Finite Group Databasebeta

← #20287405

Subgroups — #20287405

Order 6784.

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

normal subgroups only

This group has 336 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#16811348C211yesmaximal, characteristic, complemented
2#16811347C211yesmaximal, characteristic
3#16810979C212yesmaximal
4#16810928C212yesmaximal, complemented
5#16810900C211yesmaximal, characteristic
6#16810848C212yesmaximal
7#16810830C211yesmaximal, characteristic, complemented
8C8×Dic106C211yesmaximal, characteristic, complemented
9#16810715C212yesmaximal, complemented
10C4×#11719913C211yesmaximal, characteristic
11C53×#487C211yesmaximal, characteristic, complemented
12C2×#11719925C2212yes—
13C2×#11719913C2211yescharacteristic
14#13133215—21no—
15#13133212—22no—
16#13133208C2212yes—
17#13133207—22no—
18#13133206C2212yes—
19#13133197—21no—
20Q8×Dic53—21no—
21Q8×Dic53C2212yes—
22C4×Dic106C2211yescharacteristic
23C4×Dic106C2211yescharacteristic
24#13133182C2212yes—
25C4×D212C2211yescharacteristic
26#13133175—21no—
27#13133150C2212yes—
28#13133149C2211yescharacteristic
29C4⋊#88322C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2211yescharacteristic
32#13133141C2212yes—
33#13133139C2211yescharacteristic
34#13133137C2212yes—
35C8×Dic53C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4⋊Q8×C53C2211yescharacteristic
38#103×C53C2212yes—
39#103×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×Q8×C53C2312yes—
43C2×D212C2311yescharacteristic
44C2×C4×D53—21no—
45C2×Dic106D411yescharacteristic
46C2×Dic106C2311yescharacteristic
47C2×C8×C53C2311yescharacteristic
48C2×C8×C53C2311yescharacteristic
49C4⋊C4×C53—21no—
50C4⋊C4×C53—21no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53C2312yes—
53C42×C53C2311yescharacteristic
54#11719925—24no—
55#11719921—21no—
56#11719921—22no—
57#11719921—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53—22no—
60C4⋊Dic53D411yescharacteristic
61C4⋊Dic53C2311yescharacteristic
62C4⋊Dic53C2311yescharacteristic
63#11719919—21no—
64#11719919—22no—
65#11719919—22no—
66#11719919D412yes—
67C4×Dic53—21no—
68C4×Dic53—22no—
69C4×Dic53—22no—
70C4×Dic53C2312yes—
71C2×#88322C2312yes—
72#11719913—22no—
73C22×D53—21no—
74Q8×C53—22no—
75Q8×C53—22no—
76Q8×C53—42no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2411yescharacteristic
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53C2×D412yes—
86C2×Dic53C2×D412yes—
87D212—21no—
88D212—21no—
89C4×D53—42no—
90Dic106—22no—
91Dic106—21no—
92Dic106#4112yes—
93C8×C53—21no—
94C8×C53—21no—
95#88322—22no—
96C22×C53C22×D411yescharacteristic
97C2×D53—41no—
98C2×D53—41no—
99C2×D53—41no—
100C4×C53—21no—
101C4×C53—42no—
102C4×C53—42no—
103C4×C53#13512yes—
104C4×C53C2×#4111yescharacteristic
105C4×C532−1+411yescharacteristic
106Dic53—22no—
107Dic53—22no—
108Dic53—22no—
109Dic53—41no—
110Dic53—41no—
111#3364—531nomaximal, Sylow
112C2×C53#54811yescharacteristic
113C2×C53C2×#13511yescharacteristic
114C2×C53#57811yescharacteristic
115D53—81no—
116#557—531no—
117#556—531no—
118#488—532no—
119#487D5311yescharacteristic, complemented
120#479—532no—
121#474—531no—
122#464—532no—
123#462—531no—
124C8×Q8—531no—
125C4×Q16—532no—
126C4×SD16—531no—
127C53#336411yescharacteristic, complemented, Sylow
128C2×Q16—532no—
129C2×SD16—531no—
130C4⋊Q8—1061no—
131C4⋊Q8C2×D5311yescharacteristic
132C4⋊Q8C2×D5311yescharacteristic
133#126—1062no—
134#125—532no—
135#125—1062no—
136#124—532no—
137C22⋊Q8—1061no—
138C4×Q8—531no—
139C4×Q8—532no—
140C4×Q8—532no—
141C4×Q8—531no—
142C4×Q8—1061no—
143C4×D4—531no—
144#117—1061no—
145#107—532no—
146#106—531no—
147C4⋊C8—532no—
148C4⋊C8—531no—
149#103—532no—
150#103—531no—
151#103C2×D5312yes—
152#103C2×D5312yes—
153#102—531no—
154#102—532no—
155C4×C8—532no—
156C4×C8C2×D5311yescharacteristic
157C2×Q8—21no—
158C2×Q8—531no—
159C2×Q8—531no—
160C2×Q8C22×D5312yes—
161C2×D4—531no—
162C22×C4—1061no—
163Q16—1064no—
164SD16—1062no—
165C2×C8—532no—
166C2×C8C22×D5311yescharacteristic
167C2×C8C22×D5311yescharacteristic
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—531no—
172C4⋊C4—532no—
173C4⋊C4—531no—
174C4⋊C4—531no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4—1061no—
180C4⋊C4C22×D5312yes—
181C22⋊C4—1062no—
182C22⋊C4—1062no—
183C22⋊C4—1061no—
184C42—532no—
185C42—1062no—
186C42—1062no—
187C42—1061no—
188C42C22×D5311yescharacteristic
189C23—1061no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—532no—
194Q8—1062no—
195Q8—1061no—
196D4—1061no—
197D4—1061no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—532no—
201C2×C4—532no—
202C2×C4—1061no—
203C2×C4—1061no—
204C2×C4—1061no—
205C2×C4—2122no—
206C2×C4D4×D5311yescharacteristic
207C2×C4D4×D5311yescharacteristic
208C2×C4C23×D5311yescharacteristic
209C8—21no—
210C8—21no—
211C8—1062no—
212C22—2121no—
213C22—2121no—
214C22—2121no—
215C22C2×D4×D5311yescharacteristic
216C4—21no—
217C4—42no—
218C4—42no—
219C4—1062no—
220C4—1062no—
221C4—1062no—
222C4—2121no—
223C4—2121no—
224C4#1313323812yes—
225C4#41×D5311yescharacteristic
226C4#1313326411yescharacteristic
227C2—4241no—
228C2#1681128111yescharacteristic
229C2C2×#1313323811yescharacteristic
230C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3364
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #16811348C2semidirect
4. #16810928C2semidirect
7. #16810830C2semidirect
8. C8×Dic106C2semidirect
9. #16810715C2semidirect
11. C53×#487C2semidirect
119. #487D53semidirect
127. C53#3364semidirect