Finite Group Databasebeta

← #20285423

Subgroups — #20285423

Order 6784.

366 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811178C211yesmaximal, characteristic, complemented
2#16811158C211yesmaximal, characteristic
3#16811033C211yesmaximal, characteristic, complemented
4#16811015C211yesmaximal, characteristic, complemented
5#16810983C211yesmaximal, characteristic, complemented
6#16810863C212yesmaximal, complemented
7#16810855C212yesmaximal
8#16810727C212yesmaximal, complemented
9#16810726C212yesmaximal, complemented
10#97×D53C211yesmaximal, characteristic, complemented
11C53×#491C211yesmaximal, characteristic, complemented
12C2×#11719925C2212yes—
13C2×#11719924C2212yes—
14C2×C8×D53C2212yes—
15#13133217C2211yescharacteristic
16#13133212—21no—
17#13133208—21no—
18#13133205C2211yescharacteristic
19#13133202—21no—
20#13133197—21no—
21#13133197C2211yescharacteristic
22#13133193C2212yes—
23#13133192C2211yescharacteristic
24Q8×Dic53C2212yes—
25#13133182—21no—
26#13133182C2211yescharacteristic
27#13133179—21no—
28#13133179C2211yescharacteristic
29#13133175—21no—
30#13133175—21no—
31C42×D53C2211yescharacteristic
32#13133150C2212yes—
33#13133148C2212yes—
34#13133142C2212yes—
35#13133139C2212yes—
36#13133128C2211yescharacteristic
37#13133127C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39#125×C53C2211yescharacteristic
40#103×C53C2212yes—
41#103×C53C2212yes—
42#97×C53C2211yescharacteristic
43C2×Q8×C53C2312yes—
44C2×D212—21no—
45C2×D212C2311yescharacteristic
46C2×C4×D53—21no—
47C2×C4×D53D411yescharacteristic
48C2×C4×D53D411yescharacteristic
49C2×C4×D53C2311yescharacteristic
50C2×Dic106—21no—
51C2×Dic106C2311yescharacteristic
52C2×C8×C53C2312yes—
53C4⋊C4×C53—21no—
54C4⋊C4×C53—21no—
55C4⋊C4×C53—21no—
56C4⋊C4×C53—21no—
57C4⋊C4×C53C2311yescharacteristic
58C4⋊C4×C53C2311yescharacteristic
59C42×C53C2311yescharacteristic
60#11719925—24no—
61#11719924—24no—
62#11719921—22no—
63#11719921—21no—
64#11719921—21no—
65#11719921—21no—
66#11719921—21no—
67#11719921—21no—
68C4⋊Dic53—21no—
69C4⋊Dic53—21no—
70C4⋊Dic53—22no—
71C4⋊Dic53C2312yes—
72#11719919—22no—
73#11719919—21no—
74C4×Dic53—21no—
75C4×Dic53—22no—
76C4×Dic53D411yescharacteristic
77C4×Dic53D411yescharacteristic
78C4×Dic53C2311yescharacteristic
79C2×#88322C2312yes—
80C8×D53—24no—
81C22×D53—21no—
82C22×D53C2×D411yescharacteristic
83Q8×C53—22no—
84Q8×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53C2×D411yescharacteristic
89C2×C4×C53C2×D411yescharacteristic
90C2×C4×C53C2411yescharacteristic
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53C2×D411yescharacteristic
94C2×Dic53C2×D411yescharacteristic
95C2×Dic53C2×D411yescharacteristic
96D212—22no—
97D212—42no—
98C4×D53—22no—
99C4×D53—22no—
100C4×D53—41no—
101C4×D53—42no—
102Dic106—22no—
103Dic106—42no—
104C8×C53—22no—
105#88322#4114yes—
106C22×C53C22×D411yescharacteristic
107C2×D53—21no—
108C2×D53—21no—
109C2×D53—41no—
110C2×D53—41no—
111C2×D53—41no—
112C2×D53#13712yes—
113C4×C53—21no—
114C4×C53—21no—
