Finite Group Databasebeta

← #20280600

Subgroups — #20280600

Order 6784.

620 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 137 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×C4×C22⋊D53C211yesmaximal, characteristic, complemented
2#16810561C211yesmaximal, characteristic, complemented
3#16810558C211yesmaximal, characteristic, complemented
4#16810537C212yesmaximal, complemented
5#16810537C211yesmaximal, characteristic, complemented
6#16810531C212yesmaximal, complemented
7#16810531C211yesmaximal, characteristic, complemented
8#16810511C212yesmaximal, complemented
9#16810499C212yesmaximal, complemented
10#16810486C211yesmaximal, characteristic, complemented
11C2×#121×C53C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C4×D53C2211yescharacteristic
15C4×C22⋊D53—24no—
16C2×#11719919C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2212yes—
25C2×C4×Dic53C2211yescharacteristic
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2211yescharacteristic
31C22×D4×C53C2212yes—
32C22×D4×C53C2211yescharacteristic
33C23×C4×C53C2211yescharacteristic
34#121×C53—24no—
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2212yes—
37C24×C53C2312yes—
38C24×C53C2311yescharacteristic
39C23×D53C2311yescharacteristic
40C2×D4×C53—24no—
41C2×D4×C53—22no—
42C2×D4×C53—42no—
43C2×D4×C53—42no—
44C2×D4×C53—42no—
45C22×C4×C53—42no—
46C22×C4×C53D414yescomplemented
47C22×C4×C53C2312yes—
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2311yescharacteristic
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—24no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—42no—
55C2×C22⋊D53—42no—
56C22×Dic53C2312yes—
57C22×Dic53C2312yes—
58C22×Dic53C2311yescharacteristic
59C22×Dic53C2311yescharacteristic
60C22×Dic53C2311yescharacteristic
61C2×C4×D53—22no—
62C2×C4×D53—42no—
63C4⋊C4×C53—22no—
64C22⋊C4×C53—24no—
65C22⋊Dic53—22no—
66C22⋊Dic53—24no—
67C22⋊Dic53—42no—
68C22⋊Dic53—42no—
69C22⋊Dic53—42no—
70#11719921—22no—
71#11719921—42no—
72#11719919—22no—
73#11719919—42no—
74C4×Dic53—22no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53C2×D412yes—
87C23×C53C2411yescharacteristic
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53#4112yes—
93D4×C53—44no—
94D4×C53—48no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—44no—
101C2×C4×C53C2×D412yes—
102C2×C4×C53C2×D412yes—
103C22⋊D53—28no—
104C22⋊D53—48no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53#4112yes—
118C4×D53—44no—
119C22×C53—22no—
120C22×C53—22no—
121C22×C53—22no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53C22×D411yescharacteristic
132C22×C53C2×#4111yescharacteristic
133C22×C53C2×#4111yescharacteristic
134C22×C53H(5,𝔽2)12yes—
135C22×C53H(5,𝔽2)12yes—
136C2×D53—22no—
137C2×D53—24no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—42no—
141C4×C53—22no—
142C4×C53—22no—
143C4×C53—42no—
144Dic53—22no—
145Dic53—41no—
146Dic53—41no—
147Dic53—42no—
148Dic53—42no—
149#2635—531nomaximal, Sylow
150C2×C53—42no—
151C2×C53—82no—
152C2×C53#52511yescharacteristic
153C2×C53#53411yescharacteristic
154C2×C53#53511yescharacteristic
155C2×C53#53812yes—
156C2×C53#54012yes—
157D53—42no—
158C2×#121D5311yescharacteristic, complemented
159C2×C4×D4—531no—
160#400—531no—
161#399—531no—
162#394—531no—
163#394—532no—
164#392—531no—
165#392—532no—
166#388—532no—
167#386—532no—
168#381—531no—
169C53#263511yescharacteristic, complemented, Sylow
170C22×D4—531no—
171C22×D4—532no—
172C22×D4C2×D5312yes—
173C22×D4C2×D5311yescharacteristic
174C23×C4—531no—
175C23×C4C2×D5311yescharacteristic
176#121—24no—
177C4×D4—1064no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—532no—
180C2×C4⋊C4C2×D5311yescharacteristic
181C2×C22⋊C4—531no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—531no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4—532no—
188C2×C22⋊C4C2×D5312yes—
189C2×C42—531no—
190#95—532no—
191#95—531no—
192#95—531no—
193#95—532no—
194#95—532no—
195C24—531no—
196C24C22×D5312yes—
197C24C22×D5311yescharacteristic
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—1062no—
204C2×D4—1062no—
205C2×D4—1062no—
206C2×D4—1064no—
207C2×D4—2122no—
208C2×D4—2122no—
209C22×C4—42no—
210C22×C4—531no—
211C22×C4—532no—
212C22×C4—532no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—1062no—
216C22×C4—2122no—
217C22×C4C22⋊D5314yescomplemented
218C22×C4C22×D5312yes—
219C22×C4C22×D5311yescharacteristic
220C22×C4C22×D5311yescharacteristic
221C4⋊C4—22no—
222C4⋊C4—1062no—
223C4⋊C4—2122no—
224C22⋊C4—24no—
225C22⋊C4—1064no—
226C22⋊C4—1062no—
227C22⋊C4—1062no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C22⋊C4—2122no—
231C22⋊C4—2122no—
232C42—1062no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—532no—
245C23—1061no—
246C23—1061no—
247C23—1062no—
248C23—1062no—
249C23C2×C22⋊D5312yes—
250C23C23×D5311yescharacteristic
251D4—44no—
252D4—48no—
253D4—1068no—
254D4—2128no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—44no—
261C2×C4—532no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—2124no—
275C2×C4C2×C22⋊D5312yes—
276C2×C4C2×C22⋊D5312yes—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—1062no—
290C22—1064no—
291C22—2121no—
292C22—2121no—
293C22—2122no—
294C22C22×C22⋊D5311yescharacteristic
295C22#41×D5311yescharacteristic
296C22#41×D5311yescharacteristic
297C22#1313326212yes—
298C22#1313326212yes—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—1062no—
303C4—2121no—
304C4—2121no—
305C4—2122no—
306C4—2122no—
307C2—42no—
308C2—82no—
309C2—2122no—
310C2#1681113011yescharacteristic
311C2#1681118211yescharacteristic
312C2#1681119011yescharacteristic
313C2#1681120512yes—
314C2#1681122312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2635
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊D53C2semidirect
2. #16810561C2semidirect
3. #16810558C2semidirect
4. #16810537C2semidirect
5. #16810537C2semidirect
6. #16810531C2semidirect
7. #16810531C2semidirect
8. #16810511C2semidirect
9. #16810499C2semidirect
10. #16810486C2semidirect
11. C2×#121×C53C2semidirect
46. C22×C4×C53D4semidirect
158. C2×#121D53semidirect
169. C53#2635semidirect
217. C22×C4C22⋊D53semidirect