Finite Group Databasebeta

← #20285771

Subgroups — #20285771

Order 6784.

388 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 109 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#16811227C211yesmaximal, characteristic, complemented
2#16811193C211yesmaximal, characteristic, complemented
3#16810995C211yesmaximal, characteristic, complemented
4#16810992C211yesmaximal, characteristic, complemented
5#16810977C211yesmaximal, characteristic, complemented
6#16810836C212yesmaximal, complemented
7#16810826C212yesmaximal, complemented
8#16810765C212yesmaximal, complemented
9#16810760C212yesmaximal, complemented
10#16810685C211yesmaximal, characteristic, complemented
11C53×#496C211yesmaximal, characteristic, complemented
12C2×#11719947C2211yescharacteristic
13C2×Q8×D53C2211yescharacteristic
14C2×#11719923C2212yes—
15C2×#11719923C2212yes—
16#13133217C2211yescharacteristic
17#13133215C2211yescharacteristic
18#13133209C2211yescharacteristic
19#13133204C2211yescharacteristic
20#13133199—22no—
21#13133198—22no—
22#13133194—22no—
23#13133189—22no—
24#13133188C2212yes—
25D4×Dic53C2212yes—
26#13133174C2211yescharacteristic
27#13133146C2212yes—
28#13133142C2212yes—
29#13133142C2212yes—
30#13133141C2212yes—
31#13133138C2212yes—
32#13133129C2212yes—
33C4×#88322C2211yescharacteristic
34C2×SD16×C53C2212yes—
35C2×D8×C53C2212yes—
36#127×C53C2211yescharacteristic
37#124×C53C2211yescharacteristic
38#97×C53C2211yescharacteristic
39C2×Q8×C53C2311yescharacteristic
40C2×D4×C53—22no—
41C2×D4×C53C2311yescharacteristic
42C2×D4×C53C2312yes—
43C2×C22⋊D53—21no—
44C2×C22⋊D53—22no—
45C22×Dic53—21no—
46C22×Dic53—22no—
47Q8×D53—42no—
48#11719947—42no—
49C2×C4×D53—21no—
50C2×C4×D53C2311yescharacteristic
51C2×Dic106—21no—
52C2×Dic106C2311yescharacteristic
53C2×Dic106C2311yescharacteristic
54SD16×C53—42no—
55D8×C53—42no—
56C2×C8×C53C2312yes—
57C22⋊C4×C53—22no—
58C42×C53C2311yescharacteristic
59C22⋊Dic53—22no—
60C22⋊Dic53—22no—
61C22⋊Dic53—22no—
62#11719923—42no—
63#11719923—42no—
64#11719921D412yes—
65C4⋊Dic53—22no—
66C4⋊Dic53C2312yes—
67#11719919—22no—
68#11719919—22no—
69#11719919D412yes—
70C4×Dic53C2311yescharacteristic
71C2×#88322C2312yes—
72C23×C53—22no—
73C23×C53—21no—
74C22×D53C2×D411yescharacteristic
75Q8×C53—22no—
76D4×C53—22no—
77D4×C53—22no—
78D4×C53—22no—
79D4×C53—82no—
80C2×C4×C53—21no—
81C2×C4×C53C2×D412yes—
82C2×C4×C53C2411yescharacteristic
83C22⋊D53—42no—
84C22⋊D53—82no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—22no—
88C2×Dic53—42no—
89C2×Dic53—82no—
90C2×Dic53C2×D411yescharacteristic
91C2×Dic53C2×D412yes—
92C4×D53—22no—
93C4×D53—42no—
94Dic106—22no—
95Dic106—22no—
96Dic106—42no—
97C8×C53—22no—
98#88322—22no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53—42no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53C22×D411yescharacteristic
105C2×D53—21no—
106C2×D53—22no—
