Finite Group Databasebeta

← C5×#84125 #14425661

Subgroups — C5×#84125 #14425661

Order 1920.

1532 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1532 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
1C3×C5×#585C211yesmaximal, characteristic, complemented
2C5×#5828C213yesmaximal, complemented
3C5×#5244C214yesmaximal, complemented
4C5×#5243C214yesmaximal, complemented
5C5×#5240C213yesmaximal, complemented
6C5×#3025—31nomaximal, Hall
7C3×C5×2−1+4—23no—
8C3×C5×H(5,𝔽2)—23no—
9C3×C5×H(5,𝔽2)C2214yes—
10C2×C3×C5×#41—26no—
11C2×C3×C5×#41C2213yes—
12C22×C4×C5×D3C2211yescharacteristic, complemented
13C5×#946—23no—
14C5×#938—23no—
15C5×#937C2216yescomplemented
16C5×#935C2216yescomplemented
17C4×C5×#67—26no—
18C5×#926C2213yescomplemented
19C5×#834C22112yescomplemented
20#84125C511yesmaximal, characteristic, direct product factor, Hall
21C5×#585D311yescharacteristic, complemented
22C5×#525—33no—
23C5×#458—34no—
24C5×B(4,𝔽2)—34no—
25C5×#410—33no—
26C23×C5×D3C2311yescharacteristic
27C3×C5×#41—24no—
28C3×C5×#41—212no—
29C3×C5×#41—26no—
30C3×C5×#41—212no—
31C3×C5×#41—26no—
32C2×C3×C5×Q8—26no—
33C2×C3×C5×Q8D413yescomplemented
34C2×C3×C5×D4—26no—
35C2×C3×C5×D4—212no—
36C2×C3×C5×D4D413yescomplemented
37C2×C3×C5×D4C2316yes—
38C22×C3×C4×C5—24no—
39C22×C3×C4×C5D416yescomplemented
40C22×C3×C4×C5C2311yescharacteristic
41C2×C5×#67—26no—
42C22×C5×Dic3C2311yescharacteristic
43C2×C4×C5×D3—23no—
44C2×C4×C5×D3—23no—
45C5×#217—26no—
46C5×#217C2316yes—
47C5×(C2×D3)⋊C4—26no—
48C5×C4⋊Dic3—23no—
49C5×Dic3⋊C4—26no—
50C4×C5×Dic3—23no—
51C3×#585C2×C511yescharacteristic, complemented
52#5828C2×C513yescomplemented
53#5244C2×C514yescomplemented
54#5243C2×C514yescomplemented
55#5240C2×C513yescomplemented
56C5×2−1+4—23no—
57C5×H(5,𝔽2)—23no—
58C5×H(5,𝔽2)C2×D314yes—
59C2×C5×#41—26no—
60C2×C5×#41C2×D313yes—
61C23×C4×C5—31no—
62C5×#123—36no—
63C5×C22⋊Q8—63no—
64C5×#121—63no—
65C5×#120—36no—
66C4×C5×D4—66no—
67C5×#117—33no—
68C5×#99—312no—
69#3025—31noSylow
70C23×C3×C5—24no—
71C23×C3×C5C2×D416yes—
72C23×C3×C5C2411yescharacteristic
73C22×C5×D3—23no—
74C22×C5×D3—42no—
75C3×C5×Q8—24no—
76C3×C5×Q8—26no—
77C3×C5×D4—212no—
78C3×C5×D4—212no—
79C3×C5×D4—26no—
80C2×C3×C4×C5—212no—
81C2×C3×C4×C5—26no—
82C2×C3×C4×C5—26no—
83C2×C3×C4×C5C2×D416yes—
84C2×C3×C4×C5C2×D413yes—
85C2×C3×C4×C5C2×D413yes—
86C5×#67—412no—
87C2×C5×Dic3—26no—
88C2×C5×Dic3—23no—
89C4×C5×D3—43no—
90C4×C5×D3—41no—
91C3×2−1+4—23no—
92C3×H(5,𝔽2)—23no—
93C3×H(5,𝔽2)C22×C514yes—
94C2×C3×#41—26no—
95C2×C3×#41C22×C513yes—
96C22×C4×D3C22×C511yescharacteristic, complemented
97#946—23no—
98#938—23no—
99#937C22×C516yescomplemented
100#935C22×C516yescomplemented
101C4×#67—26no—
102#926C22×C513yescomplemented
103#834C22×C5112yescomplemented
104C24×C5—31no—
105C5×#41—24no—
106C5×#41—212no—
107C5×#41—26no—
108C5×#41—212no—
109C5×#41—26no—
110C2×C5×Q8—26no—
111C2×C5×Q8#6713yescomplemented
112C2×C5×D4—26no—
113C2×C5×D4—212no—
114C2×C5×D4—66no—
115C2×C5×D4#6713yescomplemented
116C2×C5×D4C22×D316yes—
117C22×C4×C5—24no—
118C22×C4×C5—31no—
119C22×C4×C5—63no—
120C22×C4×C5—63no—
121C22×C4×C5#6716yescomplemented
122C22×C4×C5C22×D311yescharacteristic
123C5×C4⋊C4—63no—
124C5×C4⋊C4—66no—
125C5×C22⋊C4—36no—
126C5×C22⋊C4—66no—
127C5×C22⋊C4—66no—
128C42×C5—63no—
129#585C5×D311yescharacteristic, complemented
130#525—33no—
131#458—34no—
132B(4,𝔽2)—34no—
133#410—33no—
134C22×C3×C5—26no—
135C22×C3×C5—41no—
136C22×C3×C5—46no—
137C22×C3×C5—84no—
138C22×C3×C5C22×D413yes—
139C2×C5×D3—26no—
140C2×C5×D3—43no—
141C2×C5×D3—41no—
142C3×C4×C5—26no—
143C3×C4×C5#12013yes—
144C3×C4×C5#12011yescharacteristic
145C5×Dic3—46no—
146C5×Dic3—41no—
147C23×D3C23×C511yescharacteristic
148C3×#41—24no—
149C3×#41—212no—
150C3×#41—26no—
