Finite Group Databasebeta

← #20280928

Subgroups — #20280928

Order 6784.

932 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133189C212yesmaximal, complemented
2C2×#13133185C212yesmaximal, complemented
3C2×#13133185C211yesmaximal, characteristic, complemented
4#16810537C212yesmaximal, complemented
5#16810531C212yesmaximal, complemented
6#16810531C212yesmaximal, complemented
7#16810514C212yesmaximal, complemented
8#16810503C211yesmaximal, characteristic, complemented
9C2×#127×C53C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2212yes—
14C23×Dic53C2211yescharacteristic, complemented
15#13133189—28no—
16#13133185—28no—
17#13133185—24no—
18C2×#11719919C2212yescomplemented
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2212yescomplemented
22C2×C22⋊Dic53C2211yescharacteristic, complemented
23C2×C22⋊Dic53C2212yescomplemented
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27#13133124C2212yescomplemented
28#13133124C2212yescomplemented
29C22×D4×C53C2212yes—
30C22×D4×C53C2211yescharacteristic
31C22×D4×C53C2212yescomplemented
32C22×D4×C53C2211yescharacteristic, complemented
33#127×C53—24no—
34C2×C42×C53C2211yescharacteristic, complemented
35C24×C53C2312yes—
36C24×C53C2311yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53—22no—
39C23×D53—24no—
40C23×D53—41no—
41C23×D53—42no—
42C23×D53—41no—
43C23×D53C2312yes—
44C2×D4×C53—24no—
45C2×D4×C53—24no—
46C2×D4×C53—22no—
47C2×D4×C53—24no—
48C2×D4×C53—24no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53D414yescomplemented
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2312yes—
58C2×C22⋊D53—24no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—24no—
61C2×C22⋊D53—24no—
62C2×C22⋊D53—24no—
63C2×C22⋊D53—42no—
64C2×C22⋊D53—42no—
65C2×C22⋊D53—42no—
66C2×C22⋊D53—42no—
67C22×Dic53—24no—
68C22×Dic53—41no—
69C22×Dic53C2312yes—
70C22×Dic53C2312yes—
71C22×Dic53C2312yes—
72C42×C53—22no—
73C22⋊Dic53—22no—
74C22⋊Dic53—24no—
75C22⋊Dic53—24no—
76C22⋊Dic53—41no—
77C22⋊Dic53—42no—
78#11719921—24no—
79#11719921—24no—
80#11719921D418yescomplemented
81#11719919—24no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—21no—
85C23×C53—22no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—22no—
89C23×C53—21no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53C2×D412yes—
98C23×C53C2×D412yes—
99C23×C53C2411yescharacteristic
100C22×D53—22no—
101C22×D53—22no—
102C22×D53—22no—
103C22×D53—22no—
104C22×D53—22no—
105C22×D53—22no—
106C22×D53—44no—
107C22×D53—48no—
108C22×D53—48no—
109C22×D53—44no—
110C22×D53—44no—
111C22×D53C2×D414yes—
112D4×C53—44no—
113D4×C53—48no—
114D4×C53—48no—
115D4×C53—44no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53C2×D414yes—
121C2×C4×C53C2×D412yes—
122C22⋊D53—48no—
123C22⋊D53—48no—
124C22⋊D53—48no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53—44no—
134C2×Dic53C2×D414yes—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—24no—
138C22×C53—24no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53C22×D412yes—
153C22×C53C22×D411yescharacteristic
154C22×C53H(5,𝔽2)11yescharacteristic
155C22×C53H(5,𝔽2)11yescharacteristic
156C22×C53H(5,𝔽2)12yes—
157C2×D53—24no—
158C2×D53—24no—
159C2×D53—24no—
160C2×D53—416no—
161C2×D53—42no—
162C2×D53—42no—
163C2×D53—42no—
164C2×D53—42no—
165C4×C53—22no—
166C4×C53—24no—
167Dic53—24no—
168Dic53—42no—
169Dic53—42no—
170#2669—531nomaximal, Sylow
171C2×C53—42no—
172C2×C53—42no—
173C2×C53—82no—
174C2×C53#53411yescharacteristic
175C2×C53#53512yes—
176C2×C53D4212yes—
177C2×C53D4211yescharacteristic
178C2×C53#56011yescharacteristic
179D53—44no—
180C2×#127D5311yescharacteristic, complemented
181C2×#121—532no—
182C2×#120—532no—
