Finite Group Databasebeta

← #20290103

Subgroups — #20290103

Order 6784.

334 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 62 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
1#16811595C211yesmaximal, characteristic, complemented
2C2×(C2×#6157)⋊C4C211yesmaximal, characteristic, complemented
3#16811476C212yesmaximal, complemented
4#16811475C212yesmaximal, complemented
5#16811135C211yesmaximal, characteristic
6C23×#6157—21no—
7C2×#11719940—21no—
8#13133274—21no—
9#13133274C2212yes—
10(C2×#6157)⋊C4—21no—
11(C2×#6157)⋊C4C2212yes—
12C2×#11719945C411yescharacteristic, complemented
13C22×C4×D53C2211yescharacteristic
14#13133201—21no—
15#13133197C411yescharacteristic, complemented
16#13133197C2211yescharacteristic
17#13133192C411yescharacteristic, complemented
18#13133192C2211yescharacteristic
19#13133187—21no—
20C4×D212—21no—
21C4×D212—21no—
22#117×C53C411yescharacteristic, complemented
23C23×D53D411yescharacteristic
24C22×#6157—24no—
25C22×#6157—22no—
26C22×#6157—42no—
27C22×#6157—42no—
28C22×C4×C53C2×C411yescharacteristic
29C2×C22⋊D53—21no—
30C22×Dic53D411yescharacteristic
31#11719945—42no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212C2×C411yescharacteristic, complemented
35C2×C4×D53—21no—
36C2×C4×D53—42no—
37C2×C4×D53C2×C411yescharacteristic
38C2×C4×D53D411yescharacteristic
39C2×C4×D53D411yescharacteristic
40C2×C4×D53C2311yescharacteristic
41C2×Dic106C2×C411yescharacteristic, complemented
42#11719940—42no—
43C2×#88324—22no—
44C4⋊C4×C53C2×C411yescharacteristic, complemented
45C4⋊C4×C53C2×C411yescharacteristic, complemented
46C22⋊C4×C53—21no—
47C42×C53—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51#11719921—21no—
52C4⋊Dic53—21no—
53C4⋊Dic53—21no—
54C23×C53C22⋊C411yescharacteristic
55C22×D53—22no—
56C22×D53—22no—
57C22×D53—21no—
58C22×D53—21no—
59C22×D53—21no—
60C22×D53C22⋊C411yescharacteristic
61C22×D53C2×D411yescharacteristic
62C2×#6157—24no—
63C2×#6157—48no—
64C2×#6157—42no—
65C2×#6157—42no—
66C2×C4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53C22⋊C411yescharacteristic
70C2×C4×C53C22×C411yescharacteristic
71C22⋊D53—42no—
72C2×Dic53—22no—
73C2×Dic53—21no—
74C2×Dic53C22⋊C411yescharacteristic
75C2×Dic53C2×D411yescharacteristic
76D212—41no—
77D212—42no—
78D212—41no—
79D212—41no—
80C4×D53—41no—
81C4×D53—41no—
82C4×D53—41no—
83C4×D53—41no—
84C4×D53—42no—
85Dic106—41no—
86#88324—42no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53#9912yes—
90C22×C53C2×C22⋊C411yescharacteristic
91C2×D53—24no—
92C2×D53—24no—
93C2×D53—21no—
94C2×D53—21no—
95C2×D53—21no—
96C2×D53—21no—
97C2×D53—41no—
98C2×D53—41no—
99C2×D53—41no—
100C2×D53#10412yes—
101C2×D53AGL(1,ℤ/8)11yescharacteristic
102C2×D53#13711yescharacteristic
103#6157—44no—
104C4×C53—21no—
105C4×C53—21no—
106C4×C53—41no—
107C4×C53—41no—
108Dic53—21no—
109Dic53—21no—
110Dic53—41no—
111#1504—531nomaximal, Sylow
112C2×C53—22no—
113C2×C53C2×#9911yescharacteristic
