Finite Group Databasebeta

← C52×#3484 #72221179

Subgroups — C52×#3484 #72221179

Order 3200.

7600 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C52×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C23×C52×D4C211yesmaximal, characteristic, complemented
3C52×#546C218yesmaximal, complemented
4C52×D42C218yesmaximal, complemented
5C52×#535C212yesmaximal, complemented
6C52×#534C214yesmaximal, complemented
7C52×#525C212yesmaximal, complemented
8C2×C52×#120C214yesmaximal, complemented
9C52×#518C211yesmaximal, characteristic, complemented
10C25×C52C2212yescomplemented
11C52×H(5,𝔽2)—28no—
12C2×C52×#41C2212yescomplemented
13C2×C52×#41C2214yescomplemented
14C22×C52×D4—28no—
15C22×C52×D4C2211yescharacteristic
16C22×C52×D4C2212yescomplemented
17C22×C52×D4C2211yescharacteristic, complemented
18C22×C52×D4C2218yescomplemented
19C22×C52×D4C2214yescomplemented
20C22×C52×D4C2214yescomplemented
21C22×C52×D4C2218yescomplemented
22C23×C4×C52C2211yescharacteristic, complemented
23C52×#127C2214yescomplemented
24C52×#124C2214yescomplemented
25C52×#123C2214yescomplemented
26C52×#123C2218yescomplemented
27C52×C22⋊Q8C2214yescomplemented
28C52×#121C2214yescomplemented
29C52×#121C22116yescomplemented
30C52×#121C2218yescomplemented
31C52×#120C2214yescomplemented
32C52×#120C2218yescomplemented
33C52×#120C22116yescomplemented
34C52×#120C2218yescomplemented
35C4×C52×D4C2218yescomplemented
36C4×C52×D4C2218yescomplemented
37C52×#117C2212yescomplemented
38C2×C52×C22⋊C4C2214yescomplemented
39C2×C52×C22⋊C4C2218yescomplemented
40C5×#3484C516yesmaximal, direct product factor
41C24×C52—28no—
42C24×C52—48no—
43C24×C52C2311yescharacteristic
44C24×C52C2318yes—
45C24×C52C2312yes—
46C24×C52C2312yes—
47C24×C52C2314yescomplemented
48C52×#41—216no—
49C52×#41—28no—
50C2×C52×Q8C2312yescomplemented
51C2×C52×D4—28no—
52C2×C52×D4—216no—
53C2×C52×D4—232no—
54C2×C52×D4—216no—
55C2×C52×D4—24no—
56C2×C52×D4—416no—
57C2×C52×D4D418yescomplemented
58C2×C52×D4C2318yes—
59C2×C52×D4C2314yes—
60C2×C52×D4C2314yes—
61C2×C52×D4C2318yes—
62C2×C52×D4C2314yes—
63C2×C52×D4C2318yescomplemented
64C2×C52×D4C2312yescomplemented
65C2×C52×D4C23116yescomplemented
66C22×C4×C52—24no—
67C22×C4×C52C2314yes—
68C22×C4×C52C2311yescharacteristic
69C22×C4×C52C2311yescharacteristic
70C22×C4×C52C2314yescomplemented
71C22×C4×C52C2314yescomplemented
72C22×C4×C52C2318yescomplemented
73C52×C4⋊C4C2314yescomplemented
74C52×C4⋊C4C2318yescomplemented
75C52×C22⋊C4C2314yes—
76C52×C22⋊C4C2318yes—
77C52×C22⋊C4C2318yescomplemented
78C52×C22⋊C4C23116yescomplemented
79C52×C22⋊C4C2318yescomplemented
80C42×C52C2314yescomplemented
81C2×C5×H(5,𝔽2)C2×C516yescomplemented
82C23×C5×D4C2×C516yescomplemented
83C5×#546C2×C5148yescomplemented
