Finite Group Databasebeta

← #20290556

Subgroups — #20290556

Order 6784.

618 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 90 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×#13133281C214yesmaximal, complemented
2#16811594C211yesmaximal, characteristic, complemented
3C4×C22⋊#6157C211yesmaximal, characteristic, complemented
4C22×#11719947C211yesmaximal, characteristic
5C2×#11719936C2212yescomplemented
6C2×C22⋊#6157—21no—
7C2×C4×#6157C2212yescomplemented
8#13133281—28no—
9#13133274—21no—
10C22⋊#88324—21no—
11(C2×#6157)⋊C4—21no—
12C2×#11719947—28no—
13C2×#11719947—21no—
14C2×#11719947—21no—
15C2×#11719947C412yescomplemented
16C2×#11719947C2212yes—
17C22×Dic106C411yescharacteristic, complemented
18C22×C22⋊D53—21no—
19C23×Dic53—21no—
20C22×C4×D53C2211yescharacteristic
21C22×D4×C53C411yescharacteristic, complemented
22C24×C53—21no—
23C23×D53D411yescharacteristic
24C22×#6157—22no—
25C2×D4×C53—24no—
26C2×D4×C53—21no—
27C2×D4×C53C2×C412yes—
28C22×C4×C53C2×C411yescharacteristic
29C2×C22⋊D53—28no—
30C2×C22⋊D53—22no—
31C2×C22⋊D53—22no—
32C22×Dic53—28no—
33C22×Dic53—21no—
34C22×Dic53—22no—
35C22×Dic53—22no—
36C22×Dic53D411yescharacteristic
37#11719947—28no—
38#11719947—28no—
39#11719947—24no—
40#11719947—24no—
41#11719947—24no—
42#11719947D418yes—
43C2×C4×D53—24no—
44C2×C4×D53C2×C411yescharacteristic
45C2×C4×D53D411yescharacteristic
46C2×C4×D53D411yescharacteristic
47C2×C4×D53C2311yescharacteristic
48C2×Dic106—21no—
49C2×Dic106—24no—
50C2×Dic106C2×C412yes—
51C22⋊#6157—24no—
52C2×#88324—22no—
53C4×#6157—24no—
54#11719936—24no—
55C23×C53—24no—
56C23×C53—22no—
57C23×C53—42no—
58C23×C53—42no—
59C23×C53C22⋊C411yescharacteristic
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—41no—
63C22×D53C22⋊C411yescharacteristic
64C22×D53C2×D411yescharacteristic
65C2×#6157—24no—
66C2×#6157—42no—
67C2×#6157—42no—
68D4×C53—24no—
69D4×C53—22no—
70D4×C53C22⋊C414yescomplemented
71C2×C4×C53—22no—
72C2×C4×C53C22⋊C411yescharacteristic
73C2×C4×C53C22×C411yescharacteristic
74C22⋊D53—28no—
75C22⋊D53—28no—
76C2×Dic53—24no—
77C2×Dic53—28no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—28no—
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C2×D411yescharacteristic
83C4×D53—22no—
84C4×D53—22no—
85C4×D53—22no—
86C4×D53C2×D414yes—
87Dic106—24no—
88Dic106—22no—
89Dic106C22⋊C414yescomplemented
90#88324—42no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—24no—
94C22×C53—48no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53#10412yescomplemented
98C22×C53C2×C22⋊C411yescharacteristic
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—21no—
104C2×D53—44no—
105C2×D53#12012yes—
106#6157—44no—
107C4×C53—21no—
108C4×C53C2×C22⋊C412yes—
109Dic53—24no—
110Dic53—21no—
111Dic53#12012yes—
112#1789—531nomaximal, Sylow
113C2×C53—22no—
114C2×C53—44no—
115C2×C53#37911yescharacteristic
116C2×C53C2×#10411yescharacteristic
117C2×C53#42111yescharacteristic
118D53—22no—
119D53—41no—
120C22×#41—531no—
121C2×#104—534no—
122#407—531no—
123C4×C22⋊C4—531no—
124C53#178911yescharacteristic, complemented, Sylow
125C2×#41—532no—
126C2×#41—532no—
127C2×#41—1061no—
128C2×#41—1061no—
129C2×#41—1068no—
130C22×Q8—531no—
131C22×D4—1061no—
132C22×D4#615711yescharacteristic, complemented
133C23×C4—531no—
134C23×C4—1061no—
135C2×M16—532no—
136C2×C22⋊C4—1061no—
137C2×C42—532no—
138#104—1068no—
139C22⋊C8—1061no—
140C22⋊C8—1061no—
141#95—1061no—
142C24—21no—
143C24—531no—
144#41—538no—
145#41—1064no—
146#41—1064no—
147#41—1068no—
148#41—1068no—
149#41—1064no—
150C2×Q8—532no—
151C2×Q8—1064no—
152C2×Q8—1061no—
153C2×D4—24no—
154C2×D4—21no—
155C2×D4—1062no—
156C2×D4—1062no—
157C2×D4—1068no—
158C2×D4C2×#615712yes—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—531no—
164C22×C4—1064no—
165C22×C4—1062no—
166C22×C4—1062no—
167C22×C4—1068no—
168C22×C4—1061no—
169C22×C4—1062no—
170C22×C4C2×#615711yescharacteristic
171M16—1064no—
172C2×C8—1062no—
173C22⋊C4—1064no—
174C42—1064no—
175C23—24no—
176C23—22no—
177C23—42no—
178C23—42no—
179C23—531no—
180C23—531no—
181C23—1062no—
182C23—1062no—
183C23—2121no—
184C23C22⋊#615711yescharacteristic
185Q8—534no—
186Q8—1064no—
187Q8—1062no—
188D4—24no—
189D4—22no—
190D4—1068no—
191D4—1068no—
192D4C22⋊#615714yescomplemented
193C2×C4—22no—
194C2×C4—531no—
195C2×C4—534no—
196C2×C4—531no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1068no—
201C2×C4—1064no—
202C2×C4—1068no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1064no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4C22⋊#615711yescharacteristic
209C2×C4C22×#615711yescharacteristic
210C8—2122no—
211C22—21no—
212C22—21no—
213C22—24no—
214C22—48no—
215C22—42no—
216C22—42no—
217C22—532no—
218C22—1061no—
219C22—1061no—
220C22—1061no—
221C22—1061no—
222C22—1061no—
223C22—2124no—
224C22#1313328112yescomplemented
225C22C2×C22⋊#615711yescharacteristic
226C4—21no—
227C4—532no—
228C4—1061no—
229C4—1064no—
230C4—2124no—
231C4C2×C22⋊#615712yes—
232C2—22no—
233C2—44no—
234C2—1062no—
235C2—2121no—
236C2#1681156411yescharacteristic
237C2C2×#1313328111yescharacteristic
238C2#1681162111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22×D4#6157
hypocenterC53#1789
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133281C2semidirect
2. #16811594C2semidirect
3. C4×C22⋊#6157C2semidirect
5. C2×#11719936C22semidirect
7. C2×C4×#6157C22semidirect
15. C2×#11719947C4semidirect
17. C22×Dic106C4semidirect
21. C22×D4×C53C4semidirect
70. D4×C53C22⋊C4semidirect
89. Dic106C22⋊C4semidirect
97. C22×C53#104semidirect
124. C53#1789semidirect
132. C22×D4#6157semidirect
192. D4C22⋊#6157semidirect
224. C22#13133281semidirect