Finite Group Databasebeta

← #20285345

Subgroups — #20285345

Order 6784.

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

normal subgroups only

This group has 362 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#16811236C211yesmaximal, characteristic, complemented
2#16811229C211yesmaximal, characteristic
3#16810996C211yesmaximal, characteristic, complemented
4#16810993C211yesmaximal, characteristic, complemented
5#16810989C211yesmaximal, characteristic, complemented
6#16810870C212yesmaximal, complemented
7#16810853C212yesmaximal, complemented
8#16810722C212yesmaximal, complemented
9#16810710C212yesmaximal, complemented
10#16810577C211yesmaximal, characteristic, complemented
11C53×#499C211yesmaximal, characteristic, complemented
12C2×#11719913C2212yes—
13C2×#11719913C2212yes—
14C2×#11719912C2211yescharacteristic
15C2×C8×D53C2211yescharacteristic
16#13133217C2211yescharacteristic
17#13133214C2211yescharacteristic
18#13133212—22no—
19#13133208—22no—
20#13133202—22no—
21#13133197C2212yes—
22#13133192—22no—
23#13133192C2212yes—
24#13133182C2212yes—
25#13133179C2212yes—
26#13133174C2211yescharacteristic
27#13133140C2212yes—
28#13133140C2212yes—
29#13133137C2212yes—
30#13133137C2212yes—
31#13133128C2211yescharacteristic
32#13133127C2211yescharacteristic
33C8×Dic53C2211yescharacteristic
34#125×C53C2211yescharacteristic
35#125×C53C2211yescharacteristic
36#107×C53C2212yes—
37#106×C53C2212yes—
38C4×C8×C53C2211yescharacteristic
39C2×D212—22no—
40C2×D212—22no—
41C2×D212C2312yes—
42C2×C4×D53—22no—
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#11719921D412yes—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58#11719919—22no—
59#11719919D412yes—
60C4×Dic53—22no—
61C4×Dic53C2311yescharacteristic
62C2×#88322C2311yescharacteristic
63C2×#88322C2311yescharacteristic
64#11719913—24no—
65#11719913—24no—
66#11719912—22no—
67C8×D53—22no—
68C22×D53—22no—
69C22×D53C2×D411yescharacteristic
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53C2×D412yes—
73C2×C4×C53C2411yescharacteristic
74C2×Dic53—22no—
75C2×Dic53C2×D412yes—
76C2×Dic53C2×D411yescharacteristic
77D212—22no—
78D212—22no—
79D212—82no—
80D212—82no—
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—82no—
84Dic106—22no—
85Dic106—22no—
86C8×C53#4112yes—
87C8×C53#4112yes—
88#88322—21no—
89#88322—21no—
90C22×C53C22×D411yescharacteristic
91C2×D53—21no—
92C2×D53—22no—
93C2×D53—42no—
94C2×D53—42no—
95C2×D53—42no—
96C4×C53—22no—
97C4×C53—42no—
98C4×C53—42no—
99C4×C53C2×#4111yescharacteristic
100C4×C53C2×#4111yescharacteristic
101Dic53—21no—
102Dic53—22no—
103Dic53—42no—
104#3174—531nomaximal, Sylow
105C2×C53#53211yescharacteristic
106C2×C53#57712yes—
107D53—41no—
108D53—82no—
109#541—531no—
110#540—531no—
111#499D5311yescharacteristic, complemented
112#492—531no—
113#491—531no—
114#490—531no—
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#127—531no—
127#126—1062no—
128#125—1062no—
129#125C2×D5311yescharacteristic
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#107C2×D5312yes—
139#106C2×D5312yes—
140#103—532no—
141#103—532no—
142#102—532no—
143#102—532no—
144#97—531no—
145#97—531no—
146C4×C8—531no—
147C4×C8C2×D5311yescharacteristic
148C2×Q8—532no—
149C2×D4—532no—
150C2×D4—1062no—
151C2×D4—1062no—
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—22no—
164C4⋊C4—532no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4C22×D5312yes—
169C4⋊C4C22×D5312yes—
170C22⋊C4—532no—
171C22⋊C4—1062no—
172C22⋊C4—1062no—
173C22⋊C4—1062no—
174C42—531no—
175C42—1062no—
176C42C22×D5311yescharacteristic
177C23—531no—
178C23—1062no—
179Q8—1062no—
180Q8—1062no—
181D4—1062no—
182D4—1062no—
183D4—4242no—
184D4—4242no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—532no—
188C2×C4—531no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1062no—
192C2×C4—4242no—
193C2×C4D4×D5312yes—
194C2×C4C23×D5311yescharacteristic
195C8—1061no—
196C8—1061no—
197C8#1171994912yes—
198C8#1171994912yes—
199C22—1061no—
200C22—1062no—
201C22—2122no—
202C22—2122no—
203C22—2122no—
204C22C2×D4×D5311yescharacteristic
205C4—22no—
206C4—42no—
207C4—42no—
208C4—1061no—
209C4—1062no—
210C4—2122no—
211C4C2×#1171994911yescharacteristic
212C4C2×#1171994911yescharacteristic
213C2—4242no—
214C2—2121no—
215C2#1681117411yescharacteristic
216C2#1681143012yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#499C2
socleC22×C53C22×D4
hypercenter#499D53
hypocenterC53#3174
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. #16811236C2semidirect
3. #16810996C2semidirect
4. #16810993C2semidirect
5. #16810989C2semidirect
6. #16810870C2semidirect
7. #16810853C2semidirect
8. #16810722C2semidirect
9. #16810710C2semidirect
10. #16810577C2semidirect
11. C53×#499C2semidirect
111. #499D53semidirect
120. C53#3174semidirect