Finite Group Databasebeta

← #20273255

Subgroups — #20273255

Order 6784.

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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#16811133C211yesmaximal, characteristic
2#16810585C211yesmaximal, characteristic, complemented
3#16810482C211yesmaximal, characteristic, complemented
4#16810361C211yesmaximal, characteristic, complemented
5#16810359C211yesmaximal, characteristic, complemented
6#16810357C211yesmaximal, characteristic, complemented
7C53×#327C211yesmaximal, characteristic, complemented
8C2×#11719947C411yescharacteristic, complemented
9C22×C4×D53C2211yescharacteristic
10C2×#11719912—21no—
11#13133199—21no—
12#13133197C411yescharacteristic, complemented
13#13133196C2211yescharacteristic
14#13133188C411yescharacteristic, complemented
15#13133185—21no—
16D4×Dic53—21no—
17#13133177—21no—
18#13133175C411yescharacteristic, complemented
19C2×#11719919—21no—
20C2×#11719921—21no—
21C22⋊#88322C2211yescharacteristic
22#13133129—21no—
23#13133124C2211yescharacteristic
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C23×D53D411yescharacteristic
28C2×D4×C53—21no—
29C2×D4×C53C2×C411yescharacteristic
30C22×C4×C53—21no—
31C22×C4×C53C2311yescharacteristic
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C22×Dic53—21no—
35C22×Dic53—21no—
36C22×Dic53D411yescharacteristic
37#11719947—42no—
38C2×C4×D53—42no—
39C2×C4×D53C2×C411yescharacteristic
40C2×C4×D53C2×C411yescharacteristic
41C2×C4×D53D411yescharacteristic
42C2×C4×D53D411yescharacteristic
43C2×Dic106C2×C411yescharacteristic
44C2×C8×C53—21no—
45C4⋊C4×C53C2×C411yescharacteristic
46C22⋊C4×C53—21no—
47C22⋊Dic53—21no—
48C22⋊Dic53—21no—
49#11719921—21no—
50#11719921—21no—
51#11719921—42no—
52C4⋊Dic53—21no—
53C4⋊Dic53C2×C411yescharacteristic
54#11719919—21no—
55#11719919—42no—
56C4×Dic53—21no—
57C2×#88322—21no—
58#11719912—42no—
59C23×C53—21no—
60C23×C53C2×D411yescharacteristic
61C22×D53—21no—
62C22×D53—21no—
63C22×D53—41no—
64C22×D53—41no—
65C22×D53C22⋊C411yescharacteristic
66C22×D53C22⋊C411yescharacteristic
67D4×C53—41no—
68D4×C53—41no—
69D4×C53—41no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53C22×C411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C22⋊D53—42no—
78C22⋊D53—42no—
79C2×Dic53—22no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—42no—
83C2×Dic53—41no—
84C2×Dic53—41no—
85C2×Dic53—41no—
86C2×Dic53C22⋊C411yescharacteristic
87C2×Dic53C22⋊C411yescharacteristic
88C4×D53—22no—
89C4×D53—22no—
90C4×D53—41no—
91C4×D53—41no—
92Dic106—41no—
93C8×C53—41no—
94#88322—41no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53C2×C22⋊C411yescharacteristic
101C22×C53AGL(1,ℤ/8)11yescharacteristic
102C22×C53#13711yescharacteristic
103C2×D53—21no—
104C2×D53—21no—
105C2×D53—21no—
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—21no—
109C2×D53—42no—
110C2×D53—42no—
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—41no—
114C4×C53—81no—
115Dic53—21no—
116Dic53—41no—
117Dic53—41no—
118Dic53—81no—
119Dic53#9912yes—
120#1513—531nomaximal, Sylow
121C2×C53—21no—
122C2×C53—21no—
123C2×C53—81no—
124C2×C53C2×#9911yescharacteristic
125C2×C53#41711yescharacteristic
126C2×C53#42111yescharacteristic
127D53—41no—
128D53—41no—
129#525—531no—
130#407—531no—
131#380—531no—
132#328—531no—
133#328—531no—
134#327—531no—
135#327D5311yescharacteristic, complemented
136C53#151311yescharacteristic, complemented, Sylow
137C2×#41—531no—
138C23×C4—531no—
139C2×M16—1061no—
140#123—1061no—
141C22⋊Q8—531no—
142C22⋊Q8—531no—
143#121—531no—
144#121C2×D5311yescharacteristic
145#120—1061no—
146C4×D4—1061no—
147C4×D4—1061no—
148#117—531no—
149C2×C4⋊C4—1061no—
150C2×C22⋊C4—1061no—
151C22⋊C8—531no—
152C22⋊C8—1061no—
153C22⋊C8C2×D5311yescharacteristic
154#95—531no—
155#95C2×D5311yescharacteristic
156C24—531no—
157#41—2122no—
158C2×Q8—531no—
159C2×D4—21no—
160C2×D4—1061no—
161C2×D4—1061no—
162C2×D4C4×D5311yescharacteristic
163C22×C4—21no—
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4—1061no—
170C22×C4—1061no—
171C22×C4—2122no—
172C22×C4C22×D5311yescharacteristic
173M16—2122no—
174C2×C8—21no—
175C2×C8—1061no—
176C4⋊C4—531no—
177C4⋊C4—1061no—
178C4⋊C4—1061no—
179C4⋊C4—2122no—
180C4⋊C4C4×D5311yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—1061no—
183C22⋊C4—1061no—
184C22⋊C4—1061no—
185C22⋊C4—1061no—
186C22⋊C4—2122no—
187C42—1061no—
188C23—21no—
189C23—531no—
190C23—531no—
191C23—1061no—
192C23—1061no—
193C23—2121no—
194C23—2121no—
195C23D4×D5311yescharacteristic
196Q8—2121no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—2122no—
201D4—2122no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—531no—
208C2×C4—531no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—2121no—
215C2×C4—2121no—
216C2×C4—2122no—
217C2×C4—2121no—
218C2×C4—2121no—
219C2×C4—2121no—
220C2×C4C2×C4×D5311yescharacteristic
221C2×C4D4×D5311yescharacteristic
222C8—41no—
223C8—2121no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—1061no—
230C22—1061no—
231C22—1061no—
232C22—1061no—
233C22—1061no—
234C22—1061no—
235C22—2122no—
236C22—2122no—
237C22C22⋊C4×D5311yescharacteristic
238C22#1313324111yescharacteristic
239C22#1313324611yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—81no—
244C4—532no—
245C4—1061no—
246C4—4241no—
247C4—2121no—
248C4—2121no—
249C2—21no—
250C2—21no—
251C2—81no—
252C2—2121no—
253C2—2121no—
254C2#99×D5311yescharacteristic
255C2#1681062111yescharacteristic
256C2#1681063811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53C2×C22⋊C4
hypercenter#327D53
hypocenterC53#1513
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. #16810482C2semidirect
4. #16810361C2semidirect
5. #16810359C2semidirect
6. #16810357C2semidirect
7. C53×#327C2semidirect
8. C2×#11719947C4semidirect
12. #13133197C4semidirect
14. #13133188C4semidirect
18. #13133175C4semidirect
135. #327D53semidirect
136. C53#1513semidirect