Finite Group Databasebeta

← #20287459

Subgroups — #20287459

Order 6784.

338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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
1Q8×Dic106C211yesmaximal, characteristic, complemented
2#16811347C211yesmaximal, characteristic
3#16810984C212yesmaximal
4#16810973C211yesmaximal, characteristic, complemented
5#16810914C211yesmaximal, characteristic
6#16810906C212yesmaximal, complemented
7#16810847C212yesmaximal
8#16810750C211yesmaximal, characteristic, complemented
9#16810730C212yesmaximal, complemented
10C4×#11719924C211yesmaximal, characteristic
11C53×#462C211yesmaximal, characteristic, complemented
12C2×#11719924C2211yescharacteristic
13#13133217—21no—
14#13133217C2211yescharacteristic
15#13133216—22no—
16#13133216C2212yes—
17#13133212—22no—
18#13133206C2212yes—
19#13133197—21no—
20Q8×Dic53C2212yes—
21Q8×Dic53C2212yes—
22C4×Dic106—21no—
23C4×Dic106C2211yescharacteristic
24C4×D212C2211yescharacteristic
25#13133175—21no—
26#13133149C2212yes—
27#13133148C2211yescharacteristic
28#13133146C2212yes—
29#13133141C2212yes—
30#13133141C2211yescharacteristic
31#13133141C2212yes—
32#13133139C2212yes—
33#13133137C2211yescharacteristic
34#13133127C2212yes—
35C4×#88322C2211yescharacteristic
36C2×Q16×C53C2212yes—
37C4⋊Q8×C53C2211yescharacteristic
38C4×Q8×C53C2211yescharacteristic
39C4⋊C8×C53C2211yescharacteristic
40#103×C53C2212yes—
41C2×Q8×C53C2312yes—
42C2×Q8×C53C2311yescharacteristic
43C2×D212C2311yescharacteristic
44C2×C4×D53—21no—
45C2×Dic106—21no—
46C2×Dic106—22no—
47C2×Dic106D411yescharacteristic
48Q16×C53—24no—
49C2×C8×C53C2312yes—
50C4⋊C4×C53—21no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53—21no—
53C4⋊C4×C53C2311yescharacteristic
54C42×C53—21no—
55C42×C53C2311yescharacteristic
56#11719924—22no—
57#11719921—21no—
58#11719921—22no—
59#11719921—22no—
60C4⋊Dic53—21no—
61C4⋊Dic53—22no—
62C4⋊Dic53—22no—
63C4⋊Dic53D411yescharacteristic
64C4⋊Dic53C2312yes—
65C4⋊Dic53C2311yescharacteristic
66#11719919—22no—
67#11719919D412yes—
68C4×Dic53—22no—
69C4×Dic53—22no—
70C4×Dic53C2312yes—
71C2×#88322C2311yescharacteristic
72C2×#88322C2311yescharacteristic
73C22×D53—21no—
74Q8×C53—21no—
75Q8×C53—22no—
76Q8×C53—22no—
77Q8×C53#4112yes—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2411yescharacteristic
84C2×Dic53—22no—
85C2×Dic53—21no—
86C2×Dic53C2×D412yes—
87C2×Dic53C2×D412yes—
88D212—21no—
89D212—21no—
90C4×D53—42no—
91Dic106—22no—
92Dic106—82no—
93Dic106—81no—
94C8×C53—22no—
95#88322—21no—
96#88322—21no—
97C22×C53C22×D411yescharacteristic
98C2×D53—41no—
99C2×D53—41no—
100C2×D53—41no—
101C4×C53—22no—
102C4×C53—21no—
103C4×C53—42no—
104C4×C53—41no—
105C4×C53#13712yes—
106C4×C53C2×#4111yescharacteristic
107C4×C532−1+411yescharacteristic
108Dic53—22no—
109Dic53—22no—
110Dic53—41no—
111Dic53—42no—
112#3369—531nomaximal, Sylow
113C2×C53#54811yescharacteristic
114C2×C53C2×#13711yescharacteristic
115C2×C53#57711yescharacteristic
116D53—81no—
117Q82—531no—
118#556—531no—
119#489—532no—
120#487—531no—
121#476—531no—
122#475—532no—
123#464—532no—
124#462D5311yescharacteristic, complemented
125#446—531no—
126#441—532no—
127C4×SD16—531no—
128C53#336911yescharacteristic, complemented, Sylow
129C2×Q16C2×D5312yes—
130C2×SD16—531no—
131C4⋊Q8—532no—
132C4⋊Q8—531no—
133C4⋊Q8—1061no—
134C4⋊Q8—1062no—
135C4⋊Q8C2×D5311yescharacteristic
136#126—1062no—
137#124—532no—
138C22⋊Q8—1061no—
139C4×Q8—531no—
140C4×Q8—532no—
141C4×Q8—532no—
142C4×Q8—1061no—
143C4×Q8C2×D5311yescharacteristic
144C4×D4—531no—
145#117—1061no—
146#106—532no—
147#106—531no—
148C4⋊C8—532no—
149C4⋊C8C2×D5311yescharacteristic
150#103—531no—
151#103—532no—
152#103—532no—
153#103C2×D5312yes—
154#102—532no—
155#102—531no—
156#97—532no—
157C4×C8—531no—
158C2×Q8—531no—
159C2×Q8—1062no—
160C2×Q8—1061no—
161C2×Q8C22×D5312yes—
162C2×Q8C22×D5311yescharacteristic
163C2×D4—531no—
164C22×C4—1061no—
165Q16—24no—
166SD16—1062no—
167C2×C8—531no—
168C2×C8—531no—
169C2×C8C22×D5312yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—531no—
174C4⋊C4—532no—
175C4⋊C4—532no—
176C4⋊C4—531no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C4⋊C4—1061no—
181C4⋊C4C22×D5311yescharacteristic
182C22⋊C4—1062no—
183C22⋊C4—1062no—
184C22⋊C4—1061no—
185C42—21no—
186C42—532no—
187C42—1062no—
188C42—1062no—
189C42C22×D5311yescharacteristic
190C23—1061no—
191Q8—21no—
192Q8—22no—
193Q8—22no—
194Q8—1062no—
195Q8—4241no—
196Q8—4242no—
197Q8#1171994712yes—
198D4—1061no—
199D4—1061no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—532no—
204C2×C4—532no—
205C2×C4—1061no—
206C2×C4—1062no—
207C2×C4—2122no—
208C2×C4D4×D5311yescharacteristic
209C2×C4D4×D5311yescharacteristic
210C2×C4C23×D5311yescharacteristic
211C8—22no—
212C8—1061no—
213C8—1061no—
214C22—2121no—
215C22—2121no—
216C22—2121no—
217C22C2×D4×D5311yescharacteristic
218C4—22no—
219C4—21no—
220C4—42no—
221C4—41no—
222C4—1062no—
223C4—1062no—
224C4—2121no—
225C4—2122no—
226C4#1313324912yes—
227C4C2×#1171994711yescharacteristic
228C4#1313326411yescharacteristic
229C2—4241no—
230C2#1681127911yescharacteristic
231C2C2×#1313324911yescharacteristic
232C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#462C2
socleC22×C53C22×D4
hypercenter#462D53
hypocenterC53#3369
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. Q8×Dic106C2semidirect
4. #16810973C2semidirect
6. #16810906C2semidirect
8. #16810750C2semidirect
9. #16810730C2semidirect
11. C53×#462C2semidirect
124. #462D53semidirect
128. C53#3369semidirect