151C3×#41—212no—
152C3×#41—26no—
153C2×C3×Q8—26no—
154C2×C3×Q8C5×D413yescomplemented
155C2×C3×D4—26no—
156C2×C3×D4—212no—
157C2×C3×D4C5×D413yescomplemented
158C2×C3×D4C23×C516yes—
159C22×C3×C4—24no—
160C22×C3×C4C5×D416yescomplemented
161C22×C3×C4C23×C511yescharacteristic
162C2×#67—26no—
163C22×Dic3C23×C511yescharacteristic
164C2×C4×D3—23no—
165C2×C4×D3—23no—
166#217—26no—
167#217C23×C516yes—
168(C2×D3)⋊C4—26no—
169C4⋊Dic3—23no—
170Dic3⋊C4—26no—
171C4×Dic3—23no—
172C23×C5—24no—
173C23×C5—63no—
174C23×C5—122no—
175C23×C5C2×#6716yes—
176C23×C5C23×D311yescharacteristic
177C5×Q8—24no—
178C5×Q8—26no—
179C5×D4—212no—
180C5×D4—212no—
181C5×D4—26no—
182C5×D4—1212no—
183C2×C4×C5—212no—
184C2×C4×C5—26no—
185C2×C4×C5—26no—
186C2×C4×C5—66no—
187C2×C4×C5—63no—
188C2×C4×C5—123no—
189C2×C4×C5—121no—
190C2×C4×C5C2×#6716yes—
191C2×C4×C5C2×#6713yes—
192C2×C4×C5C2×#6713yes—
1932−1+4—23no—
194H(5,𝔽2)—23no—
195H(5,𝔽2)C2×C5×D314yes—
196C2×#41—26no—
197C2×#41C2×C5×D313yes—
198C23×C4—31no—
199#123—36no—
200C22⋊Q8—63no—
201#121—63no—
202#120—36no—
203C4×D4—66no—
204#117—33no—
205#99—312no—
206C2×C3×C5—23no—
207C2×C3×C5—46no—
208C2×C3×C5C2×#12011yescharacteristic
209C5×D3—42no—
210C23×C3—24no—
211C23×C3C2×C5×D416yes—
212C23×C3C24×C511yescharacteristic
213C22×D3—23no—
214C22×D3—42no—
215C3×Q8—24no—
216C3×Q8—26no—
217C3×D4—212no—
218C3×D4—212no—
219C3×D4—26no—
220C2×C3×C4—212no—
221C2×C3×C4—26no—
222C2×C3×C4—26no—
223C2×C3×C4C2×C5×D416yes—
224C2×C3×C4C2×C5×D413yes—
225C2×C3×C4C2×C5×D413yes—
226#67—412no—
227C2×Dic3—26no—
228C2×Dic3—23no—
229C4×D3—41no—
230C4×D3—43no—
231C22×C5—26no—
232C22×C5—41no—
233C22×C5—46no—
234C22×C5—66no—
235C22×C5—84no—
236C22×C5—123no—
237C22×C5—121no—
238C22×C5C22×#6713yes—
239C4×C5—26no—
240C4×C5—126no—
241C4×C5—121no—
242C4×C5#95313yes—
243C4×C5#95311yescharacteristic
244C24—31no—
245#41—24no—
246#41—212no—
247#41—26no—
248#41—212no—
249#41—26no—
250C2×Q8—26no—
251C2×Q8C5×#6713yescomplemented
252C2×D4—26no—
253C2×D4—212no—
254C2×D4—66no—
255C2×D4C5×#6713yescomplemented
256C2×D4C22×C5×D316yes—
257C22×C4—24no—
258C22×C4—31no—
259C22×C4—63no—
260C22×C4—63no—
261C22×C4C5×#6716yescomplemented
262C22×C4C22×C5×D311yescharacteristic
263C4⋊C4—63no—
264C4⋊C4—66no—
265C22⋊C4—36no—
266C22⋊C4—66no—
267C22⋊C4—66no—
268C42—63no—
269C3×C5#302511yescharacteristic, complemented, Hall
270C22×C3—26no—
271C22×C3—41no—
272C22×C3—46no—
273C22×C3—84no—
274C22×C3C22×C5×D413yes—
275C2×D3—26no—
276C2×D3—43no—
277C2×D3—41no—
278C3×C4—26no—
279C3×C4C5×#12011yescharacteristic
280C3×C4C5×#12013yes—
281Dic3—46no—
282Dic3—41no—
283C2×C5—23no—
284C2×C5—46no—
285C2×C5—122no—
286C2×C5C2×#95311yescharacteristic
287C23—24no—
288C23—63no—
289C23—122no—
290C23C2×C5×#6716yes—
291C23C23×C5×D311yescharacteristic
292Q8—24no—
293Q8—26no—
294D4—212no—
295D4—212no—
296D4—26no—
297D4—1212no—
298C2×C4—212no—
299C2×C4—26no—
300C2×C4—26no—
301C2×C4—66no—
302C2×C4—63no—
303C2×C4—121no—
304C2×C4—123no—
305C2×C4C2×C5×#6716yes—
306C2×C4C2×C5×#6713yes—
307C2×C4C2×C5×#6713yes—
308C2×C3—23no—
309C2×C3—46no—
310C2×C3C2×C5×#12011yescharacteristic
311D3—42no—
312C5#8412511yescharacteristic, direct product factor, Sylow
313C22—26no—
314C22—41no—
315C22—46no—
316C22—66no—
317C22—84no—
318C22—123no—
319C22—121no—
320C22C22×C5×#6713yes—
321C4—26no—
322C4—126no—
323C4—121no—
324C4C5×#95311yescharacteristic
325C4C5×#95313yes—
326C3C5×#302511yescharacteristic, complemented, Sylow
327C2—23no—
328C2—46no—
329C2—122no—
330C2C2×C5×#95311yescharacteristic
subgroupG/H
centerC4×C5#953
commutator, cosocleC23×C3C24×C5
FrattiniC23C23×C5×D3
FittingC3×C5×#585C2
socleC2×C3×C5C2×#120
hypercenterC5×#585D3
hypocenterC3C5×#3025
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#585

