Finite Group Databasebeta

← C5×#73122 #14510846

Subgroups — C5×#73122 #14510846

Order 1920.

258 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C2×C5×#131⋊C3C211yesmaximal, characteristic
2C5×#5424C211yesmaximal, characteristic
3C2×C5×GU(2,𝔽3)C211yesmaximal, characteristic
4C5×#1866—31nomaximal, Hall
5C2×C5×#41⋊C3C2211yescharacteristic
6C5×#131⋊C3—21no—
7C5×GU(2,𝔽3)—21no—
8C5×#859C411yescharacteristic, complemented
9C5×#859C411yescharacteristic, complemented
10C5×Dic3⋊C8—41nomaximal
11#73122C511yesmaximal, characteristic, direct product factor, Hall
12C2×C5×#131D311yescharacteristic
13C2×C5×#108—31no—
14C5×#424—31no—
15C2×C5×#104—31no—
16C5×#416—31no—
17C5×#339—31no—
18C5×#336—31no—
19C5×#41⋊C3D411yescharacteristic
20C5×#41⋊C3Q811yescharacteristic
21C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
22C2×C3×C5×C8—41no—
23C4×C5×Dic3—41no—
24C2×C5×C3⋊C8—41no—
25C2×#131⋊C3C2×C511yescharacteristic
26#5424C2×C511yescharacteristic
27C2×GU(2,𝔽3)C2×C511yescharacteristic
28C2×C5×#41C2×D311yescharacteristic
29C5×#131—21no—
30C5×#131—61no—
31C2×C5×M16—31no—
32C2×C5×M16—31no—
33C22×C5×C8—31no—
34C22×C5×C8—31no—
35C5×#117—31no—
36C2×C42×C5—31no—
37C5×#108—62no—
38C5×C4⋊C8—61no—
39C5×C4⋊C8—61no—
40C5×#104—61no—
41C5×#104—61no—
42C5×#103—31no—
43C5×#103—31no—
44C5×#102—31no—
45C5×#102—31no—
46#1866—31noSylow
47C2×C3×C4×C5—41no—
48C2×C5×Dic3—41no—
49C2×C5×Dic3—41no—
50C5×SL(2,𝔽3)C4⋊C411yescharacteristic
51C3×C5×C8—81no—
52C5×C3⋊C8—81no—
53C2×#41⋊C3C22×C511yescharacteristic
54#131⋊C3—21no—
55GU(2,𝔽3)—21no—
56#859C4×C511yescharacteristic, complemented
57#859C4×C511yescharacteristic, complemented
58Dic3⋊C8—41no—
59C5×#41—31no—
60C5×#41—31no—
61C5×#41Dic611yescharacteristic
62C5×#41#6711yescharacteristic
63C2×C5×Q8C4×D311yescharacteristic
64C2×C5×D4—31no—
65C22×C4×C5—31no—
66C22×C4×C5—61no—
67C5×M16—32no—
68C5×M16—61no—
69C5×M16—61no—
70C5×M16—61no—
71C2×C5×C8—31no—
72C2×C5×C8—31no—
73C2×C5×C8—31no—
74C2×C5×C8—61no—
75C2×C5×C8—61no—
76C2×C5×C8—61no—
77C2×C5×C8—61no—
78C2×C5×C8—61no—
79C2×C5×C8S411yescharacteristic
80C5×C4⋊C4—31no—
81C5×C4⋊C4—31no—
82C5×C22⋊C4—61no—
83C42×C5—61no—
84C42×C5—61no—
85C42×C5—61no—
86C2×#131C5×D311yescharacteristic
87C2×#108—31no—
88#424—31no—
89C2×#104—31no—
90#416—31no—
91#339—31no—
92#336—31no—
93C22×C3×C5—41no—
94C3×C4×C5—41no—
95C3×C4×C5—41no—
96C5×Dic3—42no—
97C5×Dic3—81no—
98#41⋊C3C5×D411yescharacteristic
99#41⋊C3C5×Q811yescharacteristic
100C2×SL(2,𝔽3)C2×C4×C511yescharacteristic
101C2×C3×C8—41no—
102C4×Dic3—41no—
103C2×C3⋊C8—41no—
104C23×C5—31no—
105C5×Q8—31no—
106C5×Q8Dic3⋊C411yescharacteristic
107C5×D4—31no—
108C5×D4—31no—
109C5×D4—61no—
110C2×C4×C5—31no—
111C2×C4×C5—31no—
112C2×C4×C5—31no—
113C2×C4×C5—31no—
114C2×C4×C5—31no—
115C2×C4×C5—62no—
116C2×C4×C5—61no—
117C2×C4×C5—61no—
118C2×C4×C5—121no—
119C2×C4×C5—121no—
120C2×C4×C5C2×S411yescharacteristic
121C5×C8—21no—
122C5×C8—61no—
