Finite Group Databasebeta

← #20287380

Subgroups — #20287380

Order 6784.

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

normal subgroups only

This group has 362 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, complemented
2#16811334C211yesmaximal, characteristic, complemented
3#16810976C212yesmaximal, complemented
4#16810925C212yesmaximal, complemented
5#16810915C211yesmaximal, characteristic, complemented
6#16810837C212yesmaximal, complemented
7#16810815C211yesmaximal, characteristic
8C8×Dic106C211yesmaximal, characteristic, complemented
9C4×#11719913C211yesmaximal, characteristic, complemented
10#16810703C212yesmaximal, complemented
11C53×#486C211yesmaximal, characteristic, complemented
12C2×#11719913C2211yescharacteristic
13C2×#11719922C2212yes—
14#13133208C2212yes—
15#13133204—21no—
16#13133204C2212yes—
17#13133198—22no—
18#13133197—21no—
19#13133193—22no—
20C4×Dic106C2211yescharacteristic
21C4×Dic106C2211yescharacteristic
22C4×D212C2211yescharacteristic
23D4×Dic53—21no—
24D4×Dic53C2212yes—
25#13133179C2212yes—
26C4⋊C4×D53—21no—
27#13133150C2212yes—
28#13133149C2211yescharacteristic
29C4⋊#88322C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2211yescharacteristic
32#13133140C2212yes—
33#13133139C2211yescharacteristic
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×D212C2311yescharacteristic
47C2×C4×D53—21no—
48C2×C4×D53—22no—
49C2×Dic106D411yescharacteristic
50C2×Dic106C2311yescharacteristic
51C2×C8×C53C2311yescharacteristic
52C2×C8×C53C2311yescharacteristic
53C4⋊C4×C53—21no—
54C4⋊C4×C53C2312yes—
55C42×C53C2311yescharacteristic
56C22⋊Dic53—22no—
57C22⋊Dic53—22no—
58C22⋊Dic53—22no—
59#11719922—24no—
60#11719921—21no—
61#11719921—22no—
62C4⋊Dic53—22no—
63C4⋊Dic53D411yescharacteristic
64C4⋊Dic53C2311yescharacteristic
65C4⋊Dic53C2311yescharacteristic
66#11719919—21no—
67#11719919—22no—
68#11719919D412yes—
69C4×Dic53—21no—
70C4×Dic53C2312yes—
71C2×#88322C2312yes—
72#11719913—22no—
73C23×C53—22no—
74C22×D53—21no—
75Q8×C53—81no—
76D4×C53—22no—
77D4×C53—22no—
78D4×C53—42no—
79D4×C53—81no—
80C2×C4×C53—22no—
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2411yescharacteristic
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—82no—
88C2×Dic53C2×D412yes—
89C2×Dic53C2×D412yes—
90D212—21no—
91D212#4112yes—
92C4×D53—42no—
93C4×D53—82no—
94Dic106—22no—
95Dic106—21no—
96Dic106—21no—
97C8×C53—21no—
98C8×C53—21no—
99#88322—22no—
100C22×C53—42no—
101C22×C53—42no—
102C22×C53—42no—
103C22×C53C22×D411yescharacteristic
104C2×D53—22no—
105C2×D53—41no—
106C2×D53—41no—
107C4×C53—21no—
108C4×C53—42no—
109C4×C53#13512yes—
110C4×C53C2×#4111yescharacteristic
111C4×C532−1+411yescharacteristic
112Dic53—22no—
113Dic53—22no—
114Dic53—41no—
115Dic53—41no—
116Dic53—41no—
117#3361—531nomaximal, Sylow
118C2×C53—82no—
119C2×C53#54811yescharacteristic
120C2×C53C2×#13511yescharacteristic
121C2×C53#57611yescharacteristic
122D53—42no—
123#559—531no—
124#554—531no—
125#488—532no—
126#486D5311yescharacteristic, complemented
127#478—532no—
128#477—531no—
129#463—532no—
130#459—531no—
131C8×Q8—531no—
132C4×SD16—531no—
133C4×D8—532no—
134C53#336111yescharacteristic, complemented, Sylow
135C2×SD16—531no—
136C2×D8—532no—
137C4⋊Q8C2×D5311yescharacteristic
138#127C2×D5311yescharacteristic
139#125—532no—
140#124—532no—
141#124—1061no—
142#123—1062no—
143C22⋊Q8—1061no—
144C22⋊Q8—1062no—
145C4×Q8—531no—
146C4×Q8—531no—
147C4×D4—532no—
148C4×D4—532no—
149C4×D4—531no—
150C4×D4—1061no—
151C2×C4⋊C4—1061no—
152#107—532no—
153#106—531no—
154C4⋊C8—532no—
155C4⋊C8—531no—
156#103—531no—
157#103—532no—
158#102—532no—
159#102—531no—
160#102C2×D5312yes—
161#102C2×D5312yes—
162C4×C8—532no—
163C4×C8C2×D5311yescharacteristic
164C2×Q8—21no—
165C2×Q8—531no—
166C2×Q8—531no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—531no—
170C2×D4C22×D5312yes—
171C22×C4—1062no—
172C22×C4—1062no—
173C22×C4—1061no—
174SD16—1062no—
175D8—1064no—
176C2×C8—532no—
177C2×C8C22×D5311yescharacteristic
178C2×C8C22×D5311yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—531no—
181C4⋊C4—532no—
182C4⋊C4—531no—
183C4⋊C4—531no—
184C4⋊C4—1062no—
185C4⋊C4—1062no—
186C4⋊C4—1061no—
187C4⋊C4C22×D5312yes—
188C22⋊C4—1062no—
189C22⋊C4—1062no—
190C22⋊C4—1062no—
191C22⋊C4—1062no—
192C22⋊C4—1061no—
193C42—532no—
194C42—1061no—
195C42C22×D5311yescharacteristic
196C23—22no—
197C23—1061no—
198Q8—81no—
199Q8—1062no—
200Q8—1061no—
201Q8—1061no—
202D4—22no—
203D4—22no—
204D4—42no—
205D4—81no—
206D4—532no—
207D4—1061no—
208C2×C4—22no—
209C2×C4—532no—
210C2×C4—532no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—4242no—
215C2×C4—4242no—
216C2×C4—2122no—
217C2×C4D4×D5311yescharacteristic
218C2×C4D4×D5311yescharacteristic
219C2×C4C23×D5311yescharacteristic
220C8—21no—
221C8—21no—
222C8—1062no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22—1062no—
227C22—2121no—
228C22—2121no—
229C22C2×D4×D5311yescharacteristic
230C4—21no—
231C4—42no—
232C4—1062no—
233C4—1062no—
234C4—2121no—
235C4—2121no—
236C4—2121no—
237C4#1313323812yes—
238C4#41×D5311yescharacteristic
239C4#1313326411yescharacteristic
240C2—82no—
241C2—2122no—
242C2#1681128111yescharacteristic
243C2C2×#1313323811yescharacteristic
244C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3361
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
1. #16811359C2semidirect
2. #16811334C2semidirect
3. #16810976C2semidirect
4. #16810925C2semidirect
5. #16810915C2semidirect
6. #16810837C2semidirect
8. C8×Dic106C2semidirect
9. C4×#11719913C2semidirect
10. #16810703C2semidirect
11. C53×#486C2semidirect
126. #486D53semidirect
134. C53#3361semidirect