Finite Group Databasebeta

← #20287333

Subgroups — #20287333

Order 6784.

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

normal subgroups only

This group has 342 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#16811346C211yesmaximal, characteristic
2#16811334C211yesmaximal, characteristic
3#16811019C211yesmaximal, characteristic, complemented
4#16810966C212yesmaximal, complemented
5#16810952C212yesmaximal, complemented
6#16810909C211yesmaximal, characteristic, complemented
7#16810880C212yesmaximal, complemented
8C4×#11719925C211yesmaximal, characteristic, complemented
9#16810700C211yesmaximal, characteristic
10#16810664C212yesmaximal, complemented
11C53×#477C211yesmaximal, characteristic, complemented
12C2×#11719925C2211yescharacteristic
13C2×#11719912C2212yes—
14#13133217C2211yescharacteristic
15#13133212—22no—
16#13133208—21no—
17#13133205—21no—
18#13133197C2212yes—
19#13133193—22no—
20#13133193C2212yes—
21C4×Dic106—21no—
22C4×Dic106C2211yescharacteristic
23C4×D212C2211yescharacteristic
24#13133179—21no—
25#13133175C2212yes—
26C4⋊C4×D53C2212yes—
27#13133150C2211yescharacteristic
28#13133149C2212yes—
29#13133148C2212yes—
30#13133148C2212yes—
31#13133142C2212yes—
32#13133141C2211yescharacteristic
33#13133140C2211yescharacteristic
34#13133129C2212yes—
35C4×#88322C2211yescharacteristic
36C4⋊Q8×C53C2211yescharacteristic
37C4×Q8×C53C2211yescharacteristic
38#107×C53C2212yes—
39C4⋊C8×C53C2211yescharacteristic
40#103×C53C2212yes—
41C2×Q8×C53—21no—
42C2×Q8×C53C2311yescharacteristic
43C2×D212D411yescharacteristic
44C2×C4×D53—22no—
45C2×C4×D53C2312yes—
46C2×Dic106—21no—
47C2×Dic106C2311yescharacteristic
48C2×C8×C53C2312yes—
49C4⋊C4×C53—21no—
50C4⋊C4×C53—21no—
51C4⋊C4×C53C2312yes—
52C4⋊C4×C53C2311yescharacteristic
53C42×C53—21no—
54C42×C53C2311yescharacteristic
55#11719925—22no—
56#11719921—22no—
57#11719921—22no—
58#11719921—22no—
59#11719921D412yes—
60C4⋊Dic53—21no—
61C4⋊Dic53—22no—
62C4⋊Dic53D411yescharacteristic
63C4⋊Dic53C2311yescharacteristic
64C4⋊Dic53C2312yes—
65#11719919—21no—
66#11719919—22no—
67#11719919—22no—
68C4×Dic53—21no—
69C4×Dic53—22no—
70C2×#88322C2311yescharacteristic
71C2×#88322C2311yescharacteristic
72#11719912—24no—
73C22×D53C2×D412yes—
74Q8×C53—21no—
75Q8×C53—21no—
76Q8×C53—81no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2411yescharacteristic
83C2×Dic53—22no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53C2×D412yes—
87D212—22no—
88C4×D53—22no—
89C4×D53—22no—
90C4×D53—82no—
91Dic106—21no—
92Dic106—21no—
93Dic106—42no—
94C8×C53—22no—
95#88322—21no—
96#88322#4112yes—
97C22×C53C22×D411yescharacteristic
98C2×D53—22no—
99C2×D53—22no—
100C2×D53—22no—
101C4×C53—21no—
102C4×C53—41no—
103C4×C53—42no—
104C4×C53—41no—
105C4×C53#13712yes—
106C4×C53C2×#4111yescharacteristic
107C4×C532−1+411yescharacteristic
108Dic53—22no—
109Dic53—41no—
110Dic53—42no—
111Dic53—41no—
112#3356—531nomaximal, Sylow
113C2×C53#54711yescharacteristic
114C2×C53C2×#13711yescharacteristic
115C2×C53#57711yescharacteristic
116D53—42no—
117#556—531no—
118#554—531no—
119#498—531no—
120#485—532no—
121#483—532no—
122#477D5311yescharacteristic, complemented
123#475—531no—
124#470—532no—
125C4×Q16—531no—
126#436—531no—
127#428—532no—
128C53#335611yescharacteristic, complemented, Sylow
129C2×Q16—531no—
130C2×M16—532no—
131C4⋊Q8—531no—
132C4⋊Q8C2×D5311yescharacteristic
133#126—1062no—
134#125—1061no—
135#124—1061no—
136C22⋊Q8—532no—
137C22⋊Q8—532no—
138C22⋊Q8—1062no—
139C4×Q8—531no—
140C4×Q8—1061no—
141C4×Q8C2×D5311yescharacteristic
142C4×D4—531no—
143C4×D4—1061no—
144#117—532no—
145C2×C4⋊C4—532no—
146#107—531no—
147#107C2×D5312yes—
148#106—532no—
149#106—532no—
150#106—532no—
151C4⋊C8C2×D5311yescharacteristic
152#103—531no—
153#103—532no—
154#103—531no—
155#103C2×D5312yes—
156C22⋊C8—532no—
157C4×C8—531no—
158C2×Q8—21no—
159C2×Q8—531no—
160C2×Q8—1061no—
161C2×Q8C22×D5311yescharacteristic
162C2×D4—531no—
163C22×C4—532no—
164C22×C4—1062no—
165Q16—1062no—
166M16—1064no—
167C2×C8—531no—
168C2×C8—531no—
169C2×C8C22×D5312yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—531no—
173C4⋊C4—532no—
174C4⋊C4—531no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—1061no—
179C4⋊C4—1061no—
180C4⋊C4C22×D5311yescharacteristic
181C4⋊C4C22×D5312yes—
182C22⋊C4—532no—
183C22⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C42—21no—
187C42—1062no—
188C42—1061no—
189C42C22×D5311yescharacteristic
190C23—532no—
191Q8—21no—
192Q8—21no—
193Q8—81no—
194Q8—1061no—
195Q8—1061no—
196Q8—2122no—
197D4—1062no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—532no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—4242no—
208C2×C4D4×D5311yescharacteristic
209C2×C4D4×D5311yescharacteristic
210C2×C4C23×D5311yescharacteristic
211C8—22no—
212C8—532no—
213C8—1061no—
214C22—1062no—
215C22—1062no—
216C22—1062no—
217C22C2×D4×D5311yescharacteristic
218C4—21no—
219C4—41no—
220C4—42no—
221C4—41no—
222C4—1062no—
223C4—2121no—
224C4—2122no—
225C4—2121no—
226C4#1313324912yes—
227C4#41×D5311yescharacteristic
228C4#1313326511yescharacteristic
229C2—2122no—
230C2#1681127511yescharacteristic
231C2C2×#1313324911yescharacteristic
232C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#477C2
socleC22×C53C22×D4
hypercenter#477D53
hypocenterC53#3356
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #16811019C2semidirect
4. #16810966C2semidirect
5. #16810952C2semidirect
6. #16810909C2semidirect
7. #16810880C2semidirect
8. C4×#11719925C2semidirect
10. #16810664C2semidirect
11. C53×#477C2semidirect
122. #477D53semidirect
128. C53#3356semidirect