Finite Group Databasebeta

← #20286997

Subgroups — #20286997

Order 6784.

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

normal subgroups only

This group has 374 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#16811347C211yesmaximal, characteristic
2#16811288C211yesmaximal, characteristic
3#16810963C212yesmaximal, complemented
4#16810905C211yesmaximal, characteristic, complemented
5#16810880C212yesmaximal, complemented
6#16810830C211yesmaximal, characteristic, complemented
7#16810780C212yesmaximal, complemented
8C4×Dic212C211yesmaximal, characteristic, complemented
9#16810702C211yesmaximal, characteristic
10#16810630C212yesmaximal, complemented
11C53×#487C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2212yes—
13C2×Dic212C2211yescharacteristic
14C2×#11719925C2212yes—
15C2×#11719912C2212yes—
16#13133216—21no—
17#13133212—22no—
18#13133206—21no—
19#13133197C2212yes—
20#13133195—22no—
21#13133193C2212yes—
22Q8×Dic53—21no—
23C4×Dic106C2211yescharacteristic
24C4×Dic106C2211yescharacteristic
25C4×D212C2211yescharacteristic
26#13133179—21no—
27#13133175C2212yes—
28#13133151C2211yescharacteristic
29#13133148C2212yes—
30C4⋊#88322C2211yescharacteristic
31#13133142C2211yescharacteristic
32#13133142C2211yescharacteristic
33#13133141C2212yes—
34#13133140C2212yes—
35#13133129C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4⋊Q8×C53C2211yescharacteristic
38#103×C53C2212yes—
39#103×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×Q8×C53C2312yes—
43Q8×D53—24no—
44Q8×D53—42no—
45C2×D212D411yescharacteristic
46C2×C4×D53—22no—
47C2×C4×D53C2312yes—
48C2×Dic106—22no—
49C2×Dic106C2311yescharacteristic
50C2×Dic106C2311yescharacteristic
51C2×C8×C53C2311yescharacteristic
52C2×C8×C53C2311yescharacteristic
53C4⋊C4×C53—21no—
54C4⋊C4×C53—21no—
55C4⋊C4×C53—21no—
56C4⋊C4×C53C2312yes—
57C42×C53C2311yescharacteristic
58#11719925—42no—
59#11719921—22no—
60#11719921—22no—
61#11719921—22no—
62#11719921D412yes—
63C4⋊Dic53—22no—
64C4⋊Dic53D411yescharacteristic
65C4⋊Dic53C2311yescharacteristic
66C4⋊Dic53C2311yescharacteristic
67#11719919—21no—
68#11719919—21no—
69#11719919—22no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72C4×Dic53—22no—
73C2×#88322C2312yes—
74Dic212—22no—
75#11719912—42no—
76C22×D53C2×D412yes—
77Q8×C53—22no—
78Q8×C53—22no—
79Q8×C53—42no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2411yescharacteristic
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53C2×D412yes—
90D212—22no—
91C4×D53—22no—
92C4×D53—22no—
93C4×D53—44no—
94Dic106—21no—
95Dic106—21no—
96Dic106—21no—
97Dic106—44no—
98Dic106#4112yes—
99C8×C53—21no—
100C8×C53—21no—
101#88322—22no—
102C22×C53C22×D411yescharacteristic
103C2×D53—22no—
104C2×D53—22no—
105C2×D53—22no—
106C4×C53—21no—
107C4×C53—42no—
108C4×C53—42no—
109C4×C53#13712yes—
110C4×C53C2×#4111yescharacteristic
111C4×C53H(5,𝔽2)11yescharacteristic
112Dic53—22no—
113Dic53—22no—
114Dic53—41no—
115Dic53—41no—
116Dic53—41no—
117#3328—531nomaximal, Sylow
118C2×C53#54611yescharacteristic
119C2×C53C2×#13711yescharacteristic
120C2×C53#57811yescharacteristic
121D53—42no—
122#556—531no—
123D4×Q8—531no—
124#487D5311yescharacteristic, complemented
125#485—532no—
126#475—531no—
127#470—532no—
128#462—531no—
129#451—532no—
130C4×Q16—531no—
131#436—531no—
132#419—532no—
133C53#332811yescharacteristic, complemented, Sylow
134C22×Q8—532no—
135C2×Q16—532no—
136C2×Q16—531no—
137C2×M16—532no—
138C4⋊Q8—1061no—
139C4⋊Q8C2×D5311yescharacteristic
140C4⋊Q8C2×D5311yescharacteristic
141#126—1062no—
142#124—1061no—
143C22⋊Q8—532no—
144C22⋊Q8—532no—
145C22⋊Q8—1062no—
146C4×Q8—531no—
147C4×Q8—531no—
148C4×Q8—1061no—
149C4×D4—531no—
150C4×D4—1061no—
151#117—532no—
152#107—531no—
153#106—532no—
154C4⋊C8—531no—
155#103—532no—
156#103—531no—
157#103—531no—
158#103—532no—
159#103C2×D5312yes—
160#103C2×D5312yes—
161C22⋊C8—532no—
162C4×C8C2×D5311yescharacteristic
163C2×Q8—21no—
164C2×Q8—531no—
165C2×Q8—531no—
166C2×Q8—1062no—
167C2×Q8—1064no—
168C2×Q8—2122no—
169C2×Q8C22×D5312yes—
170C2×D4—531no—
171C22×C4—532no—
172C22×C4—1062no—
173Q16—1062no—
174Q16—2122no—
175M16—2122no—
176C2×C8—532no—
177C2×C8C22×D5311yescharacteristic
178C2×C8C22×D5311yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—531no—
183C4⋊C4—531no—
184C4⋊C4—531no—
185C4⋊C4—1062no—
186C4⋊C4—1062no—
187C4⋊C4—1061no—
188C4⋊C4—1061no—
189C4⋊C4C22×D5312yes—
190C22⋊C4—532no—
191C22⋊C4—1062no—
192C22⋊C4—1062no—
193C22⋊C4—1062no—
194C42—1062no—
195C42—1061no—
196C42—1061no—
197C42C22×D5311yescharacteristic
198C23—532no—
199Q8—22no—
200Q8—22no—
201Q8—42no—
202Q8—532no—
203Q8—1061no—
204Q8—1061no—
205Q8—1061no—
206Q8—2124no—
207D4—1062no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—532no—
211C2×C4—1061no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1061no—
215C2×C4—1061no—
216C2×C4—1061no—
217C2×C4—2124no—
218C2×C4D4×D5311yescharacteristic
219C2×C4D4×D5311yescharacteristic
220C2×C4C23×D5311yescharacteristic
221C8—21no—
222C8—21no—
223C8—1062no—
224C22—1062no—
225C22—1062no—
226C22—1062no—
227C22C2×D4×D5311yescharacteristic
228C4—21no—
229C4—42no—
230C4—42no—
231C4—1062no—
232C4—1062no—
233C4—2121no—
234C4—2121no—
235C4—2121no—
236C4#1313324612yes—
237C4#41×D5311yescharacteristic
238C4#1313326311yescharacteristic
239C2—2122no—
240C2#1681126611yescharacteristic
241C2C2×#1313324611yescharacteristic
242C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3328
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
3. #16810963C2semidirect
4. #16810905C2semidirect
5. #16810880C2semidirect
6. #16810830C2semidirect
7. #16810780C2semidirect
8. C4×Dic212C2semidirect
10. #16810630C2semidirect
11. C53×#487C2semidirect
124. #487D53semidirect
133. C53#3328semidirect