Finite Group Databasebeta

← C5×#12234 #11934954

Subgroups — C5×#12234 #11934954

Order 1280.

448 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2680C211yesmaximal, characteristic
2C5×#2060C211yesmaximal, characteristic, complemented
3C5×#1605C211yesmaximal, characteristic, complemented
4C5×#1599C211yesmaximal, characteristic
5C5×#1354C211yesmaximal, characteristic, complemented
6C5×#1345C211yesmaximal, characteristic
7C5×#1325C211yesmaximal, characteristic, complemented
8C5×#544C2211yescharacteristic
9C2×C5×#123—21no—
10C2×C5×C22⋊Q8—21no—
11C5×#451—21no—
12C5×#450—21no—
13C5×#428—21no—
14C5×#398—21no—
15C5×#397—21no—
16C5×#395C2211yescharacteristic
17C5×#394C2211yescharacteristic
18C5×#388—21no—
19C5×#385—21no—
20C5×#384C2211yescharacteristic
21C5×#343C2211yescharacteristic
22C5×#339—21no—
23C5×#328—21no—
24C5×#328—21no—
25C5×#324C2211yescharacteristic
26C5×#324C2211yescharacteristic
27#12234C511yesmaximal, characteristic, direct product factor, Sylow
28C22×C5×Q8D411yescharacteristic
29C22×C5×D4D411yescharacteristic
30C23×C4×C5—21no—
31C2×C5×Q16—41no—
32C2×C5×SD16—41no—
33C2×C5×M16—21no—
34C5×C4⋊Q8—41no—
35C5×#125—41no—
36C5×#123—44no—
37C5×C22⋊Q8—21no—
38C5×C22⋊Q8—21no—
39C5×C22⋊Q8—44no—
40C5×#117—21no—
41C2×C5×C4⋊C4—21no—
42C2×C5×C4⋊C4—21no—
43C2×C5×C4⋊C4—21no—
44C2×C5×C4⋊C4—21no—
45C2×C5×C4⋊C4—21no—
46C2×C5×C4⋊C4D411yescharacteristic
47C2×C5×C4⋊C4C2311yescharacteristic
48C2×C5×C22⋊C4—21no—
49C2×C5×C22⋊C4—21no—
50C2×C5×C22⋊C4—21no—
51C2×C5×C22⋊C4—21no—
52C2×C5×C22⋊C4—21no—
53C2×C42×C5D411yescharacteristic
54C5×#107—41no—
55C5×#106—41no—
56C5×#103—41no—
57C5×#102—41no—
58C5×C22⋊C8—21no—
59C5×C22⋊C8—21no—
60C5×#95—21no—
61C5×#95—21no—
62C5×#95—21no—
63C5×#95D411yescharacteristic
64C5×#95D411yescharacteristic
65#2680C2×C511yescharacteristic
66#2060C2×C511yescharacteristic, complemented
67#1605C2×C511yescharacteristic, complemented
68#1599C2×C511yescharacteristic
69#1354C2×C511yescharacteristic, complemented
70#1345C2×C511yescharacteristic
71#1325C2×C511yescharacteristic, complemented
72C24×C5—21no—
73C2×C5×Q8—21no—
74C2×C5×Q8—21no—
75C2×C5×Q8—42no—
76C2×C5×Q8—41no—
77C2×C5×D4—41no—
78C2×C5×D4—42no—
79C22×C4×C5—21no—
80C22×C4×C5—21no—
81C22×C4×C5—21no—
82C22×C4×C5—21no—
83C22×C4×C5—21no—
84C22×C4×C5—42no—
85C22×C4×C5—42no—
86C22×C4×C5—81no—
87C22×C4×C5C2×D411yescharacteristic
88C22×C4×C5C2×D411yescharacteristic
89C22×C4×C5C2×D411yescharacteristic
90C5×Q16—82no—
91C5×SD16—82no—
92C5×M16—22no—
93C5×M16—41no—
94C2×C5×C8—41no—
95C2×C5×C8—41no—
96C2×C5×C8—41no—
97C5×C4⋊C4—21no—
98C5×C4⋊C4—21no—
99C5×C4⋊C4—21no—
100C5×C4⋊C4—21no—
101C5×C4⋊C4—41no—
102C5×C4⋊C4—41no—
103C5×C4⋊C4—42no—
104C5×C4⋊C4—42no—
105C5×C4⋊C4—42no—
106C5×C4⋊C4—42no—
107C5×C4⋊C4—42no—
108C5×C4⋊C4—41no—
109C5×C4⋊C4—41no—
110C5×C22⋊C4—24no—
111C5×C22⋊C4—42no—
112C5×C22⋊C4—42no—
113C5×C22⋊C4—42no—
114C5×C22⋊C4—41no—
115C5×C22⋊C4—41no—
116C5×C22⋊C4—41no—
117C42×C5—21no—
118C42×C5—21no—
119C42×C5—41no—
120#544C22×C511yescharacteristic
121C2×#123—21no—
122C2×C22⋊Q8—21no—
123#451—21no—
124#450—21no—
125#428—21no—
126#398—21no—
127#397—21no—
128#395C22×C511yescharacteristic
129#394C22×C511yescharacteristic
130#388—21no—
131#385—21no—
132#384C22×C511yescharacteristic
133#343C22×C511yescharacteristic
134#339—21no—
135#328—21no—
136#328—21no—
137#324C22×C511yescharacteristic
138#324C22×C511yescharacteristic
139C23×C5—22no—
140C23×C5—41no—
141C23×C5—41no—
142C23×C5—41no—
143C23×C5—41no—
144C23×C5—41no—
145C23×C5—41no—
