Finite Group Databasebeta

← #20280933

Subgroups — #20280933

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×#13133213C211yesmaximal, characteristic, complemented
2C2×#13133191C212yesmaximal, complemented
3C2×#13133187C212yesmaximal, complemented
4C2×#13133185C211yesmaximal, characteristic, complemented
5#16810539C212yesmaximal, complemented
6#16810533C212yesmaximal, complemented
7#16810532C212yesmaximal, complemented
8#16810515C212yesmaximal, complemented
9C53×#386C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C22×D212C2212yes—
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic, complemented
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yes—
16C22×C22⋊D53C2212yescomplemented
17#13133213—24no—
18#13133191—28no—
19#13133187—28no—
20#13133185—24no—
21C2×C4⋊Dic53C2212yescomplemented
22C2×C22⋊Dic53C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yescomplemented
27C2×#11719921C2212yescomplemented
28C2×C4×Dic53C2211yescharacteristic, complemented
29#13133124C2212yescomplemented
30C22×D4×C53C2211yescharacteristic
31C23×C4×C53C2211yescharacteristic, complemented
32C2×C22⋊C4×C53C2212yes—
33C2×C22⋊C4×C53C2211yescharacteristic, complemented
34#95×C53C2212yescomplemented
35C24×C53C2311yescharacteristic
36C24×C53C2311yescharacteristic
37C23×D53—22no—
38C23×D53—24no—
39C23×D53—41no—
40C23×D53—41no—
41C23×D53—42no—
42C23×D53C2312yes—
43C23×D53C2312yes—
44C2×D4×C53—41no—
45C2×D4×C53—41no—
46C2×D4×C53D414yescomplemented
47C22×C4×C53—24no—
48C22×C4×C53—41no—
49C22×C4×C53C2312yes—
50C22×C4×C53C2311yescharacteristic
51C22×C4×C53C2312yes—
52C2×C22⋊D53—24no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—24no—
55C2×C22⋊D53—24no—
56C2×C22⋊D53—24no—
57C2×C22⋊D53—24no—
58C2×C22⋊D53—42no—
59C2×C22⋊D53—42no—
60C2×C22⋊D53—42no—
61C22×Dic53C2312yes—
62C22×Dic53C2312yes—
63C2×D212—24no—
64C2×D212—24no—
65C2×D212—24no—
66C2×D212—22no—
67C2×D212—42no—
68C2×D212—42no—
69C2×D212—42no—
70C2×D212—41no—
71C2×D212—41no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—41no—
74C22⋊Dic53—22no—
75#11719921—24no—
76#11719921—24no—
77#11719921—24no—
78#11719921—42no—
79#11719921D418yescomplemented
80C4⋊Dic53—24no—
81C4×Dic53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53C2×D412yes—
91C23×C53C2×D412yes—
92C23×C53C2411yescharacteristic
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—22no—
99C22×D53—22no—
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—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—44no—
121C2×C4×C53C2×D412yes—
122C2×C4×C53C2×D414yes—
123C22⋊D53—48no—
124C22⋊D53—48no—
125C22⋊D53—48no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53C2×D414yes—
132D212—48no—
133D212—44no—
134D212—48no—
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×C53C22×D412yes—
146C22×C53C22×D411yescharacteristic
147C22×C53H(5,𝔽2)11yescharacteristic
148C22×C53H(5,𝔽2)11yescharacteristic
149C22×C53H(5,𝔽2)12yes—
150C2×D53—24no—
151C2×D53—24no—
152C2×D53—24no—
153C2×D53—416no—
154C2×D53—42no—
155C2×D53—42no—
156C2×D53—42no—
157C2×D53—42no—
158C2×D53—42no—
159C2×D53—42no—
160C2×D53—42no—
161C2×D53—42no—
162C2×D53—42no—
163C2×D53—42no—
164C2×D53—42no—
165C4×C53—22no—
166C4×C53—24no—
167C4×C53—42no—
168Dic53—24no—
169Dic53—42no—
170#2669—531nomaximal, Sylow
171C2×C53—42no—
172C2×C53—42no—
173C2×C53#53411yescharacteristic
174C2×C53#53512yes—
175C2×C53D4211yescharacteristic
176C2×C53D4212yes—
177C2×C53#56011yescharacteristic
178D53—44no—
179D53—82no—
180C2×#127—531no—
181C2×#121—532no—
182C2×#120—531no—
183C2×#120—532no—
184#394—532no—
185#392—532no—
186#392—532no—
187#388—532no—
188#386D5311yescharacteristic, complemented
189C53#266911yescharacteristic, complemented, Sylow
