Finite Group Databasebeta

← C5×#71217 #14293762

Subgroups — C5×#71217 #14293762

Order 1920.

352 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C3×C5×#487C211yesmaximal, characteristic
2C5×#5093C211yesmaximal, characteristic
3C5×#5092C211yesmaximal, characteristic
4C5×#5080C211yesmaximal, characteristic
5C5×#5080C211yesmaximal, characteristic
6C5×#4680C211yesmaximal, characteristic
7C5×#4454C211yesmaximal, characteristic
8C5×#1716—31nomaximal, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×C4⋊Q8C2211yescharacteristic
11C2×C5×#216—22no—
12C5×#869C2211yescharacteristic
13C5×#869C2211yescharacteristic
14C3×C5×#103—22no—
15C3×C4×C5×C8C2211yescharacteristic
16C5×#817—22no—
17C5×#810—22no—
18C5×#808—22no—
19C5×#808—22no—
20C4×C5×C3⋊C8C2211yescharacteristic
21C4×C5×C3⋊C8C2211yescharacteristic
22#71217C511yesmaximal, characteristic, direct product factor, Hall
23C5×#498—31no—
24C5×#498—31no—
25C5×#498—31no—
26C5×#494—31no—
27C5×#487—31no—
28C5×#487D311yescharacteristic
29C5×#322—31no—
30C2×C3×C5×Q8—21no—
31C2×C3×C5×Q8—21no—
32C2×C5×Dic6—21no—
33C2×C5×Dic6—21no—
34C2×C3×C5×C8D412yes—
35C3×C5×C4⋊C4—21no—
36C3×C5×C4⋊C4—21no—
37C3×C5×C4⋊C4—21no—
38C3×C5×C4⋊C4—21no—
39C3×C42×C5C2311yescharacteristic
40C5×#216—44no—
41C5×C4⋊Dic3—21no—
42C5×C4⋊Dic3—21no—
43C5×C4⋊Dic3—21no—
44C5×C4⋊Dic3—21no—
45C2×C5×C3⋊C8D412yes—
46C2×C5×C3⋊C8D412yes—
47C3×#487C2×C511yescharacteristic
48#5093C2×C511yescharacteristic
49#5092C2×C511yescharacteristic
50#5080C2×C511yescharacteristic
51#5080C2×C511yescharacteristic
52#4680C2×C511yescharacteristic
53#4454C2×C511yescharacteristic
54C2×C5×Q16—62no—
55C5×C4⋊Q8—31no—
56C5×C4⋊Q8—31no—
57C5×C4⋊Q8C2×D311yescharacteristic
58C5×C4⋊Q8C2×D311yescharacteristic
59C5×#106—62no—
60C5×#106—62no—
61C5×#106—62no—
62C5×#103—22no—
63C5×#103—62no—
64C4×C5×C8—31no—
65C4×C5×C8—31no—
66C4×C5×C8C2×D311yescharacteristic
67#1716—31noSylow
68C3×C5×Q8—42no—
69C3×C5×Q8—42no—
70C2×C3×C4×C5—41no—
71C2×C3×C4×C5—41no—
72C2×C3×C4×C5C2×D411yescharacteristic
73C2×C3×C4×C5C2×D411yescharacteristic
74C2×C3×C4×C5C2×D411yescharacteristic
75C2×C5×Dic3—41no—
76C2×C5×Dic3—41no—
77C5×Dic6—42no—
78C5×Dic6—42no—
79C3×C5×C8—22no—
80C5×C3⋊C8—22no—
81C5×C3⋊C8SD1614yes—
82C3×C4⋊Q8C22×C511yescharacteristic
83C3×C4⋊Q8C22×C511yescharacteristic
84C2×#216—22no—
85#869C22×C511yescharacteristic
86#869C22×C511yescharacteristic
87C3×#103—22no—
88C3×C4×C8C22×C511yescharacteristic
89#817—22no—
90#810—22no—
91#808—22no—
92#808—22no—
93C4×C3⋊C8C22×C511yescharacteristic
94C4×C3⋊C8C22×C511yescharacteristic
95C2×C5×Q8—21no—
96C2×C5×Q8—21no—
97C2×C5×Q8—61no—
98C2×C5×Q8—61no—
99C5×Q16—124no—
100C2×C5×C8—32no—
101C2×C5×C8—32no—
102C2×C5×C8#6712yes—
103C5×C4⋊C4—21no—
104C5×C4⋊C4—21no—
105C5×C4⋊C4—21no—
106C5×C4⋊C4—21no—
107C5×C4⋊C4—61no—
108C5×C4⋊C4—61no—
109C5×C4⋊C4—61no—
110C5×C4⋊C4—61no—
111C42×C5C22×D311yescharacteristic
112#498—31no—
113#498—31no—
114#498—31no—
115#494—31no—
116#487—31no—
117#487C5×D311yescharacteristic
118#322—31no—
119C22×C3×C5#12711yescharacteristic
120C3×C4×C5—81no—
121C3×C4×C5—81no—
122C3×C4×C5C2×SD1612yes—
123C3×C4×C5#13712yes—
124C3×C4×C5#13712yes—
125C5×Dic3—81no—
126C5×Dic3—81no—
127C2×C3×Q8—21no—
