Finite Group Databasebeta

← #20278035

Subgroups — #20278035

Order 6784.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#11719949C211yesmaximal, characteristic, complemented
2C2×#13133187C212yesmaximal, complemented
3C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810534C211yesmaximal, characteristic, complemented
5#16810533C211yesmaximal, characteristic, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810501C212yesmaximal, complemented
8#16810501C212yesmaximal, complemented
9#16810499C212yesmaximal, complemented
10#16810485C211yesmaximal, characteristic, complemented
11C53×#393C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C2×#11719949—28no—
14C2×#11719949—24no—
15C22×D212C2212yes—
16C22×D212C2212yes—
17C22×D212C2211yescharacteristic, complemented
18C22×C22⋊D53C2212yes—
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2212yescomplemented
22#13133187—28no—
23C2×C22⋊Dic53C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2212yescomplemented
29C2×#11719921C2212yescomplemented
30C22⋊C4×D53—24no—
31#13133124C2212yescomplemented
32#13133124C2212yescomplemented
33#13133124C2211yescharacteristic, complemented
34C22×Q8×C53C2211yescharacteristic, complemented
35C2×C22⋊C4×C53C2212yes—
36C2×C22⋊C4×C53C2211yescharacteristic
37#95×C53C2212yescomplemented
38#95×C53C2211yescharacteristic, complemented
39C24×C53C2311yescharacteristic
40C23×D53—22no—
41C23×D53—22no—
42C23×D53—22no—
43C23×D53—24no—
44C23×D53—42no—
45C23×D53C2312yes—
46C23×D53C2311yescharacteristic
47C23×D53C2311yescharacteristic
48C2×Q8×C53—24no—
49C2×Q8×C53—22no—
50C22×C4×C53C2312yes—
51C22×C4×C53C2311yescharacteristic
52C22×C4×C53C2311yescharacteristic
53C22×C4×C53C2312yes—
54C2×C22⋊D53—24no—
55C2×C22⋊D53—24no—
56C2×C22⋊D53—42no—
57C22×Dic53C2312yes—
58C22×Dic53C2311yescharacteristic
59C22×Dic53C2311yescharacteristic
60#11719949—232no—
61C2×D212—24no—
62C2×D212—24no—
63C2×D212—24no—
64C2×D212—22no—
65C2×D212—24no—
66C2×D212—24no—
67C2×D212—42no—
68C2×D212D418yescomplemented
69C2×C4×D53—24no—
70C2×C4×D53—24no—
71C2×C4×D53—24no—
72C2×C4×D53—24no—
73C2×C4×D53—22no—
74C2×C4×D53—42no—
75C2×C4×D53D414yescomplemented
76C22⋊C4×C53—22no—
77C22⋊C4×C53—24no—
78C22⋊Dic53—22no—
79#11719921—24no—
80#11719921—22no—
81#11719921—24no—
82#11719921—22no—
83#11719921—42no—
84#11719921—42no—
85C23×C53—22no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—22no—
89C23×C53#4112yescomplemented
90C23×C53C2411yescharacteristic
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—22no—
94C22×D53—28no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—21no—
99C22×D53—22no—
100C22×D53—22no—
101C22×D53—21no—
102C22×D53—22no—
103C22×D53—22no—
104C22×D53—22no—
105C22×D53—48no—
106C22×D53—48no—
107C22×D53—44no—
108C22×D53—44no—
109C22×D53C2×D414yes—
110C22×D53C2×D412yes—
111C22×D53C2×D414yes—
112C22×D53#4112yescomplemented
113Q8×C53—28no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53C2×D414yes—
123C2×C4×C53C2×D412yes—
124C22⋊D53—48no—
125C2×Dic53—22no—
126C2×Dic53—21no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53C2×D412yes—
132D212—28no—
133D212—216no—
134D212—48no—
135C4×D53—216no—
136C4×D53—28no—
137C4×D53—44no—
138C22×C53—22no—
139C22×C53—24no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—42no—
143C22×C53C22×D411yescharacteristic
144C22×C53C22×D412yes—
145C22×C53C2×#4111yescharacteristic
146C22×C53C2×#4111yescharacteristic
147C22×C53H(5,𝔽2)12yes—
148C2×D53—22no—
149C2×D53—22no—
150C2×D53—24no—
151C2×D53—24no—
152C2×D53—24no—
153C2×D53—22no—
154C2×D53—24no—
155C2×D53—24no—
156C2×D53—24no—
157C2×D53—416no—
158C2×D53—41no—
159C2×D53—42no—
160C2×D53—42no—
161C2×D53—41no—
162C2×D53—42no—
163C2×D53—42no—
164C2×D53—42no—
165C2×D53#12014yes—
166C4×C53—24no—
167C4×C53—22no—
168C4×C53—41no—
169C4×C53—42no—
170Dic53—22no—
171Dic53—41no—
172Dic53—42no—
173#2404—531nomaximal, Sylow
174C2×C53—42no—
175C2×C53C2×#12011yescharacteristic
176C2×C53#52511yescharacteristic
177C2×C53#53511yescharacteristic
178C2×C53#54612yes—
179C2×C53#54612yes—
180D53—24no—
181D53—44no—
182D53—42no—
183C22×#41—531no—
184C2×#120—532no—
185C22×C22⋊C4—531no—
186#393—531no—
