Finite Group Databasebeta

← #20290733

Subgroups — #20290733

Order 6784.

442 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 76 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×C4×#6157C211yesmaximal, characteristic, complemented
2C2×#13133281C212yesmaximal, complemented
3#16811594C211yesmaximal, characteristic, complemented
4#16811493C212yesmaximal, complemented
5#16811133C211yesmaximal, characteristic
6C23×#6157—21no—
7C2×#11719936C2212yes—
8C2×C4×#6157—21no—
9C2×C4×#6157—24no—
10C2×C4×#6157C2212yes—
11#13133281—24no—
12#13133274—21no—
13C22⋊#88324—21no—
14C2×#11719947C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133199—21no—
17#13133197C411yescharacteristic, complemented
18#13133196C411yescharacteristic, complemented
19#13133188C411yescharacteristic, complemented
20#13133185—21no—
21D4×Dic53—21no—
22#13133177—21no—
23#13133175C2211yescharacteristic
24#121×C53C411yescharacteristic, complemented
25C23×D53D411yescharacteristic
26C22×#6157—24no—
27C22×#6157—24no—
28C22×#6157—24no—
29C22×#6157—22no—
30C2×D4×C53—21no—
31C2×D4×C53C2×C411yescharacteristic, complemented
32C22×C4×C53C2×C411yescharacteristic
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35C22×Dic53—21no—
36C22×Dic53D411yescharacteristic
37#11719947—22no—
38#11719947—22no—
39C2×C4×D53—22no—
40C2×C4×D53—22no—
41C2×C4×D53C2×C411yescharacteristic
42C2×C4×D53D411yescharacteristic
43C2×C4×D53D411yescharacteristic
44C2×C4×D53C2311yescharacteristic
45C2×Dic106C2×C411yescharacteristic, complemented
46C2×#88324—22no—
47C4×#6157—24no—
48C4×#6157—22no—
49C4×#6157D414yescomplemented
50#11719936—22no—
51#11719936—22no—
52C4⋊C4×C53C2×C411yescharacteristic, complemented
53C22⋊C4×C53—21no—
54C22⋊Dic53—21no—
55C22⋊Dic53—21no—
56#11719921—21no—
57#11719921—21no—
58C4⋊Dic53—21no—
59C4⋊Dic53C2×C411yescharacteristic, complemented
60#11719919—21no—
61C4×Dic53—21no—
62C23×C53—21no—
63C23×C53C22⋊C411yescharacteristic
64C22×D53—22no—
65C22×D53—22no—
66C22×D53—21no—
67C22×D53—21no—
68C22×D53C22⋊C411yescharacteristic
69C22×D53C2×D411yescharacteristic
70C2×#6157—216no—
71C2×#6157—24no—
72C2×#6157—24no—
73C2×#6157—24no—
74D4×C53—22no—
75D4×C53—42no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53C22⋊C411yescharacteristic
79C2×C4×C53C22×C411yescharacteristic
80C22⋊D53—42no—
81C22⋊D53—41no—
82C22⋊D53—41no—
83C2×Dic53—22no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—42no—
87C2×Dic53C22⋊C411yescharacteristic
88C2×Dic53C2×D411yescharacteristic
89C4×D53—21no—
90C4×D53—21no—
91C4×D53—22no—
92C4×D53—22no—
93C4×D53C2×D412yes—
94C4×D53#4112yes—
95Dic106—22no—
96#88324—42no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53#10412yescomplemented
103C22×C53C2×C22⋊C411yescharacteristic
104C2×D53—24no—
105C2×D53—24no—
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—21no—
109C2×D53—21no—
110C2×D53#10412yescomplemented
111C2×D53#12011yescharacteristic
112C2×D53#12311yescharacteristic
113#6157—28no—
114C4×C53—21no—
115C4×C53—41no—
116C4×C53C4×D412yes—
117Dic53—21no—
118Dic53—41no—
119Dic53—41no—
120Dic53#12112yes—
121#1896—531nomaximal, Sylow
122C2×C53—22no—
123C2×C53—81no—
124C2×C53#38611yescharacteristic
