Finite Group Databasebeta

← C5×#85417 #14511155

Subgroups — C5×#85417 #14511155

Order 1920.

368 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 68 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×#557⋊C3C211yesmaximal, characteristic
2C5×C4⋊#226C212yesmaximal
3C5×C4⋊#226C211yesmaximal, characteristic
4C5×#5388C212yesmaximal
5C4×C5×#226C211yesmaximal, characteristic
6C5×#3374—31nomaximal, Hall
7C2×C5×#226C2211yescharacteristic
8C2×C5×#226C2212yes—
9C5×#890—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2212yes—
12C5×#859C2211yescharacteristic
13#85417C511yesmaximal, characteristic, direct product factor, Hall
14C5×#557—31no—
15C5×#557D311yescharacteristic
16C5×#491—32no—
17C5×#487—31no—
18C5×#479—32no—
19C5×#475—31no—
20C5×#462—31no—
21C5×#462—32no—
22C5×#446—31no—
23C5×#441—32no—
24C4×C5×Q16—31no—
25C2×C5×SL(2,𝔽3)C2311yescharacteristic
26C5×#226—21no—
27C5×#226—22no—
28C3×C5×C4⋊C4—41no—
29C5×C4⋊Dic3—41no—
30C5×C4⋊Dic3—42no—
31C5×Dic3⋊C4—42no—
32C4×C5×Dic3—41no—
33#557⋊C3C2×C511yescharacteristic
34C4⋊#226C2×C511yescharacteristic
35C4⋊#226C2×C512yes—
36#5388C2×C512yes—
37C4×#226C2×C511yescharacteristic
38C2×C5×Q16—31no—
39C2×C5×Q16—32no—
40C5×C4⋊Q8—31no—
41C5×C4⋊Q8—31no—
42C5×C4⋊Q8—32no—
43C5×#125—32no—
44C5×#125—62no—
45C5×#125—61no—
46C4×C5×Q8—31no—
47C4×C5×Q8—31no—
48C4×C5×Q8—32no—
49C4×C5×Q8—61no—
50C4×C5×Q8C2×D311yescharacteristic
51C4×C5×Q8C2×D312yes—
52C5×#107—31no—
53C5×#106—32no—
54C5×C4⋊C8—31no—
55C5×C4⋊C8—32no—
56C5×#103—31no—
57C5×#103—32no—
58C5×#103—32no—
59C5×#103—32no—
60C5×#103—32no—
61C5×#103—31no—
62C5×#97—32no—
63C4×C5×C8—31no—
64#3374—31noSylow
65C2×C3×C4×C5—41no—
66C2×C3×C4×C5—42no—
67C2×C5×Dic3—41no—
68C2×C5×Dic3—42no—
69C2×C5×Dic3—41no—
70C5×SL(2,𝔽3)#4111yescharacteristic
71C2×#226C22×C511yescharacteristic
72C2×#226C22×C512yes—
73#890—41no—
74C4×SL(2,𝔽3)C22×C511yescharacteristic
75C4×SL(2,𝔽3)C22×C512yes—
76#859C22×C511yescharacteristic
77C2×C5×Q8—31no—
78C2×C5×Q8—31no—
79C2×C5×Q8—32no—
80C2×C5×Q8C22×D311yescharacteristic
81C5×Q16—61no—
82C5×Q16—62no—
83C5×Q16—61no—
84C5×Q16—62no—
85C2×C5×C8—31no—
86C2×C5×C8—32no—
87C2×C5×C8—31no—
88C5×C4⋊C4—31no—
89C5×C4⋊C4—32no—
90C5×C4⋊C4—32no—
91C5×C4⋊C4—31no—
92C5×C4⋊C4—61no—
93C5×C4⋊C4—61no—
94C5×C4⋊C4—62no—
95C5×C4⋊C4—61no—
96C5×C4⋊C4—62no—
97C5×C4⋊C4—61no—
98C5×C4⋊C4—62no—
99C5×C4⋊C4S411yescharacteristic
100C42×C5—32no—
101C42×C5—31no—
102C42×C5—62no—
103C42×C5—61no—
104#557—31no—
105#557C5×D311yescharacteristic
106#491—32no—
107#487—31no—
108#479—32no—
109#475—31no—
110#462—32no—
111#462—31no—
112#446—31no—
113#441—32no—
114C4×Q16—31no—
115C22×C3×C5—41no—
116C3×C4×C5—42no—
117C3×C4×C5—82no—
118C5×Dic3—81no—
119C5×Dic3—82no—
120C5×Dic3—81no—
121C2×SL(2,𝔽3)C23×C511yescharacteristic
122#226—21no—
123#226—22no—
124C3×C4⋊C4—41no—
125C4⋊Dic3—41no—
126C4⋊Dic3—42no—
