Finite Group Databasebeta

← C3×#10602743 #14316776

Subgroups — C3×#10602743 #14316776

Order 1920.

540 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 90 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×C5×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2C3×#65350C211yesmaximal, characteristic, complemented
3C3×#65321C212yesmaximal, complemented
4C3×#64761C211yesmaximal, characteristic, complemented
5C3×#64756C212yesmaximal, complemented
6#10602743C311yesmaximal, characteristic, direct product factor, Hall
7C3×#4144C2211yescharacteristic
8C3×C5×#124—21no—
9C3×C5×#124—22no—
10C3×C5×#124—21no—
11C3×C5×#124C2211yescharacteristic
12C3×C5×#120—22no—
13C3×C5×#120C2212yes—
14C3×#4099—22no—
15C3×#4088—21no—
16C3×#3966C2212yes—
17C3×#3965C2211yescharacteristic
18C3×#3938—22no—
19C3×#3937—21no—
20C3×#2199—51nomaximal, Hall
21C5×H(3,𝔽4)C2×C311yescharacteristic, complemented
22#65350C2×C311yescharacteristic, complemented
23#65321C2×C312yescomplemented
24#64761C2×C311yescharacteristic, complemented
25#64756C2×C312yescomplemented
26C24×C3×C5D412yescomplemented
27C2×C3×C5×Q8—21no—
28C2×C3×C5×Q8D411yescharacteristic, complemented
29C2×C3×C5×D4—21no—
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4C2311yescharacteristic
33C2×C3×C22⋊D5—22no—
34C2×C3×C22⋊D5—42no—
35C3×#711—41no—
36C3×D4×D5—41no—
37C3×#709—21no—
38C3×C5×C22⋊C4—22no—
39C3×C5×C22⋊C4—22no—
40C3×C5×C22⋊C4—22no—
41C3×C5×C22⋊C4—22no—
42C3×C5×C22⋊C4D412yescomplemented
43C3×C42×C5—21no—
44C3×C42×C5D411yescharacteristic, complemented
45C3×C22⋊Dic5—22no—
46C3×C22⋊Dic5—42no—
47C3×#689—41no—
48C3×#688—41no—
49C3×#681—21no—
50C3×H(3,𝔽4)D511yescharacteristic, complemented
51C3×B(4,𝔽2)—52no—
52C3×#455—51no—
53C3×#354—51no—
54C3×AΣL(1,𝔽16)—52no—
55#4144C22×C311yescharacteristic
56C5×#124—21no—
57C5×#124—22no—
58C5×#124—21no—
59C5×#124C22×C311yescharacteristic
60C5×#120—22no—
61C5×#120C22×C312yes—
62#4099—22no—
63#4088—21no—
64#3966C22×C312yes—
65#3965C22×C311yescharacteristic
66#3938—22no—
67#3937—21no—
68#2199—51noSylow
69C23×C3×C5—22no—
70C23×C3×C5—42no—
71C23×C3×C5—82no—
72C23×C3×C5C2×D412yes—
73C22×C3×D5—22no—
74C3×C5×Q8—41no—
75C3×C5×Q8—81no—
76C3×C5×D4—41no—
77C3×C5×D4—82no—
78C3×C5×D4—81no—
79C3×C5×D4—81no—
80C2×C3×C4×C5—21no—
81C2×C3×C4×C5—22no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5C2×D411yescharacteristic
84C3×C22⋊D5—42no—
85C3×C22⋊D5—41no—
86C3×C22⋊D5—84no—
87C2×C3×Dic5—22no—
88C2×C3×Dic5—42no—
89C3×D20—21no—
90C3×C4×D5—41no—
91C3×Dic10—21no—
92C3×#168—41no—
93C3×H(5,𝔽2)—51no—
94C3×#137—101no—
95C3×#124—21no—
96C3×#124—22no—
97C3×#124—21no—
98C3×#124C2×D511yescharacteristic
99C3×#120—22no—
100C3×#120—102no—
101C3×#120C2×D512yes—
102C3×#104—101no—
103C3×#100—51no—
104C3×#99—52no—
105C3×#99—102no—
106C24×C5C3×D412yescomplemented
107C2×C5×Q8—21no—
108C2×C5×Q8C3×D411yescharacteristic, complemented
109C2×C5×D4—21no—
110C2×C5×D4—21no—
111C2×C5×D4—22no—
112C2×C5×D4C23×C311yescharacteristic
