Finite Group Databasebeta

← C5×#12299 #11935019

Subgroups — C5×#12299 #11935019

Order 1280.

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

normal subgroups only

This group has 324 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×#3195C211yesmaximal, characteristic
2C5×#2239C212yesmaximal, complemented
3C5×#2091C211yesmaximal, characteristic
4C5×#1342C212yesmaximal, complemented
5C5×#1320C211yesmaximal, characteristic
6C2×C5×C22⋊Q8—21no—
7C5×#483—22no—
8C5×#470—21no—
9C5×#470C2212yes—
10C5×#451—22no—
11C5×#430—21no—
12C5×#430C2211yescharacteristic
13C5×#407—21no—
14C5×#407C2211yescharacteristic
15C5×#365—22no—
16C5×#362—22no—
17C5×#349C2212yes—
18C5×#338C2211yescharacteristic
19#12299C511yesmaximal, characteristic, direct product factor, Sylow
20C22×C5×Q8—21no—
21C23×C4×C5D411yescharacteristic
22C2×C5×Q16—21no—
23C2×C5×Q16D412yes—
24C2×C5×M16—21no—
25C2×C5×M16—21no—
26C2×C5×M16D411yescharacteristic
27C2×C5×M16C2311yescharacteristic
28C5×C22⋊Q8—22no—
29C5×C22⋊Q8—22no—
30C5×C22⋊Q8—41no—
31C5×C22⋊Q8—41no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—41no—
34C2×C5×C22⋊C4—41no—
35C5×M32—22no—
36C5×#108—22no—
37C5×#108—42no—
38C5×#106—21no—
39C5×#106D412yes—
40C5×#103—22no—
41C5×#103—22no—
42C5×C22⋊C8—22no—
43C5×C22⋊C8—42no—
44#3195C2×C511yescharacteristic
45#2239C2×C512yescomplemented
46#2091C2×C511yescharacteristic
47#1342C2×C512yescomplemented
48#1320C2×C511yescharacteristic
49C24×C5SD1611yescharacteristic
50C2×C5×Q8—22no—
51C2×C5×Q8—41no—
52C2×C5×Q8—81no—
53C2×C5×Q8—81no—
54C22×C4×C5—21no—
55C22×C4×C5—21no—
56C22×C4×C5—41no—
57C22×C4×C5—41no—
58C22×C4×C5—41no—
59C22×C4×C5—41no—
60C22×C4×C5SD1611yescharacteristic
61C22×C4×C5C2×D411yescharacteristic
62C5×Q16—42no—
63C5×Q16—81no—
64C5×M16—41no—
65C5×M16—41no—
66C5×M16—41no—
67C5×M16—41no—
68C5×M16—41no—
69C5×M16—41no—
70C2×C5×C8—22no—
71C2×C5×C8—21no—
72C2×C5×C8—41no—
73C2×C5×C8C2×D412yes—
74C5×C4⋊C4—22no—
75C5×C4⋊C4—41no—
76C5×C4⋊C4—42no—
77C5×C4⋊C4—42no—
78C5×C22⋊C4—42no—
79C5×C22⋊C4—42no—
80C5×C16—42no—
81C2×C22⋊Q8—21no—
82#483—22no—
83#470—21no—
84#470C22×C512yes—
85#451—22no—
86#430—21no—
87#430C22×C511yescharacteristic
88#407—21no—
89#407C22×C511yescharacteristic
90#365—22no—
91#362—22no—
92#349C22×C512yes—
93#338C22×C511yescharacteristic
94C23×C5—21no—
95C23×C5—41no—
96C23×C5—42no—
97C23×C5C2×SD1611yescharacteristic
98C5×Q8—42no—
99C5×Q8—42no—
100C5×Q8—82no—
101C2×C4×C5—21no—
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5—41no—
105C2×C4×C5—41no—
106C2×C4×C5—41no—
107C2×C4×C5—42no—
108C2×C4×C5—42no—
109C2×C4×C5—81no—
110C2×C4×C5—82no—
111C2×C4×C5—82no—
112C2×C4×C5#12011yescharacteristic
113C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
114C5×C8—22no—
115C5×C8—42no—
116C5×C8—41no—
117C5×C8—81no—
118C22×Q8—21no—
119C23×C4C5×D411yescharacteristic
120C2×Q16—21no—
121C2×Q16C5×D412yes—
122C2×M16—21no—
123C2×M16—21no—
124C2×M16C5×D411yescharacteristic
125C2×M16C23×C511yescharacteristic
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—41no—
132C2×C22⋊C4—41no—
133M32—22no—
134#108—22no—
135#108—42no—
136#106—21no—
137#106C5×D412yes—
138#103—22no—
139#103—22no—
140C22⋊C8—22no—
141C22⋊C8—42no—
142C22×C5—22no—
143C22×C5—21no—
144C22×C5—41no—
145C22×C5—41no—
146C22×C5—42no—
147C22×C5—41no—
148C22×C5—81no—
149C22×C5#45011yescharacteristic
150C4×C5—21no—
151C4×C5—41no—
152C4×C5—82no—
153C4×C5—82no—
154C4×C5#45311yescharacteristic
155C4×C5#45811yescharacteristic
156C24C5×SD1611yescharacteristic
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—81no—
160C2×Q8—81no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4C5×SD1611yescharacteristic
168C22×C4C2×C5×D411yescharacteristic
169Q16—42no—
170Q16—81no—
171M16—41no—
172M16—41no—
173M16—41no—
174M16—41no—
175M16—41no—
176M16—41no—
177C2×C8—22no—
178C2×C8—21no—
179C2×C8—41no—
180C2×C8C2×C5×D412yes—
181C4⋊C4—22no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C16—42no—
188C2×C5—21no—
189C2×C5—42no—
190C2×C5—41no—
191C2×C5#161311yescharacteristic
192C23—21no—
193C23—41no—
194C23—42no—
195C23C2×C5×SD1611yescharacteristic
196Q8—42no—
197Q8—42no—
198Q8—82no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4C5×#12011yescharacteristic
211C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
212C8—22no—
213C8—42no—
214C8—41no—
215C8—81no—
216C5#1229911yescharacteristic, direct product factor, Sylow
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22C5×#45011yescharacteristic
225C4—21no—
226C4—41no—
227C4—82no—
228C4—82no—
229C4C5×#45311yescharacteristic
230C4C5×#45811yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2C5×#161311yescharacteristic
subgroupG/H
center, socleC2×C5#1613
commutator, Frattini, cosocleC2×M16C23×C5
prime radicalC24×C5SD16
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×M16 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2239C2semidirect
4. C5×#1342C2semidirect
19. #12299C5direct (N × H)
45. #2239C2×C5semidirect
47. #1342C2×C5semidirect
216. C5#12299direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#31952
C5×#22392
C5×#20912
C5×#13422
C5×#13202
#122995