Finite Group Databasebeta

← #20283985

Subgroups — #20283985

Order 6784.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133263C211yesmaximal, characteristic, complemented
2C2×#13133245C211yesmaximal, characteristic, complemented
3C22×D424C211yesmaximal, characteristic, complemented
4#16811187C211yesmaximal, characteristic, complemented
5#16810770C214yesmaximal, complemented
6#16810759C214yesmaximal, complemented
7#16810629C211yesmaximal, characteristic, complemented
8#16810620C211yesmaximal, characteristic, complemented
9C53×#408C211yesmaximal, characteristic, complemented
10#13133263—24no—
11#13133263—24no—
12C2×#11719949—21no—
13C2×#11719945C2211yescharacteristic
14C2×#11719945C2212yescomplemented
15C22×D212C2211yescharacteristic
16C22×D212C2212yescomplemented
17C22×D212C2211yescharacteristic, complemented
18C2×D4×D53—21no—
19C2×D4×D53—22no—
20#13133245—24no—
21C2×#11719913C2212yescomplemented
22C2×D424—24no—
23C2×D424C2212yescomplemented
24C2×D424C2214yescomplemented
25#13133213—21no—
26#13133206—21no—
27#13133191C2214yescomplemented
28#13133185—22no—
29#13133176C2211yescharacteristic, complemented
30#13133142C2212yescomplemented
31#13133139C2212yescomplemented
32#13133139C2214yescomplemented
33C2×#41×C53C2211yescharacteristic
34C2×M16×C53C2211yescharacteristic, complemented
35C22×C8×C53C2211yescharacteristic, complemented
36C22⋊C8×C53C2214yescomplemented
37C23×D53—21no—
38C23×D53—22no—
39C23×D53—21no—
40#41×C53—22no—
41#41×C53—22no—
42C2×Q8×C53D411yescharacteristic
43C2×D4×C53D412yes—
44C2×D4×C53D411yescharacteristic
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—24no—
50#11719949—28no—
51D4×D53—216no—
52D4×D53—28no—
53#11719945—22no—
54#11719945—24no—
55#11719945—24no—
56#11719945D414yescomplemented
57C2×D212—21no—
58C2×D212—22no—
59C2×D212—22no—
60C2×D212—24no—
61C2×D212—24no—
62C2×D212—22no—
63C2×D212—22no—
64C2×D212—21no—
65C2×D212D414yescomplemented
66C2×D212C2312yes—
67C2×D212C2312yes—
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—22no—
71C2×Dic106C2311yescharacteristic
72M16×C53—22no—
73C2×C8×C53—22no—
74C2×C8×C53C2312yes—
75C2×C8×C53C2312yes—
76C22⋊Dic53—21no—
77#11719921—24no—
78C4⋊Dic53C2312yes—
79C4×Dic53—21no—
80D424—28no—
81D424—24no—
82#11719913—24no—
83C23×C53C2×D412yes—
84C23×C53C2×D411yescharacteristic
85C22×D53—21no—
86C22×D53—22no—
87C22×D53—28no—
88C22×D53—24no—
89C22×D53—22no—
90C22×D53—42no—
91C22×D53—82no—
92C22×D53—84no—
93C22×D53—82no—
94Q8×C53—22no—
95D4×C53—24no—
96D4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53C2×D412yes—
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2×D412yes—
102C2×C4×C53C2×D412yes—
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—28no—
105C22⋊D53—24no—
106C22⋊D53—84no—
107C2×Dic53—21no—
108C2×Dic53—22no—
109D212—28no—
110D212—24no—
111D212—21no—
112D212—22no—
113D212—22no—
114D212—22no—
115D212—42no—
116D212C2×D412yes—
117D212C2×D414yes—
118C4×D53—28no—
119C4×D53—24no—
120Dic106—21no—
121Dic106C2×D412yes—
122C8×C53—22no—
123C8×C53—22no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—24no—
127C22×C53#12012yes—
128C22×C53C22×D411yescharacteristic
129C2×D53—24no—
130C2×D53—22no—
131C2×D53—44no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—44no—
135C2×D53—84no—
136C2×D53—88no—
137C2×D53—84no—
138C4×C53—22no—
139C4×C53#12012yes—
140C4×C53C22×D411yescharacteristic
141C4×C53C22×D411yescharacteristic
142Dic53—22no—
143Dic53—42no—
144#3008—531nomaximal, Sylow
145C2×C53—22no—
146C2×C53—42no—
147C2×C53C2×#12011yescharacteristic
148C2×C53#57612yes—
149D53—44no—
150D53—42no—
151D53—82no—
152C2×H(5,𝔽2)—531no—
153C2×AGL(1,ℤ/8)—531no—
154C22×D8—531no—
