Finite Group Databasebeta

← #20284070

Subgroups — #20284070

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#16811421C211yesmaximal, characteristic, complemented
3#16811187C211yesmaximal, characteristic, complemented
4#16810799C214yesmaximal, complemented
5#16810789C214yesmaximal, complemented
6#16810640C211yesmaximal, characteristic, complemented
7#16810639C211yesmaximal, characteristic, complemented
8C2×#13133133C211yesmaximal, characteristic, complemented
9C53×#575C211yesmaximal, characteristic, complemented
10#13133263—22no—
11#13133263—22no—
12#13133263—22no—
13#13133263C2214yes—
14C2×#11719949—21no—
15C2×#11719945—21no—
16C2×#11719945C2211yescharacteristic
17C22×D212—21no—
18C22×D212C2211yescharacteristic
19C2×D4×D53—22no—
20C2×D4×D53—21no—
21#13133248C2212yescomplemented
22#13133247—21no—
23#13133243C2212yescomplemented
24#13133242—21no—
25#13133237—22no—
26C2×#11719917C2211yescharacteristic, complemented
27#13133213C2212yescomplemented
28#13133206C2212yescomplemented
29#13133191—22no—
30#13133185—22no—
31#13133176C2211yescharacteristic, complemented
32#13133144C2214yescomplemented
33#13133143C2214yescomplemented
34#13133133C2214yescomplemented
35C2×#41×C53C2211yescharacteristic
36C2×#41×C53C2211yescharacteristic, complemented
37#137×C53—21no—
38#137×C53C2212yescomplemented
39AGL(1,ℤ/8)×C53—21no—
40AGL(1,ℤ/8)×C53C2212yescomplemented
41#135×C53—22no—
42C2×M16×C53C2211yescharacteristic, complemented
43C23×D53—22no—
44C23×D53D412yes—
45#41×C53—21no—
46#41×C53—21no—
47#41×C53—21no—
48#41×C53—21no—
49#41×C53—21no—
50#41×C53—21no—
51#41×C53D412yescomplemented
52#41×C53C2312yes—
53C2×Q8×C53D411yescharacteristic, complemented
54C2×Q8×C53C2311yescharacteristic, complemented
55C2×D4×C53—21no—
56C2×D4×C53—21no—
57C2×D4×C53D411yescharacteristic, complemented
58C2×D4×C53C2311yescharacteristic, complemented
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—22no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—24no—
65#11719949—24no—
66#11719949—24no—
67D4×D53—28no—
68D4×D53—24no—
69D4×D53—28no—
70D4×D53—24no—
71#11719945—24no—
72#11719945—21no—
73#11719945—21no—
74#11719945—24no—
75#11719945D412yescomplemented
76#11719945C2312yes—
77C2×D212—22no—
78C2×D212—24no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—24no—
84C2×D212D412yes—
85C2×D212D411yescharacteristic, complemented
86C2×D212C2312yes—
87C2×C4×D53—22no—
88C2×C4×D53—21no—
89C2×Dic106D411yescharacteristic, complemented
90Q16×C53—22no—
91SD16×C53—22no—
92SD16×C53—22no—
93D8×C53—22no—
94M16×C53—21no—
95M16×C53C2312yescomplemented
96C2×C8×C53—21no—
97C22⋊Dic53—21no—
98#11719925—22no—
99#11719924—22no—
100#11719923—22no—
101#11719922—22no—
102#11719921—24no—
103C4⋊Dic53—21no—
104C4×Dic53C2312yescomplemented
105#11719917—21no—
106#11719917C2312yescomplemented
107C2×#88322—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53C2×D411yescharacteristic
111C22×D53—22no—
112C22×D53—21no—
113C22×D53—28no—
114C22×D53—22no—
115C22×D53—82no—
116C22×D53—84no—
117C22×D53C2×D414yes—
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53C2×D412yes—
122D4×C53—22no—
123D4×C53—22no—
124D4×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—41no—
128D4×C53—41no—
129D4×C53C2×D412yes—
130C2×C4×C53—21no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—21no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2411yescharacteristic
142C22⋊D53—28no—
143C22⋊D53—22no—
144C22⋊D53—22no—
145C22⋊D53—84no—
146C2×Dic53—21no—
147C2×Dic53—22no—
148D212—28no—
149D212—22no—
150D212—21no—
151D212—22no—
152D212—22no—
153D212—22no—
154D212C2×D412yes—
155C4×D53—28no—
156C4×D53—22no—
157C4×D53—22no—
158Dic106—21no—
159Dic106C2×D412yes—
160C8×C53—22no—
161#88322—22no—
162C22×C53—22no—
163C22×C53—41no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—41no—
167C22×C53#12011yescharacteristic
168C22×C53#12011yescharacteristic
169C22×C53C22×D411yescharacteristic
170C2×D53—24no—
171C2×D53—22no—
172C2×D53—44no—
173C2×D53—42no—
174C2×D53—44no—
175C2×D53—82no—
176C2×D53—88no—
177C4×C53—22no—
178C4×C53—41no—
179C4×C53#12011yescharacteristic
180C4×C53#12011yescharacteristic
181C4×C53C22×D411yescharacteristic
182C4×C53C22×D411yescharacteristic
183Dic53—22no—
184Dic53—42no—
185#3019—531nomaximal, Sylow
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—42no—
190C2×C53—81no—
191C2×C53C2×#12011yescharacteristic
192D53—44no—
193D53—42no—
194C2×H(5,𝔽2)—531no—
195#575—531no—
196#575D5311yescharacteristic, complemented
197#535—531no—
198#455—534no—
199#453—534no—
200#421—531no—
201#421—531no—
