Finite Group Databasebeta

← C5×#12125 #11934845

Subgroups — C5×#12125 #11934845

Order 1280.

440 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2841C211yesmaximal, characteristic, complemented
2C5×#2022C211yesmaximal, characteristic
3C5×#1613C211yesmaximal, characteristic
4C5×#1601C211yesmaximal, characteristic
5C5×#1352C211yesmaximal, characteristic, complemented
6C5×#1351C211yesmaximal, characteristic, complemented
7C5×#1317C211yesmaximal, characteristic, complemented
8C5×#534C2211yescharacteristic
9C2×C5×#123—21no—
10C5×#481—21no—
11C5×#448—21no—
12C2×C5×#99—21no—
13C5×#400—21no—
14C5×#399C2211yescharacteristic
15C5×#398—21no—
16C5×#398—21no—
17C5×#396—21no—
18C5×#395—21no—
19C5×#393C2211yescharacteristic
20C5×#381C2211yescharacteristic
21C5×#380—21no—
22C5×#342C2211yescharacteristic
23C5×#327—21no—
24C5×#327—21no—
25C5×#323C2211yescharacteristic
26C5×#323C2211yescharacteristic
27#12125C511yesmaximal, characteristic, direct product factor, Sylow
28C22×C5×Q8D411yescharacteristic
29C22×C5×D4—21no—
30C23×C4×C5D411yescharacteristic
31C2×C5×SD16—41no—
32C5×#123—42no—
33C5×#123—41no—
34C5×#123—42no—
35C5×#121—21no—
36C5×#121—21no—
37C5×#120—41no—
38C2×C5×C4⋊C4—21no—
39C2×C5×C4⋊C4—21no—
40C2×C5×C4⋊C4—21no—
41C2×C5×C4⋊C4—21no—
42C2×C5×C4⋊C4—21no—
43C2×C5×C4⋊C4—21no—
44C2×C5×C4⋊C4—41no—
45C2×C5×C4⋊C4D411yescharacteristic
46C2×C5×C22⋊C4—21no—
47C2×C5×C22⋊C4—21no—
48C2×C5×C22⋊C4—21no—
49C2×C5×C22⋊C4—41no—
50C2×C5×C22⋊C4D411yescharacteristic
51C2×C5×C22⋊C4C2311yescharacteristic
52C5×#106—41no—
53C5×#103—41no—
54C5×#102—41no—
55C5×#99—42no—
56C5×C22⋊C8—21no—
57C5×C22⋊C8—21no—
58C5×#95—21no—
59C5×#95—21no—
60C5×#95—21no—
61C5×#95—21no—
62C5×#95—21no—
63C5×#95—21no—
64C5×#95D411yescharacteristic
65C5×#95D411yescharacteristic
66#2841C2×C511yescharacteristic, complemented
67#2022C2×C511yescharacteristic
68#1613C2×C511yescharacteristic
69#1601C2×C511yescharacteristic
70#1352C2×C511yescharacteristic, complemented
71#1351C2×C511yescharacteristic, complemented
72#1317C2×C511yescharacteristic, complemented
73C24×C5—21no—
74C24×C5C2×D411yescharacteristic
75C2×C5×Q8—41no—
76C2×C5×Q8—81no—
77C2×C5×D4—21no—
78C2×C5×D4—21no—
79C2×C5×D4—41no—
80C2×C5×D4—41no—
81C2×C5×D4—41no—
82C2×C5×D4—41no—
83C2×C5×D4—41no—
84C2×C5×D4—41no—
85C22×C4×C5—21no—
86C22×C4×C5—21no—
87C22×C4×C5—21no—
88C22×C4×C5—21no—
89C22×C4×C5—21no—
90C22×C4×C5—21no—
91C22×C4×C5—21no—
92C22×C4×C5—21no—
93C22×C4×C5—41no—
94C22×C4×C5—41no—
95C22×C4×C5—41no—
96C22×C4×C5SD1612yescomplemented
97C22×C4×C5C2×D411yescharacteristic
98C22×C4×C5C2×D411yescharacteristic
99C5×SD16—82no—
100C2×C5×C8—41no—
101C2×C5×C8—41no—
102C5×C4⋊C4—21no—
103C5×C4⋊C4—21no—
104C5×C4⋊C4—41no—
105C5×C4⋊C4—44no—
106C5×C4⋊C4—81no—
107C5×C4⋊C4—81no—
108C5×C4⋊C4—81no—
109C5×C4⋊C4—81no—
110C5×C4⋊C4—81no—
111C5×C4⋊C4—81no—
112C5×C22⋊C4—22no—
113C5×C22⋊C4—41no—
114C5×C22⋊C4—41no—
115C5×C22⋊C4—41no—
116C5×C22⋊C4—44no—
117C5×C22⋊C4—41no—
118C5×C22⋊C4—41no—
119C5×C22⋊C4—81no—
120C5×C22⋊C4—81no—
121#534C22×C511yescharacteristic
122C2×#123—21no—
123#481—21no—
124#448—21no—
125C2×#99—21no—
126#400—21no—
127#399C22×C511yescharacteristic
128#398—21no—
129#398—21no—
130#396—21no—
131#395—21no—
132#393C22×C511yescharacteristic
133#381C22×C511yescharacteristic
134#380—21no—
135#342C22×C511yescharacteristic
136#327—21no—
137#327—21no—
138#323C22×C511yescharacteristic
139#323C22×C511yescharacteristic
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—41no—
143C23×C5—41no—
144C23×C5—41no—
145C23×C5—41no—
146C23×C5—41no—
147C23×C5—81no—
148C23×C5—81no—
149C23×C5#12011yescharacteristic
150C23×C5C2×SD1611yescharacteristic
151C23×C5#13711yescharacteristic
152C5×Q8—42no—
153C5×Q8—81no—
154C5×D4—81no—
155C5×D4—81no—
156C5×D4—81no—
157C5×D4—81no—
158C5×D4—81no—
159C5×D4—81no—