114C2×C53#41711yescharacteristic
115C2×C53C2×#10411yescharacteristic
116D53—21no—
117D53—22no—
118D53—21no—
119D53—81no—
120#525—531no—
121#407—531no—
122C2×#95—531no—
123#328—532no—
124#327—532no—
125C53#150411yescharacteristic, complemented, Sylow
126C2×#41—531no—
127C23×C4—531no—
128C23×C4—1061no—
129C2×M16—1061no—
130#123—1061no—
131C22⋊Q8—531no—
132C22⋊Q8—531no—
133#121—531no—
134#121—531no—
135#120—1061no—
136C4×D4—1061no—
137C4×D4—1061no—
138#117#615711yescharacteristic, complemented
139C22⋊C8—532no—
140C22⋊C8—1061no—
141#95—532no—
142#95—1061no—
143C24—531no—
144#41—2122no—
145C2×Q8—531no—
146C2×D4—531no—
147C2×D4—1061no—
148C2×D4—1061no—
149C2×D4—1061no—
150C22×C4—531no—
151C22×C4—531no—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—1062no—
156C22×C4—1064no—
157C22×C4—1061no—
158C22×C4—2122no—
159C22×C4—2122no—
160C22×C4—2122no—
161C22×C4C2×#615711yescharacteristic
162M16—2122no—
163C2×C8—1062no—
164C4⋊C4—1061no—
165C4⋊C4—1061no—
166C4⋊C4C2×#615711yescharacteristic, complemented
167C4⋊C4C2×#615711yescharacteristic, complemented
168C22⋊C4—21no—
169C22⋊C4—1061no—
170C22⋊C4—1061no—
171C22⋊C4—1061no—
172C22⋊C4—1061no—
173C42—21no—
174C23—531no—
175C23—531no—
176C23—1062no—
177C23—1061no—
178C23—1061no—
179C23—1062no—
180C23—1061no—
181C23C22⋊#615711yescharacteristic
182Q8—2121no—
183D4—2121no—
184D4—2121no—
185D4—2121no—
186D4—2122no—
187D4—2122no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—531no—
192C2×C4—531no—
193C2×C4—1062no—
194C2×C4—1064no—
195C2×C4—1061no—
196C2×C4—2121no—
197C2×C4—2121no—
198C2×C4—2121no—
199C2×C4—2121no—
200C2×C4—2122no—
201C2×C4—2122no—
202C2×C4—2128no—
203C2×C4—2122no—
204C2×C4C22⋊#615711yescharacteristic
205C2×C4C22×#615711yescharacteristic
206C8—2122no—
207C22—21no—
208C22—21no—
209C22—531no—
210C22—531no—
211C22—532no—
212C22—1064no—
213C22—1064no—
214C22—1061no—
215C22—1061no—
216C22—1061no—
217C22—1061no—
218C22—2121no—
219C22—2121no—
220C22—2121no—
221C22#1313327612yes—
222C22C2×C22⋊#615711yescharacteristic
223C4—21no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—1061no—
228C4—1061no—
229C4—2124no—
230C4—2121no—
231C2—22no—
232C2—1061no—
233C2—1061no—
234C2—1062no—
235C2—4241no—
236C2C2×#1313327611yescharacteristic
237C2#1681161411yescharacteristic
238C2D53⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#117×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#117#6157
hypocenterC53#1504
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #16811595C2semidirect
2. C2×(C2×#6157)⋊C4C2semidirect
3. #16811476C2semidirect
4. #16811475C2semidirect
12. C2×#11719945C4semidirect
15. #13133197C4semidirect
17. #13133192C4semidirect
22. #117×C53C4semidirect
34. C2×D212C2×C4semidirect
41. C2×Dic106C2×C4semidirect
44. C4⋊C4×C53C2×C4semidirect
45. C4⋊C4×C53C2×C4semidirect
125. C53#1504semidirect
138. #117#6157semidirect
166. C4⋊C4C2×#6157semidirect
167. C4⋊C4C2×#6157semidirect