Finite Group Databasebeta

← C3×#10604673 #14305932

Subgroups — C3×#10604673 #14305932

Order 1920.

616 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 156 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×D5×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#4009C214yesmaximal
3C3×#65689C211yesmaximal, characteristic
4C3×#65676C211yesmaximal, characteristic
5#10604673C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×Q8×D5C2212yes—
7C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
8C2×C3×D5⋊C8C2212yes—
9C3×C5×C4⋊Q8C411yescharacteristic, complemented
10C3×#4090C412yescomplemented
11C3×D5×C4⋊C4—22no—
12C3×#4034—22no—
13C3×C42×D5C2211yescharacteristic
14C3×#4015C411yescharacteristic, complemented
15C3×#4009—28no—
16C3×(C2×AGL(1,𝔽5))⋊C4—22no—
17C3×#4002—22no—
18C3×#1960—51nomaximal, Hall
19D5×C4⋊Q8C2×C311yescharacteristic
20C2×#4009C2×C314yes—
21#65689C2×C311yescharacteristic
22#65676C2×C311yescharacteristic
23C22×C3×AGL(1,𝔽5)—22no—
24C2×C3×C5×Q8C2×C412yes—
25C3×Q8×D5—24no—
26C3×Q8×D5—24no—
27C2×C3×C4×D5—22no—
28C2×C3×C4×D5D412yes—
29C2×C3×C4×D5C2311yescharacteristic
30C2×C3×Dic10—22no—
31C2×C3×Dic10C2×C412yes—
32C2×C3×#170—22no—
33C3×C4⋊AGL(1,𝔽5)—24no—
34C3×D5⋊C8—22no—
35C3×D5⋊C8D414yes—
36C3×C5×C4⋊C4—22no—
37C3×C42×C5C2×C411yescharacteristic
38C3×C4⋊Dic5—22no—
39C3×#683—24no—
40C3×C4×Dic5C2×C411yescharacteristic
41C3×C4×Dic5D412yes—
42C2×C3×C4⋊Q8—51no—
43C2×C3×#103—54no—
44C2×C3×#97—51no—
45C3×#382—51no—
46C2×Q8×D5C22×C312yes—
47C2×C4⋊AGL(1,𝔽5)C22×C312yes—
48C2×D5⋊C8C22×C312yes—
49C5×C4⋊Q8C3×C411yescharacteristic, complemented
50#4090C3×C412yescomplemented
51D5×C4⋊C4—22no—
52#4034—22no—
53C42×D5C22×C311yescharacteristic
54#4015C3×C411yescharacteristic, complemented
55#4009—28no—
56(C2×AGL(1,𝔽5))⋊C4—22no—
57#4002—22no—
58#1960—51noSylow
59C22×C3×D5C2×D411yescharacteristic
60C2×C3×AGL(1,𝔽5)—42no—
61C2×C3×AGL(1,𝔽5)—44no—
62C3×C5×Q8—24no—
63C2×C3×C4×C5—22no—
64C2×C3×C4×C5C22⋊C412yes—
65C2×C3×C4×C5C22×C411yescharacteristic
66C2×C3×Dic5—22no—
67C2×C3×Dic5C22⋊C412yes—
68C2×C3×Dic5C2×D411yescharacteristic
69C3×C4×D5—24no—
70C3×C4×D5—42no—
71C3×C4×D5—42no—
72C3×C4×D5C2×D412yes—
73C3×C4×D5#4112yes—
74C3×Dic10—24no—
75C3×Dic10—44no—
76C3×#170—24no—
77C22×C3×Q8—52no—
78C22×C3×C8—52no—
79C3×C4⋊Q8—52no—
80C3×C4⋊Q8—51no—
81C3×C4⋊Q8—102no—
82C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
83C2×C3×C4⋊C4—52no—
84C2×C3×C4⋊C4—102no—
85C2×C3×C42—51no—
86C3×#103—108no—
87C3×#97—102no—
88C3×#95—102no—
89C22×AGL(1,𝔽5)—22no—
90C2×C5×Q8C2×C3×C412yes—
91Q8×D5—24no—
92Q8×D5—24no—
93C2×C4×D5—22no—
94C2×C4×D5C3×D412yes—
95C2×C4×D5C23×C311yescharacteristic
96C2×Dic10—22no—
97C2×Dic10C2×C3×C412yes—
98C2×#170—22no—
99C4⋊AGL(1,𝔽5)—24no—
100D5⋊C8—22no—
101D5⋊C8C3×D414yes—
102C5×C4⋊C4—22no—
103C42×C5C2×C3×C411yescharacteristic
104C4⋊Dic5—22no—
105#683—24no—
106C4×Dic5C2×C3×C411yescharacteristic
107C4×Dic5C3×D412yes—
108C2×C4⋊Q8—51no—
109C2×#103—54no—
110C2×#97—51no—
111#382—51no—
112C22×C3×C5C2×C22⋊C411yescharacteristic
113C2×C3×D5#12411yescharacteristic
114C2×C3×D5#12711yescharacteristic
115C2×C3×D5#13714yes—
116C3×AGL(1,𝔽5)—82no—
117C3×C4×C5—22no—
118C3×C4×C5—42no—
119C3×C4×C5C4×D412yes—
120C3×Dic5—22no—
121C3×Dic5—42no—
122C3×Dic5#12112yes—
123C2×C3×Q8—52no—
124C2×C3×Q8—102no—
125C2×C3×Q8—104no—
126C2×C3×Q8—104no—
127C2×C3×Q8C2×AGL(1,𝔽5)12yes—
128C22×C3×C4—51no—
129C22×C3×C4—52no—
130C22×C3×C4—102no—
131C22×C3×C4—102no—
132C2×C3×C8—54no—
133C2×C3×C8—102no—
134C2×C3×C8—102no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—104no—
137C3×C4⋊C4—104no—
138C3×C4⋊C4—102no—
139C3×C42—52no—
140C3×C42—51no—
141C3×C42C2×AGL(1,𝔽5)11yescharacteristic
