Finite Group Databasebeta

← #20285446

Subgroups — #20285446

Order 6784.

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

normal subgroups only

This group has 370 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
1#16811158C212yesmaximal, complemented
2#16811007C211yesmaximal, characteristic, complemented
3#16810980C212yesmaximal, complemented
4#16810882C211yesmaximal, characteristic, complemented
5#16810877C211yesmaximal, characteristic
6#16810870C212yesmaximal, complemented
7#16810734C211yesmaximal, characteristic, complemented
8#16810729C211yesmaximal, characteristic, complemented
9#16810722C212yesmaximal, complemented
10C4×#11719912C211yesmaximal, characteristic, complemented
11C53×#499C211yesmaximal, characteristic, complemented
12C2×Dic212C2211yescharacteristic
13C2×#11719913C2212yes—
14C2×D424C2211yescharacteristic
15C2×#11719912C2211yescharacteristic
16C2×#11719912C2211yescharacteristic
17#13133212—22no—
18#13133205C2212yes—
19#13133202—22no—
20#13133197C2212yes—
21#13133192C2212yes—
22#13133182C2212yes—
23#13133179—22no—
24#13133179C2212yes—
25#13133175—22no—
26C42×D53C2211yescharacteristic
27#13133140C2212yes—
28#13133140C2212yes—
29#13133137C2212yes—
30#13133137C2212yes—
31#13133127C2211yescharacteristic
32#13133127C2211yescharacteristic
33C4×#88322C2211yescharacteristic
34#125×C53C2212yes—
35#107×C53C2212yes—
36#106×C53C2211yescharacteristic
37#106×C53C2211yescharacteristic
38C4×C8×C53C2211yescharacteristic
39C2×D212—22no—
40C2×D212C2312yes—
41C2×C4×D53—22no—
42C2×C4×D53D412yes—
43C2×C4×D53C2311yescharacteristic
44C2×Dic106C2312yes—
45C2×C8×C53C2311yescharacteristic
46C2×C8×C53C2311yescharacteristic
47C4⋊C4×C53—22no—
48C4⋊C4×C53—22no—
49C4⋊C4×C53C2312yes—
50C4⋊C4×C53C2312yes—
51C42×C53C2311yescharacteristic
52#11719921—22no—
53#11719921—22no—
54#11719921—22no—
55#11719921—22no—
56#11719921—22no—
57C4⋊Dic53—22no—
58#11719919—22no—
59C4×Dic53—22no—
60C4×Dic53D412yes—
61C4×Dic53C2311yescharacteristic
62C2×#88322C2311yescharacteristic
63C2×#88322C2311yescharacteristic
64Dic212—22no—
65D424—22no—
66#11719913—24no—
67#11719912—22no—
68#11719912—22no—
69C22×D53—22no—
70C22×D53C2×D411yescharacteristic
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53C2×D412yes—
74C2×C4×C53C2411yescharacteristic
75C2×Dic53—22no—
76C2×Dic53C2×D412yes—
77C2×Dic53C2×D411yescharacteristic
78D212—22no—
79D212—22no—
80D212—44no—
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—42no—
84C4×D53—44no—
85Dic106—22no—
86Dic106—22no—
87C8×C53#4112yes—
88C8×C53#4112yes—
89#88322—21no—
90#88322—21no—
91C22×C53C22×D411yescharacteristic
92C2×D53—21no—
93C2×D53—22no—
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—42no—
97C4×C53—22no—
98C4×C53—42no—
99C4×C53—42no—
100C4×C53C2×#4111yescharacteristic
101C4×C53C2×#4111yescharacteristic
102Dic53—21no—
103Dic53—22no—
104Dic53—42no—
105#3183—531nomaximal, Sylow
106C2×C53#53211yescharacteristic
107C2×C53#57512yes—
108D53—41no—
109D53—82no—
110#529—532no—
111#499D5311yescharacteristic, complemented
112#495—531no—
113#488—532no—
114#470—531no—
115#469—531no—
116#468—532no—
117#442—531no—
118#441—531no—
119#440—532no—
120C4×M16—531no—
121C53#318311yescharacteristic, complemented, Sylow
122C2×Q16—531no—
123C2×SD16—532no—
124C2×D8—531no—
125C2×M16—531no—
126C2×M16—531no—
127#126—1062no—
128#125C2×D5312yes—
129#124—532no—
130#123—1062no—
131C22⋊Q8—532no—
132#121—532no—
133C4×Q8—532no—
134C4×D4—532no—
135C4×D4—1062no—
136#117—1062no—
137C2×C42—531no—
138#107C2×D5312yes—
139#106C2×D5311yescharacteristic
140#106C2×D5311yescharacteristic
141#103—532no—
142#103—532no—
143#102—532no—
144#102—532no—
145#97—531no—
146#97—531no—
147C4×C8—531no—
148C4×C8C2×D5311yescharacteristic
149C2×Q8—532no—
150C2×D4—532no—
151C2×D4—1062no—
152C22×C4—531no—
153C22×C4—532no—
154C22×C4—1062no—
155Q16—1062no—
156SD16—1064no—
157D8—1062no—
158M16—1062no—
159M16—1062no—
160C2×C8—531no—
161C2×C8—531no—
162C2×C8C22×D5311yescharacteristic
163C2×C8C22×D5311yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4C22×D5312yes—
169C4⋊C4C22×D5312yes—
170C22⋊C4—1062no—
171C22⋊C4—1062no—
172C22⋊C4—1062no—
173C22⋊C4—1062no—
174C22⋊C4—1062no—
175C42—532no—
176C42—531no—
177C42—1062no—
178C42C22×D5311yescharacteristic
179C23—531no—
180C23—1062no—
181Q8—1062no—
182Q8—1062no—
183D4—1062no—
184D4—1062no—
185D4—2124no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—532no—
189C2×C4—531no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1062no—
193C2×C4—2122no—
194C2×C4—2124no—
195C2×C4D4×D5312yes—
196C2×C4C23×D5311yescharacteristic
197C8—1061no—
198C8—1061no—
199C8#1171994912yes—
200C8#1171994912yes—
201C22—1062no—
202C22—1061no—
203C22—2122no—
204C22—2122no—
205C22—2122no—
206C22C2×D4×D5311yescharacteristic
207C4—22no—
208C4—42no—
209C4—42no—
210C4—1062no—
211C4—1061no—
212C4—2122no—
213C4C2×#1171994911yescharacteristic
214C4C2×#1171994911yescharacteristic
215C2—4242no—
216C2—2121no—
217C2#1681117411yescharacteristic
218C2#1681142012yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#499C2
socleC22×C53C22×D4
hypercenter#499D53
hypocenterC53#3183
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
1. #16811158C2semidirect
2. #16811007C2semidirect
3. #16810980C2semidirect
4. #16810882C2semidirect
6. #16810870C2semidirect
7. #16810734C2semidirect
8. #16810729C2semidirect
9. #16810722C2semidirect
10. C4×#11719912C2semidirect
11. C53×#499C2semidirect
111. #499D53semidirect
121. C53#3183semidirect