Finite Group Databasebeta

← #14307450

Subgroups — #14307450

Order 1920.

488 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C3×#66022C211yesmaximal, characteristic
2#11752345C211yesmaximal, characteristic
3#11752340C211yesmaximal, characteristic
4#11746746C214yesmaximal
5#10604588—31nomaximal, Hall
6C2×C4×ΓL(1,𝔽16)C2212yes—
7C2×#6758C2212yes—
8C3×C5×#125C411yescharacteristic, complemented
9C3×#4042—22no—
10C3×#4038C2212yes—
11C3×#4034C412yescomplemented
12C3×#4033—22no—
13C3×C42×D5C2211yescharacteristic
14C3×#4016C411yescharacteristic, complemented
15#91235—22no—
16C4⋊(C3⋊#170)—22no—
17#71840—51nomaximal, Hall
18#66022D311yescharacteristic
19#65686—31no—
20#65681—31no—
21#64875—34no—
22C22×ΓL(1,𝔽16)—22no—
23C2×C3×C4×D5D412yes—
24C2×C3×C4×D5C2311yescharacteristic
25C2×C3×Dic10—22no—
26C2×C3⋊#170—22no—
27C4×ΓL(1,𝔽16)—22no—
28C4×ΓL(1,𝔽16)Q814yescomplemented
29#6758—24no—
30C3×C5×C4⋊C4—22no—
31C3×C5×C4⋊C4C2×C412yes—
32C3×C42×C5C2×C411yescharacteristic
33C3×#685—22no—
34C3×C4⋊Dic5C2×C412yes—
35C3×#683—22no—
36C3×#683—22no—
37C3×C4×Dic5—22no—
38C3×C4×Dic5C2×C411yescharacteristic
39C3×C4×Dic5D412yes—
40C3×#533—51no—
41#4932—51no—
42#4924—51no—
43#4532—54no—
44C2×C4×AGL(1,𝔽5)—32no—
45C2×#700—32no—
46C5×#125Dic311yescharacteristic, complemented
47#4042—22no—
48#4038C2×D312yes—
49#4034Dic312yescomplemented
50#4033—22no—
51C42×D5C2×D311yescharacteristic
52#4016Dic311yescharacteristic, complemented
53(C2×AGL(1,𝔽5))⋊C4—62no—
54C4⋊#170—62no—
55#1995—151noSylow
56C22×C3×D5C2×D411yescharacteristic
57C2×ΓL(1,𝔽16)—24no—
58C2×ΓL(1,𝔽16)—44no—
59C2×C3×C4×C5—22no—
60C2×C3×C4×C5C22⋊C412yes—
61C2×C3×C4×C5C22×C411yescharacteristic
62C2×C3×Dic5—22no—
63C2×C3×Dic5C22⋊C412yes—
64C2×C3×Dic5C2×D411yescharacteristic
65C3×C4×D5—42no—
66C3×C4×D5C2×Q812yes—
67C3×C4×D5#4112yes—
68C3×Dic10—44no—
69C3⋊#170—42no—
70C3×C4⋊Q8—52no—
71C3×#125—51no—
72C3×#125AGL(1,𝔽5)11yescharacteristic, complemented
73C3×C22⋊Q8—102no—
74C3×C4×Q8—102no—
75C3×#117—52no—
76C2×C3×C42—51no—
77C2×C4×Dic3—52no—
78C2×#208—52no—
79C3⋊#95—102no—
80C4⋊(C3⋊C8)—102no—
81C22×AGL(1,𝔽5)—62no—
82C2×C4×D5#6712yes—
83C2×C4×D5C22×D311yescharacteristic
84C2×Dic10—22no—
85C2×#170—62no—
86C4×AGL(1,𝔽5)—34no—
87C4×AGL(1,𝔽5)—62no—
88#700—64no—
89C5×C4⋊C4—22no—
90C5×C4⋊C4C2×Dic312yes—
91C42×C5C2×Dic311yescharacteristic
92#685—22no—
93C4⋊Dic5C2×Dic312yes—
94#683—22no—
95#683—22no—
96C4×Dic5—22no—
97C4×Dic5C2×Dic311yescharacteristic
98C4×Dic5#6712yes—
99#533—51no—
100#423—151no—
101#376—151no—
102#339—154no—
103C22×C3×C5C2×C22⋊C411yescharacteristic
104C2×C3×D5—22no—
105C2×C3×D5C22⋊Q812yes—
106ΓL(1,𝔽16)—44no—
107C3×C4×C5—22no—
108C3×C4×C5—42no—
109C3×C4×C5C4×Q812yes—
110C3×Dic5—22no—
111C3×Dic5—42no—
112C3×Dic5#12411yescharacteristic
113C3×Dic5C4⋊Q811yescharacteristic
114C2×C3×Q8—102no—
115C22×C3×C4—51no—
116C22×C3×C4—52no—
117C22×Dic3—102no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—52no—
120C3×C4⋊C4—102no—
121C3×C4⋊C4—102no—
122C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
123C3×C22⋊C4—102no—
124C3×C42—52no—
125C3×C42—51no—
126C3×C42—102no—
127C3×C42C2×AGL(1,𝔽5)11yescharacteristic
128C4×Dic3—54no—
129C4×Dic3—102no—
130#208—104no—
131C2×C3⋊C8—102no—