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—1061no—
210C2×#41—1061no—
211C2×#41C2×D5311yescharacteristic
212C2×#41C2×D5311yescharacteristic, complemented
213C22×D4—531no—
214C22×D4—1061no—
215C22×D4—1062no—
216C22×D4—1061no—
217#137—21no—
218#137—532no—
219#137—1061no—
220#137C2×D5312yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—532no—
223AGL(1,ℤ/8)—1061no—
224AGL(1,ℤ/8)C2×D5312yescomplemented
225#135—22no—
226#135—1062no—
227C2×M16—531no—
228C2×M16C2×D5311yescharacteristic, complemented
229#127—532no—
230#124—532no—
231#121—1062no—
232#120—1062no—
233#117—531no—
234#104—534no—
235#104—534no—
236#100—534no—
237C24—532no—
238C24—1062no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—532no—
246#41—532no—
247#41—1064no—
248#41—1064no—
249#41—1064no—
250#41—1064no—
251#41—1061no—
252#41—1061no—
253#41C22⋊D5312yescomplemented
254#41C22×D5312yes—
255C2×Q8—531no—
256C2×Q8C22⋊D5311yescharacteristic, complemented
257C2×Q8C22×D5311yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—531no—
261C2×D4—532no—
262C2×D4—532no—
263C2×D4—1064no—
264C2×D4—1062no—
265C2×D4—1064no—
266C2×D4—1068no—
267C2×D4—1068no—
268C2×D4—1064no—
269C2×D4—1062no—
270C2×D4—1061no—
271C2×D4—1064no—
272C2×D4—1064no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4C22⋊D5311yescharacteristic, complemented
278C2×D4C22×D5311yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—1062no—
282C22×C4—1061no—
283C22×C4C22×D5311yescharacteristic
284Q16—22no—
285Q16—1062no—
286SD16—22no—
287SD16—22no—
288SD16—1062no—
289SD16—1062no—
290D8—22no—
291D8—1062no—
292M16—21no—
293M16—532no—
294M16—1061no—
295M16C22×D5312yescomplemented
296C2×C8—21no—
297C2×C8—1061no—
298C4⋊C4—1061no—
299C22⋊C4—1064no—
300C22⋊C4—1061no—
301C42—532no—
302C23—21no—
303C23—21no—
304C23—534no—
305C23—1068no—
306C23—1062no—
307C23—1061no—
308C23—1062no—
309C23—4244no—
310C23—4242no—
311C23D4×D5311yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—532no—
316Q8—1061no—
317Q8C2×C22⋊D5312yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—532no—
326D4—1068no—
327D4—1068no—
328D4—1062no—
329D4—1062no—
330D4—1061no—
331D4—1062no—
332D4—1062no—
333D4—1062no—
334D4—1062no—
335D4—4244no—
336D4C2×C22⋊D5312yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—1068no—
344C2×C4—1061no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—1062no—
348C2×C4D4×D5311yescharacteristic
349C2×C4D4×D5311yescharacteristic
350C2×C4D4×D5311yescharacteristic
351C2×C4C2×C22⋊D5311yescharacteristic
352C2×C4C2×C22⋊D5311yescharacteristic
353C2×C4C23×D5311yescharacteristic
354C8—22no—
355C8—1062no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—1062no—
362C22—1064no—
363C22—4242no—
364C22—4248no—
365C22—2124no—
366C22—2122no—
367C22—2124no—
368C22#1313318511yescharacteristic
369C22#1313318511yescharacteristic
370C22C2×D4×D5311yescharacteristic
371C4—22no—
372C4—41no—
373C4—1062no—
374C4—2122no—
375C4#1313318511yescharacteristic
376C4#1313318511yescharacteristic
377C4C2×D4×D5311yescharacteristic
378C4C22×C22⋊D5311yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—2122no—
385C2—2124no—
386C2C2×#1313318511yescharacteristic
subgroupG/H
centerC2C2×#13133185
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#575C2
socleC2×C53C2×#120
hypercenter#575D53
hypocenterC53#3019
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. #16811421C2semidirect
3. #16811187C2semidirect
4. #16810799C2semidirect
5. #16810789C2semidirect
6. #16810640C2semidirect
7. #16810639C2semidirect
8. C2×#13133133C2semidirect
9. C53×#575C2semidirect
21. #13133248C22semidirect
23. #13133243C22semidirect
26. C2×#11719917C22semidirect
27. #13133213C22semidirect
28. #13133206C22semidirect
31. #13133176C22semidirect
32. #13133144C22semidirect
33. #13133143C22semidirect
34. #13133133C22semidirect
36. C2×#41×C53C22semidirect
38. #137×C53C22semidirect
40. AGL(1,ℤ/8)×C53C22semidirect
42. C2×M16×C53C22semidirect
51. #41×C53D4semidirect
53. C2×Q8×C53D4semidirect
54. C2×Q8×C53C23semidirect
57. C2×D4×C53D4semidirect
58. C2×D4×C53C23semidirect
75. #11719945D4semidirect
85. C2×D212D4semidirect
89. C2×Dic106D4semidirect
95. M16×C53C23semidirect
104. C4×Dic53C23semidirect
106. #11719917C23semidirect
196. #575D53semidirect
203. C53#3019semidirect
212. C2×#41C2×D53semidirect
220. #137C2×D53semidirect
224. AGL(1,ℤ/8)C2×D53semidirect
228. C2×M16C2×D53semidirect
253. #41C22⋊D53semidirect
256. C2×Q8C22⋊D53semidirect
257. C2×Q8C22×D53semidirect
277. C2×D4C22⋊D53semidirect
278. C2×D4C22×D53semidirect
295. M16C22×D53semidirect