Finite Group Databasebeta

← C5×#9966 #11932686

Subgroups — C5×#9966 #11932686

Order 1280.

1020 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×C5×#450C212yesmaximal
2C5×#2659C211yesmaximal, characteristic, complemented
3C2×C5×#328C212yesmaximal, complemented
4C5×#1438C211yesmaximal, characteristic
5C5×#1280C211yesmaximal, characteristic, complemented
6C23×C5×D4C2211yescharacteristic
7C22×C5×SD16—22no—
8C2×C5×C22⋊Q8C2212yes—
9C22×C5×C22⋊C4—22no—
10C5×#450—24no—
11C5×#450—24no—
12C2×C5×#102—22no—
13C2×C5×C22⋊C8C2212yes—
14C5×#398—22no—
15C5×#395—22no—
16C5×#380—22no—
17C2×C5×#95C2211yescharacteristic
18C2×C5×#95C2211yescharacteristic
19C5×#328—24no—
20#9966C511yesmaximal, characteristic, direct product factor, Sylow
21C25×C5—21no—
22C22×C5×Q8D412yes—
23C22×C5×D4—22no—
24C22×C5×D4—22no—
25C22×C5×D4—21no—
26C22×C5×D4—44no—
27C22×C5×D4D412yes—
28C23×C4×C5—21no—
29C23×C4×C5D412yes—
30C23×C4×C5C2311yescharacteristic
31C2×C5×SD16—28no—
32C2×C5×SD16—48no—
33C22×C5×C8—22no—
34C5×C22⋊Q8—22no—
35C5×C22⋊Q8—22no—
36C5×C22⋊Q8—22no—
37C5×C22⋊Q8—22no—
38C2×C5×C4⋊C4—22no—
39C2×C5×C4⋊C4—22no—
40C2×C5×C4⋊C4—22no—
41C2×C5×C4⋊C4D412yes—
42C2×C5×C22⋊C4—22no—
43C2×C5×C22⋊C4—44no—
44C2×C5×C22⋊C4—44no—
45C2×C5×C22⋊C4—44no—
46C5×#102—28no—
47C5×C22⋊C8—22no—
48C5×C22⋊C8D414yes—
49C5×#95—24no—
50C5×#95—22no—
51C5×#95—22no—
52C2×#450C2×C512yes—
53#2659C2×C511yescharacteristic, complemented
54C2×#328C2×C512yescomplemented
55#1438C2×C511yescharacteristic
56#1280C2×C511yescharacteristic, complemented
57C24×C5—24no—
58C24×C5—22no—
59C24×C5—42no—
60C24×C5—42no—
61C24×C5—42no—
62C24×C5—41no—
63C24×C5—41no—
64C24×C5—42no—
65C24×C5—42no—
66C24×C5C2×D411yescharacteristic
67C2×C5×Q8—22no—
68C2×C5×Q8—22no—
69C2×C5×Q8—44no—
70C2×C5×D4—28no—
71C2×C5×D4—24no—
72C2×C5×D4—44no—
73C2×C5×D4—44no—
74C2×C5×D4—48no—
75C2×C5×D4—44no—
76C2×C5×D4—42no—
77C22×C4×C5—22no—
78C22×C4×C5—22no—
79C22×C4×C5—22no—
80C22×C4×C5—22no—
81C22×C4×C5—22no—
82C22×C4×C5—42no—
83C22×C4×C5—42no—
84C22×C4×C5—42no—
85C22×C4×C5—42no—
86C22×C4×C5—41no—
87C22×C4×C5—41no—
88C22×C4×C5—42no—
89C22×C4×C5SD1614yescomplemented
90C22×C4×C5C2×D412yes—
91C22×C4×C5C2×D411yescharacteristic
92C22×C4×C5C2×D412yes—
93C22×C4×C5#4111yescharacteristic
94C5×SD16—416no—
95C2×C5×C8—24no—
96C2×C5×C8—44no—
97C5×C4⋊C4—22no—
98C5×C4⋊C4—22no—
99C5×C4⋊C4—42no—
100C5×C4⋊C4—42no—
101C5×C4⋊C4—82no—
102C5×C4⋊C4—82no—
103C5×C22⋊C4—416no—
104C5×C22⋊C4—42no—
105C5×C22⋊C4—42no—
106C23×D4C22×C511yescharacteristic
107C22×SD16—22no—
108C2×C22⋊Q8C22×C512yes—
109C22×C22⋊C4—22no—
110#450—24no—
111#450—24no—
112C2×#102—22no—
113C2×C22⋊C8C22×C512yes—
114#398—22no—
115#395—22no—
116#380—22no—
117C2×#95C22×C511yescharacteristic
118C2×#95C22×C511yescharacteristic
119#328—24no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—22no—
123C23×C5—24no—
124C23×C5—44no—
125C23×C5—44no—
126C23×C5—42no—
127C23×C5—44no—
128C23×C5—44no—
129C23×C5—44no—
130C23×C5—44no—
131C23×C5—42no—
132C23×C5—44no—
133C23×C5—44no—
134C23×C5—82no—
135C23×C5—82no—
136C23×C5—82no—
137C23×C5—82no—
138C23×C5—82no—
139C23×C5—82no—
140C23×C5—81no—
141C23×C5—81no—
142C23×C5—81no—
143C23×C5—81no—
144C23×C5#12011yescharacteristic
145C23×C5#12111yescharacteristic
146C23×C5#12711yescharacteristic
147C23×C5C2×SD1612yes—
148C23×C5#13712yes—
149C5×Q8—44no—
150C5×Q8—82no—
151C5×D4—28no—
152C5×D4—44no—
153C5×D4—48no—
154C2×C4×C5—24no—
155C2×C4×C5—24no—
156C2×C4×C5—41no—
157C2×C4×C5—42no—
158C2×C4×C5—41no—
159C2×C4×C5—42no—
160C2×C4×C5—42no—
161C2×C4×C5—44no—
162C2×C4×C5—44no—
163C2×C4×C5—42no—
164C2×C4×C5—42no—
165C2×C4×C5—42no—
166C2×C4×C5—42no—
167C2×C4×C5—42no—
168C2×C4×C5—44no—
169C2×C4×C5—82no—
170C2×C4×C5—82no—
171C2×C4×C5—81no—
172C2×C4×C5—81no—
173C2×C4×C5#12012yes—
