Finite Group Databasebeta

← C3×#10604965 #14313754

Subgroups — C3×#10604965 #14313754

Order 1920.

460 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 108 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×#66150C211yesmaximal, characteristic
2C3×#65761C212yesmaximal
3C3×#64929C214yesmaximal, complemented
4#10604965C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5×Q8C411yescharacteristic
6C2×C3×#709C2211yescharacteristic
7C3×D5⋊M16—22no—
8C3×#4093C412yes—
9C3×#4092—22no—
10C3×#4090C2212yes—
11C3×#4075—22no—
12C3×#4072C411yescharacteristic
13C3×#4013—22no—
14C3×#4005C2214yes—
15C3×#2135—51nomaximal, Hall
16#66150C2×C311yescharacteristic
17#65761C2×C312yes—
18#64929C2×C314yescomplemented
19C2×C3×C5×Q8—21no—
20C2×C3×C5×Q8—42no—
21C2×C3×C5×Q8C2×C412yes—
22C22×C3×C4×C5—21no—
23C22×C3×C4×C5C2×C411yescharacteristic
24C2×C3×C22⋊D5D412yes—
25C3×#709—42no—
26C2×C3×D20C2×C411yescharacteristic
27C2×C3×C4×D5D412yes—
28C2×C3×Dic10C2311yescharacteristic
29C3×#704—24no—
30C3×#700—42no—
31C3×D5⋊C8—42no—
32C3×C22⋊Dic5—21no—
33C3×#685—24no—
34C3×C4⋊Dic5—21no—
35C3×#683—24no—
36C3×C4×Dic5C2×C412yes—
37C3×#536—51no—
38C3×#413—52no—
39C3×#356—54no—
40C22×C5×Q8C3×C411yescharacteristic
41C2×#709C22×C311yescharacteristic
42D5⋊M16—22no—
43#4093C3×C412yes—
44#4092—22no—
45#4090C22×C312yes—
46#4075—22no—
47#4072C3×C411yescharacteristic
48#4013—22no—
49#4005C22×C314yes—
50#2135—51noSylow
51C23×C3×C5C22⋊C411yescharacteristic
52C22×C3×D5C22⋊C412yes—
53C3×C5×Q8—42no—
54C3×C5×Q8—42no—
55C2×C3×C4×C5—22no—
56C2×C3×C4×C5—22no—
57C2×C3×C4×C5—42no—
58C2×C3×C4×C5C22⋊C411yescharacteristic
59C2×C3×C4×C5C22×C411yescharacteristic
60C3×C22⋊D5—42no—
61C2×C3×Dic5—22no—
62C2×C3×Dic5C2×D412yes—
63C3×D20—41no—
64C3×C4×D5—22no—
65C3×C4×D5—22no—
66C3×Dic10—41no—
67C3×#170—44no—
68C2×C3×#41—51no—
69C22×C3×Q8AGL(1,𝔽5)11yescharacteristic
70C2×C3×M16—102no—
71C3×C4⋊Q8—52no—
72C3×#124—52no—
73C3×#123—102no—
74C3×C22⋊Q8—102no—
75C3×#117—51no—
76C3×#101—54no—
77C3×#100—102no—
78C2×C5×Q8—21no—
79C2×C5×Q8—42no—
80C2×C5×Q8C2×C3×C412yes—
81C22×C4×C5—21no—
82C22×C4×C5C2×C3×C411yescharacteristic
83C2×C22⋊D5C3×D412yes—
84#709—42no—
85C2×D20C2×C3×C411yescharacteristic
86C2×C4×D5C3×D412yes—
87C2×Dic10C23×C311yescharacteristic
88#704—24no—
89#700—42no—
90D5⋊C8—42no—
91C22⋊Dic5—21no—
92#685—24no—
93C4⋊Dic5—21no—
94#683—24no—
95C4×Dic5C2×C3×C412yes—
96#536—51no—
97#413—52no—
98#356—54no—
99C22×C3×C5—21no—
100C22×C3×C5—41no—
101C22×C3×C5C2×C22⋊C411yescharacteristic
102C2×C3×D5—22no—
103C2×C3×D5—42no—
104C3×C4×C5—21no—
105C3×C4×C5—42no—
106C3×C4×C5#9912yes—
107C3×Dic5—22no—
108C3×Dic5—42no—
109C3×#41—202no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×Q8—51no—
113C2×C3×Q8C2×AGL(1,𝔽5)12yes—
114C2×C3×D4—52no—
115C2×C3×D4—51no—
116C22×C3×C4—21no—
117C22×C3×C4—52no—
118C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
119C3×M16—104no—
120C3×M16—202no—
121C2×C3×C8—202no—
122C3×C4⋊C4—104no—
123C3×C4⋊C4—101no—
124C3×C22⋊C4—104no—
125C3×C22⋊C4—101no—
126C3×C42—52no—
127C23×C5C3×C22⋊C411yescharacteristic
128C22×D5C3×C22⋊C412yes—
129C5×Q8—42no—
130C5×Q8—42no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—42no—
