Finite Group Databasebeta

← #20276358

Subgroups — #20276358

Order 6784.

280 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811183C211yesmaximal, characteristic
2#100×D53C211yesmaximal, characteristic, complemented
3#99×D53C211yesmaximal, characteristic, complemented
4#16810414C211yesmaximal, characteristic, complemented
5#16810412C211yesmaximal, characteristic, complemented
6#16810410C211yesmaximal, characteristic, complemented
7C53×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8C22×C22⋊D53C411yescharacteristic, complemented
9C2×D4×D53C2211yescharacteristic
10M16×D53—21no—
11#13133201C411yescharacteristic, complemented
12#13133199—21no—
13#13133198C2211yescharacteristic
14#13133190—21no—
15#13133188—21no—
16#13133186C411yescharacteristic, complemented
17#13133185—21no—
18C22⋊C4×D53—21no—
19C22⋊C4×D53C411yescharacteristic, complemented
20#13133134C2211yescharacteristic
21#13133133—21no—
22#13133132C2211yescharacteristic
23#13133131—21no—
24#120×C53C2211yescharacteristic
25#100×C53C2211yescharacteristic
26#99×C53C2211yescharacteristic
27C24×C53C2×C411yescharacteristic, complemented
28C23×D53C2×C411yescharacteristic
29C23×D53D411yescharacteristic
30C2×D4×C53—21no—
31C2×D4×C53C2311yescharacteristic
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35C2×C22⋊D53—41no—
36C2×C22⋊D53—41no—
37C2×C22⋊D53C2×C411yescharacteristic
38C2×C22⋊D53D411yescharacteristic
39C22×Dic53C2×C411yescharacteristic, complemented
40D4×D53—41no—
41D4×D53—41no—
42C2×D212D411yescharacteristic
43C2×C4×D53—21no—
44C2×C4×D53—41no—
45C2×C4×D53D411yescharacteristic
46M16×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53C2×C411yescharacteristic, complemented
50C22⋊Dic53—21no—
51C22⋊Dic53—21no—
52C22⋊Dic53—21no—
53C22⋊Dic53C2×C411yescharacteristic, complemented
54#11719921—21no—
55#11719921—21no—
56#11719921—41no—
57C4⋊Dic53—21no—
58#11719919—21no—
59#11719917—21no—
60#11719912—41no—
61C8×D53—41no—
62C23×C53—21no—
63C23×C53—41no—
64C23×C53—81no—
65C23×C53C22×C411yescharacteristic
66C23×C53C2×D411yescharacteristic
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—21no—
70C22×D53—21no—
71C22×D53—81no—
72C22×D53—81no—
73C22×D53C22⋊C411yescharacteristic
74C22×D53C22⋊C411yescharacteristic
75C22×D53C22⋊C411yescharacteristic
76D4×C53—41no—
77D4×C53—81no—
78C2×C4×C53—21no—
79C2×C4×C53—41no—
80C2×C4×C53C2×D411yescharacteristic
81C22⋊D53—41no—
82C22⋊D53—41no—
83C22⋊D53—42no—
84C22⋊D53—41no—
85C22⋊D53—41no—
86C22⋊D53—81no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—41no—
90C2×Dic53—41no—
91C2×Dic53C22⋊C411yescharacteristic
92D212—41no—
93C4×D53—21no—
94C4×D53—21no—
95C4×D53—81no—
96C4×D53—82no—
97C8×C53—41no—
98#88322—41no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—82no—
107C22×C53C2×C22⋊C411yescharacteristic
108C2×D53—21no—
109C2×D53—21no—
110C2×D53—41no—
111C2×D53—42no—
112C2×D53—41no—
113C2×D53—41no—
114C2×D53—81no—
115C2×D53—81no—
116C2×D53—81no—
117C2×D53#9911yescharacteristic
118C2×D53#9911yescharacteristic
119C4×C53—21no—
120C4×C53—41no—
121C4×C53—81no—
122Dic53—21no—
123Dic53—41no—
124Dic53—41no—
125Dic53—81no—
126#2121—531nomaximal, Sylow
