Finite Group Databasebeta

← #20287427

Subgroups — #20287427

Order 6784.

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

normal subgroups only

This group has 396 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#16811293C211yesmaximal, characteristic, complemented
3#16810986C212yesmaximal, complemented
4#16810921C212yesmaximal, complemented
5#16810910C211yesmaximal, characteristic, complemented
6#16810817C212yesmaximal, complemented
7#16810815C211yesmaximal, characteristic, complemented
8#16810753C211yesmaximal, characteristic, complemented
9#16810732C212yesmaximal, complemented
10C4×D424C211yesmaximal, characteristic, complemented
11C53×#486C211yesmaximal, characteristic, complemented
12C2×#11719947—21no—
13C2×D424C2211yescharacteristic
14C2×#11719922C2212yes—
15#13133213C2212yes—
16#13133208C2212yes—
17#13133197—21no—
18#13133193—22no—
19#13133189—22no—
20#13133188—21no—
21C4×Dic106C2211yescharacteristic
22C4×D212C2211yescharacteristic
23C4×D212C2211yescharacteristic
24D4×Dic53C2212yes—
25#13133179C2212yes—
26C4⋊C4×D53—21no—
27#13133151C2211yescharacteristic
28#13133148C2212yes—
29C4⋊#88322C2211yescharacteristic
30#13133146C2212yes—
31#13133139C2211yescharacteristic
32#13133139C2211yescharacteristic
33#13133138C2212yes—
34#13133137C2212yes—
35#13133127C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37#127×C53C2211yescharacteristic
38#102×C53C2212yes—
39#102×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53C2312yes—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—22no—
47C22×Dic53—22no—
48#11719947—44no—
49C2×D212C2311yescharacteristic
50C2×D212C2311yescharacteristic
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—22no—
54C2×Dic106D411yescharacteristic
55C2×C8×C53C2311yescharacteristic
56C2×C8×C53C2311yescharacteristic
57C4⋊C4×C53—21no—
58C4⋊C4×C53C2312yes—
59C42×C53C2311yescharacteristic
60C22⋊Dic53—22no—
61#11719922—24no—
62#11719921—21no—
63#11719921—21no—
64#11719921—22no—
65C4⋊Dic53—22no—
66C4⋊Dic53D411yescharacteristic
67C4⋊Dic53C2311yescharacteristic
68C4⋊Dic53C2311yescharacteristic
69#11719919—22no—
70#11719919D412yes—
71C4×Dic53C2312yes—
72C2×#88322C2312yes—
73D424—22no—
74C23×C53—22no—
75C22×D53—21no—
76C22×D53—21no—
77Q8×C53—81no—
78D4×C53—22no—
79D4×C53—22no—
80D4×C53—42no—
81D4×C53—81no—
82C2×C4×C53—22no—
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2×D411yescharacteristic
85C2×C4×C53C2411yescharacteristic
86C22⋊D53—44no—
87C22⋊D53—82no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—44no—
91C2×Dic53C2×D412yes—
92C2×Dic53C2×D412yes—
93D212—21no—
94D212—21no—
95D212—21no—
96D212#4112yescomplemented
97C4×D53—42no—
98C4×D53—42no—
99C4×D53—82no—
100Dic106—22no—
101C8×C53—21no—
102C8×C53—21no—
103#88322—22no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53C22×D411yescharacteristic
108C2×D53—22no—
109C2×D53—41no—
110C2×D53—41no—
111C2×D53—41no—
112C2×D53—41no—
113C2×D53—41no—
114C4×C53—21no—
115C4×C53—42no—
116C4×C53AGL(1,ℤ/8)12yes—
117C4×C53C2×#4111yescharacteristic
118C4×C532−1+411yescharacteristic
119Dic53—22no—
120Dic53—22no—
121Dic53—41no—
122Dic53—41no—
123#3366—531nomaximal, Sylow
124C2×C53—82no—
125C2×C53#54811yescharacteristic
126C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C53#57611yescharacteristic
128D53—42no—
129D53—81no—
130#554—531no—
131#550—531no—
132#490—532no—
133#486D5311yescharacteristic, complemented
134#478—532no—
135#476—531no—
136#459—531no—
137#459—532no—
138#446—531no—
139#442—532no—
140C4×D8—531no—
141C53#336611yescharacteristic, complemented, Sylow
142C2×#41—1061no—
143C2×D8—532no—
144C2×D8—531no—
145C4⋊Q8C2×D5311yescharacteristic
146#127—532no—
147#127C2×D5311yescharacteristic
148#125—532no—
149C22⋊Q8—1061no—
150C22⋊Q8—1062no—
151#121—1061no—
152#121—1062no—
153C4×Q8—531no—
154C4×D4—532no—
155C4×D4—532no—
156C4×D4—531no—
157C4×D4—531no—
158C2×C4⋊C4—1061no—
159#107—531no—
160#106—532no—
161C4⋊C8—532no—
162C4⋊C8—531no—
163#102—532no—
164#102—531no—
165#102—531no—
166#102—532no—
167#102C2×D5312yes—
168#102C2×D5312yes—
169#97—532no—
170C4×C8C2×D5311yescharacteristic
171#41—2124no—
172C2×Q8—21no—
173C2×Q8—531no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—531no—
177C2×D4—531no—
178C2×D4—1062no—
179C2×D4—1062no—
180C2×D4C22×D5312yes—
181C22×C4—1062no—
182C22×C4—1062no—
183C22×C4—1061no—
184C22×C4—1061no—
185D8—1062no—
186D8—1064no—
187C2×C8—532no—
188C2×C8C22×D5311yescharacteristic
189C2×C8C22×D5311yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—531no—
192C4⋊C4—532no—
193C4⋊C4—531no—
194C4⋊C4—531no—
195C4⋊C4—1062no—
196C4⋊C4—1062no—
197C4⋊C4C22×D5312yes—
198C22⋊C4—1062no—
199C22⋊C4—1062no—
200C22⋊C4—1061no—
201C22⋊C4—1061no—
202C42—532no—
203C42C22×D5311yescharacteristic
204C23—22no—
205C23—1061no—
206C23—1061no—
207Q8—81no—
208Q8—1062no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—81no—
213D4—532no—
214D4—1061no—
215D4—1061no—
216D4—1061no—
217D4—4242no—
218D4—2124no—
219C2×C4—22no—
220C2×C4—532no—
221C2×C4—532no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—4242no—
225C2×C4—2124no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4D4×D5311yescharacteristic
229C2×C4D4×D5311yescharacteristic
230C2×C4C23×D5311yescharacteristic
231C8—21no—
232C8—21no—
233C8—1062no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—1062no—
238C22—2121no—
239C22—2121no—
240C22—2121no—
241C22—2121no—
242C22—2121no—
243C22C2×D4×D5311yescharacteristic
244C4—21no—
245C4—42no—
246C4—1062no—
247C4—1062no—
248C4—2121no—
249C4—2121no—
250C4#1313324412yes—
251C4#41×D5311yescharacteristic
252C4#1313326411yescharacteristic
253C2—82no—
254C2—4241no—
255C2—2122no—
256C2#1681128111yescharacteristic
257C2C2×#1313324411yescharacteristic
258C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3366
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811334C2semidirect
2. #16811293C2semidirect
3. #16810986C2semidirect
4. #16810921C2semidirect
5. #16810910C2semidirect
6. #16810817C2semidirect
7. #16810815C2semidirect
8. #16810753C2semidirect
9. #16810732C2semidirect
10. C4×D424C2semidirect
11. C53×#486C2semidirect
96. D212#41semidirect
133. #486D53semidirect
141. C53#3366semidirect