142C22×D5C2×C3×D411yescharacteristic
143C2×AGL(1,𝔽5)—42no—
144C2×AGL(1,𝔽5)—44no—
145C5×Q8—24no—
146C2×C4×C5—22no—
147C2×C4×C5C3×C22⋊C412yes—
148C2×C4×C5C22×C3×C411yescharacteristic
149C2×Dic5—22no—
150C2×Dic5C3×C22⋊C412yes—
151C2×Dic5C2×C3×D411yescharacteristic
152C4×D5—24no—
153C4×D5—42no—
154C4×D5—42no—
155C4×D5C2×C3×D412yes—
156C4×D5C3×#4112yes—
157Dic10—24no—
158Dic10—44no—
159#170—24no—
160C22×Q8—52no—
161C22×C8—52no—
162C4⋊Q8—52no—
163C4⋊Q8—51no—
164C4⋊Q8—102no—
165C4⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
166C2×C4⋊C4—52no—
167C2×C4⋊C4—102no—
168C2×C42—51no—
169#103—108no—
170#97—102no—
171#95—102no—
172C2×C3×C5#39011yescharacteristic
173C2×C3×C5#41912yes—
174C3×D5#49112yes—
175C3×D5#49712yes—
176C23×C3—51no—
177C3×Q8—24no—
178C3×Q8—104no—
179C3×Q8—204no—
180C2×C3×C4—22no—
181C2×C3×C4—51no—
182C2×C3×C4—52no—
183C2×C3×C4—52no—
184C2×C3×C4—52no—
185C2×C3×C4—104no—
186C2×C3×C4—102no—
187C2×C3×C4—202no—
188C2×C3×C4—202no—
189C2×C3×C4—202no—
190C2×C3×C4—204no—
191C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
192C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
193C3×C8—104no—
194C22×C5C2×C3×C22⋊C411yescharacteristic
195C2×D5C3×#12411yescharacteristic
196C2×D5C3×#12711yescharacteristic
197C2×D5C3×#13714yes—
198AGL(1,𝔽5)—82no—
199C4×C5—22no—
200C4×C5—42no—
201C4×C5C3×C4×D412yes—
202Dic5—22no—
203Dic5—42no—
204Dic5C3×#12112yes—
205C2×Q8—52no—
206C2×Q8—102no—
207C2×Q8—104no—
208C2×Q8—104no—
209C2×Q8C2×C3×AGL(1,𝔽5)12yes—
210C22×C4—51no—
211C22×C4—52no—
212C22×C4—102no—
213C22×C4—102no—
214C2×C8—54no—
215C2×C8—102no—
216C2×C8—102no—
217C4⋊C4—22no—
218C4⋊C4—104no—
219C4⋊C4—104no—
220C4⋊C4—102no—
221C42—52no—
222C42—51no—
223C42C2×C3×AGL(1,𝔽5)11yescharacteristic
224C3×C5#196011yescharacteristic, complemented, Hall
225C22×C3—51no—
226C22×C3—51no—
227C22×C3—54no—
228C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
229C3×C4—22no—
230C3×C4—42no—
231C3×C4—52no—
232C3×C4—102no—
233C3×C4—202no—
234C3×C4—402no—
235C3×C4D4×AGL(1,𝔽5)12yes—
236C2×C5C3×#39011yescharacteristic
237C2×C5C3×#41912yes—
238D5C3×#49112yes—
239D5C3×#49712yes—
240C23—51no—
241Q8—24no—
242Q8—104no—
243Q8—204no—
244C2×C4—22no—
245C2×C4—51no—
246C2×C4—52no—
247C2×C4—52no—
248C2×C4—52no—
249C2×C4—104no—
250C2×C4—102no—
251C2×C4—202no—
252C2×C4—202no—
253C2×C4—202no—
254C2×C4—204no—
255C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
256C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
257C8—104no—
258C2×C3—52no—
259C2×C3—52no—
260C2×C3#6578011yescharacteristic
261C2×C3#6578212yes—
262C5C3×#196011yescharacteristic, complemented, Sylow
263C22—51no—
264C22—51no—
265C22—54no—
266C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
267C4—22no—
268C4—42no—
269C4—52no—
270C4—102no—
271C4—202no—
272C4—402no—
273C4C3×D4×AGL(1,𝔽5)12yes—
274C3#1060467311yescharacteristic, direct product factor, Sylow
275C2—52no—
276C2—52no—
277C2C3×#6578011yescharacteristic
278C2C3×#6578212yes—
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C4⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C4×D5C23×C3
hypercenterC3×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1960
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
5. #10604673C3direct (N × H)
9. C3×C5×C4⋊Q8C4semidirect
10. C3×#4090C4semidirect
14. C3×#4015C4semidirect
49. C5×C4⋊Q8C3×C4semidirect
50. #4090C3×C4semidirect
54. #4015C3×C4semidirect
82. C3×C4⋊Q8AGL(1,𝔽5)semidirect
165. C4⋊Q8C3×AGL(1,𝔽5)semidirect
224. C3×C5#1960semidirect
262. C5C3×#1960semidirect
274. C3#10604673direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×C4⋊Q82
C2×C3×#40092
C3×#656892
C3×#656762
#106046733
C3×#19605