Finite Group Databasebeta

← #20280467

Subgroups — #20280467

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×#13133190C212yesmaximal, complemented
3C2×#13133175C211yesmaximal, characteristic, complemented
4#16810540C211yesmaximal, characteristic, complemented
5#16810539C212yesmaximal, complemented
6#16810533C211yesmaximal, characteristic, complemented
7#16810532C212yesmaximal, complemented
8#16810531C211yesmaximal, characteristic, complemented
9#16810523C212yesmaximal, complemented
10#16810487C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yes—
16C22×C22⋊D53C2212yes—
17C22×C4×D53C2211yescharacteristic
18#13133213—24no—
19#13133190—28no—
20#13133175—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×C4×Dic53C2211yescharacteristic
30C2×C4×Dic53C2211yescharacteristic
31#13133124C2211yescharacteristic
32#13133124C2212yes—
33C22×D4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36C2×C22⋊C4×C53C2212yes—
37#95×C53C2211yescharacteristic
38C24×C53C2312yes—
39C23×D53C2312yes—
40C23×D53C2311yescharacteristic
41C2×D4×C53—22no—
42C2×D4×C53—42no—
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—24no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53—42no—
54C2×C22⋊D53—42no—
55C2×C22⋊D53—42no—
56C22×Dic53C2312yes—
57C22×Dic53C2311yescharacteristic
58C22×Dic53C2311yescharacteristic
59C22×Dic53C2311yescharacteristic
60C2×D212—24no—
61C2×D212—22no—
62C2×D212—42no—
63C2×D212—42no—
64C2×D212—42no—
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53D414yescomplemented
68C4⋊C4×C53—22no—
69C22⋊C4×C53—24no—
70C22⋊C4×C53—42no—
71C22⋊Dic53—42no—
72#11719921—22no—
73#11719921—24no—
74#11719921—22no—
75#11719921—42no—
76#11719921—42no—
77C4⋊Dic53—22no—
78#11719919—24no—
79C4×Dic53—22no—
80C4×Dic53D414yescomplemented
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53C2411yescharacteristic
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—22no—
99C22×D53—22no—
100C22×D53C2×D412yes—
101D4×C53—44no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53C2×D412yes—
111C2×C4×C53C2×D412yes—
112C22⋊D53—48no—
113C22⋊D53—48no—
114C22⋊D53—48no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—21no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53C2×D412yes—
122C2×Dic53C2×D412yes—
123C2×Dic53C2×D412yes—
124C2×Dic53#4112yes—
125D212—44no—
126D212—48no—
127C4×D53—28no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53C22×D411yescharacteristic
136C22×C53C22×D411yescharacteristic
137C22×C53C2×#4111yescharacteristic
138C22×C53H(5,𝔽2)12yes—
139C22×C53H(5,𝔽2)12yes—
140C2×D53—22no—
141C2×D53—24no—
142C2×D53—41no—
143C2×D53—41no—
144C2×D53—42no—
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—22no—
154C4×C53—42no—
155C4×C53—41no—
156Dic53—22no—
157Dic53—41no—
158Dic53—42no—
159Dic53#12114yes—
160#2619—531nomaximal, Sylow
161C2×C53—82no—
162C2×C53C2×#12111yescharacteristic
163C2×C53#53511yescharacteristic
164C2×C53#53512yes—
165C2×C53#53511yescharacteristic
166C2×C53#54012yes—
167D53—42no—
168D53—82no—
169C2×#127—531no—
170C2×#121—532no—
171C2×#117—531no—
172#394—531no—
173#394—532no—
174#394D5311yescharacteristic, complemented
175#392—531no—
176#392—532no—
177#392—531no—
178#390—532no—
179#382—531no—
180C53#261911yescharacteristic, complemented, Sylow
181C22×D4—532no—
182C22×D4—532no—
183C22×D4—532no—
184C22×D4—532no—
185C22×D4—531no—
186C22×D4C2×D5311yescharacteristic
187C23×C4—531no—
188#127—1064no—
189#121—1068no—
190#117—1064no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—532no—
193C2×C4⋊C4C2×D5311yescharacteristic
194C2×C22⋊C4—531no—
195C2×C22⋊C4—531no—
196C2×C22⋊C4—532no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4—532no—
199C2×C22⋊C4—532no—
200C2×C22⋊C4C2×D5312yes—
201C2×C22⋊C4C2×D5312yes—
202C2×C42—531no—
203C2×C42—531no—
204#95—532no—
205#95—531no—
206#95C2×D5311yescharacteristic
207C24—531no—
208C24—532no—
209C24C22×D5312yes—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—1064no—
213C2×D4—1064no—
214C2×D4—1064no—
215C2×D4—1064no—
216C2×D4—1064no—
217C2×D4—1064no—
218C2×D4—1062no—
219C2×D4—2122no—
220C2×D4—2122no—
221C2×D4—2122no—
222C2×D4—2122no—
223C2×D4—2122no—
224C2×D4—2122no—
225C2×D4—2122no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—534no—
229C22×C4—532no—
230C22×C4—531no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4C22×D5312yes—
234C22×C4C22×D5311yescharacteristic
235C22×C4C22×D5311yescharacteristic
236C22×C4C22×D5311yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—1062no—
239C4⋊C4—1064no—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—1062no—
243C22⋊C4—1062no—
244C22⋊C4—1064no—
245C22⋊C4—2122no—
246C22⋊C4—2122no—
247C22⋊C4—2122no—
248C42—534no—
249C42—1062no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—532no—
258C23—1061no—
259C23—1061no—
260C23—1062no—
261C23—1062no—
262C23—1062no—
263C23—1062no—
264C23—1062no—
265C23—1062no—
266C23—1062no—
267C23—1062no—
268C23—1062no—
269C23C23×D5311yescharacteristic
270D4—44no—
271D4—2124no—
272D4—2128no—
273D4—2128no—
274D4—2128no—
275D4—2128no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—532no—
285C2×C4—532no—
286C2×C4—532no—
287C2×C4—532no—
288C2×C4—1068no—
289C2×C4—1062no—
290C2×C4—1061no—
291C2×C4—1062no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4D4×D5312yes—
296C2×C4D4×D5312yes—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—1062no—
305C22—1064no—
306C22—2122no—
307C22—2122no—
308C22—2122no—
309C22—2122no—
310C22—2122no—
311C22—2122no—
312C22—2122no—
313C22—2121no—
314C22—2121no—
315C22—2122no—
316C22C2×D4×D5311yescharacteristic
317C22C2×D4×D5311yescharacteristic
318C22#41×D5311yescharacteristic
319C22#1313326212yes—
320C22#1313326312yes—
321C4—22no—
322C4—22no—
323C4—41no—
324C4—42no—
325C4—534no—
326C4—1062no—
327C4—2122no—
328C4—2121no—
329C2—82no—
330C2—4242no—
331C2—2122no—
332C2#121×D5311yescharacteristic
333C2#1681118811yescharacteristic
334C2#1681118912yes—
335C2#1681119111yescharacteristic
336C2#1681122512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2619
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. C2×#13133190C2semidirect
3. C2×#13133175C2semidirect
4. #16810540C2semidirect
5. #16810539C2semidirect
6. #16810533C2semidirect
7. #16810532C2semidirect
8. #16810531C2semidirect
9. #16810523C2semidirect
10. #16810487C2semidirect
11. C53×#394C2semidirect
67. C2×C4×D53D4semidirect
80. C4×Dic53D4semidirect
174. #394D53semidirect
180. C53#2619semidirect