Finite Group Databasebeta

← #20290730

Subgroups — #20290730

Order 6784.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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×D53⋊M16C211yesmaximal, characteristic
2C2×#13133280C212yesmaximal, complemented
3#16811565C211yesmaximal, characteristic
4#16811496C212yesmaximal, complemented
5C22⋊Q8×D53C211yesmaximal, characteristic
6C2×#11719936—21no—
7D53⋊M16—24no—
8C2×#11719940—21no—
9C2×D53⋊C8C2212yes—
10C2×C4⋊#6157C2212yes—
11C2×C22⋊#6157—21no—
12#13133280—24no—
13(C2×#6157)⋊C4—21no—
14C2×Q8×D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133197C411yescharacteristic, complemented
17#13133196C411yescharacteristic, complemented
18#13133195—21no—
19#13133194—21no—
20#13133193C411yescharacteristic, complemented
21C4⋊C4×D53—21no—
22C4⋊C4×D53C2211yescharacteristic
23C22⋊C4×D53—21no—
24C22⋊Q8×C53C411yescharacteristic, complemented
25C23×D53D411yescharacteristic
26C22×#6157—22no—
27C2×Q8×C53C2×C411yescharacteristic
28C22×C4×C53C2×C411yescharacteristic
29C22×Dic53D411yescharacteristic
30Q8×D53—22no—
31Q8×D53—22no—
32C2×C4×D53—22no—
33C2×C4×D53—22no—
34C2×C4×D53—21no—
35C2×C4×D53—21no—
36C2×C4×D53C2×C411yescharacteristic
37C2×C4×D53D411yescharacteristic
38C2×C4×D53D411yescharacteristic
39C2×C4×D53C2311yescharacteristic
40C2×Dic106—21no—
41C2×Dic106C2×C411yescharacteristic
42C22⋊#6157—42no—
43#11719940—24no—
44C2×#88324—22no—
45C4⋊#6157—22no—
46C4⋊#6157—22no—
47#11719936—24no—
48D53⋊C8—22no—
49D53⋊C8D414yes—
50C4⋊C4×C53—21no—
51C4⋊C4×C53C2×C411yescharacteristic
52C22⋊C4×C53—21no—
53C22⋊Dic53—21no—
54#11719921—21no—
55#11719921—21no—
56C4⋊Dic53—21no—
57C4⋊Dic53C2×C411yescharacteristic
58#11719919—21no—
59#11719919—21no—
60#11719919—21no—
61C23×C53C22⋊C411yescharacteristic
62C22×D53—22no—
63C22×D53—21no—
64C22×D53—21no—
65C22×D53—22no—
66C22×D53C22⋊C411yescharacteristic
67C22×D53C2×D411yescharacteristic
68C2×#6157—42no—
69C2×#6157—42no—
70C2×#6157—42no—
71Q8×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53C22⋊C411yescharacteristic
76C2×C4×C53C22×C411yescharacteristic
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C2×D411yescharacteristic
83C4×D53—22no—
84C4×D53—22no—
85C4×D53—21no—
86C4×D53—21no—
87C4×D53—42no—
88C4×D53—41no—
89C4×D53—41no—
90C4×D53C2×D412yes—
91C4×D53#4112yes—
92Dic106—22no—
93Dic106—41no—
94Dic106—41no—
95#88324—24no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53C2×C22⋊C411yescharacteristic
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53#12112yes—
106C2×D53AGL(1,ℤ/8)11yescharacteristic
107C2×D53#13712yes—
108C2×D53#13711yescharacteristic
109#6157—82no—
110C4×C53—21no—
111C4×C53—41no—
112C4×C53—41no—
113C4×C53C4×D412yes—
114Dic53—21no—
115Dic53—41no—
116Dic53—41no—
117Dic53#12011yescharacteristic
118Dic53#12311yescharacteristic
119#1894—531nomaximal, Sylow
120C2×C53—41no—
121C2×C53#38611yescharacteristic
122C2×C53#41711yescharacteristic
123C2×C53#41911yescharacteristic
124D53—41no—
125D53#46812yes—
126D53#47012yes—
127C22×M16—531no—
128C2×C22⋊Q8—531no—
129C2×#103—532no—
130#380—531no—
131#340—532no—
132C53#189411yescharacteristic, complemented, Sylow
133C22×Q8—531no—
134C23×C4—531no—
135C2×M16—1064no—
136C2×M16—1061no—
137C2×M16—1061no—
138C22×C8—532no—
139C22⋊Q8—531no—
140C22⋊Q8—531no—
141C22⋊Q8—531no—
142C22⋊Q8—1061no—
143C22⋊Q8—1061no—
144C22⋊Q8#615711yescharacteristic, complemented
145C2×C4⋊C4—531no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—1061no—
148C2×C22⋊C4—1061no—
149C2×C22⋊C4—1061no—
150#103—1064no—
151#95—1061no—
152C24—531no—
153C2×Q8—531no—
154C2×Q8—1061no—
155C2×Q8—1062no—
156C2×Q8—1062no—
157C2×Q8C2×#615711yescharacteristic
158C22×C4—531no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—1062no—
164C22×C4—1062no—
165C22×C4—1061no—
166C22×C4—1061no—
167C22×C4—1062no—
168C22×C4C2×#615711yescharacteristic
169M16—1064no—
170M16—1064no—
171C2×C8—534no—
172C2×C8—1062no—
173C2×C8—1062no—
174C4⋊C4—21no—
175C4⋊C4—531no—
176C4⋊C4—1061no—
177C4⋊C4—1061no—
178C4⋊C4—1061no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C4⋊C4—1061no—
182C4⋊C4C2×#615711yescharacteristic
183C22⋊C4—21no—
184C22⋊C4—1061no—
185C22⋊C4—1061no—
186C22⋊C4—1061no—
187C22⋊C4—2122no—
188C23—531no—
189C23—531no—
190C23—1061no—
191C23—1062no—
192C23—1061no—
193C23—1062no—
194C23C22⋊#615711yescharacteristic
195Q8—22no—
196Q8—1062no—
197Q8—2121no—
198Q8—2121no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—531no—
203C2×C4—532no—
204C2×C4—532no—
205C2×C4—531no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—1062no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—1062no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—2121no—
215C2×C4—2121no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4—2122no—
219C2×C4—2122no—
220C2×C4C22⋊#615711yescharacteristic
221C2×C4C22×#615711yescharacteristic
222C8—1064no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—532no—
227C22—531no—
228C22—531no—
229C22—532no—
230C22—1061no—
231C22—1061no—
232C22—1061no—
233C22—2122no—
234C22—2122no—
235C22C2×C22⋊#615711yescharacteristic
236C4—21no—
237C4—41no—
238C4—41no—
239C4—531no—
240C4—531no—
241C4—1061no—
242C4—4242no—
243C4—2121no—
244C4—2121no—
245C4D4×#615712yes—
246C2—41no—
247C2—532no—
248C2—532no—
249C2—2121no—
250C2#1681157111yescharacteristic
251C2#1681161411yescharacteristic
252C2#1681161611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1894
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#13133280C2semidirect
4. #16811496C2semidirect
16. #13133197C4semidirect
17. #13133196C4semidirect
20. #13133193C4semidirect
24. C22⋊Q8×C53C4semidirect
132. C53#1894semidirect
144. C22⋊Q8#6157semidirect