Finite Group Databasebeta

← C5×#3300 #10616895

Subgroups — C5×#3300 #10616895

Order 640.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#552C211yesmaximal, characteristic, complemented
2C5×#548C211yesmaximal, characteristic
3C5×#488C211yesmaximal, characteristic, complemented
4C5×#485C211yesmaximal, characteristic, complemented
5C5×#480C211yesmaximal, characteristic, complemented
6C5×#479C211yesmaximal, characteristic, complemented
7C5×#467C211yesmaximal, characteristic, complemented
8C5×#465C211yesmaximal, characteristic, complemented
9C5×#464C211yesmaximal, characteristic, complemented
10C5×#452C211yesmaximal, characteristic, complemented
11C5×#451C211yesmaximal, characteristic, complemented
12C4×C5×Q16C211yesmaximal, characteristic, complemented
13C5×C8×D4C211yesmaximal, characteristic, complemented
14C5×#416C211yesmaximal, characteristic, complemented
15C2×C5×#103C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic
18C2×C5×Q16C2211yescharacteristic
19C2×C5×Q16C2211yescharacteristic
20C2×C5×SD16C2211yescharacteristic
21C22×C5×C8C2211yescharacteristic
22C22×C5×C8C2211yescharacteristic
23C5×#126—21no—
24C5×#125—21no—
25C5×#125C2211yescharacteristic
26C5×#124—21no—
27C5×#124C2211yescharacteristic
28C5×#123—21no—
29C5×C22⋊Q8—21no—
30C5×C22⋊Q8C2211yescharacteristic
31C5×C22⋊Q8C2211yescharacteristic
32C5×C22⋊Q8C2211yescharacteristic
33C5×#121—21no—
34C5×#121C2211yescharacteristic
35C4×C5×Q8C2211yescharacteristic
36C4×C5×Q8C2211yescharacteristic
37C4×C5×D4—21no—
38C4×C5×D4C2211yescharacteristic
39C5×#117—21no—
40C5×#117C2211yescharacteristic
41C2×C5×C4⋊C4C2211yescharacteristic
42C5×#107C2211yescharacteristic
43C5×#107C2211yescharacteristic
44C5×#106C2211yescharacteristic
45C5×C4⋊C8C2211yescharacteristic
46C5×#103C2211yescharacteristic
47C5×#103C2211yescharacteristic
48C5×#103C2211yescharacteristic
49C5×#103C2211yescharacteristic
50C5×#103C2211yescharacteristic
51C5×#103C2211yescharacteristic
52C5×#103C2211yescharacteristic
53C5×#102C2211yescharacteristic
54C5×#102C2211yescharacteristic
55C5×#102C2211yescharacteristic
56C5×C22⋊C8C2211yescharacteristic
57C5×C22⋊C8C2211yescharacteristic
58C4×C5×C8C2211yescharacteristic
59#3300C511yesmaximal, characteristic, direct product factor, Sylow
60C5×#41—22no—
61C5×#41—41no—
62C2×C5×Q8—21no—
63C2×C5×Q8—22no—
64C2×C5×Q8—41no—
65C2×C5×Q8C2311yescharacteristic
66C2×C5×Q8C2311yescharacteristic
67C2×C5×Q8C2311yescharacteristic
68C2×C5×D4—21no—
69C2×C5×D4—21no—
70C2×C5×D4D411yescharacteristic
71C2×C5×D4C2311yescharacteristic
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5—21no—
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C5×Q16—22no—
78C5×Q16—41no—
79C5×SD16—41no—
80C2×C5×C8—41no—
81C2×C5×C8—41no—
82C2×C5×C8C2311yescharacteristic
83C2×C5×C8C2311yescharacteristic
84C2×C5×C8C2311yescharacteristic
85C2×C5×C8C2311yescharacteristic
86C5×C4⋊C4—21no—
87C5×C4⋊C4—21no—
88C5×C4⋊C4—21no—
89C5×C4⋊C4—21no—
90C5×C4⋊C4—21no—
91C5×C4⋊C4—21no—
92C5×C4⋊C4—21no—
93C5×C4⋊C4—21no—
94C5×C4⋊C4D411yescharacteristic
95C5×C4⋊C4C2311yescharacteristic
96C5×C4⋊C4C2311yescharacteristic
97C5×C4⋊C4C2311yescharacteristic
98C5×C4⋊C4C2311yescharacteristic
99C5×C22⋊C4—21no—
100C5×C22⋊C4—21no—
101C5×C22⋊C4—21no—
