Finite Group Databasebeta

← C3×#10599749 #14309003

Subgroups — C3×#10599749 #14309003

Order 1920.

580 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 134 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×#65817C211yesmaximal, characteristic
2C3×#65232C211yesmaximal, characteristic
3C3×#65232C212yesmaximal
4C3×C5×#342C211yesmaximal, characteristic
5C3×#64747C212yesmaximal
6#10599749C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×Dic5C2211yescharacteristic
8C2×C3×C5×C22⋊C4C2212yes—
9C2×C3×C5×C22⋊C4C2211yescharacteristic
10C2×C3×C22⋊Dic5C2211yescharacteristic
11C2×C3×C22⋊Dic5C2212yes—
12C3×#4070—22no—
13C3×#4024—24no—
14C3×#3963—22no—
15C3×#3963—24no—
16C3×#2030—51nomaximal, Hall
17#65817C2×C311yescharacteristic
18#65232C2×C311yescharacteristic
19#65232C2×C312yes—
20C5×#342C2×C311yescharacteristic
21#64747C2×C312yes—
22C24×C3×C5C2311yescharacteristic
23C22×C3×C4×C5—22no—
24C22×C3×C4×C5D412yes—
25C22×C3×Dic5—21no—
26C22×C3×Dic5—22no—
27C22×C3×Dic5—22no—
28C22×C3×Dic5—21no—
29C22×C3×Dic5—21no—
30C22×C3×Dic5D414yes—
31C22×C3×Dic5Q812yescomplemented
32C2×C3×Dic10—22no—
33C3×C5×C22⋊C4—22no—
34C3×C5×C22⋊C4—22no—
35C3×C5×C22⋊C4—22no—
36C3×C22⋊Dic5—22no—
37C3×C22⋊Dic5—22no—
38C3×C22⋊Dic5—22no—
39C3×C4⋊Dic5—22no—
40C3×#683—24no—
41C3×#543—51no—
42C3×#381—52no—
43C3×#381—51no—
44C3×#342—52no—
45C3×#342D511yescharacteristic
46C23×Dic5C22×C311yescharacteristic
47C2×C5×C22⋊C4C22×C312yes—
48C2×C5×C22⋊C4C22×C311yescharacteristic
49C2×C22⋊Dic5C22×C311yescharacteristic
50C2×C22⋊Dic5C22×C312yes—
51#4070—22no—
52#4024—24no—
53#3963—22no—
54#3963—24no—
55#2030—51noSylow
56C23×C3×C5—42no—
57C23×C3×C5C2×D411yescharacteristic
58C23×C3×C5C2×D412yes—
59C23×C3×C5C2×Q811yescharacteristic
60C23×C3×C5#4112yes—
61C23×C3×C5#4111yescharacteristic
62C2×C3×C4×C5—22no—
63C2×C3×C4×C5—42no—
64C2×C3×C4×C5—44no—
65C2×C3×C4×C5—42no—
66C2×C3×Dic5—24no—
67C2×C3×Dic5—24no—
68C2×C3×Dic5—22no—
69C2×C3×Dic5—41no—
70C2×C3×Dic5—41no—
71C2×C3×Dic5—42no—
72C2×C3×Dic5—44no—
73C2×C3×Dic5—42no—
74C2×C3×Dic5—41no—
75C3×Dic10—82no—
76C23×C3×C4—51no—
77C3×C22⋊Q8—104no—
78C3×C22⋊Q8—102no—
79C2×C3×C22⋊C4—52no—
80C2×C3×C22⋊C4—51no—
81C2×C3×C22⋊C4C2×D512yes—
82C2×C3×C22⋊C4C2×D511yescharacteristic
83C3×#95—104no—
84C3×#95—102no—
85C24×C5C23×C311yescharacteristic
86C22×C4×C5—22no—
87C22×C4×C5C3×D412yes—
88C22×Dic5—21no—
89C22×Dic5—22no—
90C22×Dic5—22no—
91C22×Dic5—21no—
92C22×Dic5—21no—
93C22×Dic5C3×D414yes—
94C22×Dic5C3×Q812yescomplemented
95C2×Dic10—22no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4—22no—
98C5×C22⋊C4—22no—
99C22⋊Dic5—22no—
100C22⋊Dic5—22no—
101C22⋊Dic5—22no—
102C4⋊Dic5—22no—
103#683—24no—
104#543—51no—
105#381—52no—
106#381—51no—
107#342—52no—
108#342C3×D511yescharacteristic
109C22×C3×C5—22no—
110C22×C3×C5—24no—
111C22×C3×C5—41no—
112C22×C3×C5—42no—
113C22×C3×C5—41no—
114C22×C3×C5#12011yescharacteristic
115C22×C3×C5C22⋊Q812yes—
116C22×C3×C5C22⋊Q811yescharacteristic
117C22×C3×C5#12411yescharacteristic
118C22×C3×C5#12412yes—
119C3×C4×C5—42no—
120C3×C4×C5—82no—
121C3×Dic5—42no—
122C3×Dic5—44no—
123C3×Dic5—81no—
124C24×C3C22×D511yescharacteristic
125C2×C3×Q8—102no—
126C22×C3×C4—22no—
127C22×C3×C4—52no—
128C22×C3×C4—54no—
129C22×C3×C4—101no—
130C22×C3×C4—102no—
131C22×C3×C4—102no—
132C22×C3×C4—101no—
133C22×C3×C4—101no—
134C22×C3×C4D2012yes—
135C3×C4⋊C4—104no—
136C3×C4⋊C4—102no—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—102no—
141C3×C22⋊C4—102no—
142C3×C22⋊C4—102no—
143C23×C5—42no—
144C23×C5C2×C3×D411yescharacteristic
145C23×C5C2×C3×D412yes—
146C23×C5C2×C3×Q811yescharacteristic
147C23×C5C3×#4112yes—
148C23×C5C3×#4111yescharacteristic
149C2×C4×C5—22no—
150C2×C4×C5—42no—
151C2×C4×C5—44no—
