Finite Group Databasebeta

← C5×#2716 #10616311

Subgroups — C5×#2716 #10616311

Order 640.

450 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C5×#126C211yesmaximal, characteristic
3C5×#400C211yesmaximal, characteristic, complemented
4C5×#398C211yesmaximal, characteristic, complemented
5C5×#397C211yesmaximal, characteristic
6C5×#396C211yesmaximal, characteristic
7C5×#393C211yesmaximal, characteristic
8C5×#393C211yesmaximal, characteristic
9C5×#391C211yesmaximal, characteristic, complemented
10C5×#391C211yesmaximal, characteristic, complemented
11C5×#388C211yesmaximal, characteristic
12C5×#388C211yesmaximal, characteristic
13C5×#387C211yesmaximal, characteristic, complemented
14C5×#387C211yesmaximal, characteristic, complemented
15C5×#387C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic
18C5×C4⋊Q8—24no—
19C5×#126—24no—
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C4⋊C4C2211yescharacteristic
26C2×C5×C4⋊C4C2211yescharacteristic
27C2×C5×C4⋊C4C2211yescharacteristic
28C2×C5×C4⋊C4C2211yescharacteristic
29C2×C5×C22⋊C4C2211yescharacteristic
30C2×C5×C22⋊C4C2211yescharacteristic
31C2×C5×C22⋊C4C2211yescharacteristic
32C2×C5×C22⋊C4C2211yescharacteristic
33C2×C5×C22⋊C4C2211yescharacteristic
34C2×C5×C22⋊C4C2211yescharacteristic
35C2×C5×C22⋊C4C2211yescharacteristic
36C2×C42×C5C2211yescharacteristic
37C2×C42×C5C2211yescharacteristic
38C2×C42×C5C2211yescharacteristic
39C5×#95C2211yescharacteristic
40C5×#95C2211yescharacteristic
41C5×#95C2211yescharacteristic
42C5×#95C2211yescharacteristic
43C5×#95C2211yescharacteristic
44C5×#95C2211yescharacteristic
45C5×#95C2211yescharacteristic
46C5×#95C2211yescharacteristic
47C5×#95C2211yescharacteristic
48C5×#95C2211yescharacteristic
49C5×#95C2211yescharacteristic
50C5×#95C2211yescharacteristic
51C5×#95C2211yescharacteristic
52C5×#95C2211yescharacteristic
53#2716C511yesmaximal, characteristic, direct product factor, Sylow
54C24×C5C2311yescharacteristic
55C2×C5×Q8—22no—
56C2×C5×Q8—22no—
57C2×C5×Q8—22no—
58C2×C5×Q8—22no—
59C2×C5×Q8—41no—
60C2×C5×Q8—41no—
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C5×C4⋊C4—22no—
76C5×C4⋊C4—22no—
77C5×C4⋊C4—22no—
78C5×C4⋊C4—22no—
79C5×C4⋊C4—22no—
80C5×C4⋊C4—22no—
81C5×C4⋊C4—22no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4D414yes—
84C5×C22⋊C4—22no—
85C5×C22⋊C4—22no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4—22no—
88C5×C22⋊C4—22no—
89C5×C22⋊C4—41no—
90C5×C22⋊C4—41no—
91C42×C5—22no—
92C42×C5—22no—
93C42×C5—41no—
94C2×C4⋊Q8C2×C511yescharacteristic, complemented
95C2×#126C2×C511yescharacteristic
96#400C2×C511yescharacteristic, complemented
97#398C2×C511yescharacteristic, complemented
98#397C2×C511yescharacteristic
99#396C2×C511yescharacteristic
100#393C2×C511yescharacteristic
101#393C2×C511yescharacteristic
102#391C2×C511yescharacteristic, complemented
103#391C2×C511yescharacteristic, complemented
104#388C2×C511yescharacteristic
105#388C2×C511yescharacteristic
106#387C2×C511yescharacteristic, complemented
107#387C2×C511yescharacteristic, complemented
108#387C2×C511yescharacteristic, complemented
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5C2411yescharacteristic
117C5×Q8—44no—
118C5×Q8—44no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5C2×D412yes—
157C2×C4×C5C2×D412yes—
158C2×C4×C5C2×D412yes—
159C2×C4×C5#4112yes—
160C2×C4×C5#4112yes—
161C22×Q8C22×C511yescharacteristic
162C22×Q8C22×C511yescharacteristic
163C4⋊Q8—24no—
164#126—24no—
165C2×C4⋊C4C22×C511yescharacteristic
166C2×C4⋊C4C22×C511yescharacteristic
167C2×C4⋊C4C22×C511yescharacteristic
168C2×C4⋊C4C22×C511yescharacteristic
169C2×C4⋊C4C22×C511yescharacteristic
170C2×C4⋊C4C22×C511yescharacteristic
171C2×C4⋊C4C22×C511yescharacteristic
172C2×C4⋊C4C22×C511yescharacteristic
173C2×C4⋊C4C22×C511yescharacteristic
174C2×C22⋊C4C22×C511yescharacteristic
175C2×C22⋊C4C22×C511yescharacteristic
