Finite Group Databasebeta

← #20273229

Subgroups — #20273229

Order 6784.

346 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#121×D53C211yesmaximal, characteristic
2#16810585C211yesmaximal, characteristic, complemented
3#16810470C211yesmaximal, characteristic, complemented
4#16810358C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic, complemented
6#16810356C211yesmaximal, characteristic, complemented
7C53×#327C211yesmaximal, characteristic, complemented
8C2×D4×D53—21no—
9C2×D4×D53C411yescharacteristic, complemented
10C22×C4×D53C2211yescharacteristic
11C2×#11719912—21no—
12#13133192C411yescharacteristic, complemented
13#13133191C2211yescharacteristic
14#13133190—21no—
15#13133189—21no—
16#13133188C411yescharacteristic, complemented
17C4⋊C4×D53C411yescharacteristic, complemented
18C2×#11719921—21no—
19C22⋊C4×D53—21no—
20C2×C4×Dic53—21no—
21C22⋊#88322C2211yescharacteristic, complemented
22#13133129—21no—
23#13133124C2211yescharacteristic, complemented
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic, complemented
26#95×C53C2211yescharacteristic, complemented
27C23×D53—21no—
28C23×D53D411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53C2×C411yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53C2311yescharacteristic
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35C2×C22⋊D53—21no—
36C22×Dic53—21no—
37C22×Dic53D411yescharacteristic
38D4×D53—42no—
39D4×D53—42no—
40D4×D53—42no—
41C2×D212—21no—
42C2×D212C2×C411yescharacteristic
43C2×C4×D53—21no—
44C2×C4×D53—42no—
45C2×C4×D53C2×C411yescharacteristic
46C2×C4×D53C2×C411yescharacteristic
47C2×C4×D53D411yescharacteristic
48C2×C4×D53D411yescharacteristic
49C2×C8×C53—21no—
50C4⋊C4×C53C2×C411yescharacteristic
51C22⋊C4×C53—21no—
52C22⋊Dic53—21no—
53#11719921—24no—
54#11719921—21no—
55#11719921—21no—
56C4⋊Dic53C2×C411yescharacteristic
57#11719919—21no—
58C4×Dic53—42no—
59C2×#88322—21no—
60#11719912—42no—
61C23×C53—21no—
62C23×C53C2×D411yescharacteristic
63C22×D53—21no—
64C22×D53—21no—
65C22×D53—22no—
66C22×D53—21no—
67C22×D53—41no—
68C22×D53—42no—
69C22×D53—42no—
70C22×D53—42no—
71C22×D53C22⋊C411yescharacteristic
72C22×D53C22⋊C411yescharacteristic
73D4×C53—41no—
74D4×C53—41no—
75D4×C53—41no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53C22×C411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C22⋊D53—42no—
84C22⋊D53—41no—
85C22⋊D53—41no—
86C22⋊D53—42no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53—41no—
91C2×Dic53—41no—
92C2×Dic53C22⋊C411yescharacteristic
93C2×Dic53C22⋊C411yescharacteristic
94D212—41no—
95D212—42no—
96C4×D53—22no—
97C4×D53—22no—
98C4×D53—41no—
99C4×D53—41no—
100C4×D53—42no—
101C8×C53—41no—
102#88322—41no—
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53C2×C22⋊C411yescharacteristic
109C22×C53AGL(1,ℤ/8)11yescharacteristic
110C22×C53AGL(1,ℤ/8)11yescharacteristic
111C2×D53—21no—
112C2×D53—21no—
113C2×D53—21no—
114C2×D53—21no—
115C2×D53—21no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—41no—
119C2×D53—41no—
120C2×D53—41no—
121C2×D53—84no—
122C2×D53#10412yes—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—42no—
126C4×C53—41no—
127Dic53—21no—
128Dic53—42no—
129Dic53—41no—
130Dic53#9912yes—
131#1507—531nomaximal, Sylow
