Finite Group Databasebeta

← C5×#73002 #14510726

Subgroups — C5×#73002 #14510726

Order 1920.

240 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 24 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×#5931C211yesmaximal, characteristic
2C5×#5930C211yesmaximal, characteristic, complemented
3C5×#395⋊C3C211yesmaximal, characteristic, complemented
4C5×#1603—31nomaximal, Hall
5C5×C22⋊SL(2,𝔽3)C2211yescharacteristic
6C4×C5×S4—41nomaximal
7#73002C511yesmaximal, characteristic, direct product factor, Hall
8C5×#539—31no—
9C5×#451—31no—
10C5×#448—31no—
11C5×#395D311yescharacteristic, complemented
12C5×#328—31no—
13C5×#327—31no—
14C5×#324—31no—
15C2×C5×S4—41no—
16C4×C5×A4—41no—
17C5×SL(2,ℤ/4)—41no—
18C5×GL(2,𝔽3)—81no—
19C5×#226—81no—
20#5931C2×C511yescharacteristic
21#5930C2×C511yescharacteristic, complemented
22#395⋊C3C2×C511yescharacteristic, complemented
23C22×C5×Q8C2×D311yescharacteristic
24C2×C5×Q16—61no—
25C2×C5×SD16—61no—
26C5×#126—61no—
27C5×#125—61no—
28C5×#123—61no—
29C5×#123—61no—
30C5×C22⋊Q8—31no—
31C5×C22⋊Q8—61no—
32C5×#121—31no—
33C4×C5×D4—61no—
34C2×C5×C4⋊C4—31no—
35C5×#103—61no—
36C5×#103—61no—
37C5×C22⋊C8—31no—
38C5×C22⋊C8—31no—
39C5×#95—31no—
40#1603—31noSylow
41C2×C5×A4—41no—
42C5×S4—42no—
43C4×C5×D3—161no—
44C5×SL(2,𝔽3)—81no—
45C5×SL(2,𝔽3)—81no—
46C22⋊SL(2,𝔽3)C22×C511yescharacteristic
47C4×S4—41no—
48C2×C5×Q8—31no—
49C2×C5×Q8—61no—
50C2×C5×Q8—61no—
51C2×C5×D4—31no—
52C2×C5×D4—61no—
53C22×C4×C5—31no—
54C22×C4×C5—31no—
55C22×C4×C5—61no—
56C22×C4×C5S411yescharacteristic
57C5×Q16—121no—
58C5×Q16—121no—
59C5×SD16—121no—
60C5×SD16—121no—
61C2×C5×C8—61no—
62C2×C5×C8—61no—
63C5×C4⋊C4—31no—
64C5×C4⋊C4—31no—
65C5×C4⋊C4—61no—
66C5×C4⋊C4—61no—
67C5×C4⋊C4—61no—
68C5×C4⋊C4—61no—
69C5×C4⋊C4—61no—
70C5×C4⋊C4—61no—
71C5×C22⋊C4—61no—
72C5×C22⋊C4—61no—
73C5×C22⋊C4—61no—
74C5×C22⋊C4—61no—
75C5×C22⋊C4—61no—
76C42×C5—61no—
77#539—31no—
78#451—31no—
79#448—31no—
80#395C5×D311yescharacteristic, complemented
81#328—31no—
82#327—31no—
83#324—31no—
84C2×C5×D3—161no—
85C5×A4—41no—
86C3×C4×C5—161no—
87C5×Dic3—161no—
88C2×S4—41no—
89C4×A4—41no—
90SL(2,ℤ/4)—41no—
91GL(2,𝔽3)—81no—
92#226—81no—
93C23×C5—61no—
94C23×C5C2×S411yescharacteristic
95C5×Q8—21no—
96C5×Q8—21no—
97C5×Q8—61no—
98C5×Q8—61no—
99C5×Q8—121no—
100C5×D4—121no—
101C5×D4—121no—
102C5×D4—121no—
103C2×C4×C5—31no—
104C2×C4×C5—31no—
105C2×C4×C5—61no—
106C2×C4×C5—61no—
107C2×C4×C5—61no—
108C2×C4×C5—61no—
109C2×C4×C5—61no—
110C2×C4×C5—61no—
111C2×C4×C5—61no—
112C2×C4×C5—121no—
113C2×C4×C5—241no—
114C5×C8—121no—
115C5×C8—121no—
116C22×Q8C2×C5×D311yescharacteristic
117C2×Q16—61no—
118C2×SD16—61no—
119#126—61no—
120#125—61no—
121#123—61no—
122#123—61no—
123C22⋊Q8—31no—
124C22⋊Q8—61no—
125#121—31no—
126C4×D4—61no—
127C2×C4⋊C4—31no—
128#103—61no—
129#103—61no—
130C22⋊C8—31no—
131C22⋊C8—31no—
132#95—31no—
133C2×C3×C5—161no—
134C5×D3—162no—
135C2×A4—41no—
136S4—42no—
137C4×D3—161no—
138SL(2,𝔽3)—81no—
139SL(2,𝔽3)—81no—
140C22×C5—31no—
141C22×C5—31no—
142C22×C5—121no—
143C22×C5—121no—
144C22×C5—121no—
145C22×C5#98511yescharacteristic
146C4×C5—41no—
147C4×C5—61no—
148C4×C5—61no—
149C4×C5—121no—
150C4×C5—121no—
151C4×C5—121no—
152C4×C5—121no—
153C2×Q8—31no—
154C2×Q8—61no—
155C2×Q8—61no—
156C2×D4—31no—
157C2×D4—61no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—61no—
161C22×C4C5×S411yescharacteristic
162Q16—121no—
163Q16—121no—
164SD16—121no—
165SD16—121no—
166C2×C8—61no—
167C2×C8—61no—
168C4⋊C4—31no—
169C4⋊C4—31no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C22⋊C4—61no—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C42—61no—
182C3×C5—161noHall
183C2×D3—161no—
184A4—41no—
185C3×C4—161no—
186Dic3—161no—
187C2×C5—31no—
188C2×C5—31no—
189C2×C5—241no—
190C2×C5#1020⋊C211yescharacteristic
191C23—61no—
192C23C2×C5×S411yescharacteristic
193Q8—21no—
194Q8—21no—
195Q8—61no—
196Q8—61no—
197Q8—121no—
198D4—121no—
199D4—121no—
200D4—121no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—121no—
211C2×C4—241no—
212C8—121no—
213C8—121no—
214C2×C3—161no—
215D3—162no—
216C5#7300211yescharacteristic, direct product factor, Sylow
217C22—31no—
218C22—31no—
219C22—121no—
220C22—121no—
221C22—121no—
222C22C5×#98511yescharacteristic
223C4—41no—
224C4—61no—
225C4—61no—
226C4—121no—
227C4—121no—
228C4—121no—
229C4—121no—
230C3—161noSylow
231C2—31no—
232C2—31no—
233C2—241no—
234C2C5×#1020⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2×C5#1020⋊C2
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22×C5
FrattiniC23C2×C5×S4
FittingC5×#395D3
socleC23×C5C2×S4
prime radicalC5×#5931C2
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2×C5

Complements

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

NH ≅ G/Ntype
2. C5×#5930C2semidirect
3. C5×#395⋊C3C2semidirect
7. #73002C5direct (N × H)
11. C5×#395D3semidirect
21. #5930C2×C5semidirect
22. #395⋊C3C2×C5semidirect
80. #395C5×D3semidirect
216. C5#73002direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#59312
C5×#59302
C5×#395⋊C32
C5×#16033
C4×C5×S44
#730025