Finite Group Databasebeta

← C5×#12259 #11934979

Subgroups — C5×#12259 #11934979

Order 1280.

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

normal subgroups only

This group has 332 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×#2843C211yesmaximal, characteristic
2C5×#2060C211yesmaximal, characteristic
3C5×#1617C211yesmaximal, characteristic
4C5×#1605C211yesmaximal, characteristic
5C5×#1354C211yesmaximal, characteristic
6C5×#1353C211yesmaximal, characteristic
7C5×#1325C211yesmaximal, characteristic
8C5×#544C2211yescharacteristic
9C5×#484—21no—
10C5×#451—21no—
11C5×#428—21no—
12C5×#400—21no—
13C5×#400C2211yescharacteristic
14C5×#398—21no—
15C5×#398—21no—
16C5×#398—21no—
17C5×#395—21no—
18C5×#395C2211yescharacteristic
19C5×#384C2211yescharacteristic
20C5×#383—21no—
21C5×#343C2211yescharacteristic
22C5×#339—21no—
23C5×#328—21no—
24C5×#328—21no—
25C5×#324C2211yescharacteristic
26C5×#324C2211yescharacteristic
27#12259C511yesmaximal, characteristic, direct product factor, Sylow
28C22×C5×Q8D411yescharacteristic
29C2×C5×Q16—41no—
30C2×C5×M16—21no—
31C5×C4⋊Q8—41no—
32C5×#125—41no—
33C5×C22⋊Q8—21no—
34C5×C22⋊Q8—21no—
35C5×#117—21no—
36C2×C5×C4⋊C4—21no—
37C2×C5×C4⋊C4—21no—
38C2×C5×C4⋊C4—21no—
39C2×C5×C4⋊C4—21no—
40C2×C5×C4⋊C4—21no—
41C2×C5×C4⋊C4—21no—
42C2×C5×C4⋊C4D411yescharacteristic
43C2×C5×C4⋊C4D411yescharacteristic
44C2×C5×C4⋊C4C2311yescharacteristic
45C2×C42×C5D411yescharacteristic
46C5×#107—41no—
47C5×#107—41no—
48C5×#106—41no—
49C5×#103—41no—
50C5×#103—41no—
51C5×C22⋊C8—21no—
52C5×C22⋊C8—21no—
53C5×#95—21no—
54C5×#95—21no—
55C5×#95—21no—
56C5×#95—21no—
57C5×#95—21no—
58C5×#95—21no—
59C5×#95—21no—
60C5×#95—21no—
61C5×#95D411yescharacteristic
62C5×#95D411yescharacteristic
63#2843C2×C511yescharacteristic
64#2060C2×C511yescharacteristic
65#1617C2×C511yescharacteristic
66#1605C2×C511yescharacteristic
67#1354C2×C511yescharacteristic
68#1353C2×C511yescharacteristic
69#1325C2×C511yescharacteristic
70C2×C5×Q8—21no—
71C2×C5×Q8—21no—
72C2×C5×Q8—41no—
73C2×C5×Q8—81no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5—21no—
79C22×C4×C5—21no—
80C22×C4×C5C2×D411yescharacteristic
81C22×C4×C5C2×D411yescharacteristic
82C22×C4×C5C2×D411yescharacteristic
83C5×Q16—82no—
84C5×M16—22no—
85C5×M16—41no—
86C2×C5×C8—41no—
87C2×C5×C8—41no—
88C2×C5×C8—41no—
89C5×C4⋊C4—21no—
90C5×C4⋊C4—21no—
91C5×C4⋊C4—21no—
92C5×C4⋊C4—21no—
93C5×C4⋊C4—41no—
94C5×C4⋊C4—41no—
95C5×C4⋊C4—41no—
96C5×C4⋊C4—41no—
97C5×C4⋊C4—41no—
98C5×C4⋊C4—81no—
99C5×C4⋊C4—81no—
100C5×C4⋊C4—81no—
101C5×C4⋊C4—81no—
102C5×C4⋊C4—81no—
103C5×C4⋊C4—81no—
104C5×C22⋊C4—41no—
105C42×C5—21no—
106C42×C5—41no—
107C42×C5Q1612yescomplemented
108#544C22×C511yescharacteristic
109#484—21no—
110#451—21no—
111#428—21no—
112#400—21no—
113#400C22×C511yescharacteristic
114#398—21no—
115#398—21no—
116#398—21no—
117#395—21no—
118#395C22×C511yescharacteristic
119#384C22×C511yescharacteristic
120#383—21no—
121#343C22×C511yescharacteristic