123C5×C8—61no—
124C5×C8—61no—
125C5×C8—121no—
126C2×#41C2×C5×D311yescharacteristic
127#131—21no—
128#131—61no—
129C2×M16—31no—
130C2×M16—31no—
131C22×C8—31no—
132C22×C8—31no—
133#117—31no—
134C2×C42—31no—
135#108—62no—
136C4⋊C8—61no—
137C4⋊C8—61no—
138#104—61no—
139#104—61no—
140#103—31no—
141#103—31no—
142#102—31no—
143#102—31no—
144C2×C3×C5—41no—
145C2×C3×C5—41no—
146C2×C3×C5—41no—
147C2×C3×C4—41no—
148C2×Dic3—41no—
149C2×Dic3—41no—
150SL(2,𝔽3)C5×C4⋊C411yescharacteristic
151C3×C8—81no—
152C3⋊C8—81no—
153C22×C5—31no—
154C22×C5—31no—
155C22×C5—61no—
156C22×C5—61no—
157C22×C5C4×S411yescharacteristic
158C4×C5—31no—
159C4×C5—31no—
160C4×C5—122no—
161C4×C5—121no—
162C4×C5—121no—
163C4×C5A4⋊Q811yescharacteristic
164C4×C5GL(2,ℤ/4)11yescharacteristic
165#41—31no—
166#41—31no—
167#41C5×Dic611yescharacteristic
168#41C5×#6711yescharacteristic
169C2×Q8C4×C5×D311yescharacteristic
170C2×D4—31no—
171C22×C4—31no—
172C22×C4—61no—
173M16—32no—
174M16—61no—
175M16—61no—
176M16—61no—
177C2×C8—31no—
178C2×C8—31no—
179C2×C8—31no—
180C2×C8—61no—
181C2×C8—61no—
182C2×C8—61no—
183C2×C8—61no—
184C2×C8—61no—
185C2×C8C5×S411yescharacteristic
186C4⋊C4—31no—
187C4⋊C4—31no—
188C22⋊C4—61no—
189C42—61no—
190C42—61no—
191C42—61no—
192C3×C5—41noHall
193C22×C3—41no—
194C3×C4—41no—
195C3×C4—41no—
196Dic3—42no—
197Dic3—81no—
198C2×C5—61no—
199C2×C5—61no—
200C2×C5#540411yescharacteristic
201C2×C5#540511yescharacteristic
202C2×C5SL(2,ℤ/4)⋊C411yescharacteristic
203C23—31no—
204Q8—31no—
205Q8C5×Dic3⋊C411yescharacteristic
206D4—31no—
207D4—31no—
208D4—61no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—62no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—121no—
218C2×C4—121no—
219C2×C4C2×C5×S411yescharacteristic
220C8—21no—
221C8—61no—
222C8—61no—
223C8—61no—
224C8—121no—
225C2×C3—41no—
226C2×C3—41no—
227C2×C3—41no—
228C5#7312211yescharacteristic, direct product factor, Sylow
229C22—31no—
230C22—31no—
231C22—61no—
232C22—61no—
233C22C4×C5×S411yescharacteristic
234C4—31no—
235C4—31no—
236C4—122no—
237C4—121no—
238C4—121no—
239C4C5×A4⋊Q811yescharacteristic
240C4C5×GL(2,ℤ/4)11yescharacteristic
241C3—41noSylow
242C2—61no—
243C2—61no—
244C2C5×#540411yescharacteristic
245C2C5×#540511yescharacteristic
246C2C5×SL(2,ℤ/4)⋊C411yescharacteristic
subgroupG/H
centerC2×C4×C5C2×S4
commutatorC2×SL(2,𝔽3)C2×C4×C5
FrattiniC2×C4C2×C5×S4
FittingC2×C5×#131D3
socleC22×C5C4×S4
prime radicalC2×C5×#41⋊C3C22
cosocleC2×#41⋊C3C22×C5
hypercenterC2×C5×C8S4
hypocenterSL(2,𝔽3)C5×C4⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4×C5 ◁ C2×C5×C8

Complements

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

NH ≅ G/Ntype
8. C5×#859C4semidirect
9. C5×#859C4semidirect
11. #73122C5direct (N × H)
56. #859C4×C5semidirect
57. #859C4×C5semidirect
228. C5#73122direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#131⋊C32
C5×#54242
C2×C5×GU(2,𝔽3)2
C5×#18663
C5×Dic3⋊C84
#731225