Finite Group Databasebeta

← C5×#3321 #10616916

Subgroups — C5×#3321 #10616916

Order 640.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#555C211yesmaximal, characteristic, complemented
2C5×#548C211yesmaximal, characteristic, complemented
3C5×#489C211yesmaximal, characteristic, complemented
4C5×#485C211yesmaximal, characteristic, complemented
5C5×#482C211yesmaximal, characteristic, complemented
6C5×#475C211yesmaximal, characteristic, complemented
7C5×#469C211yesmaximal, characteristic, complemented
8C5×#465C211yesmaximal, characteristic, complemented
9C5×#463C211yesmaximal, characteristic, complemented
10C5×#449C211yesmaximal, characteristic, complemented
11C5×#448C211yesmaximal, characteristic, complemented
12C5×#440C211yesmaximal, characteristic, complemented
13C5×#435C211yesmaximal, characteristic, complemented
14C5×#419C211yesmaximal, characteristic, complemented
15C5×#416C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic
18C2×C5×SD16C2211yescharacteristic
19C2×C5×SD16C2211yescharacteristic
20C2×C5×D8C2211yescharacteristic
21C2×C5×M16C2211yescharacteristic
22C22×C5×C8C2211yescharacteristic
23C5×C4⋊Q8C2211yescharacteristic
24C5×#124—21no—
25C5×#124—21no—
26C5×#124—21no—
27C5×#124C2211yescharacteristic
28C5×C22⋊Q8—21no—
29C5×C22⋊Q8C2211yescharacteristic
30C5×#121—21no—
31C5×#121—21no—
32C5×#121C2211yescharacteristic
33C5×#121C2211yescharacteristic
34C5×#121C2211yescharacteristic
35C4×C5×Q8C2211yescharacteristic
36C4×C5×D4—21no—
37C4×C5×D4C2211yescharacteristic
38C4×C5×D4C2211yescharacteristic
39C5×#117—21no—
40C5×#117C2211yescharacteristic
41C5×#117C2211yescharacteristic
42C5×#107C2211yescharacteristic
43C5×#106C2211yescharacteristic
44C5×#106C2211yescharacteristic
45C5×C4⋊C8C2211yescharacteristic
46C5×#103C2211yescharacteristic
47C5×#103C2211yescharacteristic
48C5×#103C2211yescharacteristic
49C5×#103C2211yescharacteristic
50C5×#103C2211yescharacteristic
51C5×#103C2211yescharacteristic
52C5×#102C2211yescharacteristic
53C5×#102C2211yescharacteristic
54C5×#102C2211yescharacteristic
55C5×#102C2211yescharacteristic
56C5×C22⋊C8C2211yescharacteristic
57C5×C22⋊C8C2211yescharacteristic
58C5×#97C2211yescharacteristic
59#3321C511yesmaximal, characteristic, direct product factor, Sylow
60C5×#41—22no—
61C5×#41—41no—
62C2×C5×Q8—21no—
63C2×C5×Q8—21no—
64C2×C5×Q8—22no—
65C2×C5×Q8—41no—
66C2×C5×Q8C2311yescharacteristic
67C2×C5×Q8C2311yescharacteristic
68C2×C5×D4—21no—
69C2×C5×D4—21no—
70C2×C5×D4—21no—
71C2×C5×D4—21no—
72C2×C5×D4D411yescharacteristic
73C2×C5×D4C2311yescharacteristic
74C2×C5×D4C2311yescharacteristic
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5C2311yescharacteristic
79C22×C4×C5C2311yescharacteristic
80C5×SD16—22no—
81C5×SD16—41no—
82C5×D8—41no—
83C5×M16—41no—
84C2×C5×C8—41no—
85C2×C5×C8C2311yescharacteristic
86C2×C5×C8C2311yescharacteristic
87C2×C5×C8C2311yescharacteristic
88C2×C5×C8C2311yescharacteristic
89C5×C4⋊C4—21no—
90C5×C4⋊C4—21no—
91C5×C4⋊C4—21no—
92C5×C4⋊C4D411yescharacteristic
93C5×C4⋊C4C2311yescharacteristic
94C5×C4⋊C4C2311yescharacteristic
95C5×C4⋊C4C2311yescharacteristic
96C5×C4⋊C4C2311yescharacteristic
