2124no—
300D4—2128no—
301D4—2124no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—44no—
312C2×C4—532no—
313C2×C4—532no—
314C2×C4—1062no—
315C2×C4—1062no—
316C2×C4—1061no—
317C2×C4—1062no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—2124no—
323C22—24no—
324C22—22no—
325C22—22no—
326C22—24no—
327C22—41no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—42no—
332C22—1064no—
333C22—1064no—
334C22—1064no—
335C22—21216no—
336C22—2121no—
337C22—2122no—
338C22—2121no—
339C22—2121no—
340C22—2122no—
341C22—2122no—
342C22—2122no—
343C22—2122no—
344C22—2122no—
345C22#1313319014yes—
346C22C2×D4×D5311yescharacteristic
347C22C2×D4×D5311yescharacteristic
348C22C2×#1171994511yescharacteristic
349C22#1313326212yes—
350C22#1313326312yes—
351C4—41no—
352C4—42no—
353C4—42no—
354C4—1062no—
355C4—1062no—
356C4—2121no—
357C4—2122no—
358C2—24no—
359C2—81no—
360C2—4241no—
361C2—2124no—
362C2C2×#1313319011yescharacteristic
363C2#1681118112yes—
364C2#1681118411yescharacteristic
365C2#1681118811yescharacteristic
366C2#1681120612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2617
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. C2×#13133187C2semidirect
3. C2×#13133185C2semidirect
4. C22×#11719921C2semidirect
5. #16810552C2semidirect
6. #16810541C2semidirect
7. #16810538C2semidirect
8. #16810532C2semidirect
9. #16810502C2semidirect
10. #16810481C2semidirect
11. C53×#386C2semidirect
16. C22×C22⋊D53C22semidirect
21. C2×#11719919C22semidirect
22. C2×#11719919C22semidirect
24. C2×C22⋊Dic53C22semidirect
30. C2×#11719921C22semidirect
31. C2×#11719921C22semidirect
32. #13133124C22semidirect
33. #13133124C22semidirect
37. C2×C22⋊C4×C53C22semidirect
38. #95×C53C22semidirect
45. C23×D53D4semidirect
65. C22×Dic53D4semidirect
193. #386D53semidirect
195. C53#2617semidirect
219. C2×C22⋊C4C2×D53semidirect
222. #95C2×D53semidirect