160C2×C4×C5—22no—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—41no—
164C2×C4×C5—41no—
165C2×C4×C5—42no—
166C2×C4×C5—41no—
167C2×C4×C5—41no—
168C2×C4×C5—42no—
169C2×C4×C5—41no—
170C2×C4×C5—41no—
171C2×C4×C5—41no—
172C2×C4×C5—41no—
173C2×C4×C5—41no—
174C2×C4×C5—41no—
175C2×C4×C5—41no—
176C2×C4×C5—41no—
177C2×C4×C5—41no—
178C2×C4×C5—41no—
179C2×C4×C5—41no—
180C2×C4×C5—41no—
181C2×C4×C5—41no—
182C2×C4×C5—41no—
183C2×C4×C5—41no—
184C2×C4×C5—41no—
185C2×C4×C5—41no—
186C2×C4×C5—41no—
187C2×C4×C5—81no—
188C2×C4×C5—81no—
189C2×C4×C5—81no—
190C5×C8—81no—
191C5×C8—81no—
192C22×Q8C5×D411yescharacteristic
193C22×D4—21no—
194C23×C4C5×D411yescharacteristic
195C2×SD16—41no—
196#123—42no—
197#123—41no—
198#123—42no—
199#121—21no—
200#121—21no—
201#120—41no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C5×D411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4C5×D411yescharacteristic
215C2×C22⋊C4C23×C511yescharacteristic
216#106—41no—
217#103—41no—
218#102—41no—
219#99—42no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95C5×D411yescharacteristic
229#95C5×D411yescharacteristic
230C22×C5—21no—
231C22×C5—21no—
232C22×C5—21no—
233C22×C5—21no—
234C22×C5—41no—
235C22×C5—41no—
236C22×C5—41no—
237C22×C5—41no—
238C22×C5—81no—
239C22×C5—82no—
240C22×C5—81no—
241C22×C5—81no—
242C22×C5—81no—
243C22×C5—161no—
244C22×C5#44811yescharacteristic
245C22×C5#45311yescharacteristic
246C22×C5B(4,𝔽2)11yescharacteristic
247C4×C5—42no—
248C4×C5—41no—
249C4×C5—41no—
250C4×C5—81no—
251C4×C5—81no—
252C4×C5—81no—
253C4×C5—82no—
254C4×C5—81no—
255C4×C5—81no—
256C4×C5—81no—
257C24—21no—
258C24C2×C5×D411yescharacteristic
259C2×Q8—41no—
260C2×Q8—81no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4C5×SD1612yescomplemented
281C22×C4C2×C5×D411yescharacteristic
282C22×C4C2×C5×D411yescharacteristic
283SD16—82no—
284C2×C8—41no—
285C2×C8—41no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—44no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C22⋊C4—22no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—44no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C2×C5—21no—
306C2×C5—21no—
307C2×C5—41no—
308C2×C5—41no—
309C2×C5—161no—
310C2×C5#159611yescharacteristic
311C2×C5#220011yescharacteristic
312C2×C5#220111yescharacteristic
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—81no—
321C23—81no—
322C23C5×#12011yescharacteristic
323C23C2×C5×SD1611yescharacteristic
324C23C5×#13711yescharacteristic
325Q8—42no—
326Q8—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4—81no—
332D4—81no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C8—81no—
364C8—81no—
365C5#1212511yescharacteristic, direct product factor, Sylow
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—82no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—161no—
380C22C5×#44811yescharacteristic
381C22C5×#45311yescharacteristic
382C22C5×B(4,𝔽2)11yescharacteristic
383C4—42no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—82no—
390C4—81no—
391C4—81no—
392C4—81no—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—161no—
398C2C5×#159611yescharacteristic
399C2C5×#220011yescharacteristic
400C2C5×#220111yescharacteristic
subgroupG/H
center, socleC22×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C5
prime radicalC5×#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C5 ◁ C2×C5×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#2841C2semidirect
5. C5×#1352C2semidirect
6. C5×#1351C2semidirect
7. C5×#1317C2semidirect
27. #12125C5direct (N × H)
66. #2841C2×C5semidirect
70. #1352C2×C5semidirect
71. #1351C2×C5semidirect
72. #1317C2×C5semidirect
96. C22×C4×C5SD16semidirect
280. C22×C4C5×SD16semidirect
365. C5#12125direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#28412
C5×#20222
C5×#16132
C5×#16012
C5×#13522
C5×#13512
C5×#13172
#121255