84C5×D42C2×C5148yescomplemented
85C5×#535C2×C5112yescomplemented
86C5×#534C2×C5124yescomplemented
87C5×#525C2×C5112yescomplemented
88C2×C5×#120C2×C5124yescomplemented
89C5×#518C2×C516yescomplemented
90C23×C52—22no—
91C23×C52—24no—
92C23×C52—216no—
93C23×C52—24no—
94C23×C52—216no—
95C23×C52—28no—
96C23×C52—44no—
97C23×C52—48no—
98C23×C52—44no—
99C23×C52—48no—
100C23×C52—432no—
101C23×C52C2×D418yescomplemented
102C23×C52C2414yes—
103C23×C52C2414yes—
104C23×C52C2411yescharacteristic
105C23×C52C2414yes—
106C23×C52C2412yes—
107C52×Q8—24no—
108C52×D4—216no—
109C52×D4—28no—
110C52×D4—216no—
111C52×D4—216no—
112C52×D4—28no—
113C52×D4—24no—
114C52×D4—416no—
115C52×D4C2×D4116yescomplemented
116C2×C4×C52—22no—
117C2×C4×C52—28no—
118C2×C4×C52—28no—
119C2×C4×C52—28no—
120C2×C4×C52—48no—
121C2×C4×C52C2×D414yescomplemented
122C2×C4×C52C2418yes—
123C2×C4×C52C2414yes—
124C2×C4×C52C2412yes—
125C2×C4×C52C2412yes—
126C25×C5C22×C5112yescomplemented
127C5×H(5,𝔽2)—248no—
128C2×C5×#41C22×C5112yescomplemented
129C2×C5×#41C22×C5124yescomplemented
130C22×C5×D4—248no—
131C22×C5×D4C22×C516yes—
132C22×C5×D4C22×C5112yescomplemented
133C22×C5×D4C22×C516yescomplemented
134C22×C5×D4C22×C5148yescomplemented
135C22×C5×D4C22×C5124yescomplemented
136C22×C5×D4C22×C5124yescomplemented
137C22×C5×D4C22×C5148yescomplemented
138C23×C4×C5C22×C516yescomplemented
139C5×#127C22×C5124yescomplemented
140C5×#124C22×C5124yescomplemented
141C5×#123C22×C5124yescomplemented
142C5×#123C22×C5148yescomplemented
143C5×C22⋊Q8C22×C5124yescomplemented
144C5×#121C22×C5124yescomplemented
145C5×#121C22×C5196yescomplemented
146C5×#121C22×C5148yescomplemented
147C5×#120C22×C5124yescomplemented
148C5×#120C22×C5148yescomplemented
149C5×#120C22×C5196yescomplemented
150C5×#120C22×C5148yescomplemented
151C4×C5×D4C22×C5148yescomplemented
152C4×C5×D4C22×C5148yescomplemented
153C5×#117C22×C5112yescomplemented
154C2×C5×C22⋊C4C22×C5124yescomplemented
155C2×C5×C22⋊C4C22×C5148yescomplemented
156#3484C5211yescharacteristic, direct product factor, Sylow
157C22×C52—28no—
158C22×C52—22no—
159C22×C52—28no—
160C22×C52—22no—
161C22×C52—22no—
162C22×C52—24no—
163C22×C52—24no—
164C22×C52—28no—
165C22×C52—28no—
166C22×C52—48no—
167C22×C52—432no—
168C22×C52—48no—
169C22×C52—88no—
170C22×C52C22×D412yes—
171C22×C52C22×D418yes—
172C22×C52H(5,𝔽2)14yescomplemented
173C22×C52C2511yescharacteristic
174C4×C52—22no—
175C4×C52—24no—
176C4×C52—28no—
177C4×C52C22×D414yes—
178C24×C5—248no—
179C24×C5—448no—
180C24×C5C23×C516yes—
181C24×C5C23×C5148yes—
182C24×C5C23×C5112yes—
183C24×C5C23×C5112yes—
184C24×C5C23×C5124yescomplemented
185C5×#41—296no—
186C5×#41—248no—
187C2×C5×Q8C23×C5112yescomplemented
188C2×C5×D4—248no—
189C2×C5×D4—296no—
190C2×C5×D4—2192no—
191C2×C5×D4—296no—
