Finite Group Databasebeta

← #20287030

Subgroups — #20287030

Order 6784.

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

normal subgroups only

This group has 388 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#16811334C211yesmaximal, characteristic, complemented
2#16811276C211yesmaximal, characteristic
3#16810952C212yesmaximal, complemented
4#16810910C211yesmaximal, characteristic, complemented
5#16810880C212yesmaximal, complemented
6#16810830C211yesmaximal, characteristic, complemented
7#16810788C212yesmaximal, complemented
8C4×#11719913C211yesmaximal, characteristic, complemented
9#16810702C211yesmaximal, characteristic, complemented
10#16810623C212yesmaximal, complemented
11C53×#487C211yesmaximal, characteristic, complemented
12C2×#11719949C2212yes—
13C2×#11719925C2212yes—
14C2×#11719913C2211yescharacteristic
15C2×#11719912C2212yes—
16#13133208—21no—
17#13133202—22no—
18#13133197C2212yes—
19#13133193—22no—
20#13133193C2212yes—
21#13133192—21no—
22C4×Dic106C2211yescharacteristic
23C4×D212C2211yescharacteristic
24C4×D212C2211yescharacteristic
25#13133179—21no—
26C4⋊C4×D53—21no—
27C4⋊C4×D53C2212yes—
28#13133149C2211yescharacteristic
29#13133148C2212yes—
30C4⋊#88322C2211yescharacteristic
31#13133142C2211yescharacteristic
32#13133141C2212yes—
33#13133139C2211yescharacteristic
34#13133137C2212yes—
35#13133129C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4⋊Q8×C53C2211yescharacteristic
38#103×C53C2212yes—
39#103×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×Q8×C53C2312yes—
43#11719949—24no—
44#11719949—42no—
45C2×D212—22no—
46C2×D212D411yescharacteristic
47C2×D212C2311yescharacteristic
48C2×C4×D53—21no—
49C2×C4×D53—22no—
50C2×C4×D53—22no—
51C2×C4×D53C2312yes—
52C2×Dic106C2311yescharacteristic
53C2×C8×C53C2311yescharacteristic
54C2×C8×C53C2311yescharacteristic
55C4⋊C4×C53—21no—
56C4⋊C4×C53—21no—
57C4⋊C4×C53—21no—
58C4⋊C4×C53C2312yes—
59C42×C53C2311yescharacteristic
60#11719925—42no—
61#11719921—22no—
62#11719921—22no—
63#11719921—21no—
64#11719921D412yes—
65C4⋊Dic53—22no—
66C4⋊Dic53D411yescharacteristic
67C4⋊Dic53C2311yescharacteristic
68C4⋊Dic53C2311yescharacteristic
69#11719919—21no—
70#11719919—22no—
71#11719919—22no—
72C4×Dic53—21no—
73C2×#88322C2312yes—
74#11719913—22no—
75#11719912—42no—
76C22×D53—21no—
77C22×D53C2×D412yes—
78Q8×C53—22no—
79Q8×C53—22no—
80Q8×C53—81no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2×D411yescharacteristic
85C2×C4×C53C2411yescharacteristic
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53C2×D412yes—
90D212—22no—
91D212—21no—
92D212—44no—
93D212#4112yes—
94C4×D53—22no—
95C4×D53—22no—
96C4×D53—44no—
97C4×D53—42no—
98C4×D53—82no—
99Dic106—21no—
100Dic106—21no—
101C8×C53—21no—
102C8×C53—21no—
103#88322—22no—
104C22×C53C22×D411yescharacteristic
105C2×D53—22no—
106C2×D53—22no—
107C2×D53—22no—
108C2×D53—22no—
109C2×D53—41no—
110C2×D53—41no—
111C4×C53—21no—
112C4×C53—42no—
113C4×C53—42no—
114C4×C53#13712yes—
115C4×C53C2×#4111yescharacteristic
