Finite Group Databasebeta

← C5×#84110 #14425704

Subgroups — C5×#84110 #14425704

Order 1920.

1600 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1600 conjugacy classes of subgroups — too many to draw. Only its 322 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×C3×C5×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C5×#5804C213yesmaximal, complemented
3C5×#5244C214yesmaximal, complemented
4C5×#5243C214yesmaximal, complemented
5C2×C5×#834C213yesmaximal, complemented
6C5×#3026—31nomaximal, Hall
7C3×C5×H(5,𝔽2)—26no—
8C3×C5×H(5,𝔽2)C2214yes—
9C2×C3×C5×#41—23no—
10C22×C3×C5×D4—23no—
11C22×C3×C5×D4C2213yes—
12C22×C5×#67C2211yescharacteristic, complemented
13C5×#953—23no—
14C2×C5×#217C2213yescomplemented
15C5×#939—26no—
16C5×#937C2216yescomplemented
17C5×#935C2216yescomplemented
18C5×#929—23no—
19C5×#834C22112yescomplemented
20#84110C511yesmaximal, characteristic, direct product factor, Hall
21C2×C5×H(5,𝔽2)D311yescharacteristic, complemented
22C5×#534—33no—
23C5×#458—34no—
24C5×B(4,𝔽2)—34no—
25C2×C5×#99—33no—
26C24×C3×C5—21no—
27C24×C3×C5D413yescomplemented
28C24×C3×C5C2311yescharacteristic
29C23×C5×D3C2311yescharacteristic, complemented
30C3×C5×#41—212no—
31C3×C5×#41—212no—
32C2×C3×C5×Q8—21no—
33C2×C3×C5×D4—23no—
34C2×C3×C5×D4—212no—
35C2×C3×C5×D4—26no—
36C2×C3×C5×D4—212no—
37C2×C3×C5×D4—26no—
38C2×C3×C5×D4D416yescomplemented
39C2×C3×C5×D4C2316yes—
40C22×C3×C4×C5—23no—
41C22×C3×C4×C5D413yescomplemented
42C2×C5×#67—23no—
43C2×C5×#67—23no—
44C2×C5×#67—26no—
45C22×C5×Dic3—23no—
46C22×C5×Dic3C2311yescharacteristic, complemented
47C5×#217—23no—
48C5×#217—26no—
49C5×#217C2316yescomplemented
50C5×(C2×D3)⋊C4—26no—
51C5×Dic3⋊C4—26no—
52C2×C3×H(5,𝔽2)C2×C511yescharacteristic, complemented
53#5804C2×C513yescomplemented
54#5244C2×C514yescomplemented
55#5243C2×C514yescomplemented
56C2×#834C2×C513yescomplemented
57C5×H(5,𝔽2)—26no—
58C5×H(5,𝔽2)C2×D314yes—
59C2×C5×#41—23no—
60C22×C5×D4—23no—
61C22×C5×D4—31no—
62C22×C5×D4C2×D313yes—
63C5×#123—36no—
64C5×#123—63no—
65C5×#121—66no—
66C5×#120—36no—
67C5×#120—63no—
68C2×C5×C22⋊C4—33no—
69C5×#99—312no—
70#3026—31noSylow
71C23×C3×C5—24no—
72C23×C3×C5—23no—
73C23×C3×C5—23no—
74C23×C3×C5—81no—
75C23×C3×C5—83no—
76C23×C3×C5—82no—
77C23×C3×C5C2×D416yes—
78C23×C3×C5C2×D413yes—
79C23×C3×C5C2×D413yes—
80C23×C3×C5C2411yescharacteristic
81C22×C5×D3—23no—
82C22×C5×D3—81no—
83C3×C5×Q8—24no—
84C3×C5×D4—212no—
85C3×C5×D4—212no—