155#535—531no—
156#449—534no—
157#447—534no—
158#418—531no—
159#416—531no—
160#408D5311yescharacteristic, complemented
161C53#300811yescharacteristic, complemented, Sylow
162H(5,𝔽2)—1064no—
163H(5,𝔽2)—1064no—
164C2×#41—531no—
165C2×#41—532no—
166C2×#41—1061no—
167C2×#41C2×D5311yescharacteristic
168C22×D4—531no—
169C22×D4—531no—
170C22×D4—532no—
171C22×D4—1062no—
172C22×D4—1061no—
173AGL(1,ℤ/8)—1064no—
174C2×SD16—532no—
175C2×D8—534no—
176C2×D8—532no—
177C2×D8—1064no—
178C2×M16C2×D5311yescharacteristic, complemented
179C22×C8C2×D5311yescharacteristic, complemented
180#127—1061no—
181#124—1061no—
182#121—534no—
183#120—1062no—
184#117—531no—
185#103—532no—
186#102—532no—
187#102—534no—
188C22⋊C8C2×D5314yescomplemented
189C24—1062no—
190C24—1061no—
191C24—1061no—
192#41—22no—
193#41—22no—
194#41—534no—
195#41—1064no—
196#41—1064no—
197#41—1068no—
198#41—1062no—
199C2×Q8—531no—
200C2×Q8D21211yescharacteristic
201C2×D4—534no—
202C2×D4—532no—
203C2×D4—531no—
204C2×D4—532no—
205C2×D4—1064no—
206C2×D4—1064no—
207C2×D4—10616no—
208C2×D4—1068no—
209C2×D4—1062no—
210C2×D4—1061no—
211C2×D4—1064no—
212C2×D4—1062no—
213C2×D4—1061no—
214C2×D4—1061no—
215C2×D4—1062no—
216C2×D4—1062no—
217C2×D4—1062no—
218C2×D4D21212yes—
219C2×D4D21211yescharacteristic
220C22×C4—1062no—
221C22×C4—1061no—
222C22×C4C22×D5312yes—
223C22×C4C22×D5311yescharacteristic
224SD16—1064no—
225D8—1068no—
226D8—1064no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D5312yes—
230C2×C8C22×D5312yes—
231C4⋊C4—532no—
232C22⋊C4—1061no—
233C22⋊C4—1064no—
234C42—1061no—
235C23—1068no—
236C23—1064no—
237C23—1062no—
238C23—1061no—
239C23—1062no—
240C23—4242no—
241C23—4242no—
242C23—4244no—
243C23—2122no—
244C23C2×D21212yes—
245C23C2×D21211yescharacteristic
246Q8—22no—
247Q8—532no—
248Q8—1061no—
249D4—24no—
250D4—22no—
251D4—534no—
252D4—532no—
253D4—1068no—
254D4—1068no—
255D4—1064no—
256D4—1064no—
257D4—1061no—
258D4—1062no—
259D4—1062no—
260D4—1062no—
261D4—4244no—
262D4—2122no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—1068no—
266C2×C4—1061no—
267C2×C4—1064no—
268C2×C4—1062no—
269C2×C4C2×D21212yes—
270C2×C4C2×D21211yescharacteristic
271C2×C4D4×D5312yes—
272C2×C4D4×D5312yes—
273C2×C4C23×D5311yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—1064no—
280C22—1062no—
281C22—4244no—
282C22—4244no—
283C22—4248no—
284C22—2122no—
285C22—2124no—
286C22—2124no—
287C22—2122no—
288C22#1313318712yes—
289C22C22×D21211yescharacteristic
290C4—22no—
291C4—1062no—
292C4—2122no—
293C4#1313318712yes—
294C4C2×D4×D5311yescharacteristic
295C4C2×D4×D5311yescharacteristic
296C2—22no—
297C2—42no—
298C2—4242no—
299C2—2124no—
300C2—2122no—
301C2C2×#1313318711yescharacteristic
302C2#1681142812yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#408C2
socleC22×C53C22×D4
hypercenter#408D53
hypocenterC53#3008
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. C2×#13133245C2semidirect
3. C22×D424C2semidirect
4. #16811187C2semidirect
5. #16810770C2semidirect
6. #16810759C2semidirect
7. #16810629C2semidirect
8. #16810620C2semidirect
9. C53×#408C2semidirect
14. C2×#11719945C22semidirect
16. C22×D212C22semidirect
17. C22×D212C22semidirect
21. C2×#11719913C22semidirect
23. C2×D424C22semidirect
24. C2×D424C22semidirect
27. #13133191C22semidirect
29. #13133176C22semidirect
30. #13133142C22semidirect
31. #13133139C22semidirect
32. #13133139C22semidirect
34. C2×M16×C53C22semidirect
35. C22×C8×C53C22semidirect
36. C22⋊C8×C53C22semidirect
56. #11719945D4semidirect
65. C2×D212D4semidirect
160. #408D53semidirect
161. C53#3008semidirect
178. C2×M16C2×D53semidirect
179. C22×C8C2×D53semidirect
188. C22⋊C8C2×D53semidirect