192C2×C5×D4—224no—
193C2×C5×D4—496no—
194C2×C5×D4C5×D4148yescomplemented
195C2×C5×D4C23×C5148yes—
196C2×C5×D4C23×C5124yes—
197C2×C5×D4C23×C5148yes—
198C2×C5×D4C23×C5124yes—
199C2×C5×D4C23×C5124yes—
200C2×C5×D4C23×C5148yescomplemented
201C2×C5×D4C23×C5112yescomplemented
202C2×C5×D4C23×C5196yescomplemented
203C22×C4×C5—224no—
204C22×C4×C5C23×C5124yes—
205C22×C4×C5C23×C516yes—
206C22×C4×C5C23×C516yes—
207C22×C4×C5C23×C5124yescomplemented
208C22×C4×C5C23×C5124yescomplemented
209C22×C4×C5C23×C5148yescomplemented
210C5×C4⋊C4C23×C5124yescomplemented
211C5×C4⋊C4C23×C5148yescomplemented
212C5×C22⋊C4C23×C5124yes—
213C5×C22⋊C4C23×C5148yes—
214C5×C22⋊C4C23×C5148yescomplemented
215C5×C22⋊C4C23×C5196yescomplemented
216C5×C22⋊C4C23×C5148yescomplemented
217C42×C5C23×C5124yescomplemented
218C2×H(5,𝔽2)C2×C5211yescharacteristic, complemented
219C23×D4C2×C5211yescharacteristic, complemented
220#546C2×C5218yescomplemented
221D42C2×C5218yescomplemented
222#535C2×C5212yescomplemented
223#534C2×C5214yescomplemented
224#525C2×C5212yescomplemented
225C2×#120C2×C5214yescomplemented
226#518C2×C5211yescharacteristic, complemented
227C2×C52—22no—
228C2×C52—24no—
229C2×C52—28no—
230C2×C52—44no—
231C2×C52—44no—
232C2×C52C23×D411yescharacteristic
233C2×C52C2×H(5,𝔽2)12yes—
234C23×C5—212no—
235C23×C5—224no—
236C23×C5—296no—
237C23×C5—224no—
238C23×C5—296no—
239C23×C5—248no—
240C23×C5—424no—
241C23×C5—448no—
242C23×C5—424no—
243C23×C5—448no—
244C23×C5—4192no—
245C23×C5C2×C5×D4148yescomplemented
246C23×C5C24×C5124yes—
247C23×C5C24×C5124yes—
248C23×C5C24×C516yes—
249C23×C5C24×C5124yes—
250C23×C5C24×C5112yes—
251C5×Q8—224no—
252C5×D4—296no—
253C5×D4—248no—
254C5×D4—296no—
255C5×D4—296no—
256C5×D4—248no—
257C5×D4—224no—
258C5×D4—496no—
259C5×D4C2×C5×D4196yescomplemented
260C2×C4×C5—212no—
261C2×C4×C5—248no—
262C2×C4×C5—248no—
263C2×C4×C5—248no—
264C2×C4×C5—448no—
265C2×C4×C5C2×C5×D4124yescomplemented
266C2×C4×C5C24×C5148yes—
267C2×C4×C5C24×C5124yes—
268C2×C4×C5C24×C5112yes—
269C2×C4×C5C24×C5112yes—
270C25C22×C5212yescomplemented
271H(5,𝔽2)—28no—
272C2×#41C22×C5212yescomplemented
273C2×#41C22×C5214yescomplemented
274C22×D4—28no—
275C22×D4C22×C5211yescharacteristic
276C22×D4C22×C5212yescomplemented
277C22×D4C22×C5211yescharacteristic, complemented
278C22×D4C22×C5218yescomplemented
279C22×D4C22×C5214yescomplemented
280C22×D4C22×C5214yescomplemented
281C22×D4C22×C5218yescomplemented
282C23×C4C22×C5211yescharacteristic, complemented
283#127C22×C5214yescomplemented
284#124C22×C5214yescomplemented
285#123C22×C5214yescomplemented
286#123C22×C5218yescomplemented
287C22⋊Q8C22×C5214yescomplemented
288#121C22×C5214yescomplemented
289#121C22×C52116yescomplemented
290#121C22×C5218yescomplemented