86C3×C5×D4—212no—
87C2×C3×C4×C5—212no—
88C2×C3×C4×C5—23no—
89C2×C3×C4×C5—23no—
90C2×C3×C4×C5C2×D416yes—
91C5×#67—41no—
92C5×#67—43no—
93C5×#67—86no—
94C2×C5×Dic3—23no—
95C2×C5×Dic3—26no—
96C2×C5×Dic3—83no—
97C3×H(5,𝔽2)—26no—
98C3×H(5,𝔽2)C22×C514yes—
99C2×C3×#41—23no—
100C22×C3×D4—23no—
101C22×C3×D4C22×C513yes—
102C22×#67C22×C511yescharacteristic, complemented
103#953—23no—
104C2×#217C22×C513yescomplemented
105#939—26no—
106#937C22×C516yescomplemented
107#935C22×C516yescomplemented
108#929—23no—
109#834C22×C5112yescomplemented
110C24×C5—21no—
111C24×C5—31no—
112C24×C5#6713yescomplemented
113C24×C5C22×D311yescharacteristic
114C5×#41—212no—
115C5×#41—212no—
116C2×C5×Q8—21no—
117C2×C5×D4—23no—
118C2×C5×D4—212no—
119C2×C5×D4—26no—
120C2×C5×D4—212no—
121C2×C5×D4—26no—
122C2×C5×D4—63no—
123C2×C5×D4—63no—
124C2×C5×D4—66no—
125C2×C5×D4#6716yescomplemented
126C2×C5×D4C22×D316yes—
127C22×C4×C5—23no—
128C22×C4×C5—31no—
129C22×C4×C5—63no—
130C22×C4×C5#6713yescomplemented
131C5×C4⋊C4—66no—
132C5×C22⋊C4—36no—
133C5×C22⋊C4—63no—
134C5×C22⋊C4—66no—
135C5×C22⋊C4—66no—
136C2×H(5,𝔽2)C5×D311yescharacteristic, complemented
137#534—33no—
138#458—34no—
139B(4,𝔽2)—34no—
140C2×#99—33no—
141C22×C3×C5—26no—
142C22×C3×C5—43no—
143C22×C3×C5—41no—
144C22×C3×C5—43no—
145C22×C3×C5—46no—
146C22×C3×C5—46no—
147C22×C3×C5—83no—
148C22×C3×C5—84no—
149C22×C3×C5#12011yescharacteristic
150C22×C3×C5#12013yes—
151C22×C3×C5C22×D413yes—
152C2×C5×D3—46no—
153C2×C5×D3—41no—
154C3×C4×C5—26no—
155C5×Dic3—46no—
156C5×Dic3—41no—
157C24×C3—21no—
158C24×C3C5×D413yescomplemented
159C24×C3C23×C511yescharacteristic
160C23×D3C23×C511yescharacteristic, complemented
161C3×#41—212no—
162C3×#41—212no—
163C2×C3×Q8—21no—
164C2×C3×D4—23no—
165C2×C3×D4—212no—
166C2×C3×D4—26no—
167C2×C3×D4—212no—
168C2×C3×D4—26no—
169C2×C3×D4C5×D416yescomplemented
170C2×C3×D4C23×C516yes—
171C22×C3×C4—23no—
172C22×C3×C4C5×D413yescomplemented
173C2×#67—23no—
174C2×#67—23no—
175C2×#67—26no—
176C22×Dic3—23no—
177C22×Dic3C23×C511yescharacteristic, complemented
178#217—23no—
179#217—26no—
180#217C23×C516yescomplemented
181(C2×D3)⋊C4—26no—
182Dic3⋊C4—26no—
183C23×C5—24no—
184C23×C5—23no—
185C23×C5—23no—
186C23×C5—63no—
187C23×C5—81no—
188C23×C5—83no—
189C23×C5—82no—
190C23×C5—241no—
191C23×C5C2×#6716yes—
192C23×C5C2×#6713yes—
193C23×C5C2×#6713yes—
194C23×C5C23×D311yescharacteristic
195C5×Q8—24no—
196C5×D4—212no—
197C5×D4—212no—
