Finite Group Databasebeta

← #20286631

Subgroups — #20286631

Order 6784.

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

normal subgroups only

This group has 386 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, complemented
2#16811276C211yesmaximal, characteristic
3#16810962C212yesmaximal, complemented
4#16810900C211yesmaximal, characteristic, complemented
5#16810863C212yesmaximal, complemented
6#16810830C211yesmaximal, characteristic, complemented
7#16810788C212yesmaximal, complemented
8C4×#11719913C211yesmaximal, characteristic, complemented
9C8×D212C211yesmaximal, characteristic, complemented
10#16810618C212yesmaximal, complemented
11C53×#487C211yesmaximal, characteristic, complemented
12C2×#11719949C2212yes—
13C2×#11719925C2212yes—
14C2×#11719913C2211yescharacteristic
15C2×C8×D53C2212yes—
16#13133212—22no—
17#13133206—21no—
18#13133202—22no—
19#13133197C2212yes—
20#13133193C2212yes—
21#13133192—21no—
22Q8×Dic53—21no—
23C4×Dic106C2211yescharacteristic
24C4×D212C2211yescharacteristic
25C4×D212C2211yescharacteristic
26#13133175C2212yes—
27C4⋊C4×D53—21no—
28#13133150C2212yes—
29#13133149C2211yescharacteristic
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×D53C2312yes—
51C2×Dic106C2311yescharacteristic
52C2×C8×C53C2311yescharacteristic
53C2×C8×C53C2311yescharacteristic
54C4⋊C4×C53—21no—
55C4⋊C4×C53—21no—
56C4⋊C4×C53—21no—
57C4⋊C4×C53C2312yes—
58C42×C53C2311yescharacteristic
59#11719925—42no—
60#11719921—22no—
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—
71C4×Dic53—21no—
72C4×Dic53—22no—
73C2×#88322C2312yes—
74#11719913—22no—
75C8×D53—42no—
76C22×D53—21no—
77C22×D53C2×D412yes—
78Q8×C53—22no—
79Q8×C53—22no—
80Q8×C53—42no—
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—
98Dic106—21no—
99Dic106—21no—
100C8×C53—21no—
101C8×C53—21no—
102#88322—22no—
103C22×C53C22×D411yescharacteristic
104C2×D53—22no—
105C2×D53—22no—
106C2×D53—22no—
107C2×D53—22no—
108C2×D53—41no—
109C2×D53—41no—
110C4×C53—21no—
111C4×C53—42no—
112C4×C53—42no—
113C4×C53#13512yes—
114C4×C53C2×#4111yescharacteristic
115C4×C53H(5,𝔽2)11yescharacteristic
116Dic53—22no—
117Dic53—41no—
118Dic53—41no—
119Dic53—41no—
120#3302—531nomaximal, Sylow
121C2×C53#54611yescharacteristic
122C2×C53C2×#13511yescharacteristic
123C2×C53#57811yescharacteristic
124D53—42no—
125D53—42no—
126#556—531no—
127#547—531no—
128#487D5311yescharacteristic, complemented
129#485—532no—
130#474—531no—
131#467—532no—
132#462—531no—
133#452—532no—
134C4×SD16—531no—
135C8×D4—531no—
136#416—532no—
137C53#330211yescharacteristic, complemented, Sylow
138C2×#41—532no—
139C2×Q16—532no—
140C2×SD16—531no—
141C22×C8—532no—
142C4⋊Q8C2×D5311yescharacteristic
143C4⋊Q8C2×D5311yescharacteristic
144#126—1062no—
145#124—1061no—
146#123—1062no—
147C22⋊Q8—532no—
148C22⋊Q8—532no—
149#121—1061no—
150C4×Q8—531no—
151C4×Q8—1061no—
152C4×D4—531no—
153C4×D4—531no—
154#117—532no—
155C2×C4⋊C4—1061no—
156#107—532no—
157#106—531no—
158C4⋊C8—531no—
159#103—532no—
160#103—531no—
161#103C2×D5312yes—
162#103C2×D5312yes—
163#102—531no—
164#102—532no—
165C22⋊C8—532no—
166C4×C8C2×D5311yescharacteristic
167#41—1064no—
168#41—2122no—
169C2×Q8—21no—
170C2×Q8—531no—
171C2×Q8C22×D5312yes—
172C2×D4—531no—
173C2×D4—531no—
174C2×D4—1062no—
175C22×C4—532no—
176C22×C4—1062no—
177C22×C4—1061no—
178Q16—2122no—
179SD16—1062no—
180C2×C8—532no—
181C2×C8—2122no—
182C2×C8C22×D5311yescharacteristic
183C2×C8C22×D5311yescharacteristic
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—531no—
188C4⋊C4—531no—
189C4⋊C4—531no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—1061no—
193C4⋊C4C22×D5312yes—
194C22⋊C4—532no—
195C22⋊C4—1062no—
196C22⋊C4—1062no—
197C22⋊C4—1062no—
198C22⋊C4—1061no—
199C42—1062no—
200C42—1061no—
201C42C22×D5311yescharacteristic
202C23—532no—
203C23—1061no—
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—1061no—
208Q8—1061no—
209D4—532no—
210D4—1062no—
211D4—1061no—
212D4—2124no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—532no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1061no—
219C2×C4—1061no—
220C2×C4—1061no—
221C2×C4—2124no—
222C2×C4—2122no—
223C2×C4D4×D5311yescharacteristic
224C2×C4D4×D5311yescharacteristic
225C2×C4C23×D5311yescharacteristic
226C8—21no—
227C8—21no—
228C8—1062no—
229C22—1062no—
230C22—1062no—
231C22—1062no—
232C22—1062no—
233C22—2121no—
234C22—2121no—
235C22C2×D4×D5311yescharacteristic
236C4—21no—
237C4—42no—
238C4—42no—
239C4—1062no—
240C4—2121no—
241C4—2121no—
242C4—2121no—
243C4#1313323812yes—
244C4#41×D5311yescharacteristic
245C4#1313326311yescharacteristic
246C2—2122no—
247C2—2122no—
248C2#1681126611yescharacteristic
249C2C2×#1313323811yescharacteristic
250C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3302
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. #16811347C2semidirect
3. #16810962C2semidirect
4. #16810900C2semidirect
5. #16810863C2semidirect
6. #16810830C2semidirect
7. #16810788C2semidirect
8. C4×#11719913C2semidirect
9. C8×D212C2semidirect
10. #16810618C2semidirect
11. C53×#487C2semidirect
128. #487D53semidirect
137. C53#3302semidirect