291#120C22×C5214yescomplemented
292#120C22×C5218yescomplemented
293#120C22×C52116yescomplemented
294#120C22×C5218yescomplemented
295C4×D4C22×C5218yescomplemented
296C4×D4C22×C5218yescomplemented
297#117C22×C5212yescomplemented
298C2×C22⋊C4C22×C5214yescomplemented
299C2×C22⋊C4C22×C5218yescomplemented
300C52#348411yescharacteristic, direct product factor, Sylow
301C22×C5—248no—
302C22×C5—212no—
303C22×C5—248no—
304C22×C5—212no—
305C22×C5—212no—
306C22×C5—224no—
307C22×C5—224no—
308C22×C5—248no—
309C22×C5—248no—
310C22×C5—448no—
311C22×C5—4192no—
312C22×C5—448no—
313C22×C5—848no—
314C22×C5C22×C5×D4112yes—
315C22×C5C22×C5×D4148yes—
316C22×C5C5×H(5,𝔽2)124yescomplemented
317C22×C5C25×C516yes—
318C4×C5—212no—
319C4×C5—224no—
320C4×C5—248no—
321C4×C5C22×C5×D4124yes—
322C24—28no—
323C24—48no—
324C24C23×C5211yescharacteristic
325C24C23×C5218yes—
326C24C23×C5212yes—
327C24C23×C5212yes—
328C24C23×C5214yescomplemented
329#41—216no—
330#41—28no—
331C2×Q8C23×C5212yescomplemented
332C2×D4—28no—
333C2×D4—216no—
334C2×D4—232no—
335C2×D4—216no—
336C2×D4—24no—
337C2×D4—416no—
338C2×D4C52×D418yescomplemented
339C2×D4C23×C5218yes—
340C2×D4C23×C5214yes—
341C2×D4C23×C5218yes—
342C2×D4C23×C5214yes—
343C2×D4C23×C5214yes—
344C2×D4C23×C5218yescomplemented
345C2×D4C23×C5212yescomplemented
346C2×D4C23×C52116yescomplemented
347C22×C4—24no—
348C22×C4C23×C5214yes—
349C22×C4C23×C5211yescharacteristic
350C22×C4C23×C5211yescharacteristic
351C22×C4C23×C5214yescomplemented
352C22×C4C23×C5214yescomplemented
353C22×C4C23×C5218yescomplemented
354C4⋊C4C23×C5214yescomplemented
355C4⋊C4C23×C5218yescomplemented
356C22⋊C4C23×C5214yes—
357C22⋊C4C23×C5218yes—
358C22⋊C4C23×C5218yescomplemented
359C22⋊C4C23×C52116yescomplemented
360C22⋊C4C23×C5218yescomplemented
361C42C23×C5214yescomplemented
362C2×C5—212no—
363C2×C5—224no—
364C2×C5—248no—
365C2×C5—424no—
366C2×C5—424no—
367C2×C5C23×C5×D416yes—
368C2×C5C2×C5×H(5,𝔽2)112yes—
369C23—22no—
370C23—24no—
371C23—216no—
372C23—24no—
373C23—216no—
374C23—28no—
375C23—44no—
376C23—48no—
377C23—44no—
378C23—48no—
379C23—432no—
380C23C2×C52×D418yescomplemented
381C23C24×C5214yes—
382C23C24×C5214yes—
383C23C24×C5211yescharacteristic
384C23C24×C5214yes—
385C23C24×C5212yes—
386Q8—24no—
387D4—216no—
388D4—28no—
389D4—216no—
390D4—216no—
391D4—28no—
392D4—24no—
393D4—416no—
394D4C2×C52×D4116yescomplemented
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—48no—
400C2×C4C2×C52×D414yescomplemented
401C2×C4C24×C5218yes—
402C2×C4C24×C5214yes—
403C2×C4C24×C5212yes—
404C2×C4C24×C5212yes—
405C5C5×#348416yesdirect product factor
406C22—28no—
407C22—22no—
408C22—28no—
409C22—22no—
410C22—22no—
411C22—24no—
412C22—24no—
413C22—28no—
414C22—28no—
415C22—48no—
416C22—432no—
417C22—48no—