113C2×C22⋊D5—22no—
114C2×C22⋊D5—42no—
115#711—41no—
116D4×D5—41no—
117#709—21no—
118C5×C22⋊C4—22no—
119C5×C22⋊C4—22no—
120C5×C22⋊C4—22no—
121C5×C22⋊C4—22no—
122C5×C22⋊C4C3×D412yescomplemented
123C42×C5—21no—
124C42×C5C3×D411yescharacteristic, complemented
125C22⋊Dic5—22no—
126C22⋊Dic5—42no—
127#689—41no—
128#688—41no—
129#681—21no—
130H(3,𝔽4)C3×D511yescharacteristic, complemented
131B(4,𝔽2)—52no—
132#455—51no—
133#354—51no—
134AΣL(1,𝔽16)—52no—
135C22×C3×C5—22no—
136C22×C3×C5—42no—
137C22×C3×C5—42no—
138C22×C3×C5—42no—
139C22×C3×C5—44no—
140C22×C3×C5—42no—
141C22×C3×C5—82no—
142C22×C3×C5#12011yescharacteristic
143C2×C3×D5—41no—
144C2×C3×D5—82no—
145C3×C4×C5—21no—
146C3×C4×C5—42no—
147C3×C4×C5—41no—
148C3×C4×C5—41no—
149C3×Dic5—41no—
150C3×Dic5—82no—
151C24×C3C22⋊D512yescomplemented
152C3×#41—101no—
153C3×#41—201no—
154C2×C3×Q8—21no—
155C2×C3×Q8C22⋊D511yescharacteristic, complemented
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—102no—
160C2×C3×D4—201no—
161C2×C3×D4—202no—
162C2×C3×D4C22×D511yescharacteristic
163C3×Q16—201no—
164C3×SD16—201no—
165C3×M16—101no—
166C3×C22⋊C4—22no—
167C3×C22⋊C4—22no—
168C3×C22⋊C4—22no—
169C3×C22⋊C4—22no—
170C3×C22⋊C4—102no—
171C3×C22⋊C4—202no—
172C3×C22⋊C4C22⋊D512yescomplemented
173C3×C42—21no—
174C3×C42C22⋊D511yescharacteristic, complemented
175C23×C5—22no—
176C23×C5—42no—
177C23×C5—82no—
178C23×C5C2×C3×D412yes—
179C22×D5—22no—
180C5×Q8—41no—
181C5×Q8—81no—
182C5×D4—41no—
183C5×D4—82no—
184C5×D4—81no—
185C5×D4—81no—
186C2×C4×C5—21no—
187C2×C4×C5—22no—
188C2×C4×C5—21no—
189C2×C4×C5C2×C3×D411yescharacteristic
190C22⋊D5—42no—
191C22⋊D5—41no—
192C22⋊D5—84no—
193C2×Dic5—22no—
194C2×Dic5—42no—
195D20—21no—
196C4×D5—41no—
197Dic10—21no—
198#168—41no—
199H(5,𝔽2)—51no—
200#137—101no—
201#124—21no—
202#124—22no—
203#124—21no—
204#124C2×C3×D511yescharacteristic
205#120—22no—
206#120—102no—
207#120C2×C3×D512yes—
208#104—101no—
209#100—51no—
210#99—52no—
211#99—102no—
212C2×C3×C5—21no—
213C2×C3×C5—42no—
214C2×C3×C5—82no—
215C2×C3×C5B(4,𝔽2)11yescharacteristic
216C3×D5—81no—
217C23×C3—22no—
218C23×C3—42no—
219C23×C3—82no—
220C23×C3—102no—
221C23×C3C2×C22⋊D512yes—
222C3×Q8—41no—
223C3×Q8—81no—
224C3×Q8—101no—
225C3×D4—41no—
226C3×D4—82no—
227C3×D4—81no—
228C3×D4—81no—
229C3×D4—101no—
230C3×D4—202no—
231C3×D4—201no—
232C3×D4—404no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—21no—
236C2×C3×C4—102no—
237C2×C3×C4—201no—
238C2×C3×C4—202no—
239C2×C3×C4C2×C22⋊D511yescharacteristic
240C3×C8—201no—
241C22×C5—22no—
242C22×C5—42no—
243C22×C5—42no—
244C22×C5—42no—
245C22×C5—44no—
246C22×C5—42no—
247C22×C5—82no—
248C22×C5C3×#12011yescharacteristic
249C2×D5—41no—
250C2×D5—82no—
251C4×C5—21no—
252C4×C5—42no—
253C4×C5—41no—
254C4×C5—41no—
255Dic5—41no—
256Dic5—82no—
257C24C3×C22⋊D512yescomplemented
258#41—101no—
259#41—201no—
260C2×Q8—21no—
261C2×Q8C3×C22⋊D511yescharacteristic, complemented
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C2×D4—102no—