176C2×C22⋊C4C22×C511yescharacteristic
177C2×C22⋊C4C22×C511yescharacteristic
178C2×C22⋊C4C22×C511yescharacteristic
179C2×C22⋊C4C22×C511yescharacteristic
180C2×C22⋊C4C22×C511yescharacteristic
181C2×C42C22×C511yescharacteristic
182C2×C42C22×C511yescharacteristic
183C2×C42C22×C511yescharacteristic
184#95C22×C511yescharacteristic
185#95C22×C511yescharacteristic
186#95C22×C511yescharacteristic
187#95C22×C511yescharacteristic
188#95C22×C511yescharacteristic
189#95C22×C511yescharacteristic
190#95C22×C511yescharacteristic
191#95C22×C511yescharacteristic
192#95C22×C511yescharacteristic
193#95C22×C511yescharacteristic
194#95C22×C511yescharacteristic
195#95C22×C511yescharacteristic
196#95C22×C511yescharacteristic
197#95C22×C511yescharacteristic
198C22×C5—41no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5C22×D411yescharacteristic
206C22×C5C2×#4111yescharacteristic
207C22×C5C2×#4111yescharacteristic
208C22×C5H(5,𝔽2)11yescharacteristic
209C22×C52−1+411yescharacteristic
210C22×C52−1+411yescharacteristic
211C22×C52−1+411yescharacteristic
212C4×C5—22no—
213C4×C5—22no—
214C4×C5—22no—
215C4×C5—22no—
216C4×C5—22no—
217C4×C5—41no—
218C4×C5—41no—
219C4×C5—41no—
220C4×C5—41no—
221C4×C5—41no—
222C4×C5—41no—
223C4×C5—41no—
224C4×C5—41no—
225C4×C5—41no—
226C24C23×C511yescharacteristic
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C22×C4C23×C511yescharacteristic
234C22×C4C23×C511yescharacteristic
235C22×C4C23×C511yescharacteristic
236C22×C4C23×C511yescharacteristic
237C22×C4C23×C511yescharacteristic
238C22×C4C23×C511yescharacteristic
239C22×C4C23×C511yescharacteristic
240C22×C4C23×C511yescharacteristic
241C22×C4C23×C511yescharacteristic
242C22×C4C23×C511yescharacteristic
243C22×C4C23×C511yescharacteristic
244C22×C4C23×C511yescharacteristic
245C22×C4C23×C511yescharacteristic
246C22×C4C23×C511yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4C5×D414yes—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—22no—
264C42—22no—
265C42—41no—
266C2×C5—81no—
267C2×C5#53611yescharacteristic
268C2×C5#54111yescharacteristic
269C2×C5#54211yescharacteristic
270C2×C5#54611yescharacteristic
271C2×C5#54811yescharacteristic
272C2×C5#55611yescharacteristic
273C2×C5#56311yescharacteristic
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23C24×C511yescharacteristic
282Q8—44no—
283Q8—44no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4C2×C5×D412yes—
322C2×C4C2×C5×D412yes—
323C2×C4C2×C5×D412yes—
324C2×C4C5×#4112yes—
325C2×C4C5×#4112yes—
326C5#271611yescharacteristic, direct product factor, Sylow
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22C22×C5×D411yescharacteristic
335C22C2×C5×#4111yescharacteristic
336C22C2×C5×#4111yescharacteristic
337C22C5×H(5,𝔽2)11yescharacteristic
338C22C5×2−1+411yescharacteristic
339C22C5×2−1+411yescharacteristic
340C22C5×2−1+411yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C2—81no—
356C2C5×#53611yescharacteristic
357C2C5×#54111yescharacteristic
358C2C5×#54211yescharacteristic
359C2C5×#54611yescharacteristic
360C2C5×#54811yescharacteristic
361C2C5×#55611yescharacteristic
362C2C5×#56311yescharacteristic
subgroupG/H
center, socleC23×C5C24
commutator, Frattini, cosocleC23C24×C5
prime radicalC24×C5C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C5×C4⋊Q8C2semidirect
3. C5×#400C2semidirect
4. C5×#398C2semidirect
9. C5×#391C2semidirect
10. C5×#391C2semidirect
13. C5×#387C2semidirect
14. C5×#387C2semidirect
15. C5×#387C2semidirect
53. #2716C5direct (N × H)
94. C2×C4⋊Q8C2×C5semidirect
96. #400C2×C5semidirect
97. #398C2×C5semidirect
102. #391C2×C5semidirect
103. #391C2×C5semidirect
106. #387C2×C5semidirect
107. #387C2×C5semidirect
108. #387C2×C5semidirect
326. C5#2716direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×C4⋊Q82
C2×C5×#1262
C5×#4002
C5×#3982
C5×#3972
C5×#3962
C5×#3932
C5×#3912
C5×#3882
C5×#3872
#27165