183C2×#120—531no—
184#394—532no—
185#392—532no—
186#392—532no—
187#388—532no—
188#386—531no—
189C53#266911yescharacteristic, complemented, Sylow
190C25—531no—
191C22×D4—532no—
192C22×D4—532no—
193C22×D4—532no—
194C22×D4C2×D5312yes—
195C22×D4C2×D5311yescharacteristic
196C22×D4C2×D5312yescomplemented
197C22×D4C2×D5311yescharacteristic, complemented
198C23×C4—531no—
199#127—24no—
200#121—1068no—
201#120—1068no—
202#120—1064no—
203C2×C4⋊C4—532no—
204C2×C22⋊C4—532no—
205C2×C22⋊C4—532no—
206C2×C22⋊C4—531no—
207C2×C22⋊C4—532no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—532no—
210C2×C22⋊C4—531no—
211C2×C22⋊C4—532no—
212C2×C42C2×D5311yescharacteristic, complemented
213#95—532no—
214#95—532no—
215C24—532no—
216C24—1064no—
217C24—1062no—
218C24—2122no—
219C24—2121no—
220C24—2121no—
221C24C22×D5312yes—
222C24C22×D5311yescharacteristic
223C24C22×D5311yescharacteristic
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—1064no—
236C2×D4—1064no—
237C2×D4—1064no—
238C2×D4—1064no—
239C2×D4—1064no—
240C2×D4—2122no—
241C2×D4—2122no—
242C2×D4—2122no—
243C2×D4—2122no—
244C2×D4C22⋊D5314yescomplemented
245C22×C4—532no—
246C22×C4—532no—
247C22×C4—532no—
248C22×C4—1064no—
249C22×C4—2121no—
250C22×C4C22×D5312yes—
251C22×C4C22×D5311yescharacteristic
252C4⋊C4—1064no—
253C22⋊C4—538no—
254C22⋊C4—1064no—
255C22⋊C4—1064no—
256C22⋊C4—1062no—
257C22⋊C4—1064no—
258C22⋊C4—1064no—
259C22⋊C4—2121no—
260C22⋊C4—2122no—
261C42—22no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—21no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—534no—
278C23—1062no—
279C23—1062no—
280C23—1062no—
281C23—1062no—
282C23—1062no—
283C23—1062no—
284C23—2128no—
285C23—2124no—
286C23—2124no—
287C23—2128no—
288C23—2124no—
289C23C2×C22⋊D5312yes—
290C23C2×C22⋊D5312yes—
291C23C23×D5311yescharacteristic
292D4—44no—
293D4—48no—
294D4—48no—
295D4—44no—
296D4—2128no—
297D4—2128no—
298D4—2128no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—534no—
304C2×C4—1062no—
305C2×C4—1062no—
306C2×C4—1062no—
307C2×C4—1062no—
308C2×C4—1062no—
309C2×C4—1062no—
310C2×C4—1062no—
311C2×C4—1062no—
312C2×C4—2124no—
313C2×C4D4×D5314yes—
314C2×C4C2×C22⋊D5312yes—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—24no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—1064no—
333C22—1064no—
334C22—1064no—
335C22—21216no—
336C22—2122no—
337C22—2122no—
338C22—2122no—
339C22—2122no—
340C22C2×D4×D5312yes—
341C22C22×C22⋊D5311yescharacteristic
342C22#1313326211yescharacteristic
343C22#1313326212yes—
344C22#1313326211yescharacteristic
345C4—22no—
346C4—24no—
347C4—1064no—
348C4—2122no—
349C4—2122no—
350C2—42no—
351C2—42no—
352C2—82no—
353C2—2124no—
354C2#1681118211yescharacteristic
355C2#1681118612yes—
356C2D4×C22⋊D5312yes—
357C2#1681125811yescharacteristic
358C2#1681136011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#127×C53C2
hypercenterC2×#127D53
hypocenterC53#2669
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. C2×#13133185C2semidirect
3. C2×#13133185C2semidirect
4. #16810537C2semidirect
5. #16810531C2semidirect
6. #16810531C2semidirect
7. #16810514C2semidirect
8. #16810503C2semidirect
9. C2×#127×C53C2semidirect
14. C23×Dic53C22semidirect
18. C2×#11719919C22semidirect
21. C2×C22⋊Dic53C22semidirect
22. C2×C22⋊Dic53C22semidirect
23. C2×C22⋊Dic53C22semidirect
27. #13133124C22semidirect
28. #13133124C22semidirect
31. C22×D4×C53C22semidirect
32. C22×D4×C53C22semidirect
34. C2×C42×C53C22semidirect
55. C2×D4×C53D4semidirect
80. #11719921D4semidirect
180. C2×#127D53semidirect
189. C53#2669semidirect
196. C22×D4C2×D53semidirect
197. C22×D4C2×D53semidirect
212. C2×C42C2×D53semidirect
244. C2×D4C22⋊D53semidirect