Finite Group Databasebeta

← C5⋊#85551 #14513953

Subgroups — C5⋊#85551 #14513953

Order 1920.

340 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 23 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×#543⋊C3C211yesmaximal, characteristic, complemented
2#10602685—31nomaximal, Hall
3C22×C5⋊S4—41nomaximal
4C2×C5⋊GL(2,𝔽3)—42nomaximal
5#85551—51nomaximal, Hall
6C5×#543D311yescharacteristic, complemented
7#65335—32no—
8#65249—31no—
9#64758—32no—
10#64745—31no—
11C22×C5×A4—41no—
12C2×C5⋊S4—42no—
13C2×C5⋊S4—42no—
14C2×C5⋊S4—42no—
15C2×C5×SL(2,𝔽3)—42no—
16C2×C5×SL(2,𝔽3)—81no—
17C5⋊GL(2,𝔽3)—44no—
18#543⋊C3D511yescharacteristic, complemented
19C22×C22⋊D5—61no—
20C22×D20—31no—
21C5×C22⋊Q8—32no—
22C5×C22⋊Q8—61no—
23C2×C5×C22⋊C4—31no—
24C2×#688—62no—
25C2×#685—61no—
26#3963—31no—
27C22⋊#168—32no—
28#3941—62no—
29#1596—151noSylow
30C22×D15—161no—
31C2×C5×A4—41no—
32C2×C5×A4—41no—
33C2×C5×A4—41no—
34C5⋊S4—44no—
35C5×SL(2,𝔽3)—42no—
36C5×SL(2,𝔽3)—81no—
37C22×S4—201no—
38C2×GL(2,𝔽3)—202no—
39C24×C5S411yescharacteristic
40C23×D5—61no—
41C2×C5×Q8—21no—
42C2×C5×Q8S412yescomplemented
43C22×C4×C5—31no—
44C22×C4×C5—31no—
45C2×C22⋊D5—122no—
46C2×C22⋊D5—122no—
47C2×C22⋊D5—121no—
48C2×C22⋊D5—121no—
49C22×Dic5—61no—
50C2×D20—62no—
51C2×D20—122no—
52C5×C4⋊C4—32no—
53C5×C4⋊C4—61no—
54C5×C22⋊C4—62no—
55#688—124no—
56#685—122no—
57C2×#168—62no—
58#543D1511yescharacteristic, complemented
59#450—152no—
60#392—151no—
61#328—152no—
62#323—151no—
63C22×C3×C5—161no—
64C2×D15—162no—
65C2×D15—162no—
66C2×D15—162no—
67C5×A4—41no—
68C22×A4—41no—
69C2×S4—202no—
70C2×S4—202no—
71C2×S4—202no—
72C2×SL(2,𝔽3)—42no—
73C2×SL(2,𝔽3)—81no—
74GL(2,𝔽3)—204no—
75C23×C5—31no—
76C23×C5—31no—
77C23×C5—41no—
78C23×C5—41no—
79C23×C5GL(2,𝔽3)11yescharacteristic, complemented
80C22×D5—62no—
81C22×D5—121no—
82C22×D5—121no—
83C22×D5—121no—
84C22×D5—121no—
85C22×D5—121no—
86C22×D5—121no—
87C5×Q8—42no—
88C5×Q8—81no—
89C2×C4×C5—32no—
90C2×C4×C5—61no—
91C2×C4×C5—121no—
92C2×C4×C5—121no—
93C22⋊D5—124no—
94C22⋊D5—124no—
95C2×Dic5—62no—
96C2×Dic5—121no—
97C2×Dic5—121no—
98D20—64no—
99D20—122no—
100#168—122no—
101C22×D4—151no—
102C22×D4—301no—
103C2×SD16—302no—
104C22⋊Q8—32no—
105C22⋊Q8—61no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—301no—
108#102—302no—
109C22⋊C8—152no—
110#95—151no—
111C2×C3×C5—161no—
112C2×C3×C5—161no—
113C2×C3×C5—161no—
114D15—164no—
115C22×D3—801no—
116C2×A4—41no—
117C2×A4—41no—