122#339—21no—
123#328—21no—
124#328—21no—
125#324C22×C511yescharacteristic
126#324C22×C511yescharacteristic
127C23×C5#12011yescharacteristic
128C5×Q8—42no—
129C5×Q8—81no—
130C5×Q8—81no—
131C5×Q8—81no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—22no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—41no—
138C2×C4×C5—41no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5—42no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—41no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5C2×Q1611yescharacteristic
158C2×C4×C5#13711yescharacteristic
159C5×C8—42no—
160C5×C8—81no—
161C5×C8—81no—
162C22×Q8C5×D411yescharacteristic
163C2×Q16—41no—
164C2×M16—21no—
165C4⋊Q8—41no—
166#125—41no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169#117—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C5×D411yescharacteristic
177C2×C4⋊C4C5×D411yescharacteristic
178C2×C4⋊C4C23×C511yescharacteristic
179C2×C42C5×D411yescharacteristic
180#107—41no—
181#107—41no—
182#106—41no—
183#103—41no—
184#103—41no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95C5×D411yescharacteristic
196#95C5×D411yescharacteristic
197C22×C5—21no—
198C22×C5—21no—
199C22×C5#45111yescharacteristic
200C22×C5#45611yescharacteristic
201C22×C5B(4,𝔽2)11yescharacteristic
202C4×C5—21no—
203C4×C5—21no—
204C4×C5—42no—
205C4×C5—41no—
206C4×C5—41no—
207C4×C5—81no—
208C4×C5—81no—
209C4×C5—81no—
210C4×C5—81no—
211C4×C5—81no—
212C4×C5—81no—
213C4×C5—81no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—41no—
217C2×Q8—81no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4C2×C5×D411yescharacteristic
225C22×C4C2×C5×D411yescharacteristic
226C22×C4C2×C5×D411yescharacteristic
227Q16—82no—
228M16—22no—
229M16—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C22⋊C4—41no—
249C42—21no—
250C42—41no—
251C42C5×Q1612yescomplemented
252C2×C5—21no—
253C2×C5—21no—
254C2×C5#160511yescharacteristic
255C2×C5#220011yescharacteristic
256C2×C5#220311yescharacteristic
257C23C5×#12011yescharacteristic
258Q8—42no—
259Q8—81no—
260Q8—81no—
261Q8—81no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4C2×C5×Q1611yescharacteristic
288C2×C4C5×#13711yescharacteristic
289C8—42no—
290C8—81no—
291C8—81no—
292C5#1225911yescharacteristic, direct product factor, Sylow
293C22—21no—
294C22—21no—
295C22C5×#45111yescharacteristic
296C22C5×#45611yescharacteristic
297C22C5×B(4,𝔽2)11yescharacteristic
298C4—21no—
299C4—21no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2C5×#160511yescharacteristic
313C2C5×#220011yescharacteristic
314C2C5×#220311yescharacteristic
subgroupG/H
center, socleC22×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC23×C5#120
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
27. #12259C5direct (N × H)
107. C42×C5Q16semidirect
251. C42C5×Q16semidirect
292. C5#12259direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#28432
C5×#20602
C5×#16172
C5×#16052
C5×#13542
C5×#13532
C5×#13252
#122595