Finite Group Databasebeta

← #20285441

Subgroups — #20285441

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#16811152C212yesmaximal, complemented
2#16811025C211yesmaximal, characteristic, complemented
3#16810977C212yesmaximal, complemented
4#16810878C211yesmaximal, characteristic, complemented
5#16810873C211yesmaximal, characteristic
6#16810866C212yesmaximal, complemented
7#16810735C211yesmaximal, characteristic, complemented
8#16810730C211yesmaximal, characteristic, complemented
9#16810724C212yesmaximal, complemented
10C4×#11719912C211yesmaximal, characteristic, complemented
11C53×#495C211yesmaximal, characteristic, complemented
12C2×#11719912C2211yescharacteristic
13C2×#11719912C2211yescharacteristic
14#13133210—22no—
15#13133209C2212yes—
16#13133199—22no—
17#13133193C2212yes—
18#13133188C2212yes—
19Q8×Dic53C2212yes—
20D4×Dic53C2212yes—
21#13133177—22no—
22#13133173—22no—
23C42×D53C2211yescharacteristic
24#13133151C2212yes—
25#13133149C2211yescharacteristic
26#13133149C2211yescharacteristic
27#13133141C2212yes—
28#13133141C2212yes—
29#13133138C2212yes—
30#13133138C2212yes—
31#13133127C2211yescharacteristic
32#13133127C2211yescharacteristic
33C4×#88322C2211yescharacteristic
34C2×Q16×C53C2211yescharacteristic
35C2×SD16×C53C2212yes—
36C2×D8×C53C2211yescharacteristic
37#124×C53C2212yes—
38C4×C8×C53C2211yescharacteristic
39C2×Q8×C53C2312yes—
40C2×D4×C53C2312yes—
41C2×C22⋊D53—22no—
42C22×Dic53—22no—
43C2×C4×D53D412yes—
44C2×C4×D53C2311yescharacteristic
45Q16×C53—22no—
46SD16×C53—24no—
47D8×C53—22no—
48C2×C8×C53C2311yescharacteristic
49C2×C8×C53C2311yescharacteristic
50C22⋊C4×C53—22no—
51C22⋊C4×C53—22no—
52C42×C53C2311yescharacteristic
53C22⋊Dic53—22no—
54C22⋊Dic53—22no—
55#11719921—22no—
56C4⋊Dic53—22no—
57C4⋊Dic53C2312yes—
58C4⋊Dic53C2312yes—
59#11719919—22no—
60#11719919—22no—
61#11719919—22no—
62C4×Dic53—22no—
63C4×Dic53D412yes—
64C4×Dic53C2311yescharacteristic
65C2×#88322C2311yescharacteristic
66C2×#88322C2311yescharacteristic
67#11719912—22no—
68#11719912—22no—
69C23×C53—22no—
70C22×D53C2×D411yescharacteristic
71Q8×C53—22no—
72Q8×C53—22no—
73D4×C53—22no—
74D4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53C2×D412yes—
77C2×C4×C53C2411yescharacteristic
78C22⋊D53—44no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—44no—
82C2×Dic53C2×D412yes—
83C2×Dic53C2×D411yescharacteristic
84C4×D53—21no—
85C4×D53—21no—
86C4×D53—42no—
87C8×C53#4112yes—
88C8×C53#4112yes—
89#88322—21no—
90#88322—21no—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53C22×D411yescharacteristic
95C2×D53—21no—
96C2×D53—22no—
97C4×C53—22no—
98C4×C53—42no—
99C4×C53C2×#4111yescharacteristic
100C4×C53C2×#4111yescharacteristic
101Dic53—21no—
102Dic53—22no—
103Dic53—42no—
104Dic53—42no—
105#3183—531nomaximal, Sylow
106C2×C53—82no—
107C2×C53#53211yescharacteristic
108C2×C53#57512yes—
109D53—41no—
110#529—532no—
111#499—531no—
112#495D5311yescharacteristic, complemented
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×Q16C2×D5311yescharacteristic
123C2×SD16C2×D5312yes—
124C2×D8C2×D5311yescharacteristic
125C2×M16—531no—
126C2×M16—531no—
127#126—1062no—
128#125—532no—
129#124C2×D5312yes—
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#107—532no—
139#106—531no—
140#106—531no—
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×Q8C22×D5312yes—
150C2×D4—1062no—
151C2×D4C22×D5312yes—
152C22×C4—531no—
153C22×C4—532no—
154C22×C4—1062no—
155Q16—22no—
156SD16—24no—
157D8—22no—
158M16—1062no—
159M16—1062no—
160C2×C8—531no—
161C2×C8—531no—
162C2×C8C22×D5311yescharacteristic
163C2×C8C22×D5311yescharacteristic
164C4⋊C4—532no—
165C4⋊C4—532no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4—1062no—
169C4⋊C4—1062no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—1062no—
173C22⋊C4—1062no—
174C22⋊C4—1062no—
175C42—532no—
176C42—531no—
177C42—1062no—
178C42C22×D5311yescharacteristic
179C23—22no—
180C23—531no—
181Q8—22no—
182Q8—22no—
183D4—22no—
184D4—22no—
185D4—2124no—
186C2×C4—22no—
187C2×C4—532no—
188C2×C4—531no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—2122no—
194C2×C4—2124no—
195C2×C4D4×D5312yes—
196C2×C4C23×D5311yescharacteristic
197C8—1061no—
198C8—1061no—
199C8#1171994712yes—
200C8#1171994712yes—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—1062no—
205C22—1061no—
206C22C2×D4×D5311yescharacteristic
207C4—22no—
208C4—42no—
209C4—1062no—
210C4—1061no—
211C4—2122no—
212C4—2122no—
213C4C2×#1171994711yescharacteristic
214C4C2×#1171994711yescharacteristic
215C2—82no—
216C2—2121no—
217C2#1681117111yescharacteristic
218C2#1681142012yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#495C2
socleC22×C53C22×D4
hypercenter#495D53
hypocenterC53#3183
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #16811152C2semidirect
2. #16811025C2semidirect
3. #16810977C2semidirect
4. #16810878C2semidirect
6. #16810866C2semidirect
7. #16810735C2semidirect
8. #16810730C2semidirect
9. #16810724C2semidirect
10. C4×#11719912C2semidirect
11. C53×#495C2semidirect
112. #495D53semidirect
121. C53#3183semidirect