125C2×C53C2×#10411yescharacteristic
126C2×C53C2×#10411yescharacteristic
127D53—21no—
128D53—22no—
129D53—21no—
130#525—531no—
131C22×C42—531no—
132C2×#104—532no—
133#407—531no—
134#339—532no—
135C53#189611yescharacteristic, complemented, Sylow
136C2×#41—531no—
137C23×C4—531no—
138C23×C4—1061no—
139C2×M16—532no—
140#123—1061no—
141C22⋊Q8—531no—
142C22⋊Q8—531no—
143#121—531no—
144#121#615711yescharacteristic, complemented
145#120—1061no—
146C4×D4—1061no—
147C4×D4—1061no—
148#117—531no—
149C2×C42—532no—
150C2×C42—1064no—
151C2×C42—1061no—
152#104—1064no—
153C22⋊C8—1061no—
154C22⋊C8—1061no—
155C24—531no—
156#41—1062no—
157#41—1062no—
158C2×Q8—531no—
159C2×D4—21no—
160C2×D4—1061no—
161C2×D4—1061no—
162C2×D4C2×#615711yescharacteristic, complemented
163C22×C4—531no—
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—1062no—
169C22×C4—1062no—
170C22×C4—1061no—
171C22×C4—1062no—
172C22×C4—1064no—
173C22×C4—1064no—
174C22×C4—1064no—
175C22×C4C2×#615711yescharacteristic
176M16—1062no—
177M16—1062no—
178C2×C8—1062no—
179C4⋊C4—531no—
180C4⋊C4—1061no—
181C4⋊C4—1061no—
182C4⋊C4C2×#615711yescharacteristic, complemented
183C22⋊C4—21no—
184C22⋊C4—1061no—
185C22⋊C4—1061no—
186C22⋊C4—1061no—
187C22⋊C4—1061no—
188C42—534no—
189C42—1061no—
190C42—1062no—
191C42—1064no—
192C23—21no—
193C23—531no—
194C23—531no—
195C23—1061no—
196C23—1061no—
197C23—1062no—
198C23—1062no—
199C23C22⋊#615711yescharacteristic
200Q8—1062no—
201D4—22no—
202D4—42no—
203D4—2122no—
204D4—2121no—
205D4—2121no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—531no—
209C2×C4—532no—
210C2×C4—532no—
211C2×C4—531no—
212C2×C4—1062no—
213C2×C4—1061no—
214C2×C4—1061no—
215C2×C4—1062no—
216C2×C4—1064no—
217C2×C4—10616no—
218C2×C4—1064no—
219C2×C4—1064no—
220C2×C4—1062no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—2122no—
224C2×C4C22⋊#615711yescharacteristic
225C2×C4C22×#615711yescharacteristic
226C8—2122no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—531no—
233C22—531no—
234C22—532no—
235C22—1064no—
236C22—1064no—
237C22—1061no—
238C22—1061no—
239C22—1061no—
240C22—1061no—
241C22#1313328112yescomplemented
242C22C2×C22⋊#615711yescharacteristic
243C4—21no—
244C4—41no—
245C4—532no—
246C4—1068no—
247C4—1061no—
248C4—2121no—
249C4—2121no—
250C4D4×#615712yes—
251C2—22no—
252C2—81no—
253C2—1061no—
254C2—1061no—
255C2—1062no—
256C2#1681157111yescharacteristic
257C2C2×#1313328111yescharacteristic
258C2D53⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#121×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#121#6157
hypocenterC53#1896
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×C4×#6157C2semidirect
2. C2×#13133281C2semidirect
3. #16811594C2semidirect
4. #16811493C2semidirect
17. #13133197C4semidirect
18. #13133196C4semidirect
19. #13133188C4semidirect
24. #121×C53C4semidirect
31. C2×D4×C53C2×C4semidirect
45. C2×Dic106C2×C4semidirect
49. C4×#6157D4semidirect
52. C4⋊C4×C53C2×C4semidirect
59. C4⋊Dic53C2×C4semidirect
102. C22×C53#104semidirect
110. C2×D53#104semidirect
135. C53#1896semidirect
144. #121#6157semidirect
162. C2×D4C2×#6157semidirect
182. C4⋊C4C2×#6157semidirect
241. C22#13133281semidirect