Finite Group Databasebeta

← C5×#12251 #11934971

Subgroups — C5×#12251 #11934971

Order 1280.

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

normal subgroups only

This group has 342 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×#2822C211yesmaximal, characteristic
2C5×#2061C211yesmaximal, characteristic
3C5×#1611C212yesmaximal
4C5×#1346C212yesmaximal
5C5×#1324C211yesmaximal, characteristic
6C5×#544C2211yescharacteristic
7C2×C5×#125—21no—
8C5×#483—22no—
9C5×#428—21no—
10C5×#400—22no—
11C5×#398—22no—
12C5×#398—21no—
13C5×#398C2212yes—
14C5×#382C2211yescharacteristic
15C5×#343C2211yescharacteristic
16C5×#339—21no—
17C5×#328—22no—
18C5×#324C2212yes—
19#12251C511yesmaximal, characteristic, direct product factor, Sylow
20C2×C5×M16—21no—
21C5×C4⋊Q8—41no—
22C5×#125—42no—
23C5×#125—41no—
24C5×#125—42no—
25C5×C22⋊Q8—22no—
26C5×#117—21no—
27C2×C5×C4⋊C4—21no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4D411yescharacteristic
33C2×C5×C4⋊C4D412yes—
34C2×C5×C4⋊C4C2311yescharacteristic
35C2×C42×C5D411yescharacteristic
36C5×#107—41no—
37C5×#106—41no—
38C5×#106—42no—
39C5×#103—42no—
40C5×C22⋊C8—22no—
41C5×#95—22no—
42C5×#95—21no—
43C5×#95—22no—
44C5×#95—22no—
45C5×#95D412yes—
46#2822C2×C511yescharacteristic
47#2061C2×C511yescharacteristic
48#1611C2×C512yes—
49#1346C2×C512yes—
50#1324C2×C511yescharacteristic
51C2×C5×Q8—22no—
52C22×C4×C5—21no—
53C22×C4×C5—22no—
54C22×C4×C5—22no—
55C22×C4×C5—21no—
56C22×C4×C5C2×D411yescharacteristic
57C22×C4×C5C2×D412yes—
58C5×M16—22no—
59C5×M16—41no—
60C2×C5×C8—41no—
61C2×C5×C8—42no—
62C5×C4⋊C4—21no—
63C5×C4⋊C4—21no—
64C5×C4⋊C4—22no—
65C5×C4⋊C4—41no—
66C5×C4⋊C4—42no—
67C5×C4⋊C4—41no—
68C5×C4⋊C4—42no—
69C5×C4⋊C4—44no—
70C5×C4⋊C4—42no—
71C5×C4⋊C4—42no—
72C5×C4⋊C4—42no—
73C5×C4⋊C4—42no—
74C5×C4⋊C4—81no—
75C5×C22⋊C4—41no—
76C42×C5—21no—
77C42×C5—41no—
78C42×C5SD1612yes—
79#544C22×C511yescharacteristic
80C2×#125—21no—
81#483—22no—
82#428—21no—
83#400—22no—
84#398—21no—
85#398—22no—
86#398C22×C512yes—
87#382C22×C511yescharacteristic
88#343C22×C511yescharacteristic
89#339—21no—
90#328—22no—
91#324C22×C512yes—
92C23×C5#12011yescharacteristic
93C5×Q8—82no—
94C2×C4×C5—21no—
95C2×C4×C5—21no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—41no—
99C2×C4×C5—41no—
100C2×C4×C5—42no—
101C2×C4×C5—42no—
102C2×C4×C5—42no—
103C2×C4×C5—42no—
104C2×C4×C5—42no—
105C2×C4×C5—42no—
106C2×C4×C5—42no—
107C2×C4×C5—41no—
108C2×C4×C5—41no—
109C2×C4×C5—42no—
110C2×C4×C5—41no—
111C2×C4×C5C2×SD1611yescharacteristic
112C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
113C5×C8—42no—
114C5×C8—82no—
115C2×M16—21no—
116C4⋊Q8—41no—
117#125—42no—
118#125—41no—
119#125—42no—
120C22⋊Q8—22no—
121#117—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C5×D411yescharacteristic
128C2×C4⋊C4C5×D412yes—
129C2×C4⋊C4C23×C511yescharacteristic
130C2×C42C5×D411yescharacteristic
131#107—41no—
132#106—41no—
133#106—42no—
134#103—42no—
135C22⋊C8—22no—
136#95—22no—
137#95—21no—
138#95—22no—
139#95—22no—
140#95C5×D412yes—
141C22×C5—21no—
142C22×C5—21no—
143C22×C5#45011yescharacteristic
144C22×C5#45611yescharacteristic
145C22×C5B(4,𝔽2)11yescharacteristic
146C4×C5—21no—
147C4×C5—21no—
148C4×C5—42no—
149C4×C5—42no—
150C4×C5—82no—
151C4×C5—82no—
152C4×C5—81no—
153C4×C5—81no—
154C4×C5—81no—
155C2×Q8—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4C2×C5×D411yescharacteristic
161C22×C4C2×C5×D412yes—
162M16—22no—
163M16—41no—
164C2×C8—41no—
165C2×C8—42no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—44no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C22⋊C4—41no—
180C42—21no—
181C42—41no—
182C42C5×SD1612yes—
183C2×C5—21no—
184C2×C5—21no—
185C2×C5#160411yescharacteristic
186C2×C5#220111yescharacteristic
187C2×C5#220311yescharacteristic
188C23C5×#12011yescharacteristic
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4C2×C5×SD1611yescharacteristic
208C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
209C8—42no—
210C8—82no—
211C5#1225111yescharacteristic, direct product factor, Sylow
212C22—21no—
213C22—21no—
214C22C5×#45011yescharacteristic
215C22C5×#45611yescharacteristic
216C22C5×B(4,𝔽2)11yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4—81no—
226C2—21no—
227C2—21no—
228C2C5×#160411yescharacteristic
229C2C5×#220111yescharacteristic
230C2C5×#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
19. #12251C5direct (N × H)
211. C5#12251direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#28222
C5×#20612
C5×#16112
C5×#13462
C5×#13242
#122515