198C5×D4—212no—
199C5×D4—121no—
200C5×D4—123no—
201C5×D4—246no—
202C2×C4×C5—212no—
203C2×C4×C5—23no—
204C2×C4×C5—23no—
205C2×C4×C5—66no—
206C2×C4×C5—63no—
207C2×C4×C5—243no—
208C2×C4×C5C2×#6716yes—
209H(5,𝔽2)—26no—
210H(5,𝔽2)C2×C5×D314yes—
211C2×#41—23no—
212C22×D4—23no—
213C22×D4—31no—
214C22×D4C2×C5×D313yes—
215#123—36no—
216#123—63no—
217#121—66no—
218#120—36no—
219#120—63no—
220C2×C22⋊C4—33no—
221#99—312no—
222C2×C3×C5—23no—
223C2×C3×C5—21no—
224C2×C3×C5—23no—
225C2×C3×C5—46no—
226C2×C3×C5C2×#12011yescharacteristic
227C5×D3—81no—
228C23×C3—24no—
229C23×C3—23no—
230C23×C3—23no—
231C23×C3—81no—
232C23×C3—83no—
233C23×C3—82no—
234C23×C3C2×C5×D416yes—
235C23×C3C2×C5×D413yes—
236C23×C3C2×C5×D413yes—
237C23×C3C24×C511yescharacteristic
238C22×D3—23no—
239C22×D3—81no—
240C3×Q8—24no—
241C3×D4—212no—
242C3×D4—212no—
243C3×D4—212no—
244C2×C3×C4—212no—
245C2×C3×C4—23no—
246C2×C3×C4—23no—
247C2×C3×C4C2×C5×D416yes—
248#67—41no—
249#67—43no—
250#67—86no—
251C2×Dic3—23no—
252C2×Dic3—26no—
253C2×Dic3—83no—
254C22×C5—26no—
255C22×C5—43no—
256C22×C5—43no—
257C22×C5—41no—
258C22×C5—46no—
259C22×C5—46no—
260C22×C5—83no—
261C22×C5—84no—
262C22×C5—126no—
263C22×C5—121no—
264C22×C5#95311yescharacteristic
265C22×C5#95313yes—
266C22×C5C22×#6713yes—
267C4×C5—26no—
268C4×C5—126no—
269C4×C5—121no—
270C24—21no—
271C24—31no—
272C24C5×#6713yescomplemented
273C24C22×C5×D311yescharacteristic
274#41—212no—
275#41—212no—
276C2×Q8—21no—
277C2×D4—23no—
278C2×D4—212no—
279C2×D4—26no—
280C2×D4—212no—
281C2×D4—26no—
282C2×D4—63no—
283C2×D4—63no—
284C2×D4—66no—
285C2×D4C5×#6716yescomplemented
286C2×D4C22×C5×D316yes—
287C22×C4—23no—
288C22×C4—31no—
289C22×C4—63no—
290C22×C4C5×#6713yescomplemented
291C4⋊C4—66no—
292C22⋊C4—36no—
293C22⋊C4—63no—
294C22⋊C4—66no—
295C22⋊C4—66no—
296C3×C5#302611yescharacteristic, complemented, Hall
297C22×C3—26no—
298C22×C3—43no—
299C22×C3—41no—
300C22×C3—43no—
301C22×C3—46no—
302C22×C3—46no—
303C22×C3—83no—
304C22×C3—84no—
305C22×C3C5×#12011yescharacteristic
306C22×C3C5×#12013yes—
307C22×C3C22×C5×D413yes—
308C2×D3—46no—
309C2×D3—41no—
310C3×C4—26no—
311Dic3—46no—
312Dic3—41no—
313C2×C5—23no—
314C2×C5—21no—
315C2×C5—23no—
316C2×C5—46no—
317C2×C5—241no—
318C2×C5C2×#95311yescharacteristic
319C23—24no—
320C23—23no—
321C23—23no—
322C23—63no—
323C23—81no—
324C23—83no—
325C23—82no—
326C23—241no—
327C23C2×C5×#6716yes—
328C23C2×C5×#6713yes—
329C23C2×C5×#6713yes—
