Finite Group Databasebeta

← C5×#12124 #11934844

Subgroups — C5×#12124 #11934844

Order 1280.

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

normal subgroups only

This group has 542 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×#2842C211yesmaximal, characteristic, complemented
2C5×#2021C211yesmaximal, characteristic
3C5×#1613C212yesmaximal, complemented
4C5×#1352C212yesmaximal, complemented
5C5×#1316C211yesmaximal, characteristic, complemented
6C5×#534C2211yescharacteristic
7C2×C5×#120—21no—
8C5×#481—22no—
9C2×C5×#99—21no—
10C5×#399—21no—
11C5×#399C2212yes—
12C5×#396—22no—
13C5×#396—22no—
14C5×#379—21no—
15C5×#379C2211yescharacteristic
16C5×#342C2211yescharacteristic
17C5×#327—22no—
18C5×#323C2212yes—
19#12124C511yesmaximal, characteristic, direct product factor, Sylow
20C25×C5D411yescharacteristic
21C22×C5×D4—21no—
22C22×C5×D4—41no—
23C5×#123—41no—
24C5×#121—22no—
25C5×#120—42no—
26C5×#120—41no—
27C5×#120—42no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4D412yescomplemented
32C2×C5×C22⋊C4—21no—
33C2×C5×C22⋊C4—21no—
34C2×C5×C22⋊C4—22no—
35C2×C5×C22⋊C4—21no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4—41no—
39C2×C5×C22⋊C4D411yescharacteristic
40C2×C5×C22⋊C4C2311yescharacteristic
41C5×#106—42no—
42C5×#102—42no—
43C5×#99—42no—
44C5×C22⋊C8—22no—
45C5×#95—21no—
46C5×#95D412yescomplemented
47#2842C2×C511yescharacteristic, complemented
48#2021C2×C511yescharacteristic
49#1613C2×C512yescomplemented
50#1352C2×C512yescomplemented
51#1316C2×C511yescharacteristic, complemented
52C24×C5—21no—
53C24×C5—21no—
54C24×C5—21no—
55C24×C5—41no—
56C24×C5—41no—
57C24×C5—82no—
58C24×C5SD1612yes—
59C24×C5C2×D411yescharacteristic
60C2×C5×D4—22no—
61C2×C5×D4—41no—
62C2×C5×D4—41no—
63C2×C5×D4—41no—
64C2×C5×D4—44no—
65C2×C5×D4—41no—
66C2×C5×D4—41no—
67C2×C5×D4—41no—
68C2×C5×D4—84no—
69C22×C4×C5—21no—
70C22×C4×C5—22no—
71C22×C4×C5—22no—
72C22×C4×C5—21no—
73C22×C4×C5—41no—
74C22×C4×C5C2×D412yes—
75C2×C5×C8—42no—
76C5×C4⋊C4—22no—
77C5×C4⋊C4—42no—
78C5×C4⋊C4—82no—
79C5×C4⋊C4—82no—
80C5×C4⋊C4—81no—
81C5×C22⋊C4—22no—
82C5×C22⋊C4—41no—
83C5×C22⋊C4—42no—
84C5×C22⋊C4—44no—
85C5×C22⋊C4—41no—
86C5×C22⋊C4—41no—
87C5×C22⋊C4—82no—
88C5×C22⋊C4—82no—
89C5×C22⋊C4—82no—
90C5×C22⋊C4—81no—
91C5×C22⋊C4—81no—
92#534C22×C511yescharacteristic
93C2×#120—21no—
94#481—22no—
95C2×#99—21no—
96#399—21no—
97#399C22×C512yes—
98#396—22no—
99#396—22no—
100#379—21no—
101#379C22×C511yescharacteristic
102#342C22×C511yescharacteristic
103#327—22no—
104#323C22×C512yes—
105C23×C5—21no—
106C23×C5—21no—
107C23×C5—22no—
108C23×C5—42no—
109C23×C5—42no—
110C23×C5—42no—
111C23×C5—42no—
112C23×C5—41no—
113C23×C5—41no—
114C23×C5—44no—
115C23×C5—41no—
116C23×C5—41no—
117C23×C5—41no—
118C23×C5—81no—
119C23×C5—82no—
120C23×C5—82no—
121C23×C5—82no—
122C23×C5—84no—
123C23×C5—82no—
124C23×C5#12011yescharacteristic
125C23×C5C2×SD1611yescharacteristic
126C23×C5#13711yescharacteristic
127C5×D4—82no—
128C5×D4—81no—
129C5×D4—81no—
130C5×D4—88no—
131C5×D4—81no—
132C5×D4—81no—
133C2×C4×C5—22no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—41no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—42no—
141C2×C4×C5—42no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—42no—
145C2×C4×C5—41no—
146C2×C4×C5—42no—
147C2×C4×C5—82no—
148C5×C8—82no—