174C2×C4×C5#12112yes—
175C5×C8—44no—
176C25—21no—
177C22×Q8C5×D412yes—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—21no—
181C22×D4—44no—
182C22×D4C5×D412yes—
183C23×C4—21no—
184C23×C4C5×D412yes—
185C23×C4C23×C511yescharacteristic
186C2×SD16—28no—
187C2×SD16—48no—
188C22×C8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C5×D412yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201#102—28no—
202C22⋊C8—22no—
203C22⋊C8C5×D414yes—
204#95—24no—
205#95—22no—
206#95—22no—
207C22×C5—22no—
208C22×C5—22no—
209C22×C5—22no—
210C22×C5—22no—
211C22×C5—21no—
212C22×C5—21no—
213C22×C5—21no—
214C22×C5—21no—
215C22×C5—48no—
216C22×C5—44no—
217C22×C5—84no—
218C22×C5—84no—
219C22×C5—84no—
220C22×C5—84no—
221C22×C5—84no—
222C22×C5—82no—
223C22×C5—82no—
224C22×C5#39211yescharacteristic
225C22×C5#44812yes—
226C22×C5#45311yescharacteristic
227C22×C5#45411yescharacteristic
228C22×C5B(4,𝔽2)12yes—
229C22×C5#46112yes—
230C22×C5#49211yescharacteristic
231C22×C5#49711yescharacteristic
232C4×C5—22no—
233C4×C5—22no—
234C4×C5—44no—
235C4×C5—82no—
236C4×C5—82no—
237C4×C5—82no—
238C24—24no—
239C24—22no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24C2×C5×D411yescharacteristic
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—44no—
251C2×D4—28no—
252C2×D4—24no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—44no—
257C2×D4—42no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4C5×SD1614yescomplemented
271C22×C4C2×C5×D412yes—
272C22×C4C2×C5×D411yescharacteristic
273C22×C4C2×C5×D412yes—
274C22×C4C5×#4111yescharacteristic
275SD16—416no—
276C2×C8—24no—
277C2×C8—44no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C22⋊C4—416no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C2×C5—21no—
288C2×C5—22no—
289C2×C5—21no—
290C2×C5—84no—
291C2×C5#159612yes—
292C2×C5#159912yes—
293C2×C5#200611yescharacteristic
294C2×C5#201311yescharacteristic
295C2×C5#202111yescharacteristic
296C23—21no—
297C23—21no—
298C23—22no—
299C23—24no—
300C23—44no—
301C23—44no—
302C23—42no—
303C23—44no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—42no—
308C23—44no—
309C23—44no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23C5×#12011yescharacteristic
321C23C5×#12111yescharacteristic
322C23C5×#12711yescharacteristic
323C23C2×C5×SD1612yes—
324C23C5×#13712yes—
325Q8—44no—
326Q8—82no—
327D4—28no—
328D4—44no—
329D4—48no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4C5×#12012yes—
350C2×C4C5×#12112yes—
351C8—44no—
352C5#996611yescharacteristic, direct product factor, Sylow
353C22—21no—
354C22—21no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—21no—
360C22—21no—
361C22—48no—
362C22—44no—
363C22—84no—
364C22—84no—
365C22—84no—
366C22—82no—
367C22—84no—
368C22—84no—
369C22—82no—
370C22C5×#39211yescharacteristic
371C22C5×#44812yes—
372C22C5×#45311yescharacteristic
373C22C5×#45411yescharacteristic
374C22C5×B(4,𝔽2)12yes—
375C22C5×#46112yes—
376C22C5×#49211yescharacteristic
377C22C5×#49711yescharacteristic
378C4—22no—
379C4—22no—
380C4—44no—
381C4—82no—
382C4—82no—
383C4—82no—
384C2—21no—
385C2—22no—
386C2—21no—
387C2—84no—
388C2C5×#159612yes—
389C2C5×#159912yes—
390C2C5×#200611yescharacteristic
391C2C5×#201311yescharacteristic
392C2C5×#202111yescharacteristic
subgroupG/H
center, socleC23×C5#120
commutator, Frattini, cosocleC23×C4C23×C5
prime radicalC23×C5×D4C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2659C2semidirect
3. C2×C5×#328C2semidirect
5. C5×#1280C2semidirect
20. #9966C5direct (N × H)
53. #2659C2×C5semidirect
54. C2×#328C2×C5semidirect
56. #1280C2×C5semidirect
89. C22×C4×C5SD16semidirect
270. C22×C4C5×SD16semidirect
352. C5#9966direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4502
C5×#26592
C2×C5×#3282
C5×#14382
C5×#12802
#99665