152C2×C4×C5—42no—
153C2×Dic5—24no—
154C2×Dic5—24no—
155C2×Dic5—22no—
156C2×Dic5—41no—
157C2×Dic5—41no—
158C2×Dic5—42no—
159C2×Dic5—44no—
160C2×Dic5—42no—
161C2×Dic5—41no—
162Dic10—82no—
163C23×C4—51no—
164C22⋊Q8—104no—
165C22⋊Q8—102no—
166C2×C22⋊C4—52no—
167C2×C22⋊C4—51no—
168C2×C22⋊C4C2×C3×D512yes—
169C2×C22⋊C4C2×C3×D511yescharacteristic
170#95—104no—
171#95—102no—
172C2×C3×C5—21no—
173C2×C3×C5—22no—
174C2×C3×C5—22no—
175C2×C3×C5—21no—
176C2×C3×C5#39311yescharacteristic
177C2×C3×C5#45812yes—
178C23×C3—42no—
179C23×C3C2×D2011yescharacteristic
180C23×C3#70912yes—
181C23×C3D4×D512yes—
182C23×C3#71111yescharacteristic
183C23×C3Q8×D511yescharacteristic
184C3×Q8—402no—
185C2×C3×C4—22no—
186C2×C3×C4—42no—
187C2×C3×C4—44no—
188C2×C3×C4—42no—
189C2×C3×C4—104no—
190C2×C3×C4—104no—
191C2×C3×C4—102no—
192C2×C3×C4—201no—
193C2×C3×C4—201no—
194C2×C3×C4—202no—
195C2×C3×C4—204no—
196C2×C3×C4—202no—
197C2×C3×C4—201no—
198C22×C5—22no—
199C22×C5—24no—
200C22×C5—41no—
201C22×C5—42no—
202C22×C5—41no—
203C22×C5C3×#12011yescharacteristic
204C22×C5C3×C22⋊Q812yes—
205C22×C5C3×C22⋊Q811yescharacteristic
206C22×C5C3×#12411yescharacteristic
207C22×C5C3×#12412yes—
208C4×C5—42no—
209C4×C5—82no—
210Dic5—42no—
211Dic5—44no—
212Dic5—81no—
213C24C22×C3×D511yescharacteristic
214C2×Q8—102no—
215C22×C4—22no—
216C22×C4—52no—
217C22×C4—54no—
218C22×C4—101no—
219C22×C4—102no—
220C22×C4—102no—
221C22×C4—101no—
222C22×C4—101no—
223C22×C4C3×D2012yes—
224C4⋊C4—104no—
225C4⋊C4—102no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—102no—
230C22⋊C4—102no—
231C22⋊C4—102no—
232C3×C5#203011yescharacteristic, complemented, Hall
233C22×C3—22no—
234C22×C3—24no—
235C22×C3—42no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3#402111yescharacteristic
239C22×C3#402811yescharacteristic
240C22×C3#403112yes—
241C22×C3#404212yes—
242C22×C3#404311yescharacteristic
243C3×C4—42no—
244C3×C4—82no—
245C3×C4—204no—
246C3×C4—202no—
247C3×C4—401no—
248C2×C5—21no—
249C2×C5—22no—
250C2×C5—22no—
251C2×C5—21no—
252C2×C5C3×#39311yescharacteristic
253C2×C5C3×#45812yes—
254C23—42no—
255C23C2×C3×D2011yescharacteristic
256C23C3×#70912yes—
257C23C3×D4×D512yes—
258C23C3×#71111yescharacteristic
259C23C3×Q8×D511yescharacteristic
260Q8—402no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—104no—
266C2×C4—104no—
267C2×C4—102no—
268C2×C4—201no—
269C2×C4—201no—
270C2×C4—202no—
271C2×C4—204no—
272C2×C4—202no—
273C2×C4—201no—
274C2×C3—22no—
275C2×C3—21no—
276C2×C3—22no—
277C2×C3—21no—
278C2×C3#6496211yescharacteristic
279C2×C3#6503112yes—
280C5C3×#203011yescharacteristic, complemented, Sylow
281C22—22no—
282C22—24no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22C3×#402111yescharacteristic
287C22C3×#402811yescharacteristic
288C22C3×#403112yes—
289C22C3×#404212yes—
290C22C3×#404311yescharacteristic
291C4—42no—
292C4—82no—
293C4—204no—
294C4—202no—
295C4—401no—
296C3#1059974911yescharacteristic, direct product factor, Sylow
297C2—22no—
298C2—21no—
299C2—22no—
300C2—21no—
301C2C3×#6496211yescharacteristic
302C2C3×#6503112yes—
subgroupG/H
centerC22×C3#4028
commutator, cosocleC24×C5C23×C3
FrattiniC24C22×C3×D5
FittingC3×C5×#342C2
socleC22×C3×C5#120
prime radicalC24×C3×C5C23
hypercenterC3×#342D5
hypocenterC5C3×#2030
LCSG ▷ C24×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C3×#342

Complements

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

NH ≅ G/Ntype
6. #10599749C3direct (N × H)
31. C22×C3×Dic5Q8semidirect
94. C22×Dic5C3×Q8semidirect
232. C3×C5#2030semidirect
280. C5C3×#2030semidirect
296. C3#10599749direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658172
C3×#652322
C3×C5×#3422
C3×#647472
#105997493
C3×#20305