102C5×C22⋊C4—21no—
103C5×C22⋊C4—21no—
104C5×C22⋊C4—21no—
105C5×C22⋊C4—21no—
106C5×C22⋊C4D411yescharacteristic
107C5×C22⋊C4D411yescharacteristic
108C42×C5—21no—
109C42×C5—21no—
110C42×C5—21no—
111C42×C5C2311yescharacteristic
112#552C2×C511yescharacteristic, complemented
113#548C2×C511yescharacteristic
114#488C2×C511yescharacteristic, complemented
115#485C2×C511yescharacteristic, complemented
116#480C2×C511yescharacteristic, complemented
117#479C2×C511yescharacteristic, complemented
118#467C2×C511yescharacteristic, complemented
119#465C2×C511yescharacteristic, complemented
120#464C2×C511yescharacteristic, complemented
121#452C2×C511yescharacteristic, complemented
122#451C2×C511yescharacteristic, complemented
123C4×Q16C2×C511yescharacteristic, complemented
124C8×D4C2×C511yescharacteristic, complemented
125#416C2×C511yescharacteristic, complemented
126C2×#103C2×C511yescharacteristic, complemented
127C23×C5—21no—
128C23×C5C2×D411yescharacteristic
129C23×C5C2×D411yescharacteristic
130C5×Q8—21no—
131C5×Q8—21no—
132C5×Q8—21no—
133C5×Q8—21no—
134C5×Q8—21no—
135C5×Q8—42no—
136C5×Q8#4112yes—
137C5×D4—22no—
138C5×D4—21no—
139C5×D4—21no—
140C5×D4—42no—
141C5×D4—42no—
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—42no—
154C2×C4×C5—42no—
155C2×C4×C5—42no—
156C2×C4×C5C2×D411yescharacteristic
157C2×C4×C5C2×D411yescharacteristic
158C2×C4×C5C2×D411yescharacteristic
159C2×C4×C5C2×D411yescharacteristic
160C2×C4×C5C2411yescharacteristic
161C5×C8—21no—
162C5×C8—21no—
163C5×C8—21no—
164C5×C8—21no—
165C2×#41C22×C511yescharacteristic
166C22×Q8C22×C511yescharacteristic
167C2×Q16C22×C511yescharacteristic
168C2×Q16C22×C511yescharacteristic
169C2×SD16C22×C511yescharacteristic
170C22×C8C22×C511yescharacteristic
171C22×C8C22×C511yescharacteristic
172#126—21no—
173#125—21no—
174#125C22×C511yescharacteristic
175#124—21no—
176#124C22×C511yescharacteristic
177#123—21no—
178C22⋊Q8—21no—
179C22⋊Q8C22×C511yescharacteristic
180C22⋊Q8C22×C511yescharacteristic
181C22⋊Q8C22×C511yescharacteristic
182#121—21no—
183#121C22×C511yescharacteristic
184C4×Q8C22×C511yescharacteristic
185C4×Q8C22×C511yescharacteristic
186C4×D4—21no—
187C4×D4C22×C511yescharacteristic
188#117—21no—
189#117C22×C511yescharacteristic
190C2×C4⋊C4C22×C511yescharacteristic
191#107C22×C511yescharacteristic
192#107C22×C511yescharacteristic
193#106C22×C511yescharacteristic
194C4⋊C8C22×C511yescharacteristic
195#103C22×C511yescharacteristic
196#103C22×C511yescharacteristic
197#103C22×C511yescharacteristic
198#103C22×C511yescharacteristic
199#103C22×C511yescharacteristic
200#103C22×C511yescharacteristic
201#103C22×C511yescharacteristic
202#102C22×C511yescharacteristic
203#102C22×C511yescharacteristic
204#102C22×C511yescharacteristic
205C22⋊C8C22×C511yescharacteristic
206C22⋊C8C22×C511yescharacteristic
207C4×C8C22×C511yescharacteristic
208C22×C5—21no—
209C22×C5—21no—
210C22×C5—21no—
211C22×C5—21no—
212C22×C5—21no—
213C22×C5—41no—
214C22×C5—41no—
215C22×C5—41no—
216C22×C5#13512yes—
217C22×C5C22×D411yescharacteristic
218C4×C5—21no—
219C4×C5—21no—
220C4×C5—22no—
221C4×C5—21no—
222C4×C5—21no—
223C4×C5—41no—
224C4×C5—41no—
225C4×C5—41no—
226C4×C5—41no—
227C4×C5—41no—
228C4×C5—41no—
229C4×C5C2×#4111yescharacteristic
