Finite Group Databasebeta

← #20276363

Subgroups — #20276363

Order 6784.

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

normal subgroups only

This group has 266 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#16811188C211yesmaximal, characteristic
2#16810604C211yesmaximal, characteristic, complemented
3#16810595C211yesmaximal, characteristic, complemented
4#16810411C211yesmaximal, characteristic, complemented
5#16810410C211yesmaximal, characteristic, complemented
6#16810409C211yesmaximal, characteristic, complemented
7C53×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8C2×#11719947C2211yescharacteristic
9C22×C22⋊D53C411yescharacteristic, complemented
10M16×D53—21no—
11#13133213—21no—
12#13133203—21no—
13#13133190—21no—
14#13133189—21no—
15#13133187C411yescharacteristic, complemented
16#13133186C411yescharacteristic, complemented
17#13133185C2211yescharacteristic
18#13133173—21no—
19#13133173C411yescharacteristic, complemented
20#13133135—21no—
21#13133134C2211yescharacteristic
22#13133132C2211yescharacteristic
23#13133131—21no—
24#120×C53C2211yescharacteristic
25#100×C53C2211yescharacteristic
26#99×C53C2211yescharacteristic
27C24×C53C2×C411yescharacteristic
28C23×D53C2×C411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53C2311yescharacteristic
31C2×C22⋊D53—21no—
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
40C22×Dic53D411yescharacteristic
41#11719947—41no—
42#11719947—41no—
43C2×D212—21no—
44C2×C4×D53D411yescharacteristic
45C2×Dic106D411yescharacteristic
46M16×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53C2×C411yescharacteristic
50C22⋊Dic53—21no—
51C22⋊Dic53—21no—
52C22⋊Dic53C2×C411yescharacteristic
53#11719921—21no—
54#11719921—21no—
55#11719919—21no—
56#11719919—41no—
57C4×Dic53—21no—
58C4×Dic53—41no—
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—41no—
69C22×D53—81no—
70C22×D53C22⋊C411yescharacteristic
71D4×C53—41no—
72D4×C53—81no—
73C2×C4×C53—21no—
74C2×C4×C53—41no—
75C2×C4×C53C2×D411yescharacteristic
76C22⋊D53—41no—
77C22⋊D53—41no—
78C22⋊D53—42no—
79C22⋊D53—41no—
80C22⋊D53—41no—
81C22⋊D53—81no—
82C22⋊D53—81no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—41no—
89C2×Dic53C22⋊C411yescharacteristic
90C2×Dic53C22⋊C411yescharacteristic
91C2×Dic53C22⋊C411yescharacteristic
92D212—81no—
93C4×D53—21no—
94C4×D53—21no—
95Dic106—41no—
96C8×C53—41no—
97#88322—41no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—41no—
105C22×C53—82no—
106C22×C53C2×C22⋊C411yescharacteristic
107C2×D53—21no—
108C2×D53—41no—
109C2×D53—41no—
110C2×D53—41no—
111C2×D53—41no—
112C2×D53—82no—
113C4×C53—21no—
114C4×C53—41no—
115C4×C53—81no—
116Dic53—21no—
117Dic53—21no—
118Dic53—41no—
119Dic53—81no—
120Dic53#9911yescharacteristic
121Dic53#9911yescharacteristic
122#2122—531nomaximal, Sylow
123C2×C53—21no—
124C2×C53—41no—
125C2×C53—41no—
126C2×C53—81no—
127C2×C53C2×#9911yescharacteristic
128D53—41no—
129D53—81no—
130#535—531no—
131#413—531no—
132#410—531no—
133AΣL(1,𝔽16)—531no—
134AΣL(1,𝔽16)—531no—
135AΣL(1,𝔽16)—531no—
136AΣL(1,𝔽16)D5311yescharacteristic, complemented
137C53#212211yescharacteristic, complemented, Sylow
138C2×#41—531no—
139C22×D4—531no—
140C2×M16—1061no—
141#127—1061no—
142#124—1061no—
143#121—1061no—
144#121—1061no—
145#120—531no—
146#120—531no—
147#120—531no—
148#120C2×D5311yescharacteristic
149#117—531no—
150#117—1061no—
151#101—1061no—
152#100—531no—
153#100C2×D5311yescharacteristic
154#99—531no—
155#99—1061no—
156#99C2×D5311yescharacteristic
157C24—531no—
158C24C4×D5311yescharacteristic
159#41—2121no—
160#41—2121no—
161C2×Q8—531no—
162C2×D4—21no—
163C2×D4—531no—
164C2×D4—531no—
165C2×D4—1061no—
166C2×D4—1061no—
167C2×D4—1061no—
168C2×D4—1061no—
169C2×D4—1061no—
170C2×D4—2121no—
171C2×D4—2121no—
172C2×D4C22×D5311yescharacteristic
173C22×C4—531no—
174C22×C4—531no—
175C22×C4—531no—
176M16—21no—
177M16—1061no—
178M16—2121no—
179C2×C8—2121no—
180C4⋊C4—1061no—
181C4⋊C4—2121no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—531no—
185C22⋊C4—1061no—
186C22⋊C4—1061no—
187C22⋊C4—1061no—
188C22⋊C4—1061no—
189C22⋊C4C4×D5311yescharacteristic
190C42—1061no—
191C42—2121no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—531no—
196C23—1061no—
197C23—4241no—
198C23—2121no—
199C23C2×C4×D5311yescharacteristic
200C23D4×D5311yescharacteristic
201Q8—2121no—
202D4—41no—
203D4—81no—
204D4—4241no—
205D4—4241no—
206D4—4241no—
207D4—2121no—
208D4—2121no—
209D4—2121no—
210D4—2121no—
211D4—2122no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—531no—
215C2×C4—531no—
216C2×C4—531no—
217C2×C4—1061no—
218C2×C4—1061no—
219C2×C4—1061no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—2121no—
225C2×C4D4×D5311yescharacteristic
226C8—41no—
227C8—2121no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—1061no—
237C22—4242no—
238C22—2121no—
239C22—2121no—
240C22—2121no—
241C22—2121no—
242C22C22⋊C4×D5311yescharacteristic
243C4—21no—
244C4—41no—
245C4—81no—
246C4—531no—
247C4—531no—
248C4—1061no—
249C4—1061no—
250C4—4241no—
251C4—2121no—
252C2—21no—
253C2—41no—
254C2—41no—
255C2—81no—
256C2—4241no—
257C2—2121no—
258C2#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#2122
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. #16810604C2semidirect
3. #16810595C2semidirect
4. #16810411C2semidirect
5. #16810410C2semidirect
6. #16810409C2semidirect
7. C53×AΣL(1,𝔽16)C2semidirect
9. C22×C22⋊D53C4semidirect
15. #13133187C4semidirect
16. #13133186C4semidirect
19. #13133173C4semidirect
136. AΣL(1,𝔽16)D53semidirect
137. C53#2122semidirect