Finite Group Databasebeta

← #20276355

Subgroups — #20276355

Order 6784.

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

normal subgroups only

This group has 264 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#16811131C211yesmaximal, characteristic
2#16810604C211yesmaximal, characteristic, complemented
3#16810595C211yesmaximal, characteristic, complemented
4#16810413C211yesmaximal, characteristic, complemented
5#16810411C211yesmaximal, characteristic, complemented
6#16810409C211yesmaximal, characteristic, complemented
7C53×#352C211yesmaximal, characteristic, complemented
8C2×#11719947C2211yescharacteristic
9C22×C4×D53C411yescharacteristic, complemented
10M16×D53—21no—
11#13133200C411yescharacteristic, complemented
12#13133195—21no—
13#13133190—21no—
14#13133187C411yescharacteristic, complemented
15#13133185C2211yescharacteristic
16#13133179—21no—
17#13133177—21no—
18#13133173—21no—
19#13133173C411yescharacteristic, complemented
20#13133135—21no—
21#13133134C2211yescharacteristic
22#13133132C2211yescharacteristic
23#13133131—21no—
24#123×C53C2211yescharacteristic
25#100×C53C2211yescharacteristic
26#99×C53C2211yescharacteristic
27C23×D53C2×C411yescharacteristic
28C2×D4×C53C2311yescharacteristic
29C22×C4×C53C2×C411yescharacteristic
30C2×C22⋊D53—21no—
31C2×C22⋊D53C2×C411yescharacteristic
32C2×C22⋊D53D411yescharacteristic
33C22×Dic53C2×C411yescharacteristic
34C22×Dic53D411yescharacteristic
35#11719947—41no—
36#11719947—41no—
37C2×D212—21no—
38C2×C4×D53—21no—
39C2×C4×D53—21no—
40C2×C4×D53—41no—
41C2×C4×D53—41no—
42C2×C4×D53D411yescharacteristic
43C2×Dic106D411yescharacteristic
44M16×C53—21no—
45C4⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53C2×C411yescharacteristic
49C22⋊Dic53—21no—
50C22⋊Dic53C2×C411yescharacteristic
51#11719921—21no—
52#11719921—21no—
53#11719921—21no—
54#11719919—21no—
55#11719919—21no—
56#11719919—41no—
57C4×Dic53—21no—
58C4×Dic53—41no—
59#11719917—21no—
60#11719912—41no—
61C8×D53—41no—
62C23×C53C22×C411yescharacteristic
63C23×C53C2×D411yescharacteristic
64C22×D53—21no—
65C22×D53—42no—
66C22×D53—41no—
67C22×D53C22⋊C411yescharacteristic
68D4×C53—41no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53C2×D411yescharacteristic
74C22⋊D53—41no—
75C22⋊D53—41no—
76C22⋊D53—42no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—41no—
83C2×Dic53C22⋊C411yescharacteristic
84C2×Dic53C22⋊C411yescharacteristic
85C2×Dic53C22⋊C411yescharacteristic
86D212—42no—
87C4×D53—21no—
88C4×D53—21no—
89C4×D53—42no—
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×C53C2×C22⋊C411yescharacteristic
100C2×D53—22no—
101C2×D53—21no—
102C2×D53—41no—
103C2×D53—42no—
104C2×D53—41no—
105C2×D53—41no—
106C2×D53—42no—
107C4×C53—21no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—81no—
111Dic53—21no—
112Dic53—21no—
113Dic53—41no—
114Dic53—81no—
115Dic53#9911yescharacteristic
116Dic53#9911yescharacteristic
117#2120—531nomaximal, Sylow
118C2×C53—21no—
119C2×C53—41no—
120C2×C53—41no—
121C2×C53C2×#9911yescharacteristic
122D53—41no—
123D53—42no—
124#525—531no—
125#413—531no—
126#410—531no—
127#352—531no—
128#352D5311yescharacteristic, complemented
129AΣL(1,𝔽16)—531no—
130AΣL(1,𝔽16)—531no—
131C53#212011yescharacteristic, complemented, Sylow
132C2×#41—531no—
133C23×C4—531no—
134C2×M16—1061no—
135#123—531no—
136#123C2×D5311yescharacteristic
137C22⋊Q8—1061no—
138#121—1061no—
139#120—531no—
140#120—531no—
141C4×D4—1061no—
142C4×D4—1061no—
143#117—531no—
144#117—1061no—
145#101—1061no—
146#100—531no—
147#100C2×D5311yescharacteristic
148#99—531no—
149#99—1061no—
150#99C2×D5311yescharacteristic
151C24—531no—
152#41—2121no—
153#41—2121no—
154C2×Q8—531no—
155C2×D4—531no—
156C2×D4—531no—
157C2×D4—1061no—
158C2×D4—1061no—
159C2×D4C22×D5311yescharacteristic
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—1061no—
164C22×C4—1061no—
165C22×C4—2121no—
166C22×C4—2121no—
167C22×C4C4×D5311yescharacteristic
168M16—21no—
169M16—1061no—
170M16—2121no—
171C2×C8—2121no—
172C4⋊C4—21no—
173C4⋊C4—1061no—
174C4⋊C4—1061no—
175C4⋊C4—2121no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—531no—
179C22⋊C4—1061no—
180C22⋊C4—1061no—
181C22⋊C4—1061no—
182C22⋊C4—1061no—
183C22⋊C4C4×D5311yescharacteristic
184C42—1061no—
185C42—2121no—
186C23—531no—
187C23—1061no—
188C23—2121no—
189C23—2122no—
190C23C2×C4×D5311yescharacteristic
191C23D4×D5311yescharacteristic
192Q8—2121no—
193D4—41no—
194D4—2122no—
195D4—2121no—
196D4—2121no—
197D4—2122no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—531no—
203C2×C4—531no—
204C2×C4—531no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—1061no—
212C2×C4—2121no—
213C2×C4—2121no—
214C2×C4—2122no—
215C2×C4—2121no—
216C2×C4D4×D5311yescharacteristic
217C8—41no—
218C8—2121no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—1062no—
224C22—1061no—
225C22—2121no—
226C22—2121no—
227C22—2122no—
228C22—2122no—
229C22—2121no—
230C22C22⋊C4×D5311yescharacteristic
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—531no—
236C4—531no—
237C4—1061no—
238C4—1061no—
239C4—4241no—
240C4—2121no—
241C2—21no—
242C2—41no—
243C2—41no—
244C2—2122no—
245C2—2121no—
246C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#352C2
socleC2×C53C2×#99
hypercenter#352D53
hypocenterC53#2120
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #16810604C2semidirect
3. #16810595C2semidirect
4. #16810413C2semidirect
5. #16810411C2semidirect
6. #16810409C2semidirect
7. C53×#352C2semidirect
9. C22×C4×D53C4semidirect
11. #13133200C4semidirect
14. #13133187C4semidirect
19. #13133173C4semidirect
128. #352D53semidirect
131. C53#2120semidirect