230C4×C5H(5,𝔽2)11yescharacteristic
231#41—22no—
232#41—41no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8C23×C511yescharacteristic
237C2×Q8C23×C511yescharacteristic
238C2×Q8C23×C511yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4C5×D411yescharacteristic
242C2×D4C23×C511yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4C23×C511yescharacteristic
247C22×C4C23×C511yescharacteristic
248Q16—22no—
249Q16—41no—
250SD16—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8C23×C511yescharacteristic
254C2×C8C23×C511yescharacteristic
255C2×C8C23×C511yescharacteristic
256C2×C8C23×C511yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4C5×D411yescharacteristic
266C4⋊C4C23×C511yescharacteristic
267C4⋊C4C23×C511yescharacteristic
268C4⋊C4C23×C511yescharacteristic
269C4⋊C4C23×C511yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4C5×D411yescharacteristic
278C22⋊C4C5×D411yescharacteristic
279C42—21no—
280C42—21no—
281C42—21no—
282C42C23×C511yescharacteristic
283C2×C5—22no—
284C2×C5—41no—
285C2×C5—81no—
286C2×C5#54611yescharacteristic
287C2×C5C2×#13511yescharacteristic
288C2×C5#57811yescharacteristic
289C23—21no—
290C23C2×C5×D411yescharacteristic
291C23C2×C5×D411yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—42no—
298Q8C5×#4112yes—
299D4—22no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—42no—
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—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4C2×C5×D411yescharacteristic
319C2×C4C2×C5×D411yescharacteristic
320C2×C4C2×C5×D411yescharacteristic
321C2×C4C2×C5×D411yescharacteristic
322C2×C4C24×C511yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—21no—
327C5#330011yescharacteristic, direct product factor, Sylow
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22C5×#13512yes—
337C22C22×C5×D411yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4C2×C5×#4111yescharacteristic
350C4C5×H(5,𝔽2)11yescharacteristic
351C2—22no—
352C2—41no—
353C2—81no—
354C2C5×#54611yescharacteristic
355C2C2×C5×#13511yescharacteristic
356C2C5×#57811yescharacteristic
subgroupG/H
center, socleC22×C5C22×D4
commutator, Frattini, cosocleC2×C4C24×C5
prime radicalC5×#548C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C4×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#552C2semidirect
3. C5×#488C2semidirect
4. C5×#485C2semidirect
5. C5×#480C2semidirect
6. C5×#479C2semidirect
7. C5×#467C2semidirect
8. C5×#465C2semidirect
9. C5×#464C2semidirect
10. C5×#452C2semidirect
11. C5×#451C2semidirect
12. C4×C5×Q16C2semidirect
13. C5×C8×D4C2semidirect
14. C5×#416C2semidirect
15. C2×C5×#103C2semidirect
59. #3300C5direct (N × H)
112. #552C2×C5semidirect
114. #488C2×C5semidirect
115. #485C2×C5semidirect
116. #480C2×C5semidirect
117. #479C2×C5semidirect
118. #467C2×C5semidirect
119. #465C2×C5semidirect
120. #464C2×C5semidirect
121. #452C2×C5semidirect
122. #451C2×C5semidirect
123. C4×Q16C2×C5semidirect
124. C8×D4C2×C5semidirect
125. #416C2×C5semidirect
126. C2×#103C2×C5semidirect
327. C5#3300direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#5522
C5×#5482
C5×#4882
C5×#4852
C5×#4802
C5×#4792
C5×#4672
C5×#4652
C5×#4642
C5×#4522
C5×#4512
C4×C5×Q162
C5×C8×D42
C5×#4162
C2×C5×#1032
#33005