330C23C23×C5×D311yescharacteristic
331Q8—24no—
332D4—212no—
333D4—212no—
334D4—212no—
335D4—121no—
336D4—123no—
337D4—246no—
338C2×C4—212no—
339C2×C4—23no—
340C2×C4—23no—
341C2×C4—66no—
342C2×C4—63no—
343C2×C4—243no—
344C2×C4C2×C5×#6716yes—
345C2×C3—23no—
346C2×C3—21no—
347C2×C3—23no—
348C2×C3—46no—
349C2×C3C2×C5×#12011yescharacteristic
350D3—81no—
351C5#8411011yescharacteristic, direct product factor, Sylow
352C22—26no—
353C22—43no—
354C22—43no—
355C22—41no—
356C22—46no—
357C22—46no—
358C22—83no—
359C22—84no—
360C22—126no—
361C22—121no—
362C22C5×#95311yescharacteristic
363C22C5×#95313yes—
364C22C22×C5×#6713yes—
365C4—26no—
366C4—126no—
367C4—121no—
368C3C5×#302611yescharacteristic, complemented, Sylow
369C2—23no—
370C2—21no—
371C2—23no—
372C2—46no—
373C2—241no—
374C2C2×C5×#95311yescharacteristic
subgroupG/H
centerC2×C5C2×#953
commutator, cosocleC23×C3C24×C5
FrattiniC23C23×C5×D3
FittingC2×C3×C5×H(5,𝔽2)C2
socleC2×C3×C5C2×#120
hypercenterC2×C5×H(5,𝔽2)D3
hypocenterC3C5×#3026
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C24×C5 ◁ C2×C5×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×H(5,𝔽2)C2semidirect
2. C5×#5804C2semidirect
3. C5×#5244C2semidirect
4. C5×#5243C2semidirect
5. C2×C5×#834C2semidirect
12. C22×C5×#67C22semidirect
14. C2×C5×#217C22semidirect
16. C5×#937C22semidirect
17. C5×#935C22semidirect
19. C5×#834C22semidirect
20. #84110C5direct (N × H)
21. C2×C5×H(5,𝔽2)D3semidirect
27. C24×C3×C5D4semidirect
29. C23×C5×D3C23semidirect
38. C2×C3×C5×D4D4semidirect
41. C22×C3×C4×C5D4semidirect
46. C22×C5×Dic3C23semidirect
49. C5×#217C23semidirect
52. C2×C3×H(5,𝔽2)C2×C5semidirect
53. #5804C2×C5semidirect
54. #5244C2×C5semidirect
55. #5243C2×C5semidirect
56. C2×#834C2×C5semidirect
102. C22×#67C22×C5semidirect
104. C2×#217C22×C5semidirect
106. #937C22×C5semidirect
107. #935C22×C5semidirect
109. #834C22×C5semidirect
112. C24×C5#67semidirect
125. C2×C5×D4#67semidirect
130. C22×C4×C5#67semidirect
136. C2×H(5,𝔽2)C5×D3semidirect
158. C24×C3C5×D4semidirect
160. C23×D3C23×C5semidirect
169. C2×C3×D4C5×D4semidirect
172. C22×C3×C4C5×D4semidirect
177. C22×Dic3C23×C5semidirect
180. #217C23×C5semidirect
272. C24C5×#67semidirect
285. C2×D4C5×#67semidirect
290. C22×C4C5×#67semidirect
296. C3×C5#3026semidirect
351. C5#84110direct (N × H)
368. C3C5×#3026semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×H(5,𝔽2)2
C5×#58042
C5×#52442
C5×#52432
C2×C5×#8342
C5×#30263
#841105