134C2×C4×C5C3×C22⋊C411yescharacteristic
135C2×C4×C5C22×C3×C411yescharacteristic
136C22⋊D5—42no—
137C2×Dic5—22no—
138C2×Dic5C2×C3×D412yes—
139D20—41no—
140C4×D5—22no—
141C4×D5—22no—
142Dic10—41no—
143#170—44no—
144C2×#41—51no—
145C22×Q8C3×AGL(1,𝔽5)11yescharacteristic
146C2×M16—102no—
147C4⋊Q8—52no—
148#124—52no—
149#123—102no—
150C22⋊Q8—102no—
151#117—51no—
152#101—54no—
153#100—102no—
154C2×C3×C5—21no—
155C2×C3×C5—41no—
156C2×C3×C5C2×#9911yescharacteristic
157C3×D5—42no—
158C23×C3—52no—
159C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
160C3×Q8—42no—
161C3×Q8—42no—
162C3×Q8—201no—
163C3×D4—201no—
164C3×D4—202no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—42no—
168C2×C3×C4—52no—
169C2×C3×C4—102no—
170C2×C3×C4—102no—
171C2×C3×C4—102no—
172C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
173C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
174C3×C8—204no—
175C22×C5—21no—
176C22×C5—41no—
177C22×C5C2×C3×C22⋊C411yescharacteristic
178C2×D5—22no—
179C2×D5—42no—
180C4×C5—21no—
181C4×C5—42no—
182C4×C5C3×#9912yes—
183Dic5—22no—
184Dic5—42no—
185#41—202no—
186C2×Q8—21no—
187C2×Q8—42no—
188C2×Q8—51no—
189C2×Q8C2×C3×AGL(1,𝔽5)12yes—
190C2×D4—52no—
191C2×D4—51no—
192C22×C4—21no—
193C22×C4—52no—
194C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
195M16—104no—
196M16—202no—
197C2×C8—202no—
198C4⋊C4—104no—
199C4⋊C4—101no—
200C22⋊C4—104no—
201C22⋊C4—101no—
202C42—52no—
203C3×C5#213511yescharacteristic, complemented, Hall
204C22×C3—21no—
205C22×C3—41no—
206C22×C3—102no—
207C22×C3—202no—
208C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
209C3×C4—21no—
210C3×C4—42no—
211C3×C4—102no—
212C3×C4—202no—
213C3×C4#401112yes—
214C2×C5—21no—
215C2×C5—41no—
216C2×C5C2×C3×#9911yescharacteristic
217D5—42no—
218C23—52no—
219C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
220Q8—42no—
221Q8—42no—
222Q8—201no—
223D4—201no—
224D4—202no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—52no—
229C2×C4—102no—
230C2×C4—102no—
231C2×C4—102no—
232C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
233C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
234C8—204no—
235C2×C3—21no—
236C2×C3—41no—
237C2×C3—202no—
238C2×C3C2×#401111yescharacteristic
239C5C3×#213511yescharacteristic, complemented, Sylow
240C22—21no—
241C22—41no—
242C22—102no—
243C22—202no—
244C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
245C4—21no—
246C4—42no—
247C4—102no—
248C4—202no—
249C4C3×#401112yes—
250C3#1060496511yescharacteristic, direct product factor, Sylow
251C2—21no—
252C2—41no—
253C2—202no—
254C2C2×C3×#401111yescharacteristic
subgroupG/H
centerC2×C3C2×#4011
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×Q8C4
socleC2×C3×C5C2×#99
prime radicalC2×C3×#709C22
cosocleC2×Dic10C23×C3
hypercenterC22×C3×Q8AGL(1,𝔽5)
hypocenterC5C3×#2135
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#64929C2semidirect
4. #10604965C3direct (N × H)
18. #64929C2×C3semidirect
203. C3×C5#2135semidirect
239. C5C3×#2135semidirect
250. C3#10604965direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661502
C3×#657612
C3×#649292
#106049653
C3×#21355