190C25—531no—
191C22×D4—532no—
192C22×D4—532no—
193C22×D4—532no—
194C22×D4—532no—
195C22×D4—532no—
196C22×D4—531no—
197C22×D4C2×D5311yescharacteristic
198C23×C4C2×D5311yescharacteristic, complemented
199#127—1064no—
200#121—1068no—
201#120—1068no—
202#120—1064no—
203C2×C4⋊C4—532no—
204C2×C22⋊C4—532no—
205C2×C22⋊C4—531no—
206C2×C22⋊C4—532no—
207C2×C22⋊C4—532no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—532no—
210C2×C22⋊C4C2×D5312yes—
211C2×C22⋊C4C2×D5311yescharacteristic, complemented
212C2×C42—531no—
213#95—532no—
214#95C2×D5312yescomplemented
215C24—532no—
216C24—532no—
217C24—1064no—
218C24—1062no—
219C24—2121no—
220C24—2121no—
221C24—2122no—
222C24C22×D5311yescharacteristic
223C24C22×D5311yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—1064no—
227C2×D4—1064no—
228C2×D4—1064no—
229C2×D4—1064no—
230C2×D4—1064no—
231C2×D4—1064no—
232C2×D4—1064no—
233C2×D4—1064no—
234C2×D4—1062no—
235C2×D4—1064no—
236C2×D4—2122no—
237C2×D4—2121no—
238C2×D4—2121no—
239C2×D4—2122no—
240C2×D4—2122no—
241C2×D4—2122no—
242C2×D4—2122no—
243C2×D4—2122no—
244C2×D4D21214yescomplemented
245C22×C4—24no—
246C22×C4—41no—
247C22×C4—532no—
248C22×C4—532no—
249C22×C4C22×D5312yes—
250C22×C4C22×D5311yescharacteristic
251C22×C4C22×D5312yes—
252C4⋊C4—1064no—
253C22⋊C4—24no—
254C22⋊C4—41no—
255C22⋊C4—538no—
256C22⋊C4—1064no—
257C22⋊C4—1062no—
258C22⋊C4—1064no—
259C22⋊C4—1064no—
260C22⋊C4—2122no—
261C42—1062no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—534no—
271C23—1062no—
272C23—1062no—
273C23—1062no—
274C23—1062no—
275C23—1062no—
276C23—1062no—
277C23—1062no—
278C23—1062no—
279C23—1062no—
280C23—1062no—
281C23—1062no—
282C23—1062no—
283C23—1062no—
284C23—2128no—
285C23—2124no—
286C23—2124no—
287C23—2128no—
288C23—2124no—
289C23C2×D21212yes—
290C23C2×D21212yes—
291C23C23×D5311yescharacteristic
292D4—44no—
293D4—2128no—
294D4—2124no—
295D4—2128no—
296D4—2128no—
297D4—2128no—
298D4—2128no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—44no—
307C2×C4—534no—
308C2×C4—1062no—
309C2×C4—1062no—
310C2×C4—1062no—
311C2×C4—1062no—
312C2×C4—1062no—
313C2×C4C2×D21212yes—
314C2×C4D4×D5314yes—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—24no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—1064no—
326C22—1064no—
327C22—1064no—
328C22—21216no—
329C22—2122no—
330C22—2122no—
331C22—2122no—
332C22—2122no—
333C22—2122no—
334C22—2122no—
335C22—2122no—
336C22—2122no—
337C22—2122no—
338C22—2122no—
339C22—2122no—
340C22C2×D4×D5312yes—
341C22C22×D21211yescharacteristic
342C22#1313326211yescharacteristic
343C22#1313326211yescharacteristic
344C22#1313326312yes—
345C4—22no—
346C4—24no—
347C4—42no—
348C4—1064no—
349C4—2122no—
350C2—42no—
351C2—42no—
352C2—4242no—
353C2—2124no—
354C2#1681118511yescharacteristic
355C2#1681118912yes—
356C2#1681125511yescharacteristic
357C2D4×D21212yes—
358C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2669
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. C2×#13133191C2semidirect
3. C2×#13133187C2semidirect
4. C2×#13133185C2semidirect
5. #16810539C2semidirect
6. #16810533C2semidirect
7. #16810532C2semidirect
8. #16810515C2semidirect
9. C53×#386C2semidirect
13. C22×D212C22semidirect
16. C22×C22⋊D53C22semidirect
21. C2×C4⋊Dic53C22semidirect
26. C2×#11719921C22semidirect
27. C2×#11719921C22semidirect
28. C2×C4×Dic53C22semidirect
29. #13133124C22semidirect
31. C23×C4×C53C22semidirect
33. C2×C22⋊C4×C53C22semidirect
34. #95×C53C22semidirect
46. C2×D4×C53D4semidirect
79. #11719921D4semidirect
188. #386D53semidirect
189. C53#2669semidirect
198. C23×C4C2×D53semidirect
211. C2×C22⋊C4C2×D53semidirect
214. #95C2×D53semidirect
244. C2×D4D212semidirect