149C25C5×D411yescharacteristic
150C22×D4—21no—
151C22×D4—41no—
152#123—41no—
153#121—22no—
154#120—42no—
155#120—41no—
156#120—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C5×D412yescomplemented
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4C5×D411yescharacteristic
169C2×C22⋊C4C23×C511yescharacteristic
170#106—42no—
171#102—42no—
172#99—42no—
173C22⋊C8—22no—
174#95—21no—
175#95C5×D412yescomplemented
176C22×C5—22no—
177C22×C5—21no—
178C22×C5—21no—
179C22×C5—41no—
180C22×C5—41no—
181C22×C5—42no—
182C22×C5—42no—
183C22×C5—42no—
184C22×C5—44no—
185C22×C5—42no—
186C22×C5—81no—
187C22×C5—82no—
188C22×C5—82no—
189C22×C5—84no—
190C22×C5—82no—
191C22×C5—82no—
192C22×C5—81no—
193C22×C5—82no—
194C22×C5—161no—
195C22×C5#44811yescharacteristic
196C22×C5#45311yescharacteristic
197C22×C5B(4,𝔽2)11yescharacteristic
198C4×C5—42no—
199C4×C5—81no—
200C4×C5—82no—
201C4×C5—82no—
202C4×C5—81no—
203C4×C5—82no—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—41no—
208C24—41no—
209C24—82no—
210C24C5×SD1612yes—
211C24C2×C5×D411yescharacteristic
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—44no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—84no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4C2×C5×D412yes—
227C2×C8—42no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—81no—
233C22⋊C4—22no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—44no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—82no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C2×C5—21no—
245C2×C5—21no—
246C2×C5—41no—
247C2×C5—41no—
248C2×C5—82no—
249C2×C5—161no—
250C2×C5#159611yescharacteristic
251C2×C5#220012yes—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—41no—
261C23—44no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—81no—
266C23—82no—
267C23—82no—
268C23—82no—
269C23—84no—
270C23—82no—
271C23C5×#12011yescharacteristic
272C23C2×C5×SD1611yescharacteristic
273C23C5×#13711yescharacteristic
274D4—82no—
275D4—81no—
276D4—81no—
277D4—88no—
278D4—81no—
279D4—81no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—82no—
295C8—82no—
296C5#1212411yescharacteristic, direct product factor, Sylow
297C22—22no—
298C22—21no—
299C22—21no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—44no—
306C22—42no—
307C22—81no—
308C22—82no—
309C22—82no—
310C22—84no—
311C22—82no—
312C22—82no—
313C22—81no—
314C22—82no—
315C22—161no—
316C22C5×#44811yescharacteristic
317C22C5×#45311yescharacteristic
318C22C5×B(4,𝔽2)11yescharacteristic
319C4—42no—
320C4—81no—
321C4—82no—
322C4—82no—
323C4—81no—
324C4—82no—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—41no—
329C2—82no—
330C2—161no—
331C2C5×#159611yescharacteristic
332C2C5×#220012yes—
subgroupG/H
center, socleC22×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C5
prime radicalC5×#2021C2
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×#2842C2semidirect
3. C5×#1613C2semidirect
4. C5×#1352C2semidirect
5. C5×#1316C2semidirect
19. #12124C5direct (N × H)
31. C2×C5×C4⋊C4D4semidirect
46. C5×#95D4semidirect
47. #2842C2×C5semidirect
49. #1613C2×C5semidirect
50. #1352C2×C5semidirect
51. #1316C2×C5semidirect
160. C2×C4⋊C4C5×D4semidirect
175. #95C5×D4semidirect
296. C5#12124direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#28422
C5×#20212
C5×#16132
C5×#13522
C5×#13162
#121245