Finite Group Databasebeta

← C5×#85414 #14511152

Subgroups — C5×#85414 #14511152

Order 1920.

378 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 72 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×#5389C212yesmaximal
3C4×C5×#226C212yesmaximal
4C4×C5×#226C211yesmaximal, characteristic
5C5×#5386C211yesmaximal, characteristic
6C5×#3383—31nomaximal, Hall
7C2×C5×#226C2211yescharacteristic
8C5×#887—41nomaximal
9C4×C5×SL(2,𝔽3)C2211yescharacteristic
10C4×C5×SL(2,𝔽3)C2212yes—
11C5×#859C2212yes—
12C5×#859C2211yescharacteristic
13#85414C511yesmaximal, characteristic, direct product factor, Hall
14C5×#557—31no—
15C5×#557D311yescharacteristic
16C5×#500—31no—
17C5×#499—32no—
18C5×#479—32no—
19C5×#479—32no—
20C5×#475—31no—
21C5×#475—31no—
22C5×C8×Q8—31no—
23C4×C5×Q16—31no—
24C4×C5×Q16—32no—
25C2×C5×Dic6—41no—
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C5×#226Q812yescomplemented
28C3×C5×C4⋊C4—41no—
29C5×Dic3⋊C4—42no—
30C4×C5×Dic3—42no—
31C4×C5×Dic3—41no—
32#557⋊C3C2×C511yescharacteristic
33#5389C2×C512yes—
34C4×#226C2×C512yes—
35C4×#226C2×C511yescharacteristic
36#5386C2×C511yescharacteristic
37C2×C5×Q16—31no—
38C5×C4⋊Q8—31no—
39C5×C4⋊Q8—31no—
40C5×C4⋊Q8—61no—
41C5×#125—32no—
42C5×#125—32no—
43C5×#125—62no—
44C4×C5×Q8—31no—
45C4×C5×Q8—31no—
46C4×C5×Q8—32no—
47C4×C5×Q8—61no—
48C4×C5×Q8C2×D311yescharacteristic
49C4×C5×Q8C2×D312yes—
50C5×#107—31no—
51C5×#107—32no—
52C5×#107—32no—
53C5×#106—32no—
54C5×#106—32no—
55C5×C4⋊C8—31no—
56C5×C4⋊C8—32no—
57C5×#103—32no—
58C5×#103—31no—
59C5×#103—32no—
60C5×#103—31no—
61C4×C5×C8—32no—
62C4×C5×C8—31no—
63#3383—31noSylow
64C2×C3×C4×C5—41no—
65C2×C3×C4×C5—42no—
66C2×C5×Dic3—41no—
67C2×C5×Dic3—42no—
68C2×C5×Dic3—41no—
69C5×Dic6—44no—
70C5×SL(2,𝔽3)C2×Q811yescharacteristic
71C2×#226C22×C511yescharacteristic
72#887—41no—
73C4×SL(2,𝔽3)C22×C511yescharacteristic
74C4×SL(2,𝔽3)C22×C512yes—
75#859C22×C512yes—
76#859C22×C511yescharacteristic
77C2×C5×Q8—31no—
78C2×C5×Q8—31no—
79C2×C5×Q8—61no—
80C2×C5×Q8C22×D311yescharacteristic
81C5×Q16—32no—
82C5×Q16—32no—
83C2×C5×C8—31no—
84C2×C5×C8—32no—
85C2×C5×C8—31no—
86C5×C4⋊C4—31no—
87C5×C4⋊C4—32no—
88C5×C4⋊C4—32no—
89C5×C4⋊C4—31no—
90C5×C4⋊C4—32no—
91C5×C4⋊C4—61no—
92C5×C4⋊C4—61no—
93C5×C4⋊C4—62no—
94C5×C4⋊C4—62no—
95C5×C4⋊C4—61no—
96C5×C4⋊C4—62no—
97C5×C4⋊C4S411yescharacteristic
98C42×C5—32no—
99C42×C5—31no—
100C42×C5—62no—
101C42×C5—61no—
102#557—31no—
103#557C5×D311yescharacteristic
104#500—31no—
105#499—32no—
106#479—32no—
107#479—32no—
108#475—31no—
109#475—31no—
110C8×Q8—31no—
111C4×Q16—32no—
112C4×Q16—31no—
113C22×C3×C5—41no—
114C3×C4×C5—42no—
115C3×C4×C5—82no—
116C5×Dic3—42no—
117C5×Dic3—44no—
118C5×Dic3—81no—
119C2×Dic6—41no—
120C2×SL(2,𝔽3)C23×C511yescharacteristic
121#226C5×Q812yescomplemented
122C3×C4⋊C4—41no—
