Finite Group Databasebeta

← C3×#10604986 #14296924

Subgroups — C3×#10604986 #14296924

Order 1920.

1128 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 180 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
1C22×C3×Q8×D5C211yesmaximal, characteristic
2C2×C3×#4005C214yesmaximal, complemented
3C3×#65755C212yesmaximal
4#10604986C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5×Q8C411yescharacteristic
6C2×C3×Q8×D5—24no—
7C2×C3×Q8×D5—28no—
8C22×C3×C4×D5C412yes—
9C22×C3×C4×D5C2211yescharacteristic
10C22×C3×Dic10C411yescharacteristic
11C22×C3×Dic10C2212yes—
12C2×C3×#704C2214yes—
13C3×#4005—28no—
14C3×#4003—24no—
15C3×#1784—51nomaximal, Hall
16C22×Q8×D5C2×C311yescharacteristic
17C2×#4005C2×C314yescomplemented
18#65755C2×C312yes—
19C23×C3×D5C2×C411yescharacteristic
20C2×C3×C5×Q8—22no—
21C2×C3×C5×Q8—24no—
22C22×C3×C4×C5C2×C412yes—
23C22×C3×C4×C5C2×C411yescharacteristic
24C22×C3×Dic5C2×C412yes—
25C22×C3×Dic5C2311yescharacteristic
26C3×Q8×D5—232no—
27C2×C3×C4×D5—24no—
28C2×C3×C4×D5—24no—
29C2×C3×C4×D5—24no—
30C2×C3×C4×D5—24no—
31C2×C3×C4×D5D414yes—
32C2×C3×Dic10—22no—
33C2×C3×Dic10—24no—
34C2×C3×Dic10—24no—
35C2×C3×Dic10—24no—
36C2×C3×Dic10D418yes—
37C3×#704—28no—
38C2×C3×#170—24no—
39C23×C3×Q8—51no—
40C2×C3×#101—54no—
41C3×#407—52no—
42C22×C5×Q8C3×C411yescharacteristic
43C2×Q8×D5—24no—
44C2×Q8×D5—28no—
45C22×C4×D5C3×C412yes—
46C22×C4×D5C22×C311yescharacteristic
47C22×Dic10C3×C411yescharacteristic
48C22×Dic10C22×C312yes—
49C2×#704C22×C314yes—
50#4005—28no—
51#4003—24no—
52#1784—51noSylow
53C23×C3×C5C22×C411yescharacteristic
54C22×C3×D5—22no—
55C22×C3×D5—24no—
56C22×C3×D5C22⋊C412yes—
57C3×C5×Q8—28no—
58C2×C3×C4×C5—22no—
59C2×C3×C4×C5—22no—
60C2×C3×C4×C5—22no—
61C2×C3×C4×C5C22⋊C412yes—
62C2×C3×C4×C5C22⋊C414yes—
63C2×C3×Dic5—22no—
64C2×C3×Dic5—22no—
65C2×C3×Dic5C22⋊C414yes—
66C2×C3×Dic5C2×D412yes—
67C2×C3×Dic5C2×D414yes—
68C3×C4×D5—216no—
69C3×C4×D5—28no—
70C3×Dic10—28no—
71C3×Dic10—216no—
72C3×#170—44no—
73C22×C3×Q8—52no—
74C22×C3×Q8—51no—
75C22×C3×Q8—104no—
76C22×C3×Q8—108no—
77C22×C3×Q8AGL(1,𝔽5)11yescharacteristic
78C23×C3×C4—51no—
79C23×C3×C4—52no—
80C2×C3×M16—54no—
81C3×#101—108no—
82C3×C22⋊C8—104no—
83C23×D5C2×C3×C411yescharacteristic
84C2×C5×Q8—22no—
85C2×C5×Q8—24no—
86C22×C4×C5C2×C3×C411yescharacteristic
87C22×C4×C5C2×C3×C412yes—
88C22×Dic5C2×C3×C412yes—
89C22×Dic5C23×C311yescharacteristic
90Q8×D5—232no—
91C2×C4×D5—24no—
92C2×C4×D5—24no—
93C2×C4×D5—24no—
94C2×C4×D5—24no—
95C2×C4×D5C3×D414yes—
96C2×Dic10—22no—
97C2×Dic10—24no—
98C2×Dic10—24no—
99C2×Dic10—24no—
100C2×Dic10C3×D418yes—
101#704—28no—
102C2×#170—24no—
103C23×Q8—51no—
104C2×#101—54no—
105#407—52no—
106C22×C3×C5—22no—
107C22×C3×C5C2×C22⋊C412yes—
108C22×C3×C5C2×C22⋊C411yescharacteristic
109C2×C3×D5—28no—
110C2×C3×D5—22no—
111C2×C3×D5—22no—
112C2×C3×D5#10114yescomplemented
113C3×C4×C5—22no—
114C3×C4×C5—24no—
115C3×Dic5—24no—
116C3×Dic5#12014yes—
117C24×C3—51no—
118C2×C3×Q8—24no—
119C2×C3×Q8—22no—
120C2×C3×Q8—58no—
121C2×C3×Q8—104no—
122C2×C3×Q8—104no—
123C2×C3×Q8—104no—
124C2×C3×Q8—1032no—
125C2×C3×Q8—102no—
126C22×C3×C4—51no—
127C22×C3×C4—52no—
128C22×C3×C4—54no—
129C22×C3×C4—104no—
130C22×C3×C4—104no—
131C22×C3×C4—104no—
132C22×C3×C4—104no—
133C22×C3×C4C2×AGL(1,𝔽5)12yes—
134C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
135C3×M16—108no—
136C2×C3×C8—104no—
137C23×C5C22×C3×C411yescharacteristic
