Finite Group Databasebeta

← #20290558

Subgroups — #20290558

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×#13133282C214yesmaximal, complemented
2#16811594C211yesmaximal, characteristic, complemented
3C4×C22⋊#6157C211yesmaximal, characteristic, complemented
4C22×#11719949C211yesmaximal, characteristic
5C2×#11719936C2212yescomplemented
6C2×C22⋊#6157—21no—
7C2×C4×#6157C2212yescomplemented
8#13133282—28no—
9#13133274—21no—
10C22⋊#88324—21no—
11(C2×#6157)⋊C4—21no—
12C2×#11719949—28no—
13C2×#11719949—21no—
14C2×#11719949—21no—
15C2×#11719949C412yescomplemented
16C2×#11719949C2212yes—
17C22×D212—21no—
18C22×D212C411yescharacteristic, complemented
19C22×C4×D53—21no—
20C22×C4×D53C2211yescharacteristic
21C22×Q8×C53C411yescharacteristic, complemented
22C23×D53—21no—
23C23×D53D411yescharacteristic
24C22×#6157—22no—
25C2×Q8×C53—24no—
26C2×Q8×C53—21no—
27C2×Q8×C53C2×C412yes—
28C22×C4×C53—21no—
29C22×C4×C53C2×C411yescharacteristic
30C22×Dic53D411yescharacteristic
31#11719949—28no—
32#11719949—28no—
33#11719949—24no—
34#11719949—24no—
35#11719949—24no—
36#11719949D418yes—
37C2×D212—28no—
38C2×D212—22no—
39C2×D212—22no—
40C2×D212—21no—
41C2×D212—24no—
42C2×D212C2×C412yes—
43C2×C4×D53—28no—
44C2×C4×D53—22no—
45C2×C4×D53—22no—
46C2×C4×D53—24no—
47C2×C4×D53C2×C411yescharacteristic
48C2×C4×D53D411yescharacteristic
49C2×C4×D53D411yescharacteristic
50C2×C4×D53C2311yescharacteristic
51C22⋊#6157—24no—
52C2×#88324—22no—
53C4×#6157—24no—
54#11719936—24no—
55C23×C53C22⋊C411yescharacteristic
56C22×D53—24no—
57C22×D53—22no—
58C22×D53—22no—
59C22×D53—22no—
60C22×D53—41no—
61C22×D53—42no—
62C22×D53—42no—
63C22×D53C22⋊C411yescharacteristic
64C22×D53C2×D411yescharacteristic
65C2×#6157—24no—
66C2×#6157—42no—
67C2×#6157—42no—
68Q8×C53—24no—
69Q8×C53—22no—
70Q8×C53C22⋊C414yescomplemented
71C2×C4×C53—24no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53C22⋊C411yescharacteristic
75C2×C4×C53C22×C411yescharacteristic
76C2×Dic53—22no—
77C2×Dic53C22⋊C411yescharacteristic
78C2×Dic53C2×D411yescharacteristic
79D212—28no—
80D212—24no—
81D212—28no—
82D212—22no—
83D212C22⋊C414yescomplemented
84C4×D53—28no—
85C4×D53—22no—
86C4×D53—22no—
87C4×D53—28no—
88C4×D53—22no—
89C4×D53C2×D414yes—
90#88324—42no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53#10412yescomplemented
94C22×C53C2×C22⋊C411yescharacteristic
95C2×D53—21no—
96C2×D53—21no—
97C2×D53—21no—
98C2×D53—24no—
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—44no—
102C2×D53—48no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53#12012yes—
106#6157—44no—
107C4×C53—24no—
108C4×C53—21no—
109C4×C53C2×C22⋊C412yes—
110Dic53—21no—
111Dic53#12012yes—
112#1789—531nomaximal, Sylow
113C2×C53—22no—
114C2×C53#37911yescharacteristic
115C2×C53C2×#10411yescharacteristic
116C2×C53#42111yescharacteristic
117D53—22no—
118D53—41no—
119D53—44no—
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#615711yescharacteristic, complemented
131C22×D4—531no—
132C22×D4—1061no—
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—531no—
143C24—1061no—
144#41—538no—
145#41—1064no—
146#41—1064no—
147#41—1068no—
148#41—1068no—
149#41—1064no—
150C2×Q8—24no—
151C2×Q8—21no—
152C2×Q8C2×#615712yes—
153C2×D4—532no—
154C2×D4—1064no—
155C2×D4—1062no—
156C2×D4—1062no—
157C2×D4—1068no—
158C2×D4—1061no—
159C22×C4—21no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—531no—
164C22×C4—531no—
165C22×C4—1064no—
166C22×C4—1062no—
167C22×C4—1062no—
168C22×C4—1068no—
169C22×C4—1062no—
170C22×C4C2×#615711yescharacteristic
171M16—1064no—
172C2×C8—1062no—
173C22⋊C4—1064no—
174C42—1064no—
175C23—531no—
176C23—531no—
177C23—1062no—
178C23—1064no—
179C23—1062no—
180C23—1062no—
181C23—2121no—
182C23—2122no—
183C23—2122no—
184C23C22⋊#615711yescharacteristic
185Q8—24no—
186Q8—22no—
187Q8C22⋊#615714yescomplemented
188D4—534no—
189D4—1068no—
190D4—1068no—
191D4—1064no—
192D4—1062no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—531no—
197C2×C4—534no—
198C2×C4—531no—
199C2×C4—1062no—
200C2×C4—1068no—
201C2×C4—1062no—
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—532no—
214C22—1061no—
215C22—1061no—
216C22—1064no—
217C22—1061no—
218C22—1061no—
219C22—1061no—
220C22—2124no—
221C22—2128no—
222C22—2122no—
223C22—2122no—
224C22#1313328212yescomplemented
225C22C2×C22⋊#615711yescharacteristic
226C4—24no—
227C4—21no—
228C4—532no—
229C4—1061no—
230C4—2124no—
231C4C2×C22⋊#615712yes—
232C2—22no—
233C2—1062no—
234C2—2121no—
235C2—2124no—
236C2#1681156411yescharacteristic
237C2C2×#1313328211yescharacteristic
238C2#1681162211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22×Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22×Q8#6157
hypocenterC53#1789
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
1. C2×#13133282C2semidirect
2. #16811594C2semidirect
3. C4×C22⋊#6157C2semidirect
5. C2×#11719936C22semidirect
7. C2×C4×#6157C22semidirect
15. C2×#11719949C4semidirect
18. C22×D212C4semidirect
21. C22×Q8×C53C4semidirect
70. Q8×C53C22⋊C4semidirect
83. D212C22⋊C4semidirect
93. C22×C53#104semidirect
124. C53#1789semidirect
130. C22×Q8#6157semidirect
187. Q8C22⋊#6157semidirect
224. C22#13133282semidirect