Finite Group Databasebeta

← #20284074

Subgroups — #20284074

Order 6784.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
2#16811423C211yesmaximal, characteristic, complemented
3#16811421C211yesmaximal, characteristic, complemented
4#16811192C211yesmaximal, characteristic, complemented
5#16810802C214yesmaximal, complemented
6#16810792C214yesmaximal, complemented
7#16810641C211yesmaximal, characteristic, complemented
8C2×#13133133C211yesmaximal, characteristic, complemented
9C53×#421C211yesmaximal, characteristic, complemented
10#13133263—22no—
11#13133263—22no—
12#13133263—22no—
13#13133263C2214yes—
14C2×#11719949—21no—
15C2×#11719945—21no—
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic, complemented
18C22×D212—21no—
19C22×D212C2211yescharacteristic
20C2×D4×D53—22no—
21C2×D4×D53—21no—
22#13133250—21no—
23#13133250C2212yescomplemented
24#13133248—21no—
25#13133247C2212yescomplemented
26#13133245—21no—
27#13133245C2212yescomplemented
28#13133243—21no—
29#13133242C2212yescomplemented
30#13133239—22no—
31#13133237—22no—
32C2×#11719917C2211yescharacteristic, complemented
33#13133214C2212yescomplemented
34#13133205C2212yescomplemented
35#13133192—22no—
36#13133187—22no—
37#13133145C2214yescomplemented
38#13133133C2214yescomplemented
39C2×#41×C53C2211yescharacteristic
40C2×M16×C53C2211yescharacteristic, complemented
41#117×C53C2211yescharacteristic, complemented
42#104×C53C2214yescomplemented
43C23×D53—22no—
44C23×D53D412yes—
45#41×C53—21no—
46#41×C53—21no—
47#41×C53D412yescomplemented
48#41×C53C2312yes—
49C2×Q8×C53D411yescharacteristic, complemented
50C2×D4×C53—21no—
51C2×D4×C53D411yescharacteristic, complemented
52C22×C4×C53—21no—
53C22×C4×C53C2311yescharacteristic
54C2×C22⋊D53—22no—
55C2×C22⋊D53—21no—
56C2×C22⋊D53—21no—
57#11719949—24no—
58#11719949—24no—
59D4×D53—28no—
60D4×D53—24no—
61D4×D53—28no—
62D4×D53—24no—
63#11719945—24no—
64#11719945—21no—
65#11719945—21no—
66#11719945—21no—
67#11719945—21no—
68#11719945—21no—
69#11719945—21no—
70#11719945—24no—
71#11719945D412yescomplemented
72#11719945C2312yes—
73C2×D212—22no—
74C2×D212—24no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—24no—
80C2×D212—24no—
81C2×D212D412yes—
82C2×D212D411yescharacteristic, complemented
83C2×D212C2312yes—
84C2×D212C2311yescharacteristic, complemented
85C2×C4×D53—22no—
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×Dic106D411yescharacteristic, complemented
89C2×Dic106C2311yescharacteristic, complemented
90M16×C53—21no—
91M16×C53C2312yescomplemented
92C2×C8×C53—21no—
93C4⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C42×C53C2312yescomplemented
96#11719925—22no—
97#11719924—22no—
98#11719923—22no—
99#11719922—22no—
100#11719921—24no—
101#11719917—21no—
102#11719917C2312yescomplemented
103C2×#88322—21no—
104Dic212—22no—
105D424—22no—
106#11719913—22no—
107#11719913—22no—
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110C22×D53—22no—
111C22×D53—21no—
112C22×D53—28no—
113C22×D53—22no—
114C22×D53—21no—
115C22×D53—82no—
116C22×D53—84no—
117C22×D53C2×D414yes—
118Q8×C53—21no—
