Finite Group Databasebeta

← #20285342

Subgroups — #20285342

Order 6784.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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#16811236C211yesmaximal, characteristic, complemented
2#16811227C211yesmaximal, characteristic
3#16811022C211yesmaximal, characteristic, complemented
4#16810995C211yesmaximal, characteristic, complemented
5#16810992C211yesmaximal, characteristic, complemented
6#16810868C212yesmaximal, complemented
7#16810851C212yesmaximal, complemented
8#16810720C212yesmaximal, complemented
9#16810714C212yesmaximal, complemented
10#16810580C211yesmaximal, characteristic, complemented
11C53×#490C211yesmaximal, characteristic, complemented
12C2×#11719923C2212yes—
13C2×#11719923C2212yes—
14C2×#11719912C2211yescharacteristic
15C2×C8×D53C2211yescharacteristic
16#13133217C2211yescharacteristic
17#13133212—22no—
18#13133209C2211yescharacteristic
19#13133208—22no—
20#13133198—22no—
21#13133197C2212yes—
22#13133189—22no—
23#13133188C2212yes—
24#13133182C2212yes—
25D4×Dic53C2212yes—
26#13133174C2211yescharacteristic
27#13133150C2212yes—
28#13133148C2212yes—
29#13133142C2212yes—
30#13133142C2212yes—
31#13133127C2211yescharacteristic
32C4×#88322C2211yescharacteristic
33C8×Dic53C2211yescharacteristic
34#127×C53C2211yescharacteristic
35#125×C53C2211yescharacteristic
36#102×C53C2212yes—
37#102×C53C2212yes—
38#97×C53C2211yescharacteristic
39C2×D4×C53—22no—
40C2×D4×C53C2312yes—
41C2×C22⋊D53—22no—
42C22×Dic53—22no—
43C2×C4×D53C2311yescharacteristic
44C2×Dic106C2312yes—
45C2×C8×C53C2311yescharacteristic
46C2×C8×C53C2311yescharacteristic
47C4⋊C4×C53—22no—
48C4⋊C4×C53C2312yes—
49C42×C53C2311yescharacteristic
50C22⋊Dic53—22no—
51C22⋊Dic53—22no—
52#11719923—24no—
53#11719923—24no—
54#11719921—22no—
55#11719921D412yes—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53C2312yes—
59#11719919—22no—
60#11719919—22no—
61#11719919D412yes—
62C4×Dic53—22no—
63C4×Dic53C2311yescharacteristic
64C2×#88322C2311yescharacteristic
65C2×#88322C2311yescharacteristic
66#11719912—22no—
67C8×D53—22no—
68C23×C53—22no—
69C22×D53C2×D411yescharacteristic
70D4×C53—22no—
71D4×C53—22no—
72D4×C53—82no—
73C2×C4×C53—22no—
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2411yescharacteristic
76C22⋊D53—82no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—82no—
80C2×Dic53C2×D411yescharacteristic
81C2×Dic53C2×D412yes—
82C4×D53—21no—
83C4×D53—21no—
84Dic106—22no—
85Dic106—22no—
86C8×C53—21no—
87C8×C53—21no—
88#88322#4112yes—
89#88322#4112yes—
90C22×C53—42no—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53C22×D411yescharacteristic
94C2×D53—21no—
95C2×D53—22no—
96C4×C53—22no—
97C4×C53—42no—
98C4×C53C2×#4111yescharacteristic
99C4×C53C2×#4111yescharacteristic
100Dic53—21no—
101Dic53—22no—
102Dic53—42no—
103Dic53—42no—
104#3174—531nomaximal, Sylow
105C2×C53—82no—
106C2×C53#53211yescharacteristic
107C2×C53#57712yes—
108D53—41no—
109#541—531no—
110#540—531no—
111#499—531no—
112#492—531no—
113#491—531no—
114#490D5311yescharacteristic, complemented
115#468—532no—
116#465—532no—
117#440—532no—
118C4×SD16—532no—
119#405—531no—
120C53#317411yescharacteristic, complemented, Sylow
121C2×SD16—532no—
122C2×SD16—532no—
123C2×M16—531no—
124C22×C8—531no—
125C4⋊Q8—531no—
126#127C2×D5311yescharacteristic
127#126—1062no—
128#125—531no—
129#125—1062no—
130#125C2×D5311yescharacteristic
131#123—1062no—
132C22⋊Q8—532no—
133#121—532no—
134#121—1062no—
135C4×Q8—532no—
136C4×D4—532no—
137#117—531no—
138#107—532no—
139#106—532no—
140#103—532no—
141#103—532no—
142#102C2×D5312yes—
143#102C2×D5312yes—
144#97—531no—
145#97C2×D5311yescharacteristic
146C4×C8—531no—
147C4×C8—531no—
148C2×Q8—532no—
149C2×D4—22no—
150C2×D4—1062no—
151C2×D4C22×D5312yes—
152C22×C4—531no—
153C22×C4—1062no—
154SD16—1064no—
155SD16—1064no—
156M16—1062no—
157C2×C8—531no—
158C2×C8—531no—
159C2×C8—1062no—
160C2×C8C22×D5311yescharacteristic
161C2×C8C22×D5311yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—532no—
164C4⋊C4—532no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4—1062no—
169C4⋊C4C22×D5312yes—
170C22⋊C4—532no—
171C22⋊C4—1062no—
172C22⋊C4—1062no—
173C22⋊C4—1062no—
174C42—531no—
175C42—1062no—
176C42C22×D5311yescharacteristic
177C23—22no—
178C23—531no—
179Q8—1062no—
180Q8—1062no—
181D4—22no—
182D4—22no—
183D4—82no—
184D4—4242no—
185C2×C4—22no—
186C2×C4—532no—
187C2×C4—531no—
188C2×C4—1061no—
189C2×C4—1061no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—4242no—
193C2×C4D4×D5312yes—
194C2×C4C23×D5311yescharacteristic
195C8—21no—
196C8—21no—
197C8—532no—
198C8—532no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—1061no—
203C22—1062no—
204C22C2×D4×D5311yescharacteristic
205C4—22no—
206C4—42no—
207C4—1061no—
208C4—1062no—
209C4—2122no—
210C4—2122no—
211C4#41×D5311yescharacteristic
212C4#41×D5311yescharacteristic
213C2—82no—
214C2—2121no—
215C2#1681117211yescharacteristic
216C2#1681143312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#490C2
socleC22×C53C22×D4
hypercenter#490D53
hypocenterC53#3174
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #16811236C2semidirect
3. #16811022C2semidirect
4. #16810995C2semidirect
5. #16810992C2semidirect
6. #16810868C2semidirect
7. #16810851C2semidirect
8. #16810720C2semidirect
9. #16810714C2semidirect
10. #16810580C2semidirect
11. C53×#490C2semidirect
114. #490D53semidirect
120. C53#3174semidirect