Finite Group Databasebeta

← C5×#85415 #14511153

Subgroups — C5×#85415 #14511153

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×#5391C211yesmaximal, characteristic
3C5×#5388C212yesmaximal
4C4×C5×#226C211yesmaximal, characteristic
5C5×#5386C212yesmaximal
6C5×#3387—31nomaximal, Hall
7C2×C5×#226C2211yescharacteristic
8C5×#888—41nomaximal
9C4×C5×SL(2,𝔽3)C2212yes—
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C5×#859C2211yescharacteristic
12C5×#859C2212yes—
13#85415C511yesmaximal, characteristic, direct product factor, Hall
14C5×Q82—31no—
15C5×#557D311yescharacteristic
16C5×#501—32no—
17C5×#498—31no—
18C5×#479—32no—
19C5×#477—31no—
20C5×#475—31no—
21C5×#475—32no—
22C5×#446—31no—
23C5×#441—32no—
24C4×C5×Q16—31no—
25C2×C5×Dic6—42no—
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C5×#226Q812yescomplemented
28C3×C5×C4⋊C4—41no—
29C5×C4⋊Dic3—41no—
30C5×Dic3⋊C4—42no—
31C4×C5×Dic3—41no—
32#557⋊C3C2×C511yescharacteristic
33#5391C2×C511yescharacteristic
34#5388C2×C512yes—
35C4×#226C2×C511yescharacteristic
36#5386C2×C512yes—
37C2×C5×Q16—31no—
38C5×C4⋊Q8—31no—
39C5×C4⋊Q8—32no—
40C5×C4⋊Q8—31no—
41C5×C4⋊Q8—62no—
42C5×C4⋊Q8—61no—
43C5×#125—32no—
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×#106—32no—
53C5×#106—32no—
54C5×#106—32no—
55C5×C4⋊C8—31no—
56C5×C4⋊C8—32no—
57C5×#103—31no—
58C5×#103—32no—
59C5×#103—31no—
60C5×#103—32no—
61C5×#97—32no—
62C4×C5×C8—31no—
63#3387—31noSylow
64C2×C3×C4×C5—41no—
65C2×C3×C4×C5—42no—
66C2×C5×Dic3—41no—
67C2×C5×Dic3—41no—
68C2×C5×Dic3—42no—
69C5×Dic6—84no—
70C5×SL(2,𝔽3)C2×Q811yescharacteristic
71C2×#226C22×C511yescharacteristic
72#888—41no—
73C4×SL(2,𝔽3)C22×C512yes—
74C4×SL(2,𝔽3)C22×C511yescharacteristic
75#859C22×C511yescharacteristic
76#859C22×C512yes—
77C2×C5×Q8—31no—
78C2×C5×Q8—31no—
79C2×C5×Q8—61no—
80C2×C5×Q8—62no—
81C2×C5×Q8C22×D311yescharacteristic
82C5×Q16—32no—
83C5×Q16—32no—
84C2×C5×C8—31no—
85C2×C5×C8—31no—
86C2×C5×C8—32no—
87C5×C4⋊C4—31no—
88C5×C4⋊C4—32no—
89C5×C4⋊C4—32no—
90C5×C4⋊C4—31no—
91C5×C4⋊C4—32no—
92C5×C4⋊C4—61no—
93C5×C4⋊C4—61no—
94C5×C4⋊C4—62no—
95C5×C4⋊C4—62no—
96C5×C4⋊C4—61no—
97C5×C4⋊C4S411yescharacteristic
98C42×C5—32no—
99C42×C5—31no—
100C42×C5—62no—
101C42×C5—61no—
102Q82—31no—
103#557C5×D311yescharacteristic
104#501—32no—
105#498—31no—
106#479—32no—
107#477—31no—
108#475—31no—
109#475—32no—
110#446—31no—
111#441—32no—
112C4×Q16—31no—
113C22×C3×C5—41no—
114C3×C4×C5—42no—
115C3×C4×C5—82no—
116C5×Dic3—42no—
117C5×Dic3—42no—
118C5×Dic3—82no—
119C2×Dic6—42no—
120C2×SL(2,𝔽3)C23×C511yescharacteristic
121#226C5×Q812yescomplemented
122C3×C4⋊C4—41no—
123C4⋊Dic3—41no—
124Dic3⋊C4—42no—
125C4×Dic3—41no—
126C5×Q8—31no—
127C5×Q8—32no—