Complements

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

NH ≅ G/Ntype
1. C3×C5×#585C2semidirect
2. C5×#5828C2semidirect
3. C5×#5244C2semidirect
4. C5×#5243C2semidirect
5. C5×#5240C2semidirect
12. C22×C4×C5×D3C22semidirect
15. C5×#937C22semidirect
16. C5×#935C22semidirect
18. C5×#926C22semidirect
19. C5×#834C22semidirect
20. #84125C5direct (N × H)
21. C5×#585D3semidirect
33. C2×C3×C5×Q8D4semidirect
36. C2×C3×C5×D4D4semidirect
39. C22×C3×C4×C5D4semidirect
51. C3×#585C2×C5semidirect
52. #5828C2×C5semidirect
53. #5244C2×C5semidirect
54. #5243C2×C5semidirect
55. #5240C2×C5semidirect
96. C22×C4×D3C22×C5semidirect
99. #937C22×C5semidirect
100. #935C22×C5semidirect
102. #926C22×C5semidirect
103. #834C22×C5semidirect
111. C2×C5×Q8#67semidirect
115. C2×C5×D4#67semidirect
121. C22×C4×C5#67semidirect
129. #585C5×D3semidirect
154. C2×C3×Q8C5×D4semidirect
157. C2×C3×D4C5×D4semidirect
160. C22×C3×C4C5×D4semidirect
251. C2×Q8C5×#67semidirect
255. C2×D4C5×#67semidirect
261. C22×C4C5×#67semidirect
269. C3×C5#3025semidirect
312. C5#84125direct (N × H)
326. C3C5×#3025semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5852
C5×#58282
C5×#52442
C5×#52432
C5×#52402
C5×#30253
#841255