187#393D5311yescharacteristic, complemented
188#392—531no—
189#392—531no—
190#386—532no—
191#386—532no—
192#386—532no—
193#381—531no—
194C53#240411yescharacteristic, complemented, Sylow
195C25—531no—
196C2×#41—1064no—
197C2×#41—1068no—
198C22×Q8C2×D5311yescharacteristic, complemented
199C22×D4—532no—
200C22×D4—532no—
201C22×D4—531no—
202C22×D4—532no—
203C23×C4—531no—
204C23×C4—532no—
205C23×C4—531no—
206#120—1068no—
207C2×C22⋊C4—531no—
208C2×C22⋊C4—531no—
209C2×C22⋊C4—531no—
210C2×C22⋊C4—532no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4—532no—
213C2×C22⋊C4—532no—
214C2×C22⋊C4—1064no—
215C2×C22⋊C4C2×D5311yescharacteristic
216C2×C22⋊C4C2×D5312yes—
217#95—532no—
218#95—531no—
219#95—532no—
220#95C2×D5311yescharacteristic, complemented
221#95C2×D5312yescomplemented
222C24—531no—
223C24—531no—
224C24—532no—
225C24—1062no—
226C24—1062no—
227C24—1064no—
228C24—1062no—
229C24—2122no—
230C24C22×D5311yescharacteristic
231#41—10632no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×D4—538no—
235C2×D4—1064no—
236C2×D4—1064no—
237C2×D4—1064no—
238C2×D4—1064no—
239C2×D4—1064no—
240C2×D4—1064no—
241C2×D4—1064no—
242C2×D4—1062no—
243C2×D4—2122no—
244C2×D4—2122no—
245C22×C4—531no—
246C22×C4—534no—
247C22×C4—532no—
248C22×C4—531no—
249C22×C4—1064no—
250C22×C4—1064no—
251C22×C4—1064no—
252C22×C4—1062no—
253C22×C4—1064no—
254C22×C4—2122no—
255C22×C4C22×D5312yes—
256C22×C4C22×D5311yescharacteristic
257C22×C4C22×D5311yescharacteristic
258C22×C4C22×D5312yes—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—1062no—
262C22⋊C4—1064no—
263C22⋊C4—1062no—
264C22⋊C4—1062no—
265C22⋊C4—1064no—
266C22⋊C4—2122no—
267C22⋊C4—2122no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—534no—
273C23—532no—
274C23—534no—
275C23—532no—
276C23—1068no—
277C23—1061no—
278C23—1061no—
279C23—1062no—
280C23—1061no—
281C23—1062no—
282C23—1062no—
283C23—1062no—
284C23—1062no—
285C23—1062no—
286C23—1061no—
287C23—1062no—
288C23—1062no—
289C23—1062no—
290C23—2128no—
291C23—2124no—
292C23—2128no—
293C23—2124no—
294C23#1171994912yescomplemented
295C23C23×D5311yescharacteristic
296Q8—28no—
297D4—10616no—
298D4—1068no—
299D4—2128no—
300D4—2128no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—532no—
310C2×C4—1068no—
311C2×C4—10616no—
312C2×C4—1062no—
313C2×C4—1061no—
314C2×C4—1062no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—2124no—
319C2×C4D4×D5312yes—
320C2×C4D4×D5314yes—
321C22—24no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—534no—
327C22—1064no—
328C22—1064no—
329C22—1062no—
330C22—1062no—
331C22—1064no—
332C22—1064no—
333C22—1064no—
334C22—1064no—
335C22—1062no—
336C22—21216no—
337C22—2122no—
338C22—2122no—
339C22—2122no—
340C22—2121no—
341C22—2122no—
342C22—2121no—
343C22—2122no—
344C22C2×D4×D5312yes—
345C22C2×D4×D5311yescharacteristic
346C22#41×D5311yescharacteristic
347C22C2×#1171994911yescharacteristic
348C22#1313326312yes—
349C4—22no—
350C4—24no—
351C4—42no—
352C4—41no—
353C4—1062no—
354C4—2122no—
355C4—2121no—
356C2—42no—
357C2—1064no—
358C2—2122no—
359C2—2124no—
360C2#120×D5311yescharacteristic
361C2#1681113611yescharacteristic
362C2#1681119111yescharacteristic
363C2#1681126412yes—
364C2#1681126612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2404
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#11719949C2semidirect
2. C2×#13133187C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. #16810534C2semidirect
5. #16810533C2semidirect
6. #16810532C2semidirect
7. #16810501C2semidirect
8. #16810501C2semidirect
9. #16810499C2semidirect
10. #16810485C2semidirect
11. C53×#393C2semidirect
17. C22×D212C22semidirect
21. C22×C4×D53C22semidirect
28. C2×#11719921C22semidirect
29. C2×#11719921C22semidirect
31. #13133124C22semidirect
32. #13133124C22semidirect
33. #13133124C22semidirect
34. C22×Q8×C53C22semidirect
37. #95×C53C22semidirect
38. #95×C53C22semidirect
68. C2×D212D4semidirect
75. C2×C4×D53D4semidirect
89. C23×C53#41semidirect
112. C22×D53#41semidirect
187. #393D53semidirect
194. C53#2404semidirect
198. C22×Q8C2×D53semidirect
220. #95C2×D53semidirect
221. #95C2×D53semidirect
294. C23#11719949semidirect