Finite Group Databasebeta

← C5×#69146 #14316640

Subgroups — C5×#69146 #14316640

Order 1920.

708 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C5×#5717C211yesmaximal, characteristic, complemented
2C5×#5077C211yesmaximal, characteristic, complemented
3C5×#4741C212yesmaximal, complemented
4C3×C5×#353C211yesmaximal, characteristic, complemented
5C5×#4471C212yesmaximal, complemented
6C5×#2196—31nomaximal, Hall
7C5×#1004C2211yescharacteristic, complemented
8C2×C5×D3×D4—22no—
9C3×C5×#127C2211yescharacteristic
10C5×#938—22no—
11C5×#937—21no—
12C5×#937C2212yes—
13C5×#932—21no—
14C5×#882—22no—
15C4×C5×D12—21no—
16C3×C5×#99C2212yescomplemented
17C5×#833C2211yescharacteristic, complemented
18C5×#806—22no—
19C5×#805—21no—
20#69146C511yesmaximal, characteristic, direct product factor, Hall
21C5×D42—31no—
22C5×B(4,𝔽2)—32no—
23C5×#453—31no—
24C5×#353D311yescharacteristic, complemented
25C5×AΣL(1,𝔽16)—32no—
26C23×C5×D3—21no—
27C23×C5×D3D412yescomplemented
28C2×C3×C5×D4—22no—
29C2×C3×C5×D4D411yescharacteristic, complemented
30C2×C3×C5×D4C2311yescharacteristic
31C2×C5×#67—22no—
32C2×C5×#67—22no—
33C2×C5×#67—22no—
34C5×#237—41no—
35C5×D3×D4—41no—
36C5×D3×D4—48no—
37C5×#235—21no—
38C2×C5×D12—21no—
39C2×C5×D12—42no—
40C2×C4×C5×D3—22no—
41C3×C5×C22⋊C4—22no—
42C3×C42×C5D411yescharacteristic, complemented
43C5×#217—21no—
44C5×#217D412yescomplemented
45C5×#214—41no—
46C5×#213—41no—
47C5×(C2×D3)⋊C4—22no—
48C5×(C2×D3)⋊C4—42no—
49C5×C4⋊Dic3—21no—
50C5×#208—21no—
51#5717C2×C511yescharacteristic, complemented
52#5077C2×C511yescharacteristic, complemented
53#4741C2×C512yescomplemented
54C3×#353C2×C511yescharacteristic, complemented
55#4471C2×C512yescomplemented
56C5×H(5,𝔽2)—31no—
57C22×C5×D4—62no—
58C5×AGL(1,ℤ/8)—61no—
59C5×#127C2×D311yescharacteristic
60C5×#121—62no—
61C5×#120—32no—
62C5×#120—62no—
63C5×#120—61no—
64C4×C5×D4—61no—
65C5×#104—61no—
66C5×#100—31no—
67C5×#99—62no—
68C5×#99C2×D312yescomplemented
69#2196—31noSylow
70C23×C3×C5—21no—
71C23×C3×C5C2×D412yes—
72C22×C5×D3—22no—
73C22×C5×D3—22no—
74C22×C5×D3—44no—
75C22×C5×D3—44no—
76C22×C5×D3—42no—
77C22×C5×D3—81no—
78C3×C5×D4—41no—
79C3×C5×D4—44no—
80C3×C5×D4—41no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—42no—
83C2×C3×C4×C5C2×D411yescharacteristic
84C5×#67—44no—
85C5×#67—44no—
86C5×#67—42no—
87C5×#67—41no—
88C2×C5×Dic3—22no—
89C2×C5×Dic3—22no—
90C5×D12—24no—
91C5×D12—21no—
92C5×D12—84no—
93C4×C5×D3—44no—
94C4×C5×D3—41no—
95C5×Dic6—21no—
96C5×C3⋊C8—41no—
97#1004C22×C511yescharacteristic, complemented
98C2×D3×D4—22no—
99C3×#127C22×C511yescharacteristic
100#938—22no—
101#937—21no—
102#937C22×C512yes—
103#932—21no—
104#882—22no—
105C4×D12—21no—
106C3×#99C22×C512yescomplemented
107#833C22×C511yescharacteristic, complemented
108#806—22no—
109#805—21no—
110C24×C5—32no—
111C24×C5—61no—
112C5×#41—61no—
113C5×#41—121no—
114C2×C5×D4—22no—
115C2×C5×D4—61no—
116C2×C5×D4—62no—
117C2×C5×D4—62no—
118C2×C5×D4—62no—
119C2×C5×D4—122no—
120C2×C5×D4—121no—
121C2×C5×D4—128no—
122C2×C5×D4D1211yescharacteristic, complemented
123C2×C5×D4C22×D311yescharacteristic
124C22×C4×C5—62no—
125C5×SD16—121no—
126C5×D8—121no—
127C5×M16—61no—
128C5×C4⋊C4—61no—
129C5×C22⋊C4—22no—
130C5×C22⋊C4—32no—
131C5×C22⋊C4—61no—
132C5×C22⋊C4—62no—
133C5×C22⋊C4—122no—
134C42×C5D1211yescharacteristic, complemented
135D42—31no—
136B(4,𝔽2)—32no—
137#453—31no—
138#353C5×D311yescharacteristic, complemented
139AΣL(1,𝔽16)—32no—
140C22×C3×C5—22no—
141C22×C3×C5—42no—
142C22×C3×C5—42no—
143C22×C3×C5—41no—
144C22×C3×C5#12011yescharacteristic
145C2×C5×D3—24no—
146C2×C5×D3—42no—
147C2×C5×D3—42no—
148C2×C5×D3—44no—
149C2×C5×D3—41no—
150C2×C5×D3—82no—
151C2×C5×D3—84no—
152C2×C5×D3—82no—
153C3×C4×C5—22no—
154C3×C4×C5—21no—
155C3×C4×C5—82no—
156C5×Dic3—42no—
157C5×Dic3—41no—
158C23×D3—21no—
159C23×D3C5×D412yescomplemented
160C2×C3×D4—22no—
161C2×C3×D4C5×D411yescharacteristic, complemented
162C2×C3×D4C23×C511yescharacteristic
163C2×#67—22no—
164C2×#67—22no—
165C2×#67—22no—
166#237—41no—
167D3×D4—41no—
168D3×D4—48no—
169#235—21no—