97C5×C22⋊C4—21no—
98C5×C22⋊C4—21no—
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#555C2×C511yescharacteristic, complemented
113#548C2×C511yescharacteristic, complemented
114#489C2×C511yescharacteristic, complemented
115#485C2×C511yescharacteristic, complemented
116#482C2×C511yescharacteristic, complemented
117#475C2×C511yescharacteristic, complemented
118#469C2×C511yescharacteristic, complemented
119#465C2×C511yescharacteristic, complemented
120#463C2×C511yescharacteristic, complemented
121#449C2×C511yescharacteristic, complemented
122#448C2×C511yescharacteristic, complemented
123#440C2×C511yescharacteristic, complemented
124#435C2×C511yescharacteristic, complemented
125#419C2×C511yescharacteristic, complemented
126#416C2×C511yescharacteristic, complemented
127C23×C5—21no—
128C23×C5—21no—
129C23×C5C2×D411yescharacteristic
130C23×C5C2×D411yescharacteristic
131C5×Q8—21no—
132C5×Q8—21no—
133C5×Q8—21no—
134C5×Q8—42no—
135C5×Q8—81no—
136C5×Q8#4112yes—
137C5×D4—21no—
138C5×D4—21no—
139C5×D4—21no—
140C5×D4—21no—
141C5×D4—41no—
142C5×D4—42no—
143C5×D4—42no—
144C5×D4—81no—
145C5×D4—81no—
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×C5—42no—
157C2×C4×C5—42no—
158C2×C4×C5—42no—
159C2×C4×C5C2×D411yescharacteristic
160C2×C4×C5C2×D411yescharacteristic
161C2×C4×C5C2×D411yescharacteristic
162C2×C4×C5C2×D411yescharacteristic
163C2×C4×C5C2411yescharacteristic
164C5×C8—21no—
165C5×C8—21no—
166C5×C8—21no—
167C5×C8—21no—
168C2×#41C22×C511yescharacteristic
169C22×Q8C22×C511yescharacteristic
170C2×SD16C22×C511yescharacteristic
171C2×SD16C22×C511yescharacteristic
172C2×D8C22×C511yescharacteristic
173C2×M16C22×C511yescharacteristic
174C22×C8C22×C511yescharacteristic
175C4⋊Q8C22×C511yescharacteristic
176#124—21no—
177#124—21no—
178#124—21no—
179#124C22×C511yescharacteristic
180C22⋊Q8—21no—
181C22⋊Q8C22×C511yescharacteristic
182#121—21no—
183#121—21no—
184#121C22×C511yescharacteristic
185#121C22×C511yescharacteristic
186#121C22×C511yescharacteristic
187C4×Q8C22×C511yescharacteristic
188C4×D4—21no—
189C4×D4C22×C511yescharacteristic
190C4×D4C22×C511yescharacteristic
191#117—21no—
192#117C22×C511yescharacteristic
193#117C22×C511yescharacteristic
194#107C22×C511yescharacteristic
195#106C22×C511yescharacteristic
196#106C22×C511yescharacteristic
197C4⋊C8C22×C511yescharacteristic
198#103C22×C511yescharacteristic
199#103C22×C511yescharacteristic
200#103C22×C511yescharacteristic
201#103C22×C511yescharacteristic
202#103C22×C511yescharacteristic
203#103C22×C511yescharacteristic
204#102C22×C511yescharacteristic
205#102C22×C511yescharacteristic
206#102C22×C511yescharacteristic
207#102C22×C511yescharacteristic
208C22⋊C8C22×C511yescharacteristic
209C22⋊C8C22×C511yescharacteristic
210#97C22×C511yescharacteristic
211C22×C5—21no—
212C22×C5—21no—
213C22×C5—21no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5—41no—
223C22×C5C22×D411yescharacteristic
224C4×C5—21no—
225C4×C5—21no—
226C4×C5—22no—
227C4×C5—21no—
228C4×C5—21no—
229C4×C5—41no—
230C4×C5—41no—
231C4×C5—41no—
232C4×C5—41no—
233C4×C5—41no—
234C4×C5C2×#4111yescharacteristic
235C4×C5H(5,𝔽2)11yescharacteristic
