Finite Group Databasebeta

← #20279642

Subgroups — #20279642

Order 6784.

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

normal subgroups only

This group has 900 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
1C2×#13133202C212yesmaximal, complemented
2C2×#13133192C211yesmaximal, characteristic, complemented
3C2×#13133187C212yesmaximal, complemented
4C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
5#16810558C211yesmaximal, characteristic, complemented
6#16810533C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic, complemented
8#16810524C212yesmaximal, complemented
9#16810523C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×D212C2212yes—
14C22×D212C2212yes—
15C22×D212C2211yescharacteristic, complemented
16C22×C22⋊D53C2212yes—
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19#13133202—28no—
20#13133192—24no—
21#13133187—28no—
22C2×#11719919C2212yescomplemented
23C2×C22⋊Dic53C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2212yes—
28C2×#11719921C2212yescomplemented
29C2×#11719921C2212yescomplemented
30C2×#11719921C2212yescomplemented
31C22⋊C4×D53—24no—
32C2×C4×Dic53C2212yescomplemented
33#13133124C2211yescharacteristic, complemented
34C2×C4⋊C4×C53C2212yescomplemented
35C2×C4⋊C4×C53C2211yescharacteristic, complemented
36C2×C22⋊C4×C53C2212yes—
37C2×C22⋊C4×C53C2211yescharacteristic
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
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2312yes—
50C22×C4×C53C2311yescharacteristic
51C22×C4×C53C2312yes—
52C2×C22⋊D53—24no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—42no—
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2312yes—
57C22×Dic53C2311yescharacteristic
58C2×D212—24no—
59C2×D212—24no—
60C2×D212—24no—
61C2×D212—24no—
62C2×D212—24no—
63C2×D212—22no—
64C2×D212—42no—
65C2×D212—42no—
66C2×C4×D53—22no—
67C2×C4×D53—24no—
68C2×C4×D53—42no—
69C2×C4×D53D414yescomplemented
70C4⋊C4×C53—22no—
71C4⋊C4×C53—24no—
72C22⋊C4×C53—22no—
73C22⋊C4×C53—24no—
74C22⋊Dic53—22no—
75#11719921—22no—
76#11719921—24no—
77#11719921—22no—
78#11719921—24no—
79#11719921—24no—
80#11719921—24no—
81#11719921D418yescomplemented
82#11719919—24no—
83C4×Dic53—42no—
84C23×C53—22no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—22no—
88C23×C53#4112yes—
89C23×C53C2411yescharacteristic
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—22no—
93C22×D53—28no—
94C22×D53—22no—
95C22×D53—21no—
96C22×D53—21no—
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—48no—
106C22×D53—48no—
107C22×D53—44no—
108C22×D53—44no—
109C22×D53C2×D412yes—
110C22×D53C2×D414yes—
111C22×D53#4112yes—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53C2×D412yes—
121C2×C4×C53C2×D414yes—
122C22⋊D53—48no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—21no—
128C2×Dic53C2×D412yes—
129C2×Dic53C2×D414yes—
130D212—48no—
131D212—44no—
132D212—48no—
133C4×D53—44no—
134C4×D53—44no—
135C22×C53—22no—
136C22×C53—24no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—42no—
140C22×C53C22×D411yescharacteristic
141C22×C53C22×D412yes—
142C22×C53C2×#4111yescharacteristic
143C22×C53C2×#4111yescharacteristic
144C22×C53H(5,𝔽2)12yes—
145C2×D53—22no—
146C2×D53—22no—
147C2×D53—24no—
148C2×D53—24no—
149C2×D53—22no—
150C2×D53—24no—
151C2×D53—416no—
152C2×D53—41no—
153C2×D53—41no—
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#12714yes—
163C4×C53—22no—
164C4×C53—24no—
165C4×C53—41no—
166C4×C53—42no—
167Dic53—22no—
168Dic53—24no—
169Dic53—41no—
170#2539—531nomaximal, Sylow
171C2×C53—42no—
172C2×C53#52511yescharacteristic
173C2×C53C2×#12711yescharacteristic
174C2×C53#53411yescharacteristic
175C2×C53#54612yes—
176C2×C53#54612yes—
177D53—24no—
178D53—42no—
179D53—82no—
180C2×#123—532no—
181C2×#121—531no—
182C2×#120—532no—
183C22×C22⋊C4—531no—
184#399—531no—
185#399D5311yescharacteristic, complemented
186#392—531no—
187#392—531no—
188#390—532no—
189#390—532no—
