Finite Group Databasebeta

← #20287416

Subgroups — #20287416

Order 6784.

348 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811359C211yesmaximal, characteristic
2#16811348C211yesmaximal, characteristic, complemented
3#16810976C212yesmaximal, complemented
4#16810921C212yesmaximal, complemented
5#16810905C211yesmaximal, characteristic, complemented
6#16810844C212yesmaximal, complemented
7#16810820C211yesmaximal, characteristic
8C8×Dic106C211yesmaximal, characteristic, complemented
9C4×Dic212C211yesmaximal, characteristic, complemented
10#16810714C212yesmaximal, complemented
11C53×#486C211yesmaximal, characteristic
12C2×Dic212C2211yescharacteristic
13C2×#11719923C2212yes—
14#13133215—21no—
15#13133208C2212yes—
16#13133207—22no—
17#13133204—21no—
18#13133204C2212yes—
19#13133198—22no—
20Q8×Dic53—21no—
21C4×Dic106C2211yescharacteristic
22C4×Dic106C2211yescharacteristic
23C4×Dic106C2211yescharacteristic
24#13133182C2212yes—
25D4×Dic53—21no—
26D4×Dic53C2212yes—
27#13133151C2211yescharacteristic
28#13133148C2212yes—
29C4⋊#88322C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2211yescharacteristic
32#13133142C2211yescharacteristic
33#13133140C2212yes—
34#13133138C2212yes—
35C8×Dic53C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37#127×C53C2211yescharacteristic
38#102×C53C2212yes—
39#102×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53C2312yes—
45C22×Dic53—22no—
46C2×Dic106D411yescharacteristic
47C2×Dic106C2311yescharacteristic
48C2×Dic106C2311yescharacteristic
49C2×C8×C53C2311yescharacteristic
50C2×C8×C53C2311yescharacteristic
51C4⋊C4×C53—21no—
52C4⋊C4×C53C2312yes—
53C42×C53C2311yescharacteristic
54C22⋊Dic53—22no—
55C22⋊Dic53—22no—
56C22⋊Dic53—22no—
57#11719923—24no—
58C4⋊Dic53—22no—
59C4⋊Dic53D411yescharacteristic
60C4⋊Dic53C2311yescharacteristic
61C4⋊Dic53C2311yescharacteristic
62#11719919—21no—
63#11719919—21no—
64#11719919—22no—
65#11719919—22no—
66#11719919D412yes—
67C4×Dic53—21no—
68C4×Dic53—21no—
69C4×Dic53—22no—
70C4×Dic53C2312yes—
71C2×#88322C2312yes—
72Dic212—22no—
73C23×C53—22no—
74Q8×C53—42no—
75D4×C53—22no—
76D4×C53—22no—
77D4×C53—42no—
78D4×C53—81no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2411yescharacteristic
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—82no—
88C2×Dic53C2×D412yes—
89C2×Dic53C2×D412yes—
90Dic106—22no—
91Dic106—21no—
92Dic106—21no—
93Dic106—21no—
94Dic106#4112yes—
95C8×C53—21no—
96C8×C53—21no—
97#88322—22no—
98C22×C53—42no—
99C22×C53—42no—
100C22×C53—42no—
101C22×C53C22×D411yescharacteristic
102C4×C53—21no—
103C4×C53—42no—
104C4×C53#13512yes—
105C4×C53C2×#4111yescharacteristic
106C4×C532−1+411yescharacteristic
107Dic53—22no—
108Dic53—22no—
109Dic53—22no—
110Dic53—41no—
111Dic53—41no—
112Dic53—41no—
113#3365—531nomaximal, Sylow
114C2×C53—82no—
115C2×C53#54811yescharacteristic
116C2×C53C2×#13511yescharacteristic
117C2×C53#57711yescharacteristic
118#559—531no—
119#557—531no—
120#488—532no—
121#486D5311yescharacteristic
122#478—532no—
123#475—531no—
124#464—532no—
125#460—531no—
126C8×Q8—531no—
127C4×Q16—531no—
128C4×SD16—532no—
129C53#336511yescharacteristic, complemented, Sylow
130C2×Q16—531no—
131C2×SD16—532no—
132C4⋊Q8—1061no—
133C4⋊Q8C2×D5311yescharacteristic
134#127C2×D5311yescharacteristic
135#125—532no—
136#125—1062no—
137#124—532no—
138#124—1061no—
139#123—1062no—
140C4×Q8—531no—
141C4×Q8—532no—
142C4×Q8—531no—
143C4×Q8—531no—
144C4×Q8—1061no—
145C4×D4—532no—
146C4×D4—1061no—
147#107—531no—
148#106—532no—
149C4⋊C8—532no—
150C4⋊C8—531no—
151#103—531no—
152#103—531no—
153#103—532no—
154#102—532no—
155#102C2×D5312yes—
156#102C2×D5312yes—
157C4×C8—532no—
158C4×C8C2×D5311yescharacteristic
159C2×Q8—21no—
160C2×Q8—531no—
161C2×Q8—531no—
162C2×Q8—531no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4C22×D5312yes—
166C22×C4—1062no—
167Q16—1062no—
168SD16—1064no—
169C2×C8—532no—
170C2×C8C22×D5311yescharacteristic
171C2×C8C22×D5311yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—531no—
174C4⋊C4—532no—
175C4⋊C4—531no—
176C4⋊C4—531no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C4⋊C4—1061no—
181C4⋊C4—1061no—
182C4⋊C4C22×D5312yes—
183C22⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C42—532no—
187C42—1062no—
188C42—1061no—
189C42—1061no—
190C42C22×D5311yescharacteristic
191C23—22no—
192Q8—42no—
193Q8—532no—
194Q8—1062no—
195Q8—1061no—
196Q8—1061no—
197Q8—1061no—
198D4—22no—
199D4—22no—
200D4—42no—
201D4—81no—
202C2×C4—22no—
203C2×C4—532no—
204C2×C4—532no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1061no—
209C2×C4—4242no—
210C2×C4D4×D5311yescharacteristic
211C2×C4D4×D5311yescharacteristic
212C2×C4C23×D5311yescharacteristic
213C8—21no—
214C8—21no—
215C8—1062no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22C2×D4×D5311yescharacteristic
220C4—21no—
221C4—42no—
222C4—1062no—
223C4—1062no—
224C4—1062no—
225C4—2121no—
226C4—2121no—
227C4—2121no—
228C4#1313323612yes—
229C4#41×D5311yescharacteristic
230C4#1313326411yescharacteristic
231C2—82no—
232C2#1681128111yescharacteristic
233C2C2×#1313323611yescharacteristic
234C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3365
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #16811348C2semidirect
3. #16810976C2semidirect
4. #16810921C2semidirect
5. #16810905C2semidirect
6. #16810844C2semidirect
8. C8×Dic106C2semidirect
9. C4×Dic212C2semidirect
10. #16810714C2semidirect
129. C53#3365semidirect