170C2×D12—21no—
171C2×D12—42no—
172C2×C4×D3—22no—
173C3×C22⋊C4—22no—
174C3×C42C5×D411yescharacteristic, complemented
175#217—21no—
176#217C5×D412yescomplemented
177#214—41no—
178#213—41no—
179(C2×D3)⋊C4—22no—
180(C2×D3)⋊C4—42no—
181C4⋊Dic3—21no—
182#208—21no—
183C23×C5—21no—
184C23×C5—62no—
185C23×C5—62no—
186C23×C5—122no—
187C23×C5—124no—
188C23×C5—124no—
189C23×C5—241no—
190C23×C5D3×D412yes—
191C5×Q8—61no—
192C5×D4—41no—
193C5×D4—41no—
194C5×D4—44no—
195C5×D4—64no—
196C5×D4—61no—
197C5×D4—122no—
198C5×D4—124no—
199C5×D4—124no—
200C5×D4—121no—
201C5×D4—244no—
202C2×C4×C5—21no—
203C2×C4×C5—42no—
204C2×C4×C5—62no—
205C2×C4×C5—62no—
206C2×C4×C5—124no—
207C2×C4×C5—121no—
208C2×C4×C5C2×D1211yescharacteristic
209C5×C8—121no—
210H(5,𝔽2)—31no—
211C22×D4—62no—
212AGL(1,ℤ/8)—61no—
213#127C2×C5×D311yescharacteristic
214#121—62no—
215#120—32no—
216#120—62no—
217#120—61no—
218C4×D4—61no—
219#104—61no—
220#100—31no—
221#99—62no—
222#99C2×C5×D312yescomplemented
223C2×C3×C5—21no—
224C2×C3×C5—42no—
225C2×C3×C5—81no—
226C2×C3×C5B(4,𝔽2)11yescharacteristic
227C5×D3—44no—
228C5×D3—81no—
229C23×C3—21no—
230C23×C3C2×C5×D412yes—
231C22×D3—22no—
232C22×D3—22no—
233C22×D3—44no—
234C22×D3—44no—
235C22×D3—42no—
236C22×D3—81no—
237C3×D4—41no—
238C3×D4—41no—
239C3×D4—44no—
240C2×C3×C4—21no—
241C2×C3×C4—42no—
242C2×C3×C4C2×C5×D411yescharacteristic
243#67—44no—
244#67—44no—
245#67—42no—
246#67—41no—
247C2×Dic3—22no—
248C2×Dic3—22no—
249D12—24no—
250D12—21no—
251D12—84no—
252C4×D3—44no—
253C4×D3—41no—
254Dic6—21no—
255C3⋊C8—41no—
256C22×C5—22no—
257C22×C5—42no—
258C22×C5—42no—
259C22×C5—41no—
260C22×C5—64no—
261C22×C5—122no—
262C22×C5—124no—
263C22×C5—122no—
264C22×C5—121no—
265C22×C5—242no—
266C22×C5—242no—
267C22×C5—244no—
268C22×C5#88211yescharacteristic
269C4×C5—22no—
270C4×C5—21no—
271C4×C5—82no—
272C4×C5—122no—
273C4×C5—121no—
274C24—32no—
275C24—61no—
276#41—61no—
277#41—121no—
278C2×D4—22no—
279C2×D4—61no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—62no—
283C2×D4—122no—
284C2×D4—121no—
285C2×D4—128no—
286C2×D4C5×D1211yescharacteristic, complemented
287C2×D4C22×C5×D311yescharacteristic
288C22×C4—62no—
289SD16—121no—
290D8—121no—
291M16—61no—
292C4⋊C4—61no—
293C22⋊C4—22no—
294C22⋊C4—32no—
295C22⋊C4—61no—
296C22⋊C4—62no—
297C22⋊C4—122no—
298C42C5×D1211yescharacteristic, complemented
299C3×C5#219611yescharacteristic, complemented, Hall
300C22×C3—22no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3C5×#12011yescharacteristic
305C2×D3—24no—
306C2×D3—42no—
307C2×D3—42no—
308C2×D3—44no—
309C2×D3—41no—
310C2×D3—82no—
311C2×D3—84no—
312C2×D3—82no—
313C3×C4—22no—
314C3×C4—21no—
315C3×C4—82no—
316Dic3—42no—
317Dic3—41no—
318C2×C5—21no—
319C2×C5—42no—
320C2×C5—81no—
321C2×C5—124no—
322C2×C5—241no—
323C2×C5#474111yescharacteristic
324C23—21no—
325C23—62no—
326C23—62no—
327C23—122no—
328C23—124no—
329C23—124no—
330C23—241no—
331C23C5×D3×D412yes—
332Q8—61no—
333D4—41no—
334D4—41no—
335D4—44no—
336D4—64no—
337D4—61no—
338D4—122no—
339D4—124no—
340D4—124no—
341D4—121no—
342D4—244no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—124no—
348C2×C4—121no—
349C2×C4C2×C5×D1211yescharacteristic
350C8—121no—
351C2×C3—21no—
352C2×C3—42no—
353C2×C3—81no—
354C2×C3C5×B(4,𝔽2)11yescharacteristic
355D3—44no—
356D3—81no—
357C5#6914611yescharacteristic, direct product factor, Sylow
358C22—22no—
359C22—42no—
360C22—42no—
361C22—41no—
362C22—64no—
363C22—122no—
364C22—124no—
365C22—122no—
366C22—121no—
367C22—242no—
368C22—242no—
369C22—244no—
370C22C5×#88211yescharacteristic
371C4—22no—
372C4—21no—
373C4—82no—
374C4—122no—
375C4—121no—
376C3C5×#219611yescharacteristic, complemented, Sylow
377C2—21no—
378C2—42no—
379C2—81no—
380C2—124no—
381C2—241no—
382C2C5×#474111yescharacteristic
subgroupG/H
centerC2×C5#4741
commutator, cosocleC2×C3×D4C23×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×#353C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC5×#353D3
hypocenterC3C5×#2196
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×D4 ◁ C5×#353

