Finite Group Databasebeta

← #20280599

Subgroups — #20280599

Order 6784.

620 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133191C211yesmaximal, characteristic, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810559C211yesmaximal, characteristic, complemented
4#16810540C211yesmaximal, characteristic, complemented
5#16810538C212yesmaximal, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810531C212yesmaximal, complemented
8#16810514C212yesmaximal, complemented
9#16810502C212yesmaximal, complemented
10#16810486C211yesmaximal, characteristic, complemented
11C2×C4×D4×C53C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C22⋊D53C2212yes—
15#13133191—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2211yescharacteristic
28#13133124C2211yescharacteristic
29C22×D4×C53C2211yescharacteristic
30C23×C4×C53C2211yescharacteristic
31C23×C4×C53C2211yescharacteristic
32C4×D4×C53—24no—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36C2×C42×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C24×C53C2311yescharacteristic
39C23×D53C2312yes—
40C2×D4×C53—22no—
41C2×D4×C53—22no—
42C2×D4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—42no—
45C22×C4×C53—42no—
46C22×C4×C53D414yescomplemented
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2311yescharacteristic
50C22×C4×C53C2311yescharacteristic
51C22×C4×C53C2311yescharacteristic
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×Dic53C2312yes—
59C22×Dic53C2312yes—
60C22×Dic53C2312yes—
61C2×D212—22no—
62C2×D212—42no—
63C4⋊C4×C53—22no—
64C22⋊C4×C53—22no—
65C22⋊C4×C53—22no—
66C42×C53—22no—
67C22⋊Dic53—42no—
68C22⋊Dic53—42no—
69#11719921—24no—
70#11719921—24no—
71#11719921—42no—
72#11719921—42no—
73C4⋊Dic53—22no—
74#11719919—42no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53C2×D412yes—
84C23×C53#4112yes—
85C23×C53C2411yescharacteristic
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93D4×C53—28no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—44no—
103C2×C4×C53—44no—
104C2×C4×C53C2×D412yes—
105C2×C4×C53C2×D412yes—
106C2×C4×C53#4112yes—
107C22⋊D53—48no—
108C22⋊D53—48no—
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—22no—
118D212—44no—
119C22×C53—22no—
120C22×C53—22no—
121C22×C53—22no—
122C22×C53—22no—
123C22×C53—24no—
124C22×C53—41no—
125C22×C53—41no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53C22×D411yescharacteristic
130C22×C53C2×#4111yescharacteristic
131C22×C53C2×#4111yescharacteristic
132C22×C53H(5,𝔽2)12yes—
133C22×C53H(5,𝔽2)12yes—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—42no—
140C2×D53—42no—
141C4×C53—22no—
142C4×C53—22no—
143C4×C53—22no—
144C4×C53—41no—
145C4×C53—41no—
146Dic53—42no—
147Dic53—42no—
148Dic53—42no—
149#2635—531nomaximal, Sylow
150C2×C53—42no—
151C2×C53—42no—
152C2×C53#52511yescharacteristic
153C2×C53#53411yescharacteristic
154C2×C53#53511yescharacteristic
155C2×C53#53812yes—
156C2×C53#54012yes—
157D53—82no—
158C2×#121—531no—
159C2×C4×D4D5311yescharacteristic, complemented
160#400—531no—
161#399—531no—
162#394—531no—
163#394—532no—
164#392—532no—
165#392—531no—
166#388—532no—
167#386—532no—
168#381—531no—
169C53#263511yescharacteristic, complemented, Sylow
170C22×D4—532no—
171C22×D4—532no—
172C22×D4—531no—
173C22×D4C2×D5311yescharacteristic
174C23×C4C2×D5311yescharacteristic
175C23×C4C2×D5311yescharacteristic
176#121—1064no—
177C4×D4—24no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—532no—
180C2×C4⋊C4C2×D5311yescharacteristic
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4C2×D5311yescharacteristic
188C2×C22⋊C4C2×D5311yescharacteristic
189C2×C42C2×D5311yescharacteristic
190#95—532no—
191#95—531no—
192#95—531no—
193#95—532no—
194#95—532no—
195C24—532no—
196C24C22×D5311yescharacteristic
197C24C22×D5311yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—1064no—
202C2×D4—1064no—
203C2×D4—1062no—
204C2×D4—2122no—
205C2×D4—2122no—
206C2×D4—2122no—
207C2×D4—2122no—
208C2×D4—2122no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—532no—
213C22×C4—532no—
214C22×C4—532no—
215C22×C4C22⋊D5314yescomplemented
216C22×C4C22×D5311yescharacteristic
217C22×C4C22×D5311yescharacteristic
218C22×C4C22×D5311yescharacteristic
219C22×C4C22×D5311yescharacteristic
220C22×C4C22×D5311yescharacteristic
221C4⋊C4—22no—
222C4⋊C4—1062no—
223C4⋊C4—2122no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C22⋊C4—2122no—
231C22⋊C4—2122no—
232C42—22no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—1062no—
242C23—1062no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—1062no—
247C23—1062no—
248C23#1171994512yes—
249C23C2×C22⋊D5312yes—
250C23C23×D5311yescharacteristic
251D4—28no—
252D4—2124no—
253D4—2128no—
254D4—2128no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—1062no—
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#1171994512yes—
275C2×C4C2×C22⋊D5312yes—
276C2×C4C2×C22⋊D5312yes—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—24no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—2122no—
288C22—2122no—
289C22—2122no—
290C22—2122no—
291C22—2122no—
292C22—2122no—
293C22—2122no—
294C22C22×C22⋊D5311yescharacteristic
295C22C2×#1171994511yescharacteristic
296C22C2×#1171994511yescharacteristic
297C22#1313326212yes—
298C22#1313326312yes—
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—2122no—
305C4—2122no—
306C4—2122no—
307C2—42no—
308C2—42no—
309C2—4242no—
310C2#1681113411yescharacteristic
311C2#1681118211yescharacteristic
312C2#1681118711yescharacteristic
313C2#1681120612yes—
314C2#1681122412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C4×D4×C53C2
hypercenterC2×C4×D4D53
hypocenterC53#2635
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133191C2semidirect
2. #16810562C2semidirect
3. #16810559C2semidirect
4. #16810540C2semidirect
5. #16810538C2semidirect
6. #16810532C2semidirect
7. #16810531C2semidirect
8. #16810514C2semidirect
9. #16810502C2semidirect
10. #16810486C2semidirect
11. C2×C4×D4×C53C2semidirect
46. C22×C4×C53D4semidirect
159. C2×C4×D4D53semidirect
169. C53#2635semidirect
215. C22×C4C22⋊D53semidirect