Finite Group Databasebeta

← #20287144

Subgroups — #20287144

Order 6784.

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

normal subgroups only

This group has 340 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#16811346C211yesmaximal, characteristic
3#16811026C211yesmaximal, characteristic, complemented
4#16810967C212yesmaximal, complemented
5#16810962C212yesmaximal, complemented
6#16810909C211yesmaximal, characteristic, complemented
7#16810863C212yesmaximal, complemented
8C4×#11719925C211yesmaximal, characteristic, complemented
9#16810688C211yesmaximal, characteristic
10#16810658C212yesmaximal, complemented
11C53×#475C211yesmaximal, characteristic, complemented
12C2×#11719925C2211yescharacteristic
13C2×C8×D53C2212yes—
14#13133217C2211yescharacteristic
15#13133212—22no—
16#13133212—22no—
17#13133206—21no—
18#13133205—21no—
19#13133197C2212yes—
20#13133193C2212yes—
21Q8×Dic53—21no—
22C4×Dic106—21no—
23C4×Dic106C2211yescharacteristic
24C4×D212C2211yescharacteristic
25#13133175C2212yes—
26#13133175C2212yes—
27#13133150C2212yes—
28#13133150C2212yes—
29#13133150C2211yescharacteristic
30#13133149C2212yes—
31#13133142C2212yes—
32#13133141C2211yescharacteristic
33#13133140C2211yescharacteristic
34#13133129C2212yes—
35C4×#88322C2211yescharacteristic
36C4⋊Q8×C53C2211yescharacteristic
37C4×Q8×C53C2211yescharacteristic
38#106×C53C2212yes—
39C4⋊C8×C53C2211yescharacteristic
40#103×C53C2212yes—
41C2×Q8×C53—21no—
42C2×Q8×C53C2311yescharacteristic
43C2×D212D411yescharacteristic
44C2×C4×D53C2312yes—
45C2×Dic106—21no—
46C2×Dic106C2311yescharacteristic
47C2×C8×C53C2312yes—
48C4⋊C4×C53—21no—
49C4⋊C4×C53—21no—
50C4⋊C4×C53C2312yes—
51C4⋊C4×C53C2311yescharacteristic
52C42×C53—21no—
53C42×C53C2311yescharacteristic
54#11719925—22no—
55#11719921—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—
67C4×Dic53—21no—
68C4×Dic53—22no—
69C4×Dic53—22no—
70C2×#88322C2311yescharacteristic
71C2×#88322C2311yescharacteristic
72C8×D53—24no—
73C22×D53C2×D412yes—
74Q8×C53—21no—
75Q8×C53—21no—
76Q8×C53—42no—
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—
90Dic106—21no—
91Dic106—21no—
92Dic106—42no—
93C8×C53—22no—
94#88322—21no—
95#88322#4112yes—
96C22×C53C22×D411yescharacteristic
97C2×D53—22no—
98C2×D53—22no—
99C2×D53—22no—
100C4×C53—21no—
101C4×C53—41no—
102C4×C53—42no—
103C4×C53—41no—
104C4×C53#13512yes—
105C4×C53C2×#4111yescharacteristic
106C4×C532−1+411yescharacteristic
107Dic53—22no—
108Dic53—41no—
109Dic53—42no—
110Dic53—41no—
111#3342—531nomaximal, Sylow
112C2×C53#54711yescharacteristic
113C2×C53C2×#13511yescharacteristic
114C2×C53#57811yescharacteristic
115D53—42no—
116#556—531no—
117#556—531no—
118#500—531no—
119#485—532no—
120#485—532no—
121#475—531no—
122#475D5311yescharacteristic, complemented
123#467—532no—
124C4×Q16—531no—
125C8×D4—531no—
126#427—532no—
127C53#334211yescharacteristic, complemented, Sylow
128C2×Q16—531no—
129C22×C8—532no—
130C4⋊Q8—531no—
131C4⋊Q8C2×D5311yescharacteristic
132#126—1062no—
133#126—1062no—
134#124—1061no—
135#124—1061no—
136C22⋊Q8—532no—
137C22⋊Q8—532no—
138C4×Q8—531no—
139C4×Q8—1061no—
140C4×Q8—1061no—
141C4×Q8C2×D5311yescharacteristic
142C4×D4—531no—
143#117—532no—
144#117—532no—
145#107—531no—
146#107—532no—
147#107—532no—
148#106—532no—
149#106C2×D5312yes—
150C4⋊C8C2×D5311yescharacteristic
151#103—531no—
152#103—532no—
153#103—531no—
154#103C2×D5312yes—
155C22⋊C8—532no—
156C4×C8—531no—
157C2×Q8—21no—
158C2×Q8—531no—
159C2×Q8—1061no—
160C2×Q8C22×D5311yescharacteristic
161C2×D4—531no—
162C22×C4—532no—
163Q16—1062no—
164C2×C8—531no—
165C2×C8—531no—
166C2×C8—1064no—
167C2×C8C22×D5312yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—531no—
171C4⋊C4—531no—
172C4⋊C4—532no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C4⋊C4—1061no—
176C4⋊C4—1061no—
177C4⋊C4C22×D5311yescharacteristic
178C4⋊C4C22×D5312yes—
179C22⋊C4—532no—
180C22⋊C4—1062no—
181C22⋊C4—1062no—
182C22⋊C4—1062no—
183C22⋊C4—1062no—
184C42—21no—
185C42—1062no—
186C42—1062no—
187C42—1061no—
188C42C22×D5311yescharacteristic
189C23—532no—
190Q8—21no—
191Q8—21no—
192Q8—42no—
193Q8—1061no—
194Q8—1061no—
195Q8—2122no—
196D4—1062no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—532no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1061no—
204C2×C4—1062no—
205C2×C4—1061no—
206C2×C4D4×D5311yescharacteristic
207C2×C4D4×D5311yescharacteristic
208C2×C4C23×D5311yescharacteristic
209C8—22no—
210C8—532no—
211C8—1061no—
212C22—1062no—
213C22—1062no—
214C22—1062no—
215C22C2×D4×D5311yescharacteristic
216C4—21no—
217C4—41no—
218C4—42no—
219C4—41no—
220C4—1062no—
221C4—2121no—
222C4—2122no—
223C4—2121no—
224C4#1313323812yes—
225C4#41×D5311yescharacteristic
226C4#1313326511yescharacteristic
227C2—2122no—
228C2#1681127511yescharacteristic
229C2C2×#1313323811yescharacteristic
230C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#475C2
socleC22×C53C22×D4
hypercenter#475D53
hypocenterC53#3342
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #16811026C2semidirect
4. #16810967C2semidirect
5. #16810962C2semidirect
6. #16810909C2semidirect
7. #16810863C2semidirect
8. C4×#11719925C2semidirect
10. #16810658C2semidirect
11. C53×#475C2semidirect
122. #475D53semidirect
127. C53#3342semidirect