418C22—88no—
419C22C22×C52×D412yes—
420C22C22×C52×D418yes—
421C22C52×H(5,𝔽2)14yescomplemented
422C22C25×C5211yescharacteristic
423C4—22no—
424C4—24no—
425C4—28no—
426C4C22×C52×D414yes—
427C2—22no—
428C2—24no—
429C2—28no—
430C2—44no—
431C2—44no—
432C2C23×C52×D411yescharacteristic
433C2C2×C52×H(5,𝔽2)12yes—
subgroupG/H
center, socleC22×C52C25
commutator, Frattini, cosocleC22C25×C52
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C52 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C52×H(5,𝔽2)C2semidirect
2. C23×C52×D4C2semidirect
3. C52×#546C2semidirect
4. C52×D42C2semidirect
5. C52×#535C2semidirect
6. C52×#534C2semidirect
7. C52×#525C2semidirect
8. C2×C52×#120C2semidirect
9. C52×#518C2semidirect
10. C25×C52C22semidirect
12. C2×C52×#41C22semidirect
13. C2×C52×#41C22semidirect
16. C22×C52×D4C22semidirect
17. C22×C52×D4C22semidirect
18. C22×C52×D4C22semidirect
19. C22×C52×D4C22semidirect
20. C22×C52×D4C22semidirect
21. C22×C52×D4C22semidirect
22. C23×C4×C52C22semidirect
23. C52×#127C22semidirect
24. C52×#124C22semidirect
25. C52×#123C22semidirect
26. C52×#123C22semidirect
27. C52×C22⋊Q8C22semidirect
28. C52×#121C22semidirect
29. C52×#121C22semidirect
30. C52×#121C22semidirect
31. C52×#120C22semidirect
32. C52×#120C22semidirect
33. C52×#120C22semidirect
34. C52×#120C22semidirect
35. C4×C52×D4C22semidirect
36. C4×C52×D4C22semidirect
37. C52×#117C22semidirect
38. C2×C52×C22⋊C4C22semidirect
39. C2×C52×C22⋊C4C22semidirect
40. C5×#3484C5direct (N × H)
47. C24×C52C23semidirect
50. C2×C52×Q8C23semidirect
57. C2×C52×D4D4semidirect
63. C2×C52×D4C23semidirect
64. C2×C52×D4C23semidirect
65. C2×C52×D4C23semidirect
70. C22×C4×C52C23semidirect
71. C22×C4×C52C23semidirect
72. C22×C4×C52C23semidirect
73. C52×C4⋊C4C23semidirect
74. C52×C4⋊C4C23semidirect
77. C52×C22⋊C4C23semidirect
78. C52×C22⋊C4C23semidirect
79. C52×C22⋊C4C23semidirect
80. C42×C52C23semidirect
81. C2×C5×H(5,𝔽2)C2×C5semidirect
82. C23×C5×D4C2×C5semidirect
83. C5×#546C2×C5semidirect
84. C5×D42C2×C5semidirect
85. C5×#535C2×C5semidirect
86. C5×#534C2×C5semidirect
87. C5×#525C2×C5semidirect
88. C2×C5×#120C2×C5semidirect
89. C5×#518C2×C5semidirect
101. C23×C52C2×D4semidirect
115. C52×D4C2×D4semidirect
121. C2×C4×C52C2×D4semidirect
126. C25×C5C22×C5semidirect
128. C2×C5×#41C22×C5semidirect
129. C2×C5×#41C22×C5semidirect
132. C22×C5×D4C22×C5semidirect
133. C22×C5×D4C22×C5semidirect
134. C22×C5×D4C22×C5semidirect
135. C22×C5×D4C22×C5semidirect
136. C22×C5×D4C22×C5semidirect
137. C22×C5×D4C22×C5semidirect
138. C23×C4×C5C22×C5semidirect
139. C5×#127C22×C5semidirect
140. C5×#124C22×C5semidirect
141. C5×#123C22×C5semidirect
142. C5×#123C22×C5semidirect
143. C5×C22⋊Q8C22×C5semidirect
144. C5×#121C22×C5semidirect