119Q8×C53C2×D412yes—
120D4×C53—22no—
121D4×C53—22no—
122D4×C53—21no—
123D4×C53C2×D412yes—
124C2×C4×C53—21no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—28no—
135C22⋊D53—22no—
136C22⋊D53—22no—
137C22⋊D53—41no—
138C22⋊D53—41no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141D212—28no—
142D212—22no—
143D212—21no—
144D212—22no—
145D212—22no—
146D212—22no—
147D212—21no—
148D212—21no—
149D212—84no—
150D212C2×D412yes—
151C4×D53—28no—
152C4×D53—22no—
153C4×D53—22no—
154C4×D53—41no—
155C4×D53—41no—
156Dic106—21no—
157Dic106—21no—
158Dic106—21no—
159Dic106C2×D412yes—
160C8×C53—22no—
161#88322—22no—
162C22×C53—22no—
163C22×C53—41no—
164C22×C53—42no—
165C22×C53#12011yescharacteristic
166C22×C53#12011yescharacteristic
167C22×C53C22×D411yescharacteristic
168C2×D53—24no—
169C2×D53—22no—
170C2×D53—44no—
171C2×D53—42no—
172C2×D53—44no—
173C2×D53—42no—
174C2×D53—41no—
175C2×D53—82no—
176C2×D53—88no—
177C4×C53—22no—
178C4×C53—42no—
179C4×C53#12011yescharacteristic
180C4×C53#12011yescharacteristic
181C4×C53C22×D411yescharacteristic
182C4×C53C22×D411yescharacteristic
183Dic53—22no—
184Dic53—41no—
185#3019—531nomaximal, Sylow
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—42no—
190C2×C53C2×#12011yescharacteristic
191D53—44no—
192D53—42no—
193D53—81no—
194C2×H(5,𝔽2)—531no—
195#575—531no—
196#575—531no—
197#535—531no—
198#455—534no—
199#453—534no—
200#421—531no—
201#421D5311yescharacteristic, complemented
202C2×#100—531no—
203C53#301911yescharacteristic, complemented, Sylow
204H(5,𝔽2)—534no—
205H(5,𝔽2)—1062no—
206H(5,𝔽2)—1062no—
207H(5,𝔽2)—1062no—
208C2×#41—531no—
209C2×#41—531no—
210C2×#41—1061no—
211C2×#41—1061no—
212C2×#41C2×D5311yescharacteristic
213C22×D4—531no—
214C22×D4—1061no—
215C22×D4—1062no—
216C22×D4—1061no—
217#137—532no—
218#137—532no—
219#137—1061no—
220#137—1061no—
221AGL(1,ℤ/8)—532no—
222AGL(1,ℤ/8)—532no—
223AGL(1,ℤ/8)—1061no—
224AGL(1,ℤ/8)—1061no—
225#135—1062no—
226#135—1062no—
227C2×M16—531no—
228C2×M16C2×D5311yescharacteristic, complemented
229#127—532no—
230#124—532no—
231#121—1062no—
232#120—1062no—
233#117C2×D5311yescharacteristic, complemented
234#104—534no—
235#104C2×D5314yescomplemented
236#100—534no—
237C24—532no—
238C24—1062no—
239#41—21no—
240#41—21no—
241#41—532no—
242#41—532no—
243#41—1064no—
244#41—1064no—
245#41—1064no—
246#41—1064no—
247#41—1061no—
248#41—1061no—
249#41—1061no—
250#41—1061no—
251#41—1061no—
252#41—1061no—
253#41D21212yescomplemented
254#41C22×D5312yes—
255C2×Q8—531no—
256C2×Q8—531no—
257C2×Q8D21211yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—531no—
260C2×D4—532no—
261C2×D4—532no—
262C2×D4—531no—
263C2×D4—1064no—
264C2×D4—1062no—
265C2×D4—1064no—
266C2×D4—1064no—
267C2×D4—1068no—
268C2×D4—1068no—
269C2×D4—1064no—
270C2×D4—1062no—
271C2×D4—1061no—
272C2×D4—1064no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4D21211yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—1062no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4C22×D5311yescharacteristic
284Q16—1062no—
285Q16—1062no—
286SD16—1062no—
287SD16—1062no—
288SD16—1062no—
