Finite Group Databasebeta

← #20280647

Subgroups — #20280647

Order 6784.

576 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133195C211yesmaximal, characteristic, complemented
2C2×#13133192C211yesmaximal, characteristic
3#16810561C211yesmaximal, characteristic, complemented
4#16810552C212yesmaximal, complemented
5#16810541C211yesmaximal, characteristic, complemented
6#16810540C211yesmaximal, characteristic, complemented
7#16810536C212yesmaximal, complemented
8#16810501C212yesmaximal, complemented
9#16810501C212yesmaximal, complemented
10#95×D53C211yesmaximal, characteristic, complemented
11C53×#400C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×D212C2212yes—
14C22×D212C2211yescharacteristic
15C22×C4×D53C2212yes—
16C22×C4×D53C2211yescharacteristic
17#13133195—24no—
18#13133192—24no—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34#95×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2312yes—
38C23×D53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×D212—24no—
49C2×D212—24no—
50C2×D212—24no—
51C2×D212—22no—
52C2×D212—42no—
53C2×C4×D53—24no—
54C2×C4×D53—42no—
55C2×C4×D53—42no—
56C2×C4×D53—42no—
57C2×C4×D53D414yes—
58C2×Dic106—22no—
59C2×Dic106—42no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—22no—
62#11719921—22no—
63#11719921—22no—
64#11719921—24no—
65#11719921—42no—
66#11719921—42no—
67#11719921—42no—
68#11719921—42no—
69#11719919—22no—
70#11719919—22no—
71C23×C53C2411yescharacteristic
72C22×D53—22no—
73C22×D53—21no—
74C22×D53—21no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—22no—
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—22no—
82C22×D53C2×D412yes—
83C22×D53#4114yes—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53C2×D412yes—
97C2×Dic53—22no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53C2×D412yes—
107D212—44no—
108D212—48no—
109C4×D53—44no—
110C4×D53—48no—
111Dic106—44no—
112C22×C53C22×D411yescharacteristic
113C22×C53C2×#4112yes—
114C22×C53H(5,𝔽2)12yes—
115C22×C53H(5,𝔽2)11yescharacteristic
116C22×C532−1+411yescharacteristic
117C2×D53—22no—
118C2×D53—22no—
119C2×D53—24no—
120C2×D53—24no—
121C2×D53—42no—
122C2×D53—42no—
123C2×D53—42no—
124C2×D53—42no—
125C2×D53—42no—
126C2×D53—42no—
127C2×D53—42no—
128C2×D53#12414yes—
129C4×C53—22no—
130C4×C53—42no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—42no—
134Dic53—22no—
135Dic53—41no—
136Dic53—41no—
137Dic53—42no—
138#2639—531nomaximal, Sylow
139C2×C53C2×#12411yescharacteristic
140C2×C53#53411yescharacteristic
141C2×C53#53711yescharacteristic
142C2×C53#53812yes—
143C2×C53#53912yes—
144D53—24no—
145D53—82no—
146C2×C22⋊Q8—531no—
147C2×#121—531no—
148#400—531no—
149#400D5311yescharacteristic, complemented
150#397—532no—
151#394—531no—
152#394—531no—
153#393—532no—
154#386—532no—
155#386—532no—
156C2×#95—531no—
157C53#263911yescharacteristic, complemented, Sylow
158C22×Q8—531no—
159C22×D4—532no—
160C22×D4—531no—
161C23×C4—531no—
162C23×C4—532no—
163C22⋊Q8—1064no—
164#121—1064no—
165C2×C4⋊C4—531no—
166C2×C4⋊C4—531no—
167C2×C4⋊C4C2×D5311yescharacteristic
168C2×C4⋊C4C2×D5311yescharacteristic
169C2×C22⋊C4—532no—
170C2×C22⋊C4—531no—
171C2×C22⋊C4—531no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4—532no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176#95—532no—
177#95—531no—
178#95—532no—
179#95—532no—
180#95C2×D5311yescharacteristic
181#95C2×D5312yes—
182#95C2×D5312yes—
183C24—531no—
184C24—532no—
185C2×Q8—1062no—
186C2×Q8—2122no—
187C2×D4—1064no—
188C2×D4—1064no—
189C2×D4—1064no—
190C2×D4—1062no—
191C2×D4—2122no—
192C22×C4—531no—
193C22×C4—534no—
194C22×C4—532no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—1064no—
198C22×C4—2122no—
199C22×C4—2122no—
200C22×C4—2122no—
201C22×C4C22×D5312yes—
202C22×C4C22×D5312yes—
203C22×C4C22×D5311yescharacteristic
204C22×C4C22×D5311yescharacteristic
205C22×C4C22×D5311yescharacteristic
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—1062no—
209C4⋊C4—1062no—
210C22⋊C4—1064no—
211C22⋊C4—1062no—
212C22⋊C4—1062no—
213C22⋊C4—2122no—
214C22⋊C4—2122no—
215C22⋊C4—2122no—
216C22⋊C4—2122no—
217C23—532no—
218C23—534no—
219C23—1062no—
220C23—1061no—
221C23—1061no—
222C23—1062no—
223C23—1062no—
224C23—1062no—
225C23—1062no—
226C23—1062no—
227C23—1062no—
228C23—1062no—
229C23C23×D5311yescharacteristic
230Q8—2124no—
231D4—2124no—
232D4—2128no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—532no—
246C2×C4—1062no—
247C2×C4—1061no—
248C2×C4—1061no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1061no—
255C2×C4—2124no—
256C2×C4—2128no—
257C2×C4D4×D5312yes—
258C22—534no—
259C22—1062no—
260C22—1062no—
261C22—1064no—
262C22—1064no—
263C22—2122no—
264C22—2122no—
265C22—2122no—
266C22—2122no—
267C22—2122no—
268C22—2122no—
269C22—2122no—
270C22C2×D4×D5311yescharacteristic
271C22#41×D5312yes—
272C22#1313326312yes—
273C22#1313326311yescharacteristic
274C22#1313326511yescharacteristic
275C4—22no—
276C4—42no—
277C4—41no—
278C4—41no—
279C4—42no—
280C4—1062no—
281C4—2121no—
282C4—2121no—
283C4—2122no—
284C2—1064no—
285C2—4242no—
286C2#124×D5311yescharacteristic
287C2#1681118411yescharacteristic
288C2#1681120311yescharacteristic
289C2#1681121212yes—
290C2#1681122112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2639
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#13133195C2semidirect
3. #16810561C2semidirect
4. #16810552C2semidirect
5. #16810541C2semidirect
6. #16810540C2semidirect
7. #16810536C2semidirect
8. #16810501C2semidirect
9. #16810501C2semidirect
10. #95×D53C2semidirect
11. C53×#400C2semidirect
149. #400D53semidirect
157. C53#2639semidirect