Finite Group Databasebeta

← C3×#10599947 #14316782

Subgroups — C3×#10599947 #14316782

Order 1920.

540 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 82 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×#66017C211yesmaximal, characteristic, complemented
2C3×#65350C211yesmaximal, characteristic, complemented
3C3×#65030C212yesmaximal, complemented
4C3×C5×#354C211yesmaximal, characteristic, complemented
5C3×#64691C212yesmaximal, complemented
6#10599947C311yesmaximal, characteristic, direct product factor, Hall
7C3×#4144C2211yescharacteristic
8C3×C5×#124C2211yescharacteristic
9C3×#4093—21no—
10C3×#4088—21no—
11C3×#4083C2212yes—
12C3×#4031—22no—
13C3×#4028—22no—
14C3×#4028—22no—
15C3×#4021—21no—
16C3×C5×#99C2212yes—
17C3×#3965C2211yescharacteristic
18C3×#3938—22no—
19C3×#3937—21no—
20C3×#2199—51nomaximal, Hall
21#66017C2×C311yescharacteristic, complemented
22#65350C2×C311yescharacteristic, complemented
23#65030C2×C312yescomplemented
24C5×#354C2×C311yescharacteristic, complemented
25#64691C2×C312yescomplemented
26C23×C3×D5D412yescomplemented
27C2×C3×C5×Q8D411yescharacteristic, complemented
28C2×C3×C5×D4C2311yescharacteristic
29C2×C3×C22⋊D5—22no—
30C2×C3×C22⋊D5—22no—
31C3×#711—41no—
32C3×D4×D5—41no—
33C3×#709—21no—
34C2×C3×D20—21no—
35C2×C3×D20—21no—
36C2×C3×D20—42no—
37C2×C3×Dic10—21no—
38C3×C5×C22⋊C4—22no—
39C3×C5×C22⋊C4—22no—
40C3×C42×C5D411yescharacteristic, complemented
41C3×C22⋊Dic5D412yescomplemented
42C3×#689—41no—
43C3×#688—41no—
44C3×#685—22no—
45C3×#685—22no—
46C3×#685—22no—
47C3×#685—42no—
48C3×C4×Dic5—21no—
49C3×#681—21no—
50C3×H(3,𝔽4)—51no—
51C3×B(4,𝔽2)—52no—
52C3×#455—51no—
53C3×#354D511yescharacteristic, complemented
54C3×AΣL(1,𝔽16)—52no—
55#4144C22×C311yescharacteristic
56C5×#124C22×C311yescharacteristic
57#4093—21no—
58#4088—21no—
59#4083C22×C312yes—
60#4031—22no—
61#4028—22no—
62#4028—22no—
63#4021—21no—
64C5×#99C22×C312yes—
65#3965C22×C311yescharacteristic
66#3938—22no—
67#3937—21no—
68#2199—51noSylow
69C23×C3×C5C2×D412yes—
70C22×C3×D5—22no—
71C22×C3×D5—22no—
72C22×C3×D5—42no—
73C22×C3×D5—82no—
74C3×C5×Q8—41no—
75C3×C5×D4—41no—
76C2×C3×C4×C5—21no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—42no—
79C2×C3×C4×C5C2×D411yescharacteristic
80C3×C22⋊D5—42no—
81C3×C22⋊D5—41no—
82C3×C22⋊D5—82no—
83C2×C3×Dic5—22no—
84C2×C3×Dic5—22no—
85C3×D20—21no—
86C3×D20—81no—
87C3×D20—81no—
88C3×D20—84no—
89C3×C4×D5—41no—
90C3×Dic10—21no—
91C3×Dic10—81no—
92C3×#168—41no—
93C3×H(5,𝔽2)—51no—
94C3×#137—101no—
95C3×#124—102no—
96C3×#124—101no—
97C3×#124—101no—
98C3×#124C2×D511yescharacteristic
99C3×#120—52no—
100C3×#120—102no—
101C3×#120—102no—
102C3×#104—101no—
103C3×#100—51no—
104C3×#99—102no—
105C3×#99C2×D512yes—
106C23×D5C3×D412yescomplemented
107C2×C5×Q8C3×D411yescharacteristic, complemented
108C2×C5×D4C23×C311yescharacteristic
109C2×C22⋊D5—22no—
110C2×C22⋊D5—22no—
111#711—41no—
112D4×D5—41no—
113#709—21no—
114C2×D20—21no—
115C2×D20—21no—
116C2×D20—42no—
117C2×Dic10—21no—
118C5×C22⋊C4—22no—
119C5×C22⋊C4—22no—
120C42×C5C3×D411yescharacteristic, complemented
121C22⋊Dic5C3×D412yescomplemented
122#689—41no—
123#688—41no—
124#685—22no—
125#685—22no—
126#685—22no—
127#685—42no—
128C4×Dic5—21no—
129#681—21no—
130H(3,𝔽4)—51no—
131B(4,𝔽2)—52no—
132#455—51no—
133#354C3×D511yescharacteristic, complemented
134AΣL(1,𝔽16)—52no—
135C22×C3×C5—22no—
136C22×C3×C5—42no—
137C22×C3×C5#12011yescharacteristic
138C2×C3×D5—42no—
139C2×C3×D5—42no—
140C2×C3×D5—44no—
141C2×C3×D5—41no—
142C2×C3×D5—42no—
143C2×C3×D5—82no—
144C2×C3×D5—82no—
145C3×C4×C5—21no—
146C3×C4×C5—41no—
147C3×C4×C5—41no—
148C3×C4×C5—82no—
149C3×Dic5—42no—
150C3×Dic5—41no—
151C24×C3—52no—
152C3×#41—101no—
153C3×#41—201no—
154C2×C3×Q8—101no—
155C2×C3×Q8D2011yescharacteristic, complemented
156C2×C3×D4—101no—
157C2×C3×D4—101no—
158C2×C3×D4—102no—
159C2×C3×D4—102no—
160C2×C3×D4—202no—
161C2×C3×D4—201no—
162C2×C3×D4C22×D511yescharacteristic
163C3×Q16—201no—
164C3×SD16—201no—
165C3×M16—101no—
166C3×C22⋊C4—22no—
167C3×C22⋊C4—22no—
168C3×C22⋊C4—52no—
169C3×C22⋊C4—102no—
170C3×C22⋊C4—102no—