128C5×Q8—61no—
129C5×Q8—62no—
130C5×Q8—242no—
131C5×Q8—241no—
132C5×Q8D3×Q811yescharacteristic
133C2×C4×C5—31no—
134C2×C4×C5—32no—
135C2×C4×C5—31no—
136C2×C4×C5—61no—
137C2×C4×C5—61no—
138C2×C4×C5—62no—
139C2×C4×C5C2×S412yes—
140C2×C4×C5C2×S411yescharacteristic
141C5×C8—32no—
142C5×C8—61no—
143C5×C8—62no—
144C2×Q16—31no—
145C4⋊Q8—31no—
146C4⋊Q8—32no—
147C4⋊Q8—31no—
148C4⋊Q8—62no—
149C4⋊Q8—61no—
150#125—32no—
151C4×Q8—31no—
152C4×Q8—31no—
153C4×Q8—32no—
154C4×Q8—61no—
155C4×Q8C2×C5×D311yescharacteristic
156C4×Q8C2×C5×D312yes—
157#107—31no—
158#107—32no—
159#106—32no—
160#106—32no—
161#106—32no—
162C4⋊C8—31no—
163C4⋊C8—32no—
164#103—31no—
165#103—32no—
166#103—31no—
167#103—32no—
168#97—32no—
169C4×C8—31no—
170C2×C3×C5—41no—
171C2×C3×C5—41no—
172C2×C3×C5—41no—
173C2×C3×C4—41no—
174C2×C3×C4—42no—
175C2×Dic3—41no—
176C2×Dic3—41no—
177C2×Dic3—42no—
178Dic6—84no—
179SL(2,𝔽3)C2×C5×Q811yescharacteristic
180C22×C5C22×S411yescharacteristic
181C4×C5—22no—
182C4×C5—31no—
183C4×C5—31no—
184C4×C5—62no—
185C4×C5—62no—
186C4×C5—61no—
187C4×C5—121no—
188C4×C5—122no—
189C4×C5#98412yes—
190C2×Q8—31no—
191C2×Q8—31no—
192C2×Q8—61no—
193C2×Q8—62no—
194C2×Q8C22×C5×D311yescharacteristic
195Q16—32no—
196Q16—32no—
197C2×C8—31no—
198C2×C8—31no—
199C2×C8—32no—
200C4⋊C4—32no—
201C4⋊C4—31no—
202C4⋊C4—32no—
203C4⋊C4—31no—
204C4⋊C4—32no—
205C4⋊C4—61no—
206C4⋊C4—61no—
207C4⋊C4—62no—
208C4⋊C4—62no—
209C4⋊C4—61no—
210C4⋊C4C5×S411yescharacteristic
211C42—31no—
212C42—32no—
213C42—62no—
214C42—61no—
215C3×C5—41noHall
216C22×C3—41no—
217C3×C4—42no—
218C3×C4—82no—
219Dic3—42no—
220Dic3—42no—
221Dic3—82no—
222C2×C5Q8×S411yescharacteristic
223C2×C5C2×#98411yescharacteristic
224C2×C5#592611yescharacteristic
225Q8—31no—
226Q8—32no—
227Q8—61no—
228Q8—62no—
229Q8—242no—
230Q8—241no—
231Q8C5×D3×Q811yescharacteristic
232C2×C4—31no—
233C2×C4—32no—
234C2×C4—31no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—62no—
238C2×C4C2×C5×S412yes—
239C2×C4C2×C5×S411yescharacteristic
240C8—32no—
241C8—61no—
242C8—62no—
243C2×C3—41no—
244C2×C3—41no—
245C2×C3—41no—
246C5#8541511yescharacteristic, direct product factor, Sylow
247C22C22×C5×S411yescharacteristic
248C4—22no—
249C4—31no—
250C4—31no—
251C4—62no—
252C4—62no—
253C4—61no—
254C4—121no—
255C4—122no—
256C4C5×#98412yes—
257C3—41noSylow
258C2C5×Q8×S411yescharacteristic
259C2C2×C5×#98411yescharacteristic
260C2C5×#592611yescharacteristic
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. #85415C5direct (N × H)
27. C5×#226Q8semidirect
121. #226C5×Q8semidirect
246. C5#85415direct (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×#53912
C5×#53882
C4×C5×#2262
C5×#53862
C5×#33873
C5×#8884
#854155