236#41—22no—
237#41—41no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8C23×C511yescharacteristic
243C2×Q8C23×C511yescharacteristic
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C5×D411yescharacteristic
249C2×D4C23×C511yescharacteristic
250C2×D4C23×C511yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C23×C511yescharacteristic
255C22×C4C23×C511yescharacteristic
256SD16—22no—
257SD16—41no—
258D8—41no—
259M16—41no—
260C2×C8—41no—
261C2×C8C23×C511yescharacteristic
262C2×C8C23×C511yescharacteristic
263C2×C8C23×C511yescharacteristic
264C2×C8C23×C511yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4C5×D411yescharacteristic
269C4⋊C4C23×C511yescharacteristic
270C4⋊C4C23×C511yescharacteristic
271C4⋊C4C23×C511yescharacteristic
272C4⋊C4C23×C511yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4C5×D411yescharacteristic
283C22⋊C4C5×D411yescharacteristic
284C42—21no—
285C42—21no—
286C42—21no—
287C42C23×C511yescharacteristic
288C2×C5—41no—
289C2×C5—41no—
290C2×C5—81no—
291C2×C5—81no—
292C2×C5#54611yescharacteristic
293C2×C5#57511yescharacteristic
294C2×C5#57711yescharacteristic
295C23—21no—
296C23—21no—
297C23C2×C5×D411yescharacteristic
298C23C2×C5×D411yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—81no—
304Q8C5×#4112yes—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—41no—
310D4—42no—
311D4—42no—
312D4—81no—
313D4—81no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4C2×C5×D411yescharacteristic
328C2×C4C2×C5×D411yescharacteristic
329C2×C4C2×C5×D411yescharacteristic
330C2×C4C2×C5×D411yescharacteristic
331C2×C4C24×C511yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—21no—
336C5#332111yescharacteristic, direct product factor, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22C22×C5×D411yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4C2×C5×#4111yescharacteristic
361C4C5×H(5,𝔽2)11yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—81no—
366C2C5×#54611yescharacteristic
367C2C5×#57511yescharacteristic
368C2C5×#57711yescharacteristic
subgroupG/H
center, socleC22×C5C22×D4
commutator, Frattini, cosocleC2×C4C24×C5
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×#555C2semidirect
2. C5×#548C2semidirect
3. C5×#489C2semidirect
4. C5×#485C2semidirect
5. C5×#482C2semidirect
6. C5×#475C2semidirect
7. C5×#469C2semidirect
8. C5×#465C2semidirect
9. C5×#463C2semidirect
10. C5×#449C2semidirect
11. C5×#448C2semidirect
12. C5×#440C2semidirect
13. C5×#435C2semidirect
14. C5×#419C2semidirect
15. C5×#416C2semidirect
59. #3321C5direct (N × H)
112. #555C2×C5semidirect
113. #548C2×C5semidirect
114. #489C2×C5semidirect
115. #485C2×C5semidirect
116. #482C2×C5semidirect
117. #475C2×C5semidirect
118. #469C2×C5semidirect
119. #465C2×C5semidirect
120. #463C2×C5semidirect
121. #449C2×C5semidirect
122. #448C2×C5semidirect
123. #440C2×C5semidirect
124. #435C2×C5semidirect
125. #419C2×C5semidirect
126. #416C2×C5semidirect
336. C5#3321direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#5552
C5×#5482
C5×#4892
C5×#4852
C5×#4822
C5×#4752
C5×#4692
C5×#4652
C5×#4632
C5×#4492
C5×#4482
C5×#4402
C5×#4352
C5×#4192
C5×#4162
#33215