190C4×C22⋊C4—531no—
191C53#253911yescharacteristic, complemented, Sylow
192C25—531no—
193C22×D4—532no—
194C22×D4—532no—
195C22×D4—532no—
196C22×D4—531no—
197C23×C4—531no—
198C23×C4—531no—
199#123—1068no—
200#121—1064no—
201#120—1068no—
202C2×C4⋊C4—532no—
203C2×C4⋊C4C2×D5311yescharacteristic, complemented
204C2×C4⋊C4C2×D5312yescomplemented
205C2×C22⋊C4—532no—
206C2×C22⋊C4—531no—
207C2×C22⋊C4—531no—
208C2×C22⋊C4—531no—
209C2×C22⋊C4—532no—
210C2×C22⋊C4—532no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4—532no—
213C2×C22⋊C4—1064no—
214C2×C22⋊C4C2×D5311yescharacteristic
215C2×C22⋊C4C2×D5312yes—
216C2×C42—532no—
217#95—531no—
218#95C2×D5311yescharacteristic, complemented
219C24—531no—
220C24—531no—
221C24—532no—
222C24—1062no—
223C24—1062no—
224C24—1064no—
225C24—1062no—
226C24—2122no—
227C24C22×D5311yescharacteristic
228C2×D4—1064no—
229C2×D4—1064no—
230C2×D4—1064no—
231C2×D4—1064no—
232C2×D4—1064no—
233C2×D4—1064no—
234C2×D4—1064no—
235C2×D4—1062no—
236C2×D4—2122no—
237C2×D4—2122no—
238C2×D4—2122no—
239C22×C4—532no—
240C22×C4—531no—
241C22×C4—534no—
242C22×C4—531no—
243C22×C4—1062no—
244C22×C4—1064no—
245C22×C4—2122no—
246C22×C4C22×D5311yescharacteristic
247C22×C4C22×D5312yes—
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5312yes—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—1064no—
253C22⋊C4—22no—
254C22⋊C4—24no—
255C22⋊C4—538no—
256C22⋊C4—1062no—
257C22⋊C4—1064no—
258C22⋊C4—1064no—
259C22⋊C4—1064no—
260C22⋊C4—1062no—
261C22⋊C4—1062no—
262C22⋊C4—1064no—
263C42—2122no—
264C23—21no—
265C23—22no—
266C23—21no—
267C23—22no—
268C23—534no—
269C23—532no—
270C23—532no—
271C23—1068no—
272C23—1061no—
273C23—1061no—
274C23—1062no—
275C23—1061no—
276C23—1062no—
277C23—1061no—
278C23—1062no—
279C23—1062no—
280C23—1062no—
281C23—1062no—
282C23—1062no—
283C23—1062no—
284C23—1062no—
285C23—1062no—
286C23—2128no—
287C23—2124no—
288C23—2128no—
289C23—2124no—
290C23#1171994912yes—
291C23C23×D5311yescharacteristic
292D4—2128no—
293D4—2124no—
294D4—2128no—
295D4—2128no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—534no—
305C2×C4—532no—
306C2×C4—1062no—
307C2×C4—1062no—
308C2×C4—1062no—
309C2×C4—1061no—
310C2×C4—1062no—
311C2×C4—2124no—
312C2×C4—2124no—
313C2×C4D4×D5314yes—
314C2×C4D4×D5312yes—
315C22—24no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—534no—
321C22—1064no—
322C22—1064no—
323C22—1062no—
324C22—1062no—
325C22—1064no—
326C22—1062no—
327C22—21216no—
328C22—2122no—
329C22—2122no—
330C22—2122no—
331C22—2122no—
332C22—2122no—
333C22—2122no—
334C22—2122no—
335C22—2122no—
336C22—2121no—
337C22—2121no—
338C22C2×D4×D5312yes—
339C22C2×D4×D5311yescharacteristic
340C22#41×D5311yescharacteristic
341C22C2×#1171994911yescharacteristic
342C22#1313326312yes—
343C4—24no—
344C4—22no—
345C4—41no—
346C4—42no—
347C4—1064no—
348C4—1062no—
349C4—2121no—
350C2—42no—
351C2—1064no—
352C2—4242no—
353C2—2122no—
354C2#1681113611yescharacteristic
355C2#127×D5311yescharacteristic
356C2#1681118411yescharacteristic
357C2#1681126412yes—
358C2#1681126512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2539
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133202C2semidirect
2. C2×#13133192C2semidirect
3. C2×#13133187C2semidirect
4. C2×C22⋊C4×D53C2semidirect
5. #16810558C2semidirect
6. #16810533C2semidirect
7. #16810532C2semidirect
8. #16810524C2semidirect
9. #16810523C2semidirect
10. #16810470C2semidirect
11. C53×#399C2semidirect
15. C22×D212C22semidirect
22. C2×#11719919C22semidirect
28. C2×#11719921C22semidirect
29. C2×#11719921C22semidirect
30. C2×#11719921C22semidirect
32. C2×C4×Dic53C22semidirect
33. #13133124C22semidirect
34. C2×C4⋊C4×C53C22semidirect
35. C2×C4⋊C4×C53C22semidirect
38. #95×C53C22semidirect
69. C2×C4×D53D4semidirect
81. #11719921D4semidirect
185. #399D53semidirect
191. C53#2539semidirect
203. C2×C4⋊C4C2×D53semidirect
204. C2×C4⋊C4C2×D53semidirect
218. #95C2×D53semidirect