Finite Group Databasebeta

← C5×#18997 #11941717

Subgroups — C5×#18997 #11941717

Order 1280.

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 222 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×C5×H(3,𝔽4)C211yesmaximal, characteristic
2C5×#3215C212yesmaximal, complemented
3C5×#3140C211yesmaximal, characteristic, complemented
4C5×#1659C218yesmaximal, complemented
5C5×#1533C211yesmaximal, characteristic, complemented
6C5×#1527C212yesmaximal, complemented
7C5×H(3,𝔽4)—24no—
8C5×H(3,𝔽4)—22no—
9C5×H(3,𝔽4)C2214yes—
10C5×#533C2211yescharacteristic, complemented
11C2×C5×#124—21no—
12C2×C5×#124—22no—
13C2×C5×#124C2212yes—
14C2×C5×#124C2211yescharacteristic
15C2×C5×#120—21no—
16C2×C5×#120—22no—
17C5×#497C2214yescomplemented
18C5×#489C2214yescomplemented
19C5×#489C2214yescomplemented
20C5×#488—22no—
21C5×#464—24no—
22C5×#451—22no—
23C5×#450—22no—
24C5×#432C2212yescomplemented
25C5×#419—21no—
26C5×#418—21no—
27C4×C5×M16C2211yescharacteristic, complemented
28C5×#329C2218yescomplemented
29C5×#329C2214yescomplemented
30#18997C511yesmaximal, characteristic, direct product factor, Sylow
31C25×C5—21no—
32C22×C5×Q8D411yescharacteristic
33C22×C5×Q8D412yes—
34C22×C5×D4—22no—
35C22×C5×D4—21no—
36C22×C5×D4D411yescharacteristic
37C22×C5×D4D412yes—
38C2×C5×Q16—24no—
39C2×C5×SD16—24no—
40C2×C5×M16—21no—
41C5×C4⋊Q8C2312yescomplemented
42C5×#125—21no—
43C5×#124—24no—
44C5×#124—28no—
45C5×#124—28no—
46C5×#124—24no—
47C5×#124—22no—
48C5×#124—22no—
49C5×#124—22no—
50C5×#124D414yes—
51C5×#124D412yes—
52C5×#124C2312yes—
53C5×#124C2314yes—
54C5×C22⋊Q8—22no—
55C5×#120—24no—
56C5×#120—28no—
57C5×#120—28no—
58C5×#120—24no—
59C4×C5×Q8—22no—
60C5×#117—21no—
61C2×C5×C22⋊C4—21no—
62C2×C5×C22⋊C4—22no—
63C2×C5×C22⋊C4—22no—
64C2×C5×C22⋊C4—22no—
65C2×C5×C22⋊C4—22no—
66C2×C42×C5—21no—
67C2×C42×C5C2311yescharacteristic
68C5×C4⋊C8—22no—
69C5×#103—24no—
70C5×#103—24no—
71C5×#102—24no—
72C5×#102—24no—
73C5×C22⋊C8—22no—
74C5×#97C2312yescomplemented
75C5×#97C2314yescomplemented
76C4×C5×C8—21no—
77C2×H(3,𝔽4)C2×C511yescharacteristic
78#3215C2×C512yescomplemented
79#3140C2×C511yescharacteristic, complemented
80#1659C2×C518yescomplemented
81#1533C2×C511yescharacteristic, complemented
82#1527C2×C512yescomplemented
83C24×C5—24no—
84C24×C5—21no—
85C24×C5—22no—
86C24×C5—84no—
87C24×C5—82no—
88C2×C5×Q8—21no—
89C2×C5×Q8—22no—
90C2×C5×Q8—22no—
91C2×C5×Q8—42no—
92C2×C5×Q8—82no—
93C2×C5×Q8C2×D414yes—
94C2×C5×Q8C2×D412yes—
95C2×C5×D4—24no—
96C2×C5×D4—28no—
97C2×C5×D4—21no—
98C2×C5×D4—22no—
99C2×C5×D4—42no—
100C2×C5×D4—82no—
101C2×C5×D4—84no—
102C2×C5×D4—82no—
103C2×C5×D4C2×D414yes—
104C2×C5×D4C2×D412yes—
105C22×C4×C5—22no—
106C22×C4×C5—21no—
107C22×C4×C5C2×D411yescharacteristic
108C22×C4×C5C2×D412yes—
109C5×Q16—84no—
110C5×SD16—84no—
111C5×M16—42no—