123Dic3⋊C4—42no—
124C4×Dic3—42no—
125C4×Dic3—41no—
126C5×Q8—31no—
127C5×Q8—32no—
128C5×Q8—61no—
129C5×Q8—62no—
130C5×Q8—122no—
131C5×Q8D3×Q811yescharacteristic
132C2×C4×C5—31no—
133C2×C4×C5—31no—
134C2×C4×C5—32no—
135C2×C4×C5—61no—
136C2×C4×C5—61no—
137C2×C4×C5—62no—
138C2×C4×C5C2×S412yes—
139C2×C4×C5C2×S411yescharacteristic
140C5×C8—32no—
141C5×C8—62no—
142C5×C8—61no—
143C2×Q16—31no—
144C4⋊Q8—31no—
145C4⋊Q8—31no—
146C4⋊Q8—61no—
147#125—32no—
148#125—32no—
149#125—62no—
150C4×Q8—31no—
151C4×Q8—31no—
152C4×Q8—32no—
153C4×Q8—61no—
154C4×Q8C2×C5×D311yescharacteristic
155C4×Q8C2×C5×D312yes—
156#107—31no—
157#107—32no—
158#107—32no—
159#106—32no—
160#106—32no—
161C4⋊C8—31no—
162C4⋊C8—32no—
163#103—32no—
164#103—31no—
165#103—32no—
166#103—31no—
167C4×C8—32no—
168C4×C8—31no—
169C2×C3×C5—41no—
170C2×C3×C5—41no—
171C2×C3×C5—41no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×Dic3—42no—
175C2×Dic3—41no—
176C2×Dic3—41no—
177Dic6—44no—
178SL(2,𝔽3)C2×C5×Q811yescharacteristic
179C22×C5C22×S411yescharacteristic
180C4×C5—22no—
181C4×C5—31no—
182C4×C5—31no—
183C4×C5—62no—
184C4×C5—62no—
185C4×C5—61no—
186C4×C5—122no—
187C4×C5—121no—
188C4×C5#98512yes—
189C2×Q8—31no—
190C2×Q8—31no—
191C2×Q8—61no—
192C2×Q8C22×C5×D311yescharacteristic
193Q16—32no—
194Q16—32no—
195C2×C8—31no—
196C2×C8—32no—
197C2×C8—31no—
198C4⋊C4—31no—
199C4⋊C4—32no—
200C4⋊C4—32no—
201C4⋊C4—31no—
202C4⋊C4—32no—
203C4⋊C4—61no—
204C4⋊C4—61no—
205C4⋊C4—62no—
206C4⋊C4—62no—
207C4⋊C4—61no—
208C4⋊C4—62no—
209C4⋊C4C5×S411yescharacteristic
210C42—32no—
211C42—31no—
212C42—62no—
213C42—61no—
214C3×C5—41noHall
215C22×C3—41no—
216C3×C4—42no—
217C3×C4—82no—
218Dic3—42no—
219Dic3—44no—
220Dic3—81no—
221C2×C5Q8×S411yescharacteristic
222C2×C5C2×#98511yescharacteristic
223C2×C5#592711yescharacteristic
224Q8—31no—
225Q8—32no—
226Q8—61no—
227Q8—62no—
228Q8—122no—
229Q8C5×D3×Q811yescharacteristic
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—32no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—62no—
236C2×C4C2×C5×S412yes—
237C2×C4C2×C5×S411yescharacteristic
238C8—32no—
239C8—62no—
240C8—61no—
241C2×C3—41no—
242C2×C3—41no—
243C2×C3—41no—
244C5#8541411yescharacteristic, direct product factor, Sylow
245C22C22×C5×S411yescharacteristic
246C4—22no—
247C4—31no—
248C4—31no—
249C4—62no—
250C4—62no—
251C4—61no—
252C4—122no—
253C4—121no—
254C4C5×#98512yes—
255C3—41noSylow
256C2C5×Q8×S411yescharacteristic
257C2C2×C5×#98511yescharacteristic
258C2C5×#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)C2×C5×Q8
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. #85414C5direct (N × H)
27. C5×#226Q8semidirect
121. #226C5×Q8semidirect
244. C5#85414direct (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×#53892
C4×C5×#2262
C5×#53862
C5×#33833
C5×#8874
#854145