Finite Group Databasebeta

← #20284638

Subgroups — #20284638

Order 6784.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#16811134C211yesmaximal, characteristic
2#16810958C212yesmaximal, complemented
3#16810953C212yesmaximal, complemented
4#16810941C212yesmaximal, complemented
5#16810936C212yesmaximal, complemented
6#16810663C212yesmaximal, complemented
7#16810622C212yesmaximal, complemented
8#16810587C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C53C211yesmaximal, characteristic, complemented
10C2×#11719945C2211yescharacteristic
11C2×#11719917C2212yes—
12#13133200—21no—
13#13133196C2211yescharacteristic
14#13133196C2211yescharacteristic
15#13133191C2211yescharacteristic
16#13133191C2211yescharacteristic
17#13133186—21no—
18C4×C22⋊D53—21no—
19C4×C22⋊D53—21no—
20#13133176C2211yescharacteristic
21#13133150C2214yescomplemented
22#13133148C2214yescomplemented
23#13133140C2214yescomplemented
24#13133137C2214yescomplemented
25C22⋊#88322C2212yes—
26C22⋊#88322C2212yes—
27C23×C4×C53—21no—
28C23×C4×C53C2211yescharacteristic
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53C2212yes—
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2212yes—
35C24×C53D411yescharacteristic
36C22×C4×C53—24no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—44no—
41C22×C4×C53D411yescharacteristic
42C22×C4×C53D411yescharacteristic
43C22×C4×C53D411yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48C2×C22⋊D53—21no—
49C2×C22⋊D53—21no—
50#11719945—24no—
51C2×D212C2311yescharacteristic
52C2×C4×D53—21no—
53C2×Dic106C2311yescharacteristic
54C4⋊C4×C53—22no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53C2314yes—
58C22⋊Dic53—21no—
59C22⋊Dic53—21no—
60C22⋊Dic53—21no—
61#11719921—21no—
62#11719921—21no—
63C4⋊Dic53C2311yescharacteristic
64C4⋊Dic53C2311yescharacteristic
65#11719919—21no—
66#11719919—21no—
67C4×Dic53—21no—
68#11719917—24no—
69C2×#88322C2314yes—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53C2×D411yescharacteristic
74C23×C53C2×D411yescharacteristic
75C23×C53C2×D411yescharacteristic
76C22×D53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—24no—
83C2×C4×C53—44no—
84C2×C4×C53—44no—
85C2×C4×C53—44no—
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2×D411yescharacteristic
89C2×C4×C53#4114yes—
90C2×C4×C53C2411yescharacteristic
91C22⋊D53—42no—
92C22⋊D53—42no—
93C22⋊D53—42no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97D212—22no—
98C4×D53—42no—
99Dic106—22no—
100#88322—24no—
101C22×C53—28no—
102C22×C53—21no—
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—21no—
106C22×C53—21no—
107C22×C53—21no—
108C22×C53#12312yes—
109C22×C53AGL(1,ℤ/8)12yes—
110C22×C53#13712yes—
111C22×C53C22×D411yescharacteristic
112C2×D53—41no—
113C2×D53—41no—
114C2×D53—41no—
115C4×C53—21no—
116C4×C53—21no—
117C4×C53—44no—
118C4×C53#12312yes—
119C4×C53C2×#4112yes—
120Dic53—41no—
121Dic53—41no—
122Dic53—41no—
123#3092—531nomaximal, Sylow
124C2×C53—22no—
125C2×C53—22no—
126C2×C53—22no—
127C2×C53C2×#12311yescharacteristic
128C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C53C2×#13711yescharacteristic
130D53—81no—
131#525—531no—
132C22×C4⋊C4D5311yescharacteristic, complemented
133#484—532no—
134#483—532no—
135#481—532no—
136#480—532no—
137#428—532no—
138#417—532no—
139#407—531no—
140C53#309211yescharacteristic, complemented, Sylow
141C2×#41—531no—
142C23×C4—21no—
143C23×C4C2×D5311yescharacteristic
144C2×M16—532no—
145#123—1061no—
146C22⋊Q8—531no—
147C22⋊Q8—531no—
148#121—531no—
149#121—531no—
150#120—1061no—
151C4×D4—1061no—
152C4×D4—1061no—
153#117—531no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D5312yes—
158C2×C4⋊C4C2×D5312yes—
159C2×C4⋊C4C2×D5312yes—
160#107—534no—
161#106—534no—
162#103—534no—
163#102—534no—
164C22⋊C8—532no—
165C22⋊C8—532no—
166C24C22⋊D5311yescharacteristic
167#41—1064no—
168C2×Q8—531no—
169C2×D4—531no—
170C2×D4—1061no—
171C2×D4—1061no—
172C2×D4—1061no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—1061no—
179C22×C4C22⋊D5311yescharacteristic
180C22×C4C22⋊D5311yescharacteristic
181C22×C4C22⋊D5311yescharacteristic
182C22×C4C22×D5311yescharacteristic
183C22×C4C22×D5311yescharacteristic
184C22×C4C22×D5311yescharacteristic
185M16—1064no—
186C2×C8—534no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—531no—
191C4⋊C4—531no—
192C4⋊C4—1061no—
193C4⋊C4—1061no—
194C4⋊C4C22×D5314yes—
195C22⋊C4—1061no—
196C22⋊C4—1061no—
197C22⋊C4—1061no—
198C22⋊C4—1061no—
199C22⋊C4—1061no—
200C42—1061no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—1061no—
205C23C2×C22⋊D5311yescharacteristic
206C23C2×C22⋊D5311yescharacteristic
207C23C2×C22⋊D5311yescharacteristic
208Q8—1062no—
209D4—1062no—
210D4—2122no—
211D4—2122no—
212D4—2122no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—2122no—
226C2×C4#1171994514yes—
227C2×C4C2×C22⋊D5311yescharacteristic
228C2×C4C2×C22⋊D5311yescharacteristic
229C2×C4C2×C22⋊D5311yescharacteristic
230C2×C4C23×D5311yescharacteristic
231C8—1064no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—2121no—
240C22—2121no—
241C22—2121no—
242C22#1313320012yes—
243C22#1313324312yes—
244C22#1313324812yes—
245C22C22×C22⋊D5311yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—2121no—
250C4—2121no—
251C4—2121no—
252C4#1313320012yes—
253C4C2×#1171994512yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—4241no—
258C2C2×#1313320011yescharacteristic
259C2C2×#1313324311yescharacteristic
260C2C2×#1313324811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC22×C4⋊C4×C53C2
socleC22×C53C22×D4
hypercenterC22×C4⋊C4D53
hypocenterC53#3092
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16810958C2semidirect
3. #16810953C2semidirect
4. #16810941C2semidirect
5. #16810936C2semidirect
6. #16810663C2semidirect
7. #16810622C2semidirect
8. #16810587C2semidirect
9. C22×C4⋊C4×C53C2semidirect
21. #13133150C22semidirect
22. #13133148C22semidirect
23. #13133140C22semidirect
24. #13133137C22semidirect
132. C22×C4⋊C4D53semidirect
140. C53#3092semidirect