138C22×D5—22no—
139C22×D5—24no—
140C22×D5C3×C22⋊C412yes—
141C5×Q8—28no—
142C2×C4×C5—22no—
143C2×C4×C5—22no—
144C2×C4×C5—22no—
145C2×C4×C5C3×C22⋊C412yes—
146C2×C4×C5C3×C22⋊C414yes—
147C2×Dic5—22no—
148C2×Dic5—22no—
149C2×Dic5C3×C22⋊C414yes—
150C2×Dic5C2×C3×D412yes—
151C2×Dic5C2×C3×D414yes—
152C4×D5—216no—
153C4×D5—28no—
154Dic10—28no—
155Dic10—216no—
156#170—44no—
157C22×Q8—52no—
158C22×Q8—51no—
159C22×Q8—104no—
160C22×Q8—108no—
161C22×Q8C3×AGL(1,𝔽5)11yescharacteristic
162C23×C4—51no—
163C23×C4—52no—
164C2×M16—54no—
165#101—108no—
166C22⋊C8—104no—
167C2×C3×C5—22no—
168C2×C3×C5#37911yescharacteristic
169C2×C3×C5C2×#10112yes—
170C3×D5—24no—
171C23×C3—52no—
172C23×C3—104no—
173C23×C3—102no—
174C23×C3C22×AGL(1,𝔽5)11yescharacteristic
175C3×Q8—28no—
176C3×Q8—1016no—
177C3×Q8—108no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—54no—
182C2×C3×C4—52no—
183C2×C3×C4—54no—
184C2×C3×C4—108no—
185C2×C3×C4—1016no—
186C2×C3×C4—102no—
187C2×C3×C4—102no—
188C2×C3×C4C22⋊AGL(1,𝔽5)14yes—
189C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
190C3×C8—204no—
191C22×C5—22no—
192C22×C5C2×C3×C22⋊C412yes—
193C22×C5C2×C3×C22⋊C411yescharacteristic
194C2×D5—28no—
195C2×D5—22no—
196C2×D5—22no—
197C2×D5C3×#10114yescomplemented
198C4×C5—22no—
199C4×C5—24no—
200Dic5—24no—
201Dic5C3×#12014yes—
202C24—51no—
203C2×Q8—22no—
204C2×Q8—24no—
205C2×Q8—58no—
206C2×Q8—104no—
207C2×Q8—104no—
208C2×Q8—104no—
209C2×Q8—1032no—
210C2×Q8—102no—
211C22×C4—51no—
212C22×C4—52no—
213C22×C4—54no—
214C22×C4—104no—
215C22×C4—104no—
216C22×C4—104no—
217C22×C4—104no—
218C22×C4C2×C3×AGL(1,𝔽5)12yes—
219C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
220M16—108no—
221C2×C8—104no—
222C3×C5#178411yescharacteristic, complemented, Hall
223C22×C3—22no—
224C22×C3—54no—
225C22×C3—108no—
226C22×C3—102no—
227C22×C3—102no—
228C22×C3C2×C22⋊AGL(1,𝔽5)12yes—
229C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
230C3×C4—24no—
231C3×C4—22no—
232C3×C4—54no—
233C3×C4—104no—
234C2×C5—22no—
235C2×C5C3×#37911yescharacteristic
236C2×C5C2×C3×#10112yes—
237D5—24no—
238C23—52no—
239C23—104no—
240C23—102no—
241C23C22×C3×AGL(1,𝔽5)11yescharacteristic
242Q8—28no—
243Q8—1016no—
244Q8—108no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—52no—
249C2×C4—54no—
250C2×C4—54no—
251C2×C4—108no—
252C2×C4—1016no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
256C2×C4C3×C22⋊AGL(1,𝔽5)14yes—
257C8—204no—
258C2×C3—22no—
259C2×C3—104no—
260C2×C3D5⋊#10112yes—
261C2×C3#6580011yescharacteristic
262C5C3×#178411yescharacteristic, complemented, Sylow
263C22—22no—
264C22—54no—
265C22—108no—
266C22—102no—
267C22—102no—
268C22C2×C3×C22⋊AGL(1,𝔽5)12yes—
269C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
270C4—22no—
271C4—24no—
272C4—54no—
273C4—104no—
274C3#1060498611yescharacteristic, direct product factor, Sylow
275C2—22no—
276C2—104no—
277C2C3×D5⋊#10112yes—
278C2C3×#6580011yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5C2×C4
cosocleC22×Dic5C23×C3
hypercenterC22×C3×Q8AGL(1,𝔽5)
hypocenterC5C3×#1784
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4005C2semidirect
4. #10604986C3direct (N × H)
17. C2×#4005C2×C3semidirect
112. C2×C3×D5#101semidirect
197. C2×D5C3×#101semidirect
222. C3×C5#1784semidirect
262. C5C3×#1784semidirect
274. C3#10604986direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8×D52
C2×C3×#40052
C3×#657552
#106049863
C3×#17845