118C2×A4—41no—
119S4—204no—
120SL(2,𝔽3)—42no—
121SL(2,𝔽3)—81no—
122C22×C5—31no—
123C22×C5—31no—
124C22×C5—41no—
125C22×C5—61no—
126C22×C5—61no—
127C22×C5—121no—
128C22×C5#102011yescharacteristic
129C2×D5—122no—
130C2×D5—122no—
131C2×D5—122no—
132C2×D5—241no—
133C2×D5—241no—
134C2×D5—241no—
135C2×D5—241no—
136C4×C5—62no—
137C4×C5—121no—
138Dic5—122no—
139C24—301no—
140C24C5⋊S411yescharacteristic
141C2×Q8—21no—
142C2×Q8C5⋊S412yescomplemented
143C2×D4—302no—
144C2×D4—602no—
145C2×D4—602no—
146C2×D4—601no—
147C2×D4—602no—
148C2×D4—601no—
149C22×C4—31no—
150C22×C4—31no—
151C22×C4—301no—
152SD16—604no—
153C2×C8—302no—
154C4⋊C4—32no—
155C4⋊C4—61no—
156C22⋊C4—62no—
157C22⋊C4—602no—
158C3×C5—161noHall
159C22×C3—161no—
160C2×D3—802no—
161C2×D3—802no—
162C2×D3—802no—
163A4—41no—
164C2×C5—61no—
165C2×C5—61no—
166C2×C5#593111yescharacteristic
167C2×C5#593411yescharacteristic
168C2×C5#593511yescharacteristic
169D5—242no—
170C23—31no—
171C23—31no—
172C23—41no—
173C23—41no—
174C23—302no—
175C23—601no—
176C23—601no—
177C23—601no—
178C23—601no—
179C23—601no—
180C23—601no—
181C23C5⋊GL(2,𝔽3)11yescharacteristic, complemented
182Q8—42no—
183Q8—81no—
184D4—304no—
185D4—602no—
186D4—604no—
187D4—604no—
188C2×C4—32no—
189C2×C4—61no—
190C2×C4—121no—
191C2×C4—121no—
192C2×C4—302no—
193C2×C4—601no—
194C2×C4—601no—
195C8—602no—
196C2×C3—161no—
197C2×C3—161no—
198C2×C3—161no—
199D3—804no—
200C5#8555111yescharacteristic, complemented, Sylow
201C22—31no—
202C22—31no—
203C22—41no—
204C22—61no—
205C22—61no—
206C22—121no—
207C22—602no—
208C22—602no—
209C22—602no—
210C22—1201no—
211C22—1201no—
212C22—1201no—
213C22—1201no—
214C22C5⋊#102011yescharacteristic
215C4—62no—
216C4—121no—
217C4—602no—
218C3—161noSylow
219C2—61no—
220C2—61no—
221C2—1202no—
222C2C5⋊#593111yescharacteristic
223C2C5⋊#593411yescharacteristic
224C2C5⋊#593511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C5⋊#1020
commutator, cosocle, hypocenterC5×#543⋊C3C2
FittingC5×#543D3
socleC22×C5#1020
LCSG ▷ C5×#543⋊C3
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C5×#543⋊C3C2semidirect
6. C5×#543D3semidirect
18. #543⋊C3D5semidirect
42. C2×C5×Q8S4semidirect
58. #543D15semidirect
79. C23×C5GL(2,𝔽3)semidirect
142. C2×Q8C5⋊S4semidirect
181. C23C5⋊GL(2,𝔽3)semidirect
200. C5#85551semidirect

Maximal subgroups

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

subgroupindex
C5×#543⋊C32
#106026853
C22×C5⋊S44
C2×C5⋊GL(2,𝔽3)4
#855515