Finite Group Databasebeta

← #20287702

Subgroups — #20287702

Order 6784.

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

normal subgroups only

This group has 360 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#16811347C211yesmaximal, characteristic, complemented
2#16811343C211yesmaximal, characteristic, complemented
3#16811004C212yesmaximal
4#16811002C211yesmaximal, characteristic, complemented
5#16810848C212yesmaximal, complemented
6#16810843C212yesmaximal, complemented
7#16810829C211yesmaximal, characteristic, complemented
8#16810747C211yesmaximal, characteristic, complemented
9#16810715C212yesmaximal, complemented
10C4×#11719922C211yesmaximal, characteristic
11C53×#461C211yesmaximal, characteristic, complemented
12C2×#11719925C2212yes—
13C2×#11719925C2212yes—
14C2×#11719913C2212yes—
15C2×#11719922C2211yescharacteristic
16#13133217C2211yescharacteristic
17#13133212—22no—
18#13133210—22no—
19#13133206C2212yes—
20#13133204C2212yes—
21#13133197—21no—
22#13133196—21no—
23Q8×Dic53C2212yes—
24C4×Dic106C2211yescharacteristic
25#13133182C2212yes—
26C4×D212C2211yescharacteristic
27#13133176—21no—
28#13133175—21no—
29#13133150C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2212yes—
32#13133138C2211yescharacteristic
33#13133137C2211yescharacteristic
34C4×#88322C2211yescharacteristic
35C8×Dic53C2212yes—
36C2×SD16×C53C2212yes—
37C4⋊Q8×C53C2211yescharacteristic
38C4×D4×C53C2211yescharacteristic
39C4⋊C8×C53C2211yescharacteristic
40#103×C53C2212yes—
41C2×Q8×C53C2312yes—
42C2×D4×C53C2311yescharacteristic
43C22×C4×C53—21no—
44C2×D212C2311yescharacteristic
45C2×C4×D53—21no—
46C2×Dic106D411yescharacteristic
47C2×Dic106C2312yes—
48SD16×C53—42no—
49C2×C8×C53C2312yes—
50C4⋊C4×C53—21no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53C2311yescharacteristic
53C22⋊C4×C53—21no—
54C42×C53C2311yescharacteristic
55C22⋊Dic53—22no—
56C22⋊Dic53—22no—
57#11719925—24no—
58#11719925—42no—
59#11719922—22no—
60#11719921—21no—
61#11719921—22no—
62#11719921—22no—
63C4⋊Dic53—21no—
64C4⋊Dic53—21no—
65C4⋊Dic53—22no—
66C4⋊Dic53D411yescharacteristic
67C4⋊Dic53C2311yescharacteristic
68#11719919—22no—
69#11719919D412yes—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53C2312yes—
73C2×#88322C2311yescharacteristic
74C2×#88322C2311yescharacteristic
75#11719913—42no—
76C23×C53—21no—
77C22×D53—21no—
78Q8×C53—22no—
79Q8×C53—22no—
80D4×C53—21no—
81D4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—42no—
85C2×C4×C53C2×D411yescharacteristic
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2411yescharacteristic
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53C2×D412yes—
91C2×Dic53C2×D412yes—
92D212—21no—
93D212—21no—
94C4×D53—42no—
95Dic106—22no—
96Dic106—22no—
97Dic106—22no—
98C8×C53—22no—
99#88322—21no—
100#88322#4112yes—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53C22×D411yescharacteristic
105C2×D53—41no—
106C2×D53—41no—
107C2×D53—41no—
108C4×C53—21no—
109C4×C53—42no—
110C4×C53—41no—
111C4×C53#13512yes—