289SD16—1062no—
290D8—1062no—
291D8—1062no—
292M16—21no—
293M16—532no—
294M16—1061no—
295M16C22×D5312yescomplemented
296C2×C8—21no—
297C2×C8—1061no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—1064no—
301C42C22×D5312yescomplemented
302C23—21no—
303C23—534no—
304C23—1068no—
305C23—1062no—
306C23—1061no—
307C23—1062no—
308C23—1061no—
309C23—4244no—
310C23—4242no—
311C23D4×D5311yescharacteristic
312Q8—21no—
313Q8—532no—
314Q8—1061no—
315Q8—1061no—
316Q8—1061no—
317Q8C2×D21212yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—532no—
322D4—1068no—
323D4—1068no—
324D4—1062no—
325D4—1062no—
326D4—1061no—
327D4—1062no—
328D4—1062no—
329D4—1062no—
330D4—1061no—
331D4—1062no—
332D4—1061no—
333D4—4244no—
334D4—2121no—
335D4—2121no—
336D4C2×D21212yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—1068no—
342C2×C4—1061no—
343C2×C4—1062no—
344C2×C4—1062no—
345C2×C4—1061no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4C2×D21211yescharacteristic
349C2×C4C2×D21211yescharacteristic
350C2×C4D4×D5311yescharacteristic
351C2×C4D4×D5311yescharacteristic
352C2×C4D4×D5311yescharacteristic
353C2×C4C23×D5311yescharacteristic
354C8—22no—
355C8—1062no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—1062no—
360C22—1064no—
361C22—4242no—
362C22—4248no—
363C22—2122no—
364C22—2124no—
365C22—2122no—
366C22—2124no—
367C22—2121no—
368C22#1313318711yescharacteristic
369C22#1313318711yescharacteristic
370C22C2×D4×D5311yescharacteristic
371C4—22no—
372C4—42no—
373C4—1062no—
374C4—2121no—
375C4#1313318711yescharacteristic
376C4#1313318711yescharacteristic
377C4C2×D4×D5311yescharacteristic
378C4C22×D21211yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—4241no—
384C2—2122no—
385C2—2124no—
386C2C2×#1313318711yescharacteristic
subgroupG/H
centerC2C2×#13133187
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#421C2
socleC2×C53C2×#120
hypercenter#421D53
hypocenterC53#3019
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. #16811423C2semidirect
3. #16811421C2semidirect
4. #16811192C2semidirect
5. #16810802C2semidirect
6. #16810792C2semidirect
7. #16810641C2semidirect
8. C2×#13133133C2semidirect
9. C53×#421C2semidirect
17. C2×#11719945C22semidirect
23. #13133250C22semidirect
25. #13133247C22semidirect
27. #13133245C22semidirect
29. #13133242C22semidirect
32. C2×#11719917C22semidirect
33. #13133214C22semidirect
34. #13133205C22semidirect
37. #13133145C22semidirect
38. #13133133C22semidirect
40. C2×M16×C53C22semidirect
41. #117×C53C22semidirect
42. #104×C53C22semidirect
47. #41×C53D4semidirect
49. C2×Q8×C53D4semidirect
51. C2×D4×C53D4semidirect
71. #11719945D4semidirect
82. C2×D212D4semidirect
84. C2×D212C23semidirect
88. C2×Dic106D4semidirect
89. C2×Dic106C23semidirect
91. M16×C53C23semidirect
95. C42×C53C23semidirect
102. #11719917C23semidirect
201. #421D53semidirect
203. C53#3019semidirect
228. C2×M16C2×D53semidirect
233. #117C2×D53semidirect
235. #104C2×D53semidirect
253. #41D212semidirect
257. C2×Q8D212semidirect
278. C2×D4D212semidirect
295. M16C22×D53semidirect
301. C42C22×D53semidirect