Finite Group Databasebeta

← C5×#7817 #11930537

Subgroups — C5×#7817 #11930537

Order 1280.

410 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2734C211yesmaximal, characteristic
2C5×#2096C211yesmaximal, characteristic
3C5×#2084C212yesmaximal
4C5×#1272C211yesmaximal, characteristic
5C5×#1271C212yesmaximal
6C2×C5×C4⋊Q8—21no—
7C2×C5×#106—22no—
8C2×C5×#106—21no—
9C2×C5×#103—21no—
10C5×#398—21no—
11C5×#395—22no—
12C5×#391C2211yescharacteristic
13C5×#389C2212yes—
14C5×#387—22no—
15C5×#384C2211yescharacteristic
16C5×#340—22no—
17C5×#336C2212yes—
18C5×#336C2211yescharacteristic
19#7817C511yesmaximal, characteristic, direct product factor, Sylow
20C22×C5×Q8—21no—
21C22×C5×Q8D411yescharacteristic
22C22×C5×C8—22no—
23C22×C5×C8—21no—
24C5×C4⋊Q8—42no—
25C5×C4⋊Q8—42no—
26C2×C5×C4⋊C4—22no—
27C2×C5×C4⋊C4—22no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4D412yes—
32C2×C5×C4⋊C4D412yes—
33C2×C5×C4⋊C4D411yescharacteristic
34C2×C42×C5—21no—
35C2×C42×C5C2311yescharacteristic
36C5×#106—42no—
37C5×#106—44no—
38C5×#103—42no—
39C5×#95—21no—
40C5×#95—22no—
41C5×#95—22no—
42C5×#95—22no—
43#2734C2×C511yescharacteristic
44#2096C2×C511yescharacteristic
45#2084C2×C512yes—
46#1272C2×C511yescharacteristic
47#1271C2×C512yes—
48C2×C5×Q8—21no—
49C2×C5×Q8—42no—
50C2×C5×Q8—41no—
51C2×C5×Q8—41no—
52C2×C5×Q8—81no—
53C2×C5×Q8—81no—
54C2×C5×Q8SD1612yes—
55C22×C4×C5—22no—
56C22×C4×C5—21no—
57C22×C4×C5—22no—
58C22×C4×C5—21no—
59C22×C4×C5—41no—
60C22×C4×C5C2×D411yescharacteristic
61C22×C4×C5C2×D412yes—
62C2×C5×C8—24no—
63C2×C5×C8—22no—
64C2×C5×C8—42no—
65C2×C5×C8—41no—
66C2×C5×C8—42no—
67C2×C5×C8—41no—
68C5×C4⋊C4—22no—
69C5×C4⋊C4—42no—
70C5×C4⋊C4—41no—
71C5×C4⋊C4—44no—
72C5×C4⋊C4—41no—
73C5×C4⋊C4—41no—
74C5×C4⋊C4—42no—
75C5×C4⋊C4—42no—
76C5×C4⋊C4—42no—
77C5×C4⋊C4—42no—
78C5×C4⋊C4—81no—
79C5×C4⋊C4SD1614yes—
80C42×C5—41no—
81C42×C5—42no—
82C2×C4⋊Q8—21no—
83C2×#106—22no—
84C2×#106—21no—
85C2×#103—21no—
86#398—21no—
87#395—22no—
88#391C22×C511yescharacteristic
89#389C22×C512yes—
90#387—22no—
91#384C22×C511yescharacteristic
92#340—22no—
93#336C22×C512yes—
94#336C22×C511yescharacteristic
95C23×C5#12011yescharacteristic
96C5×Q8—42no—
97C5×Q8—81no—
98C5×Q8—84no—
99C2×C4×C5—21no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—21no—
103C2×C4×C5—24no—
104C2×C4×C5—22no—
105C2×C4×C5—42no—
106C2×C4×C5—41no—
107C2×C4×C5—42no—
108C2×C4×C5—41no—
109C2×C4×C5—42no—
110C2×C4×C5—41no—
111C2×C4×C5—42no—
112C2×C4×C5—42no—
113C2×C4×C5—42no—
114C2×C4×C5—82no—
115C2×C4×C5—81no—
116C2×C4×C5C2×SD1612yes—
117C2×C4×C5C2×SD1611yescharacteristic
118C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C5#13712yes—
120C5×C8—44no—
121C5×C8—42no—
122C22×Q8—21no—
123C22×Q8C5×D411yescharacteristic
124C22×C8—22no—
125C22×C8—21no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4C5×D412yes—
134C2×C4⋊C4C5×D412yes—
135C2×C4⋊C4C5×D411yescharacteristic
136C2×C42—21no—
137C2×C42C23×C511yescharacteristic
138#106—42no—
139#106—44no—
140#103—42no—
141#95—21no—
142#95—22no—
143#95—22no—
144#95—22no—
145C22×C5#44812yes—
146C22×C5#45011yescharacteristic
147C22×C5#45311yescharacteristic
148C22×C5#45612yes—
149C22×C5#45811yescharacteristic
150C4×C5—22no—
151C4×C5—22no—
152C4×C5—21no—
153C4×C5—21no—
154C4×C5—42no—
155C4×C5—44no—
156C4×C5—82no—
157C4×C5—81no—
158C4×C5—161no—
159C2×Q8—21no—
160C2×Q8—42no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—81no—
164C2×Q8—81no—
165C2×Q8C5×SD1612yes—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4C2×C5×D411yescharacteristic
172C22×C4C2×C5×D412yes—
173C2×C8—22no—
174C2×C8—24no—
175C2×C8—42no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—41no—
179C4⋊C4—22no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—81no—
190C4⋊C4C5×SD1614yes—
191C42—41no—
192C42—42no—
193C2×C5#161112yes—
194C2×C5#161311yescharacteristic
195C2×C5#162311yescharacteristic
196C2×C5#165112yes—
197C2×C5#166011yescharacteristic
198C23C5×#12011yescharacteristic
199Q8—42no—
200Q8—81no—
201Q8—84no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—81no—
219C2×C4C2×C5×SD1612yes—
220C2×C4C2×C5×SD1611yescharacteristic
221C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
222C2×C4C5×#13712yes—
223C8—44no—
224C8—42no—
225C5#781711yescharacteristic, direct product factor, Sylow
226C22C5×#44812yes—
227C22C5×#45011yescharacteristic
228C22C5×#45311yescharacteristic
229C22C5×#45612yes—
230C22C5×#45811yescharacteristic
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4—81no—
239C4—161no—
240C2C5×#161112yes—
241C2C5×#161311yescharacteristic
242C2C5×#162311yescharacteristic
243C2C5×#165112yes—
244C2C5×#166011yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C5#120
commutator, Frattini, cosocleC2×C42C23×C5
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
19. #7817C5direct (N × H)
225. C5#7817direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#27342
C5×#20962
C5×#20842
C5×#12722
C5×#12712
#78175