112C2×C5×C8—22no—
113C2×C5×C8—24no—
114C5×C4⋊C4—22no—
115C5×C4⋊C4—22no—
116C5×C4⋊C4—42no—
117C5×C22⋊C4—28no—
118C5×C22⋊C4—24no—
119C5×C22⋊C4—24no—
120C5×C22⋊C4—24no—
121C5×C22⋊C4—24no—
122C5×C22⋊C4—24no—
123C5×C22⋊C4—24no—
124C5×C22⋊C4—24no—
125C5×C22⋊C4—41no—
126C42×C5—24no—
127C42×C5—41no—
128C42×C5C2×D412yes—
129C42×C5C2×D411yescharacteristic
130C42×C5C2411yescharacteristic
131H(3,𝔽4)—24no—
132H(3,𝔽4)—22no—
133H(3,𝔽4)C22×C514yes—
134#533C22×C511yescharacteristic, complemented
135C2×#124—21no—
136C2×#124—22no—
137C2×#124C22×C511yescharacteristic
138C2×#124C22×C512yes—
139C2×#120—21no—
140C2×#120—22no—
141#497C22×C514yescomplemented
142#489C22×C514yescomplemented
143#489C22×C514yescomplemented
144#488—22no—
145#464—24no—
146#451—22no—
147#450—22no—
148#432C22×C512yescomplemented
149#419—21no—
150#418—21no—
151C4×M16C22×C511yescharacteristic, complemented
152#329C22×C518yescomplemented
153#329C22×C514yescomplemented
154C23×C5—22no—
155C23×C5—24no—
156C23×C5—44no—
157C23×C5—44no—
158C23×C5—44no—
159C23×C5—42no—
160C23×C5—44no—
161C23×C5—84no—
162C23×C5—88no—
163C23×C5—816no—
164C23×C5#12011yescharacteristic
165C5×Q8—44no—
166C5×Q8—42no—
167C5×Q8—44no—
168C5×Q8—81no—
169C5×Q8—82no—
170C5×D4—44no—
171C5×D4—42no—
172C5×D4—81no—
173C5×D4—82no—
174C5×D4—84no—
175C5×D4—88no—
176C2×C4×C5—22no—
177C2×C4×C5—22no—
178C2×C4×C5—24no—
179C2×C4×C5—42no—
180C2×C4×C5—44no—
181C2×C4×C5—42no—
182C2×C4×C5—42no—
183C2×C4×C5#12011yescharacteristic
184C2×C4×C5#12012yes—
185C2×C4×C5C22×D411yescharacteristic
186C2×C4×C5C22×D412yes—
187C5×C8—42no—
188C5×C8—44no—
189C25—21no—
190C22×Q8C5×D411yescharacteristic
191C22×Q8C5×D412yes—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4C5×D411yescharacteristic
195C22×D4C5×D412yes—
196C2×Q16—24no—
197C2×SD16—24no—
198C2×M16—21no—
199C4⋊Q8C23×C512yescomplemented
200#125—21no—
201#124—24no—
202#124—28no—
203#124—24no—
204#124—28no—
205#124—22no—
206#124—22no—
207#124—22no—
208#124C5×D414yes—
209#124C5×D412yes—
210#124C23×C512yes—
211#124C23×C514yes—
212C22⋊Q8—22no—
213#120—24no—
214#120—28no—
215#120—24no—
216#120—28no—
217C4×Q8—22no—
218#117—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—21no—
225C2×C42C23×C511yescharacteristic
226C4⋊C8—22no—
227#103—24no—
228#103—24no—
229#102—24no—
230#102—24no—
231C22⋊C8—22no—
232#97C23×C512yescomplemented
233#97C23×C514yescomplemented
234C4×C8—21no—
235C22×C5—22no—
236C22×C5—42no—
237C22×C5—44no—
238C22×C5—44no—
239C22×C5—44no—
240C22×C5—44no—
241C22×C5—84no—
242C22×C5—88no—
243C22×C5—816no—
244C22×C5#45512yes—
245C22×C5C2×#12011yescharacteristic
246C4×C5—21no—
247C4×C5—21no—
248C4×C5—22no—
249C4×C5—22no—
250C4×C5—42no—
251C4×C5—44no—
252C4×C5—82no—
253C24—24no—
254C24—21no—
255C24—22no—
256C24—84no—
257C24—82no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—42no—
