Finite Group Databasebeta

← #20275017

Subgroups — #20275017

Order 6784.

376 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 97 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#16811134C211yesmaximal, characteristic
2C2×#13133143C212yesmaximal, complemented
3C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
4#16810384C212yesmaximal, complemented
5C22×M16×C53C211yesmaximal, characteristic, complemented
6C2×#11719945C2211yescharacteristic
7#13133200—21no—
8#13133196C411yescharacteristic, complemented
9#13133196C411yescharacteristic, complemented
10#13133191C411yescharacteristic, complemented
11#13133191C411yescharacteristic, complemented
12#13133186—21no—
13C4×C22⋊D53—21no—
14C4×C22⋊D53—21no—
15#13133176C2211yescharacteristic
16C2×C22⋊Dic53—21no—
17C2×C4×Dic53C2212yes—
18#13133143—24no—
19#13133124—21no—
20C23×C4×C53C2211yescharacteristic
21C2×M16×C53—24no—
22C2×M16×C53—21no—
23C2×M16×C53C2212yes—
24C22×C8×C53—21no—
25C24×C53D411yescharacteristic
26C22×C4×C53—22no—
27C22×C4×C53—22no—
28C22×C4×C53C2×C411yescharacteristic
29C22×C4×C53C2×C411yescharacteristic
30C22×C4×C53D411yescharacteristic
31C22×C4×C53D411yescharacteristic
32C22×C4×C53D411yescharacteristic
33C22×C4×C53C2311yescharacteristic
34C2×C22⋊D53—21no—
35C2×C22⋊D53—21no—
36C2×C22⋊D53—21no—
37C22×Dic53—22no—
38#11719945—22no—
39#11719945—22no—
40C2×D212C2×C411yescharacteristic
41C2×C4×D53—21no—
42C2×Dic106C2×C411yescharacteristic
43M16×C53—24no—
44M16×C53—22no—
45M16×C53D414yescomplemented
46C2×C8×C53—24no—
47C2×C8×C53—22no—
48C22⋊Dic53—24no—
49C22⋊Dic53—21no—
50C22⋊Dic53—21no—
51C22⋊Dic53—21no—
52#11719921—21no—
53#11719921—21no—
54C4⋊Dic53C2×C411yescharacteristic
55C4⋊Dic53C2×C411yescharacteristic
56#11719919—21no—
57#11719919—21no—
58C4×Dic53—21no—
59C4×Dic53—22no—
60C4×Dic53—22no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53—41no—
64C23×C53C22⋊C411yescharacteristic
65C23×C53C22⋊C411yescharacteristic
66C23×C53C2×D411yescharacteristic
67C22×D53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53C22⋊C411yescharacteristic
76C2×C4×C53C22⋊C411yescharacteristic
77C2×C4×C53C22×C411yescharacteristic
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53#4112yes—
81C22⋊D53—42no—
82C22⋊D53—42no—
83C22⋊D53—42no—
84C2×Dic53—24no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90D212—22no—
91C4×D53—41no—
92C4×D53—41no—
93Dic106—22no—
94C8×C53—24no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—44no—
103C22×C53#10412yescomplemented
104C22×C53C2×C22⋊C411yescharacteristic
105C22×C53#12112yes—
106C2×D53—41no—
107C2×D53—41no—
108C2×D53—41no—
109C4×C53—21no—
110C4×C53—21no—
111C4×C53C4×D412yes—
112C4×C53#12011yescharacteristic
113C4×C53#12311yescharacteristic
114Dic53—44no—
115Dic53—41no—
116Dic53—41no—
117Dic53—41no—
118#1871—531nomaximal, Sylow
119C2×C53—22no—
120C2×C53—22no—
121C2×C53—41no—
122C2×C53#38611yescharacteristic
123C2×C53C2×#10411yescharacteristic
124C2×C53#42111yescharacteristic
125D53—81no—
126C22×M16D5311yescharacteristic, complemented
127#525—531no—
128C2×#104—532no—
129C4×C22⋊C4—531no—
130#339—532no—
131C53#187111yescharacteristic, complemented, Sylow
132C2×#41—531no—
133C23×C4C2×D5311yescharacteristic
134C2×M16—21no—
135C2×M16—24no—
136C2×M16C2×D5312yes—
137C22×C8—21no—
138#123—1061no—
139C22⋊Q8—531no—
140C22⋊Q8—531no—
141#121—531no—
142#121—531no—
143#120—1061no—
144C4×D4—1061no—
145C4×D4—1061no—
146#117—531no—
147C2×C22⋊C4—1061no—
148C2×C42—532no—
149#104—1064no—
150#95—1061no—
151C24D21211yescharacteristic
152#41—1062no—
153#41—1062no—
154C2×Q8—531no—
155C2×D4—531no—
156C2×D4—1061no—
157C2×D4—1061no—
158C2×D4—1061no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—1062no—
162C22×C4—1061no—
163C22×C4C4×D5311yescharacteristic
164C22×C4C4×D5311yescharacteristic
165C22×C4D21211yescharacteristic
166C22×C4C22⋊D5311yescharacteristic
167C22×C4C22⋊D5311yescharacteristic
168C22×C4C22×D5311yescharacteristic
169M16—24no—
170M16—22no—
171M16C22⋊D5314yescomplemented
172C2×C8—24no—
173C2×C8—22no—
174C4⋊C4—531no—
175C4⋊C4—531no—
176C4⋊C4—1061no—
177C4⋊C4—1061no—
178C22⋊C4—1064no—
179C22⋊C4—1061no—
180C22⋊C4—1061no—
181C22⋊C4—1061no—
182C22⋊C4—1061no—
183C22⋊C4—1061no—
184C42—1061no—
185C42—1062no—
186C42—1062no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—1061no—
191C23#1171992111yescharacteristic
192C23#1171992111yescharacteristic
193C23C2×D21211yescharacteristic
194Q8—1062no—
195D4—1062no—
196D4—2122no—
197D4—2122no—
198D4—2122no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—1064no—
207C2×C4—1061no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4—2121no—
213C2×C4—2121no—
214C2×C4#1171992111yescharacteristic
215C2×C4#1171992111yescharacteristic
216C2×C4C2×C4×D5311yescharacteristic
217C2×C4#1171994512yes—
218C2×C4C2×C22⋊D5312yes—
219C2×C4C2×C22⋊D5311yescharacteristic
220C8—24no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—2121no—
230C22—2121no—
231C22—2121no—
232C22#1313314312yescomplemented
233C22C2×#1171992111yescharacteristic
234C22#1313319112yes—
235C4—21no—
236C4—21no—
237C4—2124no—
238C4—2121no—
239C4—2121no—
240C4—2121no—
241C4C4×C22⋊D5312yes—
242C4#1313318611yescharacteristic
243C4#1313320011yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—4241no—
248C2#1681050211yescharacteristic
249C2C2×#1313314311yescharacteristic
250C2#1681063911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC22×M16×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC22×M16D53
hypocenterC53#1871
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#13133143C2semidirect
3. C4×C22⋊Dic53C2semidirect
4. #16810384C2semidirect
5. C22×M16×C53C2semidirect
8. #13133196C4semidirect
9. #13133196C4semidirect
10. #13133191C4semidirect
11. #13133191C4semidirect
45. M16×C53D4semidirect
103. C22×C53#104semidirect
126. C22×M16D53semidirect
131. C53#1871semidirect
171. M16C22⋊D53semidirect
232. C22#13133143semidirect