Finite Group Databasebeta

← #20284651

Subgroups — #20284651

Order 6784.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#13133176C211yesmaximal, characteristic, complemented
2#16810944C214yesmaximal, complemented
3#16810939C212yesmaximal, complemented
4#16810934C212yesmaximal, complemented
5#16810663C212yesmaximal, complemented
6#16810628C212yesmaximal, complemented
7#16810587C211yesmaximal, characteristic, complemented
8C2×#121×C53C211yesmaximal, characteristic
9C2×#11719917C2212yes—
10#13133176—21no—
11#13133176—21no—
12#13133176C2212yes—
13#13133176C2212yescomplemented
14C2×C4⋊Dic53C2211yescharacteristic, complemented
15C2×C4⋊Dic53C2211yescharacteristic, complemented
16C2×C22⋊Dic53—21no—
17C2×C4×Dic53—21no—
18#13133150C2214yescomplemented
19#13133148C2214yescomplemented
20#13133138C2214yescomplemented
21#13133138C2214yescomplemented
22C22⋊#88322C2212yescomplemented
23C22⋊#88322C2212yescomplemented
24C22×D4×C53—21no—
25C22×D4×C53C2211yescharacteristic
26C23×C4×C53C2211yescharacteristic
27#121×C53—21no—
28#121×C53—21no—
29#121×C53C2212yes—
30#121×C53C2212yes—
31C2×C4⋊C4×C53C2211yescharacteristic
32C2×C22⋊C4×C53—21no—
33C24×C53—21no—
34C24×C53D411yescharacteristic
35C2×D4×C53—22no—
36C2×D4×C53—22no—
37C2×D4×C53—21no—
38C2×D4×C53—24no—
39C2×D4×C53—44no—
40C2×D4×C53C2312yes—
41C22×C4×C53—24no—
42C22×C4×C53—21no—
43C22×C4×C53D411yescharacteristic
44C22×C4×C53D411yescharacteristic
45C22×C4×C53D411yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C22×Dic53—21no—
50C22×Dic53—21no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53C2312yes—
53C22⋊C4×C53—22no—
54C22⋊C4×C53—22no—
55C22⋊Dic53—22no—
56C22⋊Dic53—22no—
57C4⋊Dic53—21no—
58C4⋊Dic53—21no—
59C4⋊Dic53C2312yes—
60C4⋊Dic53C2312yes—
61C4×Dic53—22no—
62C4×Dic53—22no—
63#11719917—24no—
64C2×#88322C2314yes—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53—41no—
69C23×C53—42no—
70C23×C53—42no—
71C23×C53—42no—
72C23×C53C2×D411yescharacteristic
73C23×C53C2×D411yescharacteristic
74C23×C53C2×D411yescharacteristic
75D4×C53—24no—
76D4×C53—44no—
77D4×C53—44no—
78D4×C53—42no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—42no—
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2×D411yescharacteristic
89C2×C4×C53#4114yes—
90C2×C4×C53C2411yescharacteristic
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95#88322—24no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—21no—
103C22×C53—44no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53—84no—
108C22×C53#12312yes—
109C22×C53AGL(1,ℤ/8)12yes—
110C22×C53C22×D411yescharacteristic
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—42no—
114C4×C53#12312yes—
115C4×C53C2×#4112yes—
116Dic53—42no—
117Dic53—42no—
118#3094—531nomaximal, Sylow
119C2×C53—22no—
120C2×C53—22no—
121C2×C53—41no—
122C2×C53—82no—
123C2×C53C2×#12311yescharacteristic
124C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C53#57511yescharacteristic
126C2×#121D5311yescharacteristic
127C2×#117—531no—
128#482—534no—
129#481—532no—
130#480—532no—
131#428—532no—
132#418—532no—
133#407—531no—
134C53#309411yescharacteristic, complemented, Sylow
135C22×D4—21no—
136C22×D4C2×D5311yescharacteristic
137C23×C4C2×D5311yescharacteristic
138C2×M16—532no—
139#121—21no—
140#121—21no—
141#121C2×D5312yes—
142#121C2×D5312yes—
143#117—532no—
144#117—532no—
145#117—1061no—
146#117—1061no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4C2×D5311yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—1061no—
152C2×C42—1061no—
153#107—534no—
154#106—534no—
155#102—534no—
156#102—534no—
157C22⋊C8—532no—
158C22⋊C8—532no—
159C24—21no—
160C24C22⋊D5311yescharacteristic
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—24no—
164C2×D4—21no—
165C2×D4—44no—
166C2×D4C22×D5312yes—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—1061no—
170C22×C4—1061no—
171C22×C4C22⋊D5311yescharacteristic
172C22×C4C22⋊D5311yescharacteristic
173C22×C4C22⋊D5311yescharacteristic
174C22×C4C22×D5311yescharacteristic
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5311yescharacteristic
177M16—1064no—
178C2×C8—534no—
179C4⋊C4—21no—
180C4⋊C4—532no—
181C4⋊C4—532no—
182C4⋊C4—1061no—
183C4⋊C4—1061no—
184C4⋊C4C22×D5312yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—1062no—
188C22⋊C4—1062no—
189C42—1062no—
190C42—1062no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23C2×C22⋊D5311yescharacteristic
199C23C2×C22⋊D5311yescharacteristic
200C23C2×C22⋊D5311yescharacteristic
201D4—24no—
202D4—44no—
203D4—44no—
204D4—42no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4#1171994714yes—
217C2×C4C2×C22⋊D5311yescharacteristic
218C2×C4C2×C22⋊D5311yescharacteristic
219C2×C4C2×C22⋊D5311yescharacteristic
220C2×C4C23×D5311yescharacteristic
221C8—1064no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—84no—
234C22#1313319812yes—
235C22#1313324312yes—
236C22C22×C22⋊D5311yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—2122no—
241C4—2122no—
242C4#1313319812yes—
243C4C2×#1171994712yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—82no—
248C2C2×#1313319811yescharacteristic
249C2C2×#1313324311yescharacteristic
250C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#121×C53C2
socleC22×C53C22×D4
hypercenterC2×#121D53
hypocenterC53#3094
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133176C2semidirect
2. #16810944C2semidirect
3. #16810939C2semidirect
4. #16810934C2semidirect
5. #16810663C2semidirect
6. #16810628C2semidirect
7. #16810587C2semidirect
13. #13133176C22semidirect
14. C2×C4⋊Dic53C22semidirect
15. C2×C4⋊Dic53C22semidirect
18. #13133150C22semidirect
19. #13133148C22semidirect
20. #13133138C22semidirect
21. #13133138C22semidirect
22. C22⋊#88322C22semidirect
23. C22⋊#88322C22semidirect
134. C53#3094semidirect