Complements

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

NH ≅ G/Ntype
1. C5×#5717C2semidirect
2. C5×#5077C2semidirect
3. C5×#4741C2semidirect
4. C3×C5×#353C2semidirect
5. C5×#4471C2semidirect
7. C5×#1004C22semidirect
16. C3×C5×#99C22semidirect
17. C5×#833C22semidirect
20. #69146C5direct (N × H)
24. C5×#353D3semidirect
27. C23×C5×D3D4semidirect
29. C2×C3×C5×D4D4semidirect
42. C3×C42×C5D4semidirect
44. C5×#217D4semidirect
51. #5717C2×C5semidirect
52. #5077C2×C5semidirect
53. #4741C2×C5semidirect
54. C3×#353C2×C5semidirect
55. #4471C2×C5semidirect
68. C5×#99C2×D3semidirect
97. #1004C22×C5semidirect
106. C3×#99C22×C5semidirect
107. #833C22×C5semidirect
122. C2×C5×D4D12semidirect
134. C42×C5D12semidirect
138. #353C5×D3semidirect
159. C23×D3C5×D4semidirect
161. C2×C3×D4C5×D4semidirect
174. C3×C42C5×D4semidirect
176. #217C5×D4semidirect
222. #99C2×C5×D3semidirect
286. C2×D4C5×D12semidirect
298. C42C5×D12semidirect
299. C3×C5#2196semidirect
357. C5#69146direct (N × H)
376. C3C5×#2196semidirect

Maximal subgroups

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

subgroupindex
C5×#57172
C5×#50772
C5×#47412
C3×C5×#3532
C5×#44712
C5×#21963
#691465