116C4×C53H(5,𝔽2)11yescharacteristic
117Dic53—22no—
118Dic53—41no—
119Dic53—41no—
120Dic53—41no—
121#3331—531nomaximal, Sylow
122C2×C53#54611yescharacteristic
123C2×C53C2×#13711yescharacteristic
124C2×C53#57711yescharacteristic
125D53—42no—
126D53—42no—
127#554—531no—
128#547—531no—
129#487D5311yescharacteristic, complemented
130#483—532no—
131#476—531no—
132#470—532no—
133#462—531no—
134#452—532no—
135C4×SD16—531no—
136#436—531no—
137#417—532no—
138C53#333111yescharacteristic, complemented, Sylow
139C2×#41—532no—
140C2×Q16—532no—
141C2×SD16—531no—
142C2×M16—532no—
143C4⋊Q8C2×D5311yescharacteristic
144C4⋊Q8C2×D5311yescharacteristic
145#125—1061no—
146#123—1062no—
147C22⋊Q8—532no—
148C22⋊Q8—532no—
149C22⋊Q8—1062no—
150#121—1061no—
151C4×Q8—531no—
152C4×D4—531no—
153C4×D4—531no—
154C4×D4—1061no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—1061no—
157#106—531no—
158#106—532no—
159C4⋊C8—531no—
160#103—532no—
161#103—531no—
162#103C2×D5312yes—
163#103C2×D5312yes—
164#102—531no—
165#102—532no—
166C22⋊C8—532no—
167C4×C8C2×D5311yescharacteristic
168#41—1064no—
169#41—2122no—
170C2×Q8—21no—
171C2×Q8—531no—
172C2×Q8C22×D5312yes—
173C2×D4—531no—
174C2×D4—531no—
175C2×D4—1062no—
176C22×C4—532no—
177C22×C4—1062no—
178C22×C4—1062no—
179C22×C4—1061no—
180Q16—2122no—
181SD16—1062no—
182M16—2122no—
183C2×C8—532no—
184C2×C8C22×D5311yescharacteristic
185C2×C8C22×D5311yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—531no—
190C4⋊C4—531no—
191C4⋊C4—531no—
192C4⋊C4—1062no—
193C4⋊C4—1062no—
194C4⋊C4—1062no—
195C4⋊C4—1061no—
196C4⋊C4C22×D5312yes—
197C22⋊C4—532no—
198C22⋊C4—1062no—
199C22⋊C4—1062no—
200C22⋊C4—1061no—
201C42—1061no—
202C42C22×D5311yescharacteristic
203C23—532no—
204C23—1061no—
205Q8—22no—
206Q8—22no—
207Q8—81no—
208Q8—1061no—
209Q8—1061no—
210D4—532no—
211D4—1062no—
212D4—1061no—
213D4—2124no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—532no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1061no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—4242no—
223C2×C4—2124no—
224C2×C4—2122no—
225C2×C4D4×D5311yescharacteristic
226C2×C4D4×D5311yescharacteristic
227C2×C4C23×D5311yescharacteristic
228C8—21no—
229C8—21no—
230C8—1062no—
231C22—1062no—
232C22—1062no—
233C22—1062no—
234C22—1062no—
235C22—2121no—
236C22—2121no—
237C22C2×D4×D5311yescharacteristic
238C4—21no—
239C4—42no—
240C4—42no—
241C4—1062no—
242C4—2121no—
243C4—2121no—
244C4—2121no—
245C4#1313324912yes—
246C4#41×D5311yescharacteristic
247C4#1313326311yescharacteristic
248C2—2122no—
249C2—2122no—
250C2#1681126611yescharacteristic
251C2C2×#1313324911yescharacteristic
252C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3331
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #16811334C2semidirect
3. #16810952C2semidirect
4. #16810910C2semidirect
5. #16810880C2semidirect
6. #16810830C2semidirect
7. #16810788C2semidirect
8. C4×#11719913C2semidirect
9. #16810702C2semidirect
10. #16810623C2semidirect
11. C53×#487C2semidirect
129. #487D53semidirect
138. C53#3331semidirect