Finite Group Databasebeta

← #20290728

Subgroups — #20290728

Order 6784.

380 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×D53⋊C8C211yesmaximal, characteristic
2C2×#13133280C212yesmaximal, complemented
3#16811565C211yesmaximal, characteristic
4#16811496C212yesmaximal, complemented
5C22⋊Q8×D53C211yesmaximal, characteristic
6C2×D53⋊C8—24no—
7C2×D53⋊C8—21no—
8C2×D53⋊C8C2212yes—
9C22×#88324—21no—
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—
43C2×#88324—24no—
44C2×#88324—22no—
45C4⋊#6157—22no—
46C4⋊#6157—22no—
47D53⋊C8—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—22no—
64C22×D53C22⋊C411yescharacteristic
65C22×D53SD1612yes—
66C22×D53Q1612yes—
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—42no—
89C4×D53C2×D412yes—
90C4×D53#4112yes—
91Dic106—22no—
92Dic106—41no—
93Dic106—41no—
94#88324—24no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53#10312yescomplemented
98C22×C53C2×C22⋊C411yescharacteristic
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—24no—
102C2×D53—24no—
103C2×D53#10312yescomplemented
104C2×D53#12112yes—
105C2×D53C2×SD1611yescharacteristic
106C2×D53C2×Q1611yescharacteristic
107C2×D53#13512yes—
108#6157—82no—
109C4×C53—21no—
110C4×C53—41no—
111C4×C53—41no—
112C4×C53C4×D412yes—
113Dic53—21no—
114Dic53—41no—
115Dic53—41no—
116Dic53#12011yescharacteristic
117Dic53#12311yescharacteristic
118#1892—531nomaximal, Sylow
119C2×C53—22no—
120C2×C53#38611yescharacteristic
121C2×C53C2×#10311yescharacteristic
122C2×C53#41611yescharacteristic
123D53—22no—
124D53#46512yes—
125D53#46712yes—
126C23×C8—531no—
127C2×C22⋊Q8—531no—
128C2×#103—532no—
129#380—531no—
130#340—532no—
131C53#189211yescharacteristic, complemented, Sylow
132C22×Q8—531no—
133C23×C4—531no—
134C22×C8—532no—
135C22×C8—1064no—
136C22×C8—1061no—
137C22×C8—1061no—
138C22⋊Q8—531no—
139C22⋊Q8—531no—
140C22⋊Q8—531no—
141C22⋊Q8—1061no—
142C22⋊Q8—1061no—
143C22⋊Q8#615711yescharacteristic, complemented
144C2×C4⋊C4—531no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—1061no—
147C2×C22⋊C4—1061no—
148C2×C22⋊C4—1061no—
149#103—1064no—
150#95—1061no—
151C24—531no—
152C2×Q8—531no—
153C2×Q8—1061no—
154C2×Q8—1062no—
155C2×Q8—1062no—
156C2×Q8C2×#615711yescharacteristic
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—1062no—
163C22×C4—1062no—
164C22×C4—1061no—
165C22×C4—1061no—
166C22×C4—1062no—
167C22×C4C2×#615711yescharacteristic
168C2×C8—534no—
169C2×C8—1062no—
170C2×C8—1062no—
171C2×C8—1064no—
172C2×C8—1064no—
173C4⋊C4—21no—
174C4⋊C4—531no—
175C4⋊C4—1061no—
176C4⋊C4—1061no—
177C4⋊C4—1061no—
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C4⋊C4—1061no—
181C4⋊C4C2×#615711yescharacteristic
182C22⋊C4—21no—
183C22⋊C4—1061no—
184C22⋊C4—1061no—
185C22⋊C4—1061no—
186C22⋊C4—2122no—
187C23—531no—
188C23—532no—
189C23—532no—
190C23—531no—
191C23—1062no—
192C23—1062no—
193C23C22⋊#615711yescharacteristic
194Q8—22no—
195Q8—1062no—
196Q8—2121no—
197Q8—2121no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—531no—
202C2×C4—532no—
203C2×C4—532no—
204C2×C4—531no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1062no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—1062no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4C22⋊#615711yescharacteristic
219C2×C4C22×#615711yescharacteristic
220C8—1064no—
221C22—21no—
222C22—21no—
223C22—532no—
224C22—531no—
225C22—531no—
226C22—532no—
227C22—532no—
228C22—1064no—
229C22—1064no—
230C22—1061no—
231C22—1061no—
232C22#1313328012yescomplemented
233C22C2×C22⋊#615711yescharacteristic
234C4—21no—
235C4—41no—
236C4—41no—
237C4—531no—
238C4—531no—
239C4—1061no—
240C4—4242no—
241C4—2121no—
242C4—2121no—
243C4D4×#615712yes—
244C2—22no—
245C2—532no—
246C2—532no—
247C2—1062no—
248C2#1681157111yescharacteristic
249C2C2×#1313328011yescharacteristic
250C2#1681161311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1892
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
97. C22×C53#103semidirect
103. C2×D53#103semidirect
131. C53#1892semidirect
143. C22⋊Q8#6157semidirect
232. C22#13133280semidirect