Finite Group Databasebeta

← #20287410

Subgroups — #20287410

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#16811358C211yesmaximal, characteristic
2#16811348C211yesmaximal, characteristic, complemented
3#16810979C212yesmaximal, complemented
4#16810971C211yesmaximal, characteristic
5#16810922C212yesmaximal, complemented
6#16810848C212yesmaximal, complemented
7#16810824C211yesmaximal, characteristic
8#16810747C211yesmaximal, characteristic, complemented
9C4×#11719925C211yesmaximal, characteristic, complemented
10#16810710C212yesmaximal, complemented
11C53×#475C211yesmaximal, characteristic, complemented
12C2×#11719925C2211yescharacteristic
13C2×#11719913C2212yes—
14#13133215—21no—
15#13133214C2211yescharacteristic
16#13133208C2212yes—
17#13133207—22no—
18#13133206C2212yes—
19#13133205—21no—
20#13133202—22no—
21Q8×Dic53—21no—
22C4×Dic106C2211yescharacteristic
23C4×Dic106C2211yescharacteristic
24#13133182C2212yes—
25C4×D212—21no—
26#13133179C2212yes—
27#13133150C2211yescharacteristic
28#13133146C2212yes—
29#13133141C2211yescharacteristic
30#13133140C2211yescharacteristic
31#13133139C2212yes—
32#13133137C2212yes—
33#13133137C2212yes—
34C4×#88322C2211yescharacteristic
35C8×Dic53C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4×Q8×C53C2211yescharacteristic
38#106×C53C2212yes—
39C4⋊C8×C53C2211yescharacteristic
40#103×C53C2212yes—
41C2×Q8×C53—21no—
42C2×Q8×C53C2311yescharacteristic
43C2×D212—21no—
44C2×D212—21no—
45C2×D212C2312yes—
46C2×C4×D53—22no—
47C2×Dic106D411yescharacteristic
48C2×Dic106C2311yescharacteristic
49C2×C8×C53C2312yes—
50C4⋊C4×C53—21no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53C2312yes—
53C4⋊C4×C53C2311yescharacteristic
54C42×C53—21no—
55C42×C53C2311yescharacteristic
56#11719925—22no—
57#11719921—22no—
58#11719921—22no—
59#11719921—22no—
60C4⋊Dic53—22no—
61C4⋊Dic53D411yescharacteristic
62C4⋊Dic53C2311yescharacteristic
63#11719919—21no—
64#11719919—22no—
65#11719919—22no—
66#11719919D412yes—
67C4×Dic53—21no—
68C4×Dic53—22no—
69C4×Dic53C2312yes—
70C2×#88322C2311yescharacteristic
71C2×#88322C2311yescharacteristic
72#11719913—24no—
73C22×D53—22no—
74Q8×C53—21no—
75Q8×C53—21no—
76Q8×C53—42no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2411yescharacteristic
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53C2×D412yes—
86C2×Dic53C2×D412yes—
87D212—22no—
88D212—22no—
89D212—42no—
90D212—81no—
91C4×D53—82no—
92Dic106—22no—
93Dic106—21no—
94Dic106#4112yes—
95C8×C53—22no—
96#88322—21no—
97#88322—21no—
98C22×C53C22×D411yescharacteristic
99C2×D53—42no—
100C2×D53—42no—
101C2×D53—42no—
102C4×C53—21no—
103C4×C53—41no—
104C4×C53—42no—
105C4×C53—41no—
106C4×C53#13512yes—
107C4×C53C2×#4111yescharacteristic
108C4×C532−1+411yescharacteristic
109Dic53—22no—
110Dic53—22no—
111Dic53—22no—
112Dic53—41no—
113#3365—531nomaximal, Sylow
114C2×C53#54811yescharacteristic
115C2×C53C2×#13511yescharacteristic
116C2×C53#57711yescharacteristic
117D53—82no—
118#559—531no—
119#557—531no—
120#488—532no—
121#486—531no—
122#478—532no—
123#475D5311yescharacteristic, complemented
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#127—531no—
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—1061no—
144C4×Q8C2×D5311yescharacteristic
145C4×D4—532no—
146C4×D4—1061no—
147#107—531no—
148#106C2×D5312yes—
149C4⋊C8—532no—
150C4⋊C8C2×D5311yescharacteristic
151#103—531no—
152#103—531no—
153#103C2×D5312yes—
154#102—532no—
155#102—532no—
156#102—532no—
157C4×C8—532no—
158C4×C8—531no—
159C2×Q8—21no—
160C2×Q8—531no—
161C2×Q8—531no—
162C2×Q8C22×D5311yescharacteristic
163C2×D4—532no—
164C2×D4—1061no—
165C2×D4—1061no—
166C22×C4—1062no—
167Q16—1062no—
168SD16—1064no—
169C2×C8—531no—
170C2×C8—531no—
171C2×C8C22×D5312yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—531no—
175C4⋊C4—532no—
176C4⋊C4—531no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C4⋊C4—1061no—
181C4⋊C4C22×D5312yes—
182C4⋊C4C22×D5311yescharacteristic
183C22⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C42—21no—
187C42—532no—
188C42—1062no—
189C42—1061no—
190C42C22×D5311yescharacteristic
191C23—1062no—
192Q8—21no—
193Q8—21no—
194Q8—42no—
195Q8—532no—
196Q8—1062no—
197Q8—1061no—
198D4—1062no—
199D4—1062no—
200D4—4241no—
201D4—2122no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—532no—
206C2×C4—532no—
207C2×C4—1061no—
208C2×C4—1061no—
209C2×C4—4242no—
210C2×C4D4×D5311yescharacteristic
211C2×C4D4×D5311yescharacteristic
212C2×C4C23×D5311yescharacteristic
213C8—22no—
214C8—1061no—
215C8—1061no—
216C22—2122no—
217C22—2122no—
218C22—2122no—
219C22C2×D4×D5311yescharacteristic
220C4—21no—
221C4—41no—
222C4—42no—
223C4—41no—
224C4—1062no—
225C4—1062no—
226C4—1062no—
227C4—2121no—
228C4#1313323812yes—
229C4#41×D5311yescharacteristic
230C4#1313326511yescharacteristic
231C2—4242no—
232C2#1681128311yescharacteristic
233C2C2×#1313323811yescharacteristic
234C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#475C2
socleC22×C53C22×D4
hypercenter#475D53
hypocenterC53#3365
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #16811348C2semidirect
3. #16810979C2semidirect
5. #16810922C2semidirect
6. #16810848C2semidirect
8. #16810747C2semidirect
9. C4×#11719925C2semidirect
10. #16810710C2semidirect
11. C53×#475C2semidirect
123. #475D53semidirect
129. C53#3365semidirect