146C23×C5#12011yescharacteristic
147C5×Q8—42no—
148C5×Q8—81no—
149C5×Q8—81no—
150C5×Q8—81no—
151C5×D4—42no—
152C5×D4—81no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—22no—
156C2×C4×C5—22no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5—41no—
164C2×C4×C5—41no—
165C2×C4×C5—41no—
166C2×C4×C5—41no—
167C2×C4×C5—41no—
168C2×C4×C5—41no—
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—84no—
177C2×C4×C5#13511yescharacteristic
178C2×C4×C5#13711yescharacteristic
179C5×C8—42no—
180C5×C8—81no—
181C5×C8—81no—
182C22×Q8C5×D411yescharacteristic
183C22×D4C5×D411yescharacteristic
184C23×C4—21no—
185C2×Q16—41no—
186C2×SD16—41no—
187C2×M16—21no—
188C4⋊Q8—41no—
189#125—41no—
190#123—44no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—44no—
194#117—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C5×D411yescharacteristic
201C2×C4⋊C4C23×C511yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42C5×D411yescharacteristic
208#107—41no—
209#106—41no—
210#103—41no—
211#102—41no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95C5×D411yescharacteristic
218#95C5×D411yescharacteristic
219C22×C5—21no—
220C22×C5—21no—
221C22×C5—42no—
222C22×C5—42no—
223C22×C5—42no—
224C22×C5—81no—
225C22×C5—81no—
226C22×C5—81no—
227C22×C5—81no—
228C22×C5#45211yescharacteristic
229C22×C5#45611yescharacteristic
230C22×C5B(4,𝔽2)11yescharacteristic
231C4×C5—21no—
232C4×C5—21no—
233C4×C5—42no—
234C4×C5—41no—
235C4×C5—41no—
236C4×C5—42no—
237C4×C5—81no—
238C4×C5—81no—
239C4×C5—81no—
240C4×C5—81no—
241C4×C5—81no—
242C24—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×D4—41no—
248C2×D4—42no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—81no—
257C22×C4C2×C5×D411yescharacteristic
258C22×C4C2×C5×D411yescharacteristic
259C22×C4C2×C5×D411yescharacteristic
260Q16—82no—
261SD16—82no—
262M16—22no—
263M16—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—24no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—21no—
288C42—21no—
289C42—41no—
290C2×C5—21no—
291C2×C5—21no—
292C2×C5—82no—
293C2×C5#160311yescharacteristic
294C2×C5#219611yescharacteristic
295C2×C5#220311yescharacteristic
296C23—22no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23C5×#12011yescharacteristic
304Q8—42no—
305Q8—81no—
306Q8—81no—
307Q8—81no—
308D4—42no—
309D4—81no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—84no—
334C2×C4C5×#13511yescharacteristic
335C2×C4C5×#13711yescharacteristic
336C8—42no—
337C8—81no—
338C8—81no—
339C5#1223411yescharacteristic, direct product factor, Sylow
340C22—21no—
341C22—21no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22C5×#45211yescharacteristic
350C22C5×#45611yescharacteristic
351C22C5×B(4,𝔽2)11yescharacteristic
352C4—21no—
353C4—21no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C2—21no—
364C2—21no—
365C2—82no—
366C2C5×#160311yescharacteristic
367C2C5×#219611yescharacteristic
368C2C5×#220311yescharacteristic
subgroupG/H
center, socleC22×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC22×C5×D4D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C5 ◁ C2×C5×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2060C2semidirect
3. C5×#1605C2semidirect
5. C5×#1354C2semidirect
7. C5×#1325C2semidirect
27. #12234C5direct (N × H)
66. #2060C2×C5semidirect
67. #1605C2×C5semidirect
69. #1354C2×C5semidirect
71. #1325C2×C5semidirect
339. C5#12234direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#26802
C5×#20602
C5×#16052
C5×#15992
C5×#13542
C5×#13452
C5×#13252
#122345