128C2×C3×Q8—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×C3×C8C5×D412yes—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C42C23×C511yescharacteristic
137#216—44no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C2×C3⋊C8C5×D412yes—
143C2×C3⋊C8C5×D412yes—
144C5×Q8—42no—
145C5×Q8—42no—
146C5×Q8—122no—
147C5×Q8—122no—
148C2×C4×C5—41no—
149C2×C4×C5—41no—
150C2×C4×C5—121no—
151C2×C4×C5—121no—
152C2×C4×C5D3×D411yescharacteristic
153C2×C4×C5D3×D411yescharacteristic
154C2×C4×C5C2×#6711yescharacteristic
155C5×C8—22no—
156C5×C8—34no—
157C5×C8—62no—
158C2×Q16—62no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8C2×C5×D311yescharacteristic
162C4⋊Q8C2×C5×D311yescharacteristic
163#106—62no—
164#106—62no—
165#106—62no—
166#103—22no—
167#103—62no—
168C4×C8—31no—
169C4×C8—31no—
170C4×C8C2×C5×D311yescharacteristic
171C2×C3×C5#49211yescharacteristic
172C2×C3×C5#49711yescharacteristic
173C2×C3×C5#49711yescharacteristic
174C3×Q8—42no—
175C3×Q8—42no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4C2×C5×D411yescharacteristic
179C2×C3×C4C2×C5×D411yescharacteristic
180C2×C3×C4C2×C5×D411yescharacteristic
181C2×Dic3—41no—
182C2×Dic3—41no—
183Dic6—42no—
184Dic6—42no—
185C3×C8—22no—
186C3⋊C8—22no—
187C3⋊C8C5×SD1614yes—
188C22×C5#94011yescharacteristic
189C4×C5—81no—
190C4×C5—81no—
191C4×C5—241no—
192C4×C5—241no—
193C4×C5D3×SD1612yes—
194C4×C5#91812yes—
195C4×C5#95112yes—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—61no—
199C2×Q8—61no—
200Q16—124no—
201C2×C8—32no—
202C2×C8—32no—
203C2×C8C5×#6712yes—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—61no—
209C4⋊C4—61no—
210C4⋊C4—61no—
211C4⋊C4—61no—
212C42C22×C5×D311yescharacteristic
213C3×C5#171611yescharacteristic, complemented, Hall
214C22×C3C5×#12711yescharacteristic
215C3×C4—81no—
216C3×C4—81no—
217C3×C4C2×C5×SD1612yes—
218C3×C4C5×#13712yes—
219C3×C4C5×#13712yes—
220Dic3—81no—
221Dic3—81no—
222C2×C5#506011yescharacteristic
223C2×C5#517511yescharacteristic
224C2×C5#519011yescharacteristic
225Q8—42no—
226Q8—42no—
227Q8—122no—
228Q8—122no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—121no—
232C2×C4—121no—
233C2×C4C5×D3×D411yescharacteristic
234C2×C4C5×D3×D411yescharacteristic
235C2×C4C2×C5×#6711yescharacteristic
236C8—22no—
237C8—34no—
238C8—62no—
239C2×C3C5×#49211yescharacteristic
240C2×C3C5×#49711yescharacteristic
241C2×C3C5×#49711yescharacteristic
242C5#7121711yescharacteristic, direct product factor, Sylow
243C22C5×#94011yescharacteristic
244C4—81no—
245C4—81no—
246C4—241no—
247C4—241no—
248C4C5×D3×SD1612yes—
249C4C5×#91812yes—
250C4C5×#95112yes—
251C3C5×#171611yescharacteristic, complemented, Sylow
252C2C5×#506011yescharacteristic
253C2C5×#517511yescharacteristic
254C2C5×#519011yescharacteristic
subgroupG/H
centerC22×C5#940
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×#487C2
socle, prime radicalC22×C3×C5#127
hypercenterC5×#487D3
hypocenterC3C5×#1716
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×#487

Complements

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

NH ≅ G/Ntype
22. #71217C5direct (N × H)
213. C3×C5#1716semidirect
242. C5#71217direct (N × H)
251. C3C5×#1716semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#4872
C5×#50932
C5×#50922
C5×#50802
C5×#46802
C5×#44542
C5×#17163
#712175