Finite Group Databasebeta

← #20280464

Subgroups — #20280464

Order 6784.

728 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 143 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×#13133189C212yesmaximal, complemented
3C2×#13133173C211yesmaximal, characteristic, complemented
4#16810541C211yesmaximal, characteristic, complemented
5#16810538C212yesmaximal, complemented
6#16810537C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic, complemented
8#16810531C212yesmaximal, complemented
9#16810523C212yesmaximal, complemented
10#16810487C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yes—
16C23×Dic53C2211yescharacteristic
17#13133213—24no—
18#13133189—28no—
19#13133173—24no—
20C2×#11719919C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×C4×Dic53C2211yescharacteristic
30C2×C4×Dic53C2211yescharacteristic
31#13133124C2212yes—
32#13133124C2211yescharacteristic
33C22×D4×C53C2212yes—
34C22×D4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2212yes—
37#95×C53C2211yescharacteristic
38C24×C53C2312yes—
39C24×C53C2311yescharacteristic
40C23×D53C2312yes—
41C2×D4×C53—24no—
42C2×D4×C53—22no—
43C2×D4×C53—42no—
44C2×D4×C53—42no—
45C2×D4×C53—42no—
46C22×C4×C53C2312yes—
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—24no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—24no—
52C2×C22⋊D53—24no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—42no—
55C2×C22⋊D53—42no—
56C2×C22⋊D53—42no—
57C2×C22⋊D53—42no—
58C22×Dic53—22no—
59C22×Dic53—22no—
60C22×Dic53D414yescomplemented
61C22×Dic53C2312yes—
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C2×D212—22no—
67C2×D212—42no—
68C22⋊C4×C53—22no—
69C22⋊C4×C53—42no—
70C22⋊Dic53—22no—
71C22⋊Dic53—24no—
72C22⋊Dic53—42no—
73#11719921—24no—
74#11719921—42no—
75#11719921—42no—
76#11719919—22no—
77#11719919—22no—
78#11719919—24no—
79C4×Dic53—22no—
80C4×Dic53D414yescomplemented
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53C2×D412yes—
93C23×C53C2411yescharacteristic
94C22×D53—22no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—22no—
99C22×D53—22no—
100C22×D53—22no—
101D4×C53—44no—
102D4×C53—48no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53C2×D412yes—
110C22⋊D53—48no—
111C22⋊D53—48no—
112C22⋊D53—48no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—22no—
116C2×Dic53—28no—
117C2×Dic53—22no—
118C2×Dic53—21no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53C2×D412yes—
123C2×Dic53C2×D412yes—
124C2×Dic53C2×D412yes—
125C2×Dic53C2×D412yes—
126C2×Dic53#4112yes—
127D212—44no—
128C22×C53—22no—
129C22×C53—24no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53C22×D411yescharacteristic
141C22×C53C22×D411yescharacteristic
142C22×C53C2×#4111yescharacteristic
143C22×C53H(5,𝔽2)12yes—
144C22×C53H(5,𝔽2)12yes—
145C2×D53—42no—
146C2×D53—42no—
147C2×D53—42no—
148C2×D53—42no—
149C2×D53—42no—
150C2×D53—42no—
151C2×D53—42no—
152C4×C53—22no—
153C4×C53—41no—
154C4×C53—42no—
155Dic53—22no—
156Dic53—22no—
157Dic53—41no—
158Dic53—42no—
159Dic53#12114yes—
160#2619—531nomaximal, Sylow
161C2×C53—42no—
162C2×C53—82no—
163C2×C53C2×#12111yescharacteristic
164C2×C53#53511yescharacteristic
165C2×C53#53511yescharacteristic
166C2×C53#53512yes—
167C2×C53#54012yes—
168D53—82no—
169C2×#127—531no—
170C2×#121—532no—
171C2×#117—531no—
172#394—531no—
173#394—532no—
174#394—531no—
175#392—532no—
176#392—531no—
177#392D5311yescharacteristic, complemented
178#390—532no—
179#382—531no—
180C53#261911yescharacteristic, complemented, Sylow
181C22×D4—532no—
182C22×D4—532no—
183C22×D4—532no—
184C22×D4—531no—
185C22×D4C2×D5312yes—
186C22×D4C2×D5311yescharacteristic
187C23×C4—531no—
188#127—1064no—
189#121—1068no—
190#117—1064no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—532no—
194C2×C22⋊C4—531no—
195C2×C22⋊C4—532no—
196C2×C22⋊C4—532no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4—532no—
199C2×C22⋊C4—532no—
200C2×C22⋊C4C2×D5311yescharacteristic
201C2×C22⋊C4C2×D5312yes—
202C2×C42—531no—
203C2×C42—531no—
204#95—532no—
205#95—531no—
206#95C2×D5311yescharacteristic
207C24—532no—
208C24C22×D5312yes—
209C24C22×D5311yescharacteristic
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—1064no—
216C2×D4—1064no—
217C2×D4—1064no—
218C2×D4—1064no—
219C2×D4—1064no—
220C2×D4—1062no—
221C2×D4—2122no—
222C2×D4—2122no—
223C2×D4—2122no—
224C2×D4—2122no—
225C2×D4—2122no—
226C22×C4—531no—
227C22×C4—534no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—532no—
231C22×C4—531no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4C22×D5312yes—
235C22×C4C22×D5311yescharacteristic
236C22×C4C22×D5311yescharacteristic
237C4⋊C4—1062no—
238C4⋊C4—1064no—
239C4⋊C4—1062no—
240C22⋊C4—22no—
241C22⋊C4—42no—
242C22⋊C4—1064no—
243C22⋊C4—1062no—
244C22⋊C4—1064no—
245C22⋊C4—2122no—
246C22⋊C4—2122no—
247C22⋊C4—2122no—
248C42—534no—
249C42—1062no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—1062no—
262C23—1062no—
263C23—1062no—
264C23—1062no—
265C23—1062no—
266C23—1062no—
267C23—1062no—
268C23D4×D5312yes—
269C23C23×D5311yescharacteristic
270D4—44no—
271D4—48no—
272D4—2124no—
273D4—2128no—
274D4—2128no—
275D4—2128no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—532no—
283C2×C4—532no—
284C2×C4—532no—
285C2×C4—532no—
286C2×C4—532no—
287C2×C4—1068no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1062no—
291C2×C4—1061no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4D4×D5312yes—
297C22—22no—
298C22—24no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—2122no—
310C22—2122no—
311C22—2122no—
312C22—2122no—
313C22—2122no—
314C22—2122no—
315C22—2122no—
316C22C2×D4×D5311yescharacteristic
317C22C2×D4×D5311yescharacteristic
318C22#41×D5311yescharacteristic
319C22#1313326212yes—
320C22#1313326212yes—
321C4—22no—
322C4—41no—
323C4—42no—
324C4—534no—
325C4—1062no—
326C4—1062no—
327C4—2121no—
328C4—2122no—
329C2—42no—
330C2—82no—
331C2—4242no—
332C2#121×D5311yescharacteristic
333C2#1681118611yescharacteristic
334C2#1681118811yescharacteristic
335C2#1681118812yes—
336C2#1681122312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2619
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. C2×#13133189C2semidirect
3. C2×#13133173C2semidirect
4. #16810541C2semidirect
5. #16810538C2semidirect
6. #16810537C2semidirect
7. #16810532C2semidirect
8. #16810531C2semidirect
9. #16810523C2semidirect
10. #16810487C2semidirect
11. C53×#392C2semidirect
60. C22×Dic53D4semidirect
80. C4×Dic53D4semidirect
177. #392D53semidirect
180. C53#2619semidirect