107C4×C53—22no—
108C4×C53—41no—
109C4×C53H(5,𝔽2)12yes—
110Dic53—21no—
111Dic53—22no—
112Dic53—41no—
113Dic53—41no—
114Dic53—42no—
115#3221—531nomaximal, Sylow
116C2×C53—81no—
117C2×C53—82no—
118C2×C53#53511yescharacteristic
119C2×C53#57712yes—
120D53—41no—
121#540—531no—
122#536—531no—
123#496D5311yescharacteristic, complemented
124#492—531no—
125#491—531no—
126#488—531no—
127#463—532no—
128#461—532no—
129#449—532no—
130#448—532no—
131#433—531no—
132C53#322111yescharacteristic, complemented, Sylow
133C2×#41—531no—
134C22×Q8—531no—
135C2×SD16—532no—
136C2×SD16—532no—
137C2×SD16C2×D5312yes—
138C2×D8C2×D5312yes—
139C4⋊Q8—531no—
140C4⋊Q8—531no—
141#127C2×D5311yescharacteristic
142#125—531no—
143#124—531no—
144#124C2×D5311yescharacteristic
145#123—1062no—
146#123—1062no—
147C22⋊Q8—1062no—
148#121—532no—
149#121—1062no—
150C4×D4—532no—
151#117—531no—
152C4⋊C8—532no—
153#103—532no—
154#103—532no—
155#103—532no—
156#102—532no—
157C22⋊C8—532no—
158#97C2×D5311yescharacteristic
159C4×C8—531no—
160#41—2122no—
161C2×Q8—531no—
162C2×Q8—531no—
163C2×Q8—1061no—
164C2×Q8—2122no—
165C2×Q8C22×D5311yescharacteristic
166C2×D4—22no—
167C2×D4—1062no—
168C2×D4—1061no—
169C2×D4C22×D5311yescharacteristic
170C2×D4C22×D5312yes—
171C22×C4—531no—
172C22×C4—1061no—
173C22×C4—1062no—
174C22×C4—1061no—
175SD16—42no—
176SD16—2122no—
177SD16—2122no—
178D8—42no—
179C2×C8—532no—
180C2×C8C22×D5312yes—
181C4⋊C4—532no—
182C4⋊C4—532no—
183C4⋊C4—1062no—
184C4⋊C4—1062no—
185C4⋊C4—1062no—
186C22⋊C4—22no—
187C22⋊C4—532no—
188C22⋊C4—1062no—
189C22⋊C4—1062no—
190C22⋊C4—1062no—
191C42—531no—
192C42C22×D5311yescharacteristic
193C23—21no—
194C23—22no—
195C23—531no—
196Q8—22no—
197Q8—1062no—
198Q8—1062no—
199Q8—2122no—
200D4—22no—
201D4—22no—
202D4—22no—
203D4—82no—
204D4—4242no—
205D4—2122no—
206C2×C4—21no—
207C2×C4—532no—
208C2×C4—531no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—4242no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4D4×D5312yes—
217C2×C4C23×D5311yescharacteristic
218C8—22no—
219C8—1062no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—41no—
224C22—42no—
225C22—1062no—
226C22—1061no—
227C22C2×D4×D5311yescharacteristic
228C4—22no—
229C4—41no—
230C4—1062no—
231C4—1061no—
232C4—2122no—
233C4—2121no—
234C4—2121no—
235C4#1313326212yes—
236C2—82no—
237C2—81no—
238C2—2121no—
239C2#1681118611yescharacteristic
240C2#1681143312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#496C2
socleC22×C53C22×D4
hypercenter#496D53
hypocenterC53#3221
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #16811227C2semidirect
2. #16811193C2semidirect
3. #16810995C2semidirect
4. #16810992C2semidirect
5. #16810977C2semidirect
6. #16810836C2semidirect
7. #16810826C2semidirect
8. #16810765C2semidirect
9. #16810760C2semidirect
10. #16810685C2semidirect
11. C53×#496C2semidirect
123. #496D53semidirect
132. C53#3221semidirect