145. C5×#121C22×C5semidirect
146. C5×#121C22×C5semidirect
147. C5×#120C22×C5semidirect
148. C5×#120C22×C5semidirect
149. C5×#120C22×C5semidirect
150. C5×#120C22×C5semidirect
151. C4×C5×D4C22×C5semidirect
152. C4×C5×D4C22×C5semidirect
153. C5×#117C22×C5semidirect
154. C2×C5×C22⋊C4C22×C5semidirect
155. C2×C5×C22⋊C4C22×C5semidirect
156. #3484C52direct (N × H)
172. C22×C52H(5,𝔽2)semidirect
184. C24×C5C23×C5semidirect
187. C2×C5×Q8C23×C5semidirect
194. C2×C5×D4C5×D4semidirect
200. C2×C5×D4C23×C5semidirect
201. C2×C5×D4C23×C5semidirect
202. C2×C5×D4C23×C5semidirect
207. C22×C4×C5C23×C5semidirect
208. C22×C4×C5C23×C5semidirect
209. C22×C4×C5C23×C5semidirect
210. C5×C4⋊C4C23×C5semidirect
211. C5×C4⋊C4C23×C5semidirect
214. C5×C22⋊C4C23×C5semidirect
215. C5×C22⋊C4C23×C5semidirect
216. C5×C22⋊C4C23×C5semidirect
217. C42×C5C23×C5semidirect
218. C2×H(5,𝔽2)C2×C52semidirect
219. C23×D4C2×C52semidirect
220. #546C2×C52semidirect
221. D42C2×C52semidirect
222. #535C2×C52semidirect
223. #534C2×C52semidirect
224. #525C2×C52semidirect
225. C2×#120C2×C52semidirect
226. #518C2×C52semidirect
245. C23×C5C2×C5×D4semidirect
259. C5×D4C2×C5×D4semidirect
265. C2×C4×C5C2×C5×D4semidirect
270. C25C22×C52semidirect
272. C2×#41C22×C52semidirect
273. C2×#41C22×C52semidirect
276. C22×D4C22×C52semidirect
277. C22×D4C22×C52semidirect
278. C22×D4C22×C52semidirect
279. C22×D4C22×C52semidirect
280. C22×D4C22×C52semidirect
281. C22×D4C22×C52semidirect
282. C23×C4C22×C52semidirect
283. #127C22×C52semidirect
284. #124C22×C52semidirect
285. #123C22×C52semidirect
286. #123C22×C52semidirect
287. C22⋊Q8C22×C52semidirect
288. #121C22×C52semidirect
289. #121C22×C52semidirect
290. #121C22×C52semidirect
291. #120C22×C52semidirect
292. #120C22×C52semidirect
293. #120C22×C52semidirect
294. #120C22×C52semidirect
295. C4×D4C22×C52semidirect
296. C4×D4C22×C52semidirect
297. #117C22×C52semidirect
298. C2×C22⋊C4C22×C52semidirect
299. C2×C22⋊C4C22×C52semidirect
300. C52#3484direct (N × H)
316. C22×C5C5×H(5,𝔽2)semidirect
328. C24C23×C52semidirect
331. C2×Q8C23×C52semidirect
338. C2×D4C52×D4semidirect
344. C2×D4C23×C52semidirect
345. C2×D4C23×C52semidirect
346. C2×D4C23×C52semidirect
351. C22×C4C23×C52semidirect
352. C22×C4C23×C52semidirect
353. C22×C4C23×C52semidirect
354. C4⋊C4C23×C52semidirect
355. C4⋊C4C23×C52semidirect
358. C22⋊C4C23×C52semidirect
359. C22⋊C4C23×C52semidirect
360. C22⋊C4C23×C52semidirect
361. C42C23×C52semidirect
380. C23C2×C52×D4semidirect
394. D4C2×C52×D4semidirect
400. C2×C4C2×C52×D4semidirect
405. C5C5×#3484direct (N × H)
421. C22C52×H(5,𝔽2)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C52×H(5,𝔽2)2
C23×C52×D42
C52×#5462
C52×D422
C52×#5352
C52×#5342
C52×#5252
C2×C52×#1202
C52×#5182
C5×#34845