266C2×D4—201no—
267C2×D4—202no—
268C2×D4C22×C3×D511yescharacteristic
269Q16—201no—
270SD16—201no—
271M16—101no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—102no—
277C22⋊C4—202no—
278C22⋊C4C3×C22⋊D512yescomplemented
279C42—21no—
280C42C3×C22⋊D511yescharacteristic, complemented
281C3×C5#219911yescharacteristic, complemented, Hall
282C22×C3—22no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3—42no—
286C22×C3—44no—
287C22×C3—42no—
288C22×C3—82no—
289C22×C3—201no—
290C22×C3—402no—
291C22×C3#409911yescharacteristic
292C3×C4—21no—
293C3×C4—42no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—201no—
297C3×C4—402no—
298C2×C5—21no—
299C2×C5—42no—
300C2×C5—82no—
301C2×C5C3×B(4,𝔽2)11yescharacteristic
302D5—81no—
303C23—22no—
304C23—42no—
305C23—82no—
306C23—102no—
307C23C2×C3×C22⋊D512yes—
308Q8—41no—
309Q8—81no—
310Q8—101no—
311D4—41no—
312D4—82no—
313D4—81no—
314D4—81no—
315D4—101no—
316D4—202no—
317D4—201no—
318D4—404no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—102no—
323C2×C4—201no—
324C2×C4—202no—
325C2×C4C2×C3×C22⋊D511yescharacteristic
326C8—201no—
327C2×C3—21no—
328C2×C3—42no—
329C2×C3—82no—
330C2×C3—401no—
331C2×C3#6553311yescharacteristic
332C5C3×#219911yescharacteristic, complemented, Sylow
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—44no—
338C22—42no—
339C22—82no—
340C22—201no—
341C22—402no—
342C22C3×#409911yescharacteristic
343C4—21no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—201no—
348C4—402no—
349C3#1060274311yescharacteristic, direct product factor, Sylow
350C2—21no—
351C2—42no—
352C2—82no—
353C2—401no—
354C2C3×#6553311yescharacteristic
subgroupG/H
centerC2×C3#65533
commutator, cosocleC2×C5×D4C23×C3
FrattiniC2×D4C22×C3×D5
FittingC3×C5×H(3,𝔽4)C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC3×H(3,𝔽4)D5
hypocenterC5C3×#2199
LCSG ▷ C2×C5×D4 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C3×C5×H(3,𝔽4)C2semidirect
2. C3×#65350C2semidirect
3. C3×#65321C2semidirect
4. C3×#64761C2semidirect
5. C3×#64756C2semidirect
6. #10602743C3direct (N × H)
21. C5×H(3,𝔽4)C2×C3semidirect
22. #65350C2×C3semidirect
23. #65321C2×C3semidirect
24. #64761C2×C3semidirect
25. #64756C2×C3semidirect
26. C24×C3×C5D4semidirect
28. C2×C3×C5×Q8D4semidirect
42. C3×C5×C22⋊C4D4semidirect
44. C3×C42×C5D4semidirect
50. C3×H(3,𝔽4)D5semidirect
106. C24×C5C3×D4semidirect
108. C2×C5×Q8C3×D4semidirect
122. C5×C22⋊C4C3×D4semidirect
124. C42×C5C3×D4semidirect
130. H(3,𝔽4)C3×D5semidirect
151. C24×C3C22⋊D5semidirect
155. C2×C3×Q8C22⋊D5semidirect
172. C3×C22⋊C4C22⋊D5semidirect
174. C3×C42C22⋊D5semidirect
257. C24C3×C22⋊D5semidirect
261. C2×Q8C3×C22⋊D5semidirect
278. C22⋊C4C3×C22⋊D5semidirect
280. C42C3×C22⋊D5semidirect
281. C3×C5#2199semidirect
332. C5C3×#2199semidirect
349. C3#10602743direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×H(3,𝔽4)2
C3×#653502
C3×#653212
C3×#647612
C3×#647562
#106027433
C3×#21995