127Dic3⋊C4—42no—
128C4×Dic3—41no—
129C5×Q8—31no—
130C5×Q8—61no—
131C5×Q8—62no—
132C5×Q8—61no—
133C5×Q8—62no—
134C5×Q8—62no—
135C5×Q8#23911yescharacteristic
136C2×C4×C5—31no—
137C2×C4×C5—32no—
138C2×C4×C5—31no—
139C2×C4×C5—61no—
140C2×C4×C5—62no—
141C2×C4×C5—61no—
142C2×C4×C5C2×S412yes—
143C2×C4×C5C2×S411yescharacteristic
144C5×C8—61no—
145C5×C8—61no—
146C5×C8—62no—
147C2×Q16—31no—
148C2×Q16—32no—
149C4⋊Q8—31no—
150C4⋊Q8—31no—
151C4⋊Q8—32no—
152#125—32no—
153#125—62no—
154#125—61no—
155C4×Q8—31no—
156C4×Q8—31no—
157C4×Q8—32no—
158C4×Q8—61no—
159C4×Q8C2×C5×D311yescharacteristic
160C4×Q8C2×C5×D312yes—
161#107—31no—
162#106—32no—
163C4⋊C8—31no—
164C4⋊C8—32no—
165#103—31no—
166#103—32no—
167#103—32no—
168#103—32no—
169#103—32no—
170#103—31no—
171#97—32no—
172C4×C8—31no—
173C2×C3×C5—41no—
174C2×C3×C5—41no—
175C2×C3×C5—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×Dic3—41no—
179C2×Dic3—41no—
180C2×Dic3—42no—
181SL(2,𝔽3)C5×#4111yescharacteristic
182C22×C5C22×S411yescharacteristic
183C4×C5—22no—
184C4×C5—31no—
185C4×C5—31no—
186C4×C5—62no—
187C4×C5—61no—
188C4×C5—121no—
189C4×C5—122no—
190C4×C5—121no—
191C4×C5#98412yes—
192C2×Q8—31no—
193C2×Q8—31no—
194C2×Q8—32no—
195C2×Q8C22×C5×D311yescharacteristic
196Q16—61no—
197Q16—62no—
198Q16—61no—
199Q16—62no—
200C2×C8—31no—
201C2×C8—31no—
202C2×C8—32no—
203C4⋊C4—32no—
204C4⋊C4—31no—
205C4⋊C4—32no—
206C4⋊C4—31no—
207C4⋊C4—61no—
208C4⋊C4—61no—
209C4⋊C4—62no—
210C4⋊C4—61no—
211C4⋊C4—62no—
212C4⋊C4—61no—
213C4⋊C4—62no—
214C4⋊C4C5×S411yescharacteristic
215C42—31no—
216C42—32no—
217C42—62no—
218C42—61no—
219C3×C5—41noHall
220C22×C3—41no—
221C3×C4—42no—
222C3×C4—82no—
223Dic3—81no—
224Dic3—81no—
225Dic3—82no—
226C2×C5#591911yescharacteristic
227C2×C5C2×#98411yescharacteristic
228C2×C5#592711yescharacteristic
229Q8—31no—
230Q8—61no—
231Q8—62no—
232Q8—61no—
233Q8—62no—
234Q8—62no—
235Q8C5×#23911yescharacteristic
236C2×C4—31no—
237C2×C4—32no—
238C2×C4—31no—
239C2×C4—61no—
240C2×C4—62no—
241C2×C4—61no—
242C2×C4C2×C5×S412yes—
243C2×C4C2×C5×S411yescharacteristic
244C8—61no—
245C8—62no—
246C8—61no—
247C2×C3—41no—
248C2×C3—41no—
249C2×C3—41no—
250C5#8541711yescharacteristic, direct product factor, Sylow
251C22C22×C5×S411yescharacteristic
252C4—22no—
253C4—31no—
254C4—31no—
255C4—62no—
256C4—61no—
257C4—122no—
258C4—121no—
259C4—121no—
260C4C5×#98412yes—
261C3—41noSylow
262C2C5×#591911yescharacteristic
263C2C2×C5×#98411yescharacteristic
264C2C5×#592711yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×#557D3
prime radicalC2×C5×SL(2,𝔽3)C23
hypercenterC5×C4⋊C4S4
hypocenterSL(2,𝔽3)C5×#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
13. #85417C5direct (N × H)
250. C5#85417direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#557⋊C32
C5×C4⋊#2262
C5×#53882
C4×C5×#2262
C5×#33743
C5×#8904
#854175