127C2×C53—21no—
128C2×C53—41no—
129C2×C53—41no—
130C2×C53—81no—
131C2×C53C2×#9911yescharacteristic
132D53—21no—
133D53—21no—
134D53—41no—
135D53—41no—
136#534—531no—
137C2×#100—531no—
138C2×#99—531no—
139#352—531no—
140#352—531no—
141AΣL(1,𝔽16)—531no—
142AΣL(1,𝔽16)D5311yescharacteristic, complemented
143C53#212111yescharacteristic, complemented, Sylow
144C22×D4—531no—
145C22×D4—531no—
146C2×M16—1061no—
147#123—531no—
148#123—531no—
149#123—1061no—
150#121—1061no—
151#121—1061no—
152#120—531no—
153#120—1061no—
154#120C2×D5311yescharacteristic
155C2×C22⋊C4—531no—
156C2×C22⋊C4—1061no—
157#100—531no—
158#100—1061no—
159#100C2×D5311yescharacteristic
160#99—531no—
161#99—1061no—
162#99C2×D5311yescharacteristic
163C24—531no—
164C24—531no—
165C24C4×D5311yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—531no—
168C2×D4—531no—
169C2×D4—531no—
170C2×D4—1061no—
171C2×D4—1061no—
172C2×D4—1061no—
173C2×D4—2121no—
174C2×D4—2121no—
175C2×D4—2121no—
176C2×D4—2121no—
177C2×D4C22×D5311yescharacteristic
178C22×C4—531no—
179C22×C4—531no—
180C22×C4—1061no—
181C22×C4—2121no—
182M16—21no—
183M16—1061no—
184M16—2121no—
185C2×C8—2121no—
186C4⋊C4—1061no—
187C4⋊C4—1061no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—531no—
191C22⋊C4—1061no—
192C22⋊C4—1061no—
193C22⋊C4—1061no—
194C22⋊C4—1061no—
195C22⋊C4—1061no—
196C22⋊C4—2121no—
197C22⋊C4C4×D5311yescharacteristic, complemented
198C23—21no—
199C23—41no—
200C23—81no—
201C23—531no—
202C23—531no—
203C23—531no—
204C23—1061no—
205C23—1061no—
206C23—1061no—
207C23—1061no—
208C23—4241no—
209C23—4241no—
210C23C2×C4×D5311yescharacteristic
211C23D4×D5311yescharacteristic
212D4—41no—
213D4—81no—
214D4—4241no—
215D4—2121no—
216D4—2121no—
217D4—2121no—
218D4—2121no—
219D4—2121no—
220D4—2122no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—531no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—4241no—
229C2×C4—4242no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4D4×D5311yescharacteristic
233C8—41no—
234C8—2121no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—82no—
243C22—531no—
244C22—531no—
245C22—1061no—
246C22—1061no—
247C22—4241no—
248C22—4241no—
249C22—4241no—
250C22—2121no—
251C22—2121no—
252C22—2121no—
253C22—2122no—
254C22C22⋊C4×D5311yescharacteristic
255C4—21no—
256C4—41no—
257C4—81no—
258C4—1061no—
259C4—4241no—
260C4—2121no—
261C4—2121no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—81no—
266C2—1061no—
267C2—1061no—
268C2—2121no—
269C2—2121no—
270C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×AΣL(1,𝔽16)C2
socleC2×C53C2×#99
hypercenterAΣL(1,𝔽16)D53
hypocenterC53#2121
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #100×D53C2semidirect
3. #99×D53C2semidirect
4. #16810414C2semidirect
5. #16810412C2semidirect
6. #16810410C2semidirect
7. C53×AΣL(1,𝔽16)C2semidirect
8. C22×C22⋊D53C4semidirect
11. #13133201C4semidirect
16. #13133186C4semidirect
19. C22⋊C4×D53C4semidirect
27. C24×C53C2×C4semidirect
39. C22×Dic53C2×C4semidirect
49. C22⋊C4×C53C2×C4semidirect
53. C22⋊Dic53C2×C4semidirect
142. AΣL(1,𝔽16)D53semidirect
143. C53#2121semidirect
165. C24C4×D53semidirect
197. C22⋊C4C4×D53semidirect