112C4×C53C2×#4111yescharacteristic
113C4×C53H(5,𝔽2)11yescharacteristic
114Dic53—22no—
115Dic53—22no—
116Dic53—42no—
117Dic53—41no—
118#3393—531nomaximal, Sylow
119C2×C53—81no—
120C2×C53#55011yescharacteristic
121C2×C53C2×#13511yescharacteristic
122C2×C53#57811yescharacteristic
123D53—81no—
124#556—531no—
125#556—531no—
126#495—532no—
127#494—531no—
128#464—532no—
129#464—532no—
130#461—531no—
131#461D5311yescharacteristic, complemented
132C8×Q8—531no—
133C4×Q16—532no—
134C4×D8—531no—
135C53#339311yescharacteristic, complemented, Sylow
136C2×Q16—532no—
137C2×Q16—532no—
138C2×SD16—532no—
139C2×SD16C2×D5312yes—
140C2×D8—531no—
141C4⋊Q8—531no—
142C4⋊Q8C2×D5311yescharacteristic
143#126—1062no—
144#126—1062no—
145#124—532no—
146#124—532no—
147C22⋊Q8—1061no—
148C22⋊Q8—1061no—
149C4×Q8—531no—
150C4×Q8—532no—
151C4×Q8—532no—
152C4×D4—531no—
153C4×D4C2×D5311yescharacteristic
154#117—1061no—
155#117—1061no—
156#107—531no—
157C4⋊C8—532no—
158C4⋊C8C2×D5311yescharacteristic
159#103—532no—
160#103C2×D5312yes—
161#102—531no—
162#102—531no—
163C4×C8—532no—
164C4×C8—531no—
165C2×Q8—531no—
166C2×Q8—532no—
167C2×Q8C22×D5312yes—
168C2×D4—531no—
169C2×D4C22×D5311yescharacteristic
170C22×C4—21no—
171C22×C4—1061no—
172Q16—1064no—
173Q16—2122no—
174SD16—42no—
175SD16—2122no—
176D8—1062no—
177C2×C8—531no—
178C2×C8—531no—
179C2×C8C22×D5312yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—531no—
183C4⋊C4—532no—
184C4⋊C4—531no—
185C4⋊C4—1062no—
186C4⋊C4—1062no—
187C4⋊C4—1061no—
188C4⋊C4—1061no—
189C4⋊C4C22×D5311yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—1062no—
192C22⋊C4—1062no—
193C22⋊C4—1062no—
194C22⋊C4—1062no—
195C22⋊C4—1061no—
196C42—532no—
197C42—1062no—
198C42—1062no—
199C42C22×D5311yescharacteristic
200C23—21no—
201C23—1061no—
202Q8—22no—
203Q8—22no—
204Q8—1062no—
205Q8—1062no—
206Q8—1062no—
207D4—21no—
208D4—21no—
209D4—1061no—
210D4—1061no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—532no—
215C2×C4—532no—
216C2×C4—1061no—
217C2×C4—1062no—
218C2×C4—2122no—
219C2×C4D4×D5311yescharacteristic
220C2×C4D4×D5311yescharacteristic
221C2×C4C23×D5311yescharacteristic
222C8—22no—
223C8—532no—
224C8—1061no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—2121no—
229C22—2121no—
230C22—2121no—
231C22C2×D4×D5311yescharacteristic
232C4—21no—
233C4—42no—
234C4—41no—
235C4—1062no—
236C4—1062no—
237C4—2122no—
238C4—2121no—
239C4#1313323812yes—
240C4#41×D5311yescharacteristic
241C4#1313326211yescharacteristic
242C2—81no—
243C2—4241no—
244C2#1681129111yescharacteristic
245C2C2×#1313323811yescharacteristic
246C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#461C2
socleC22×C53C22×D4
hypercenter#461D53
hypocenterC53#3393
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #16811347C2semidirect
2. #16811343C2semidirect
4. #16811002C2semidirect
5. #16810848C2semidirect
6. #16810843C2semidirect
7. #16810829C2semidirect
8. #16810747C2semidirect
9. #16810715C2semidirect
11. C53×#461C2semidirect
131. #461D53semidirect
135. C53#3393semidirect