262C2×Q8—82no—
263C2×Q8C2×C5×D412yes—
264C2×Q8C2×C5×D414yes—
265C2×D4—24no—
266C2×D4—28no—
267C2×D4—21no—
268C2×D4—22no—
269C2×D4—42no—
270C2×D4—82no—
271C2×D4—84no—
272C2×D4—82no—
273C2×D4C2×C5×D412yes—
274C2×D4C2×C5×D414yes—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4C2×C5×D411yescharacteristic
278C22×C4C2×C5×D412yes—
279Q16—84no—
280SD16—84no—
281M16—42no—
282C2×C8—22no—
283C2×C8—24no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—42no—
287C22⋊C4—28no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—41no—
296C42—24no—
297C42—41no—
298C42C2×C5×D411yescharacteristic
299C42C2×C5×D412yes—
300C42C24×C511yescharacteristic
301C2×C5—22no—
302C2×C5—82no—
303C2×C5—84no—
304C2×C5C2×#45511yescharacteristic
305C2×C5#301912yes—
306C23—22no—
307C23—24no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—44no—
313C23—84no—
314C23—88no—
315C23—816no—
316C23C5×#12011yescharacteristic
317Q8—44no—
318Q8—42no—
319Q8—44no—
320Q8—81no—
321Q8—82no—
322D4—44no—
323D4—42no—
324D4—81no—
325D4—82no—
326D4—84no—
327D4—88no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4C5×#12011yescharacteristic
336C2×C4C5×#12012yes—
337C2×C4C22×C5×D411yescharacteristic
338C2×C4C22×C5×D412yes—
339C8—42no—
340C8—44no—
341C5#1899711yescharacteristic, direct product factor, Sylow
342C22—22no—
343C22—42no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—84no—
349C22—88no—
350C22—816no—
351C22C5×#45512yes—
352C22C2×C5×#12011yescharacteristic
353C4—21no—
354C4—21no—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—44no—
359C4—82no—
360C2—22no—
361C2—82no—
362C2—84no—
363C2C2×C5×#45511yescharacteristic
364C2C5×#301912yes—
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC2×C5×H(3,𝔽4)C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3215C2semidirect
3. C5×#3140C2semidirect
4. C5×#1659C2semidirect
5. C5×#1533C2semidirect
6. C5×#1527C2semidirect
10. C5×#533C22semidirect
17. C5×#497C22semidirect
18. C5×#489C22semidirect
19. C5×#489C22semidirect
24. C5×#432C22semidirect
27. C4×C5×M16C22semidirect
28. C5×#329C22semidirect
29. C5×#329C22semidirect
30. #18997C5direct (N × H)
41. C5×C4⋊Q8C23semidirect
74. C5×#97C23semidirect
75. C5×#97C23semidirect
78. #3215C2×C5semidirect
79. #3140C2×C5semidirect
80. #1659C2×C5semidirect
81. #1533C2×C5semidirect
82. #1527C2×C5semidirect
134. #533C22×C5semidirect
141. #497C22×C5semidirect
142. #489C22×C5semidirect
143. #489C22×C5semidirect
148. #432C22×C5semidirect
151. C4×M16C22×C5semidirect
152. #329C22×C5semidirect
153. #329C22×C5semidirect
199. C4⋊Q8C23×C5semidirect
232. #97C23×C5semidirect
233. #97C23×C5semidirect
341. C5#18997direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×H(3,𝔽4)2
C5×#32152
C5×#31402
C5×#16592
C5×#15332
C5×#15272
#189975