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—81no—
135C2×C53C2×#9911yescharacteristic
136C2×C53#41811yescharacteristic
137C2×C53C2×#10411yescharacteristic
138D53—22no—
139D53—41no—
140D53—81no—
141C2×#121—531no—
142#407—531no—
143C4×C22⋊C4—531no—
144#327—531no—
145#327—531no—
146#327—531no—
147#327D5311yescharacteristic, complemented
148C53#150711yescharacteristic, complemented, Sylow
149C22×D4—531no—
150C22×D4—1061no—
151C23×C4—531no—
152C2×M16—1061no—
153#121—531no—
154#121—531no—
155#121—531no—
156#121—1061no—
157#121—1061no—
158#121C2×D5311yescharacteristic
159C2×C4⋊C4—531no—
160C2×C22⋊C4—1061no—
161C2×C22⋊C4—1061no—
162C2×C42—1061no—
163C22⋊C8—531no—
164C22⋊C8—1061no—
165C22⋊C8C2×D5311yescharacteristic, complemented
166#95—531no—
167#95C2×D5311yescharacteristic, complemented
168C24—531no—
169C24—1061no—
170C2×D4—21no—
171C2×D4—531no—
172C2×D4—1061no—
173C2×D4—1061no—
174C2×D4—1061no—
175C2×D4—1061no—
176C2×D4—2122no—
177C2×D4—2122no—
178C2×D4—2122no—
179C2×D4C4×D5311yescharacteristic
180C22×C4—21no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—1061no—
187C22×C4—1061no—
188C22×C4—2122no—
189C22×C4C22×D5311yescharacteristic
190M16—2122no—
191C2×C8—21no—
192C2×C8—1061no—
193C4⋊C4—531no—
194C4⋊C4—1061no—
195C4⋊C4C4×D5311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—1061no—
198C22⋊C4—1064no—
199C22⋊C4—1061no—
200C22⋊C4—1061no—
201C42—2122no—
202C23—21no—
203C23—531no—
204C23—531no—
205C23—1062no—
206C23—1061no—
207C23—1061no—
208C23—1061no—
209C23—2121no—
210C23—2122no—
211C23—2122no—
212C23—2122no—
213C23D4×D5311yescharacteristic
214D4—41no—
215D4—41no—
216D4—41no—
217D4—2121no—
218D4—2122no—
219D4—2122no—
220D4—2122no—
221D4—2121no—
222D4—2121no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—531no—
229C2×C4—531no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1061no—
235C2×C4—2121no—
236C2×C4—2121no—
237C2×C4—2122no—
238C2×C4—2121no—
239C2×C4—2121no—
240C2×C4C2×C4×D5311yescharacteristic
241C2×C4D4×D5311yescharacteristic
242C8—41no—
243C8—2121no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—532no—
250C22—1061no—
251C22—1061no—
252C22—1061no—
253C22—1061no—
254C22—1061no—
255C22—4244no—
256C22—2122no—
257C22—2122no—
258C22—2121no—
259C22—2121no—
260C22—2121no—
261C22C22⋊C4×D5311yescharacteristic
262C22#1313324111yescharacteristic
263C22#1313324411yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—41no—
268C4—532no—
269C4—1061no—
270C4—2122no—
271C4—2121no—
272C2—21no—
273C2—21no—
274C2—81no—
275C2—1062no—
276C2—4241no—
277C2—2121no—
278C2#99×D5311yescharacteristic
279C2#1681062611yescharacteristic
280C2#104×D5311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53C2×C22⋊C4
hypercenter#327D53
hypocenterC53#1507
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #16810585C2semidirect
3. #16810470C2semidirect
4. #16810358C2semidirect
5. #16810357C2semidirect
6. #16810356C2semidirect
7. C53×#327C2semidirect
9. C2×D4×D53C4semidirect
12. #13133192C4semidirect
16. #13133188C4semidirect
17. C4⋊C4×D53C4semidirect
21. C22⋊#88322C22semidirect
23. #13133124C22semidirect
25. C22⋊C8×C53C22semidirect
26. #95×C53C22semidirect
147. #327D53semidirect
148. C53#1507semidirect
165. C22⋊C8C2×D53semidirect
167. #95C2×D53semidirect