115C4×C53—42no—
116C4×C53—41no—
117C4×C53—41no—
118C4×C53C2×#4112yes—
119Dic53—21no—
120Dic53—21no—
121Dic53—21no—
122Dic53—41no—
123Dic53—42no—
124#3181—531nomaximal, Sylow
125C2×C53#53211yescharacteristic
126C2×C53C2×#13711yescharacteristic
127C2×C53#57511yescharacteristic
128D53—22no—
129D53—81no—
130#533—531no—
131#529—531no—
132#501—531no—
133#497—531no—
134#491D5311yescharacteristic, complemented
135#489—531no—
136#467—532no—
137#465—532no—
138#441—532no—
139#440—532no—
140C2×#97—531no—
141C53#318111yescharacteristic, complemented, Sylow
142C2×Q16—532no—
143C2×SD16—532no—
144C22×C8—532no—
145C4⋊Q8—531no—
146C4⋊Q8C2×D5311yescharacteristic
147#126—1061no—
148#125—1061no—
149#125C2×D5311yescharacteristic
150#124—531no—
151#123—1061no—
152C22⋊Q8—532no—
153C22⋊Q8—531no—
154C22⋊Q8—1061no—
155#121—531no—
156C4×Q8—532no—
157C4×Q8—531no—
158C4×Q8—1061no—
159C4×D4—531no—
160C4×D4—1061no—
161#117—1061no—
162#117—1061no—
163C2×C42—531no—
164#107—532no—
165#106—532no—
166#103—532no—
167#103C2×D5312yes—
168#103C2×D5312yes—
169#102—532no—
170#97—532no—
171#97—531no—
172#97C2×D5311yescharacteristic
173C2×Q8—531no—
174C2×Q8—1061no—
175C2×Q8C22×D5312yes—
176C2×D4—531no—
177C2×D4—1061no—
178C22×C4—531no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—1061no—
182Q16—1064no—
183SD16—1064no—
184C2×C8—532no—
185C2×C8—1064no—
186C2×C8C22×D5312yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—532no—
192C4⋊C4—1061no—
193C4⋊C4—1062no—
194C4⋊C4—1061no—
195C4⋊C4—1062no—
196C4⋊C4—1061no—
197C4⋊C4C22×D5311yescharacteristic
198C4⋊C4C22×D5311yescharacteristic
199C22⋊C4—1061no—
200C22⋊C4—1061no—
201C22⋊C4—1062no—
202C22⋊C4—1061no—
203C22⋊C4—1061no—
204C22⋊C4—1061no—
205C42—531no—
206C42—531no—
207C42—531no—
208C42—1061no—
209C42—1062no—
210C42C22×D5311yescharacteristic
211C23—531no—
212C23—1061no—
213Q8—22no—
214Q8—22no—
215Q8—1062no—
216Q8—2122no—
217D4—1062no—
218D4—2122no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—531no—
223C2×C4—531no—
224C2×C4—531no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1061no—
228C2×C4—1062no—
229C2×C4—2121no—
230C2×C4—2122no—
231C2×C4D4×D5311yescharacteristic
232C2×C4D4×D5311yescharacteristic
233C2×C4C23×D5311yescharacteristic
234C8—22no—
235C8—534no—
236C22—532no—
237C22—1061no—
238C22—1061no—
239C22—2121no—
240C22—2121no—
241C22—2121no—
242C22C2×D4×D5311yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—1061no—
249C4—1061no—
250C4—1061no—
251C4—2122no—
252C4—2121no—
253C4#41×D5312yes—
254C2—1062no—
255C2—4241no—
256C2#1681117211yescharacteristic
257C2#137×D5311yescharacteristic
258C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#491C2
socleC22×C53C22×D4
hypercenter#491D53
hypocenterC53#3181
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16811033C2semidirect
4. #16811015C2semidirect
5. #16810983C2semidirect
6. #16810863C2semidirect
8. #16810727C2semidirect
9. #16810726C2semidirect
10. #97×D53C2semidirect
11. C53×#491C2semidirect
134. #491D53semidirect
141. C53#3181semidirect