132C22×D5C2×#6711yescharacteristic
133C2×AGL(1,𝔽5)—64no—
134C2×AGL(1,𝔽5)—124no—
135C2×C4×C5—22no—
136C2×C4×C5#21712yes—
137C2×C4×C5C22×Dic311yescharacteristic
138C2×Dic5—22no—
139C2×Dic5#21712yes—
140C2×Dic5C2×#6711yescharacteristic
141C4×D5—42no—
142C4×D5D3×Q812yes—
143C4×D5#23912yes—
144Dic10—44no—
145#170—122no—
146C2×M16—152no—
147C4⋊Q8—52no—
148#125—51no—
149#125ΓL(1,𝔽16)11yescharacteristic, complemented
150C22⋊Q8—102no—
151C4×Q8—102no—
152#117—52no—
153C2×C42—51no—
154C2×C42—152no—
155C4⋊C8—302no—
156#95—302no—
157C2×C3×C5#39111yescharacteristic
158C2×C3×C5#42112yes—
159C3×D5—22no—
160C23×C3—51no—
161C3×Q8—204no—
162C2×C3×C4—22no—
163C2×C3×C4—51no—
164C2×C3×C4—52no—
165C2×C3×C4—52no—
166C2×C3×C4—52no—
167C2×C3×C4—102no—
168C2×C3×C4—202no—
169C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
170C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
171C2×Dic3—104no—
172C2×Dic3—204no—
173C3⋊C8—202no—
174C22×C5C2×#21711yescharacteristic
175C2×D5—22no—
176C2×D5#94612yes—
177AGL(1,𝔽5)—124no—
178C4×C5—22no—
179C4×C5—42no—
180C4×C5Q8×Dic312yes—
181Dic5—22no—
182Dic5—42no—
183Dic5#94411yescharacteristic
184Dic5#94711yescharacteristic
185C2×Q8—102no—
186C22×C4—51no—
187C22×C4—52no—
188C22×C4—302no—
189M16—304no—
190C2×C8—302no—
191C4⋊C4—22no—
192C4⋊C4—52no—
193C4⋊C4—102no—
194C4⋊C4—102no—
195C4⋊C4C2×ΓL(1,𝔽16)12yes—
196C22⋊C4—102no—
197C42—52no—
198C42—51no—
199C42—102no—
200C42—154no—
201C42—302no—
202C42C2×ΓL(1,𝔽16)11yescharacteristic
203C3×C5#199511yescharacteristic, complemented, Hall
204C22×C3—52no—
205C22×C3—102no—
206C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
207C3×C4—22no—
208C3×C4—42no—
209C3×C4—51no—
210C3×C4—51no—
211C3×C4—102no—
212C3×C4—202no—
213C3×C4Q8×AGL(1,𝔽5)12yes—
214Dic3—204no—
215C2×C5#522911yescharacteristic
216C2×C5#523612yes—
217D5—22no—
218C23—51no—
219Q8—204no—
220C2×C4—22no—
221C2×C4—51no—
222C2×C4—52no—
223C2×C4—52no—
224C2×C4—52no—
225C2×C4—102no—
226C2×C4—202no—
227C2×C4—304no—
228C2×C4—604no—
229C2×C4#676312yes—
230C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
231C8—602no—
232C2×C3—102no—
233C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
234C2×C3#6579512yes—
235C5#7184011yescharacteristic, complemented, Sylow
236C22—52no—
237C22—102no—
238C22C2×#676311yescharacteristic
239C4—22no—
240C4—42no—
241C4—51no—
242C4—51no—
243C4—102no—
244C4—202no—
245C4—604no—
246C4Q8×ΓL(1,𝔽16)12yes—
247C3#1060458811yescharacteristic, complemented, Sylow
248C2—102no—
249C2#1175244911yescharacteristic
250C2#1175245412yes—
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#125C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenter#125ΓL(1,𝔽16)
hypocenterC3×C5#1995
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
8. C3×C5×#125C4semidirect
11. C3×#4034C4semidirect
14. C3×#4016C4semidirect
28. C4×ΓL(1,𝔽16)Q8semidirect
46. C5×#125Dic3semidirect
49. #4034Dic3semidirect
52. #4016Dic3semidirect
72. C3×#125AGL(1,𝔽5)semidirect
149. #125ΓL(1,𝔽16)semidirect
203. C3×C5#1995semidirect
235. C5#71840semidirect
247. C3#10604588semidirect

Maximal subgroups

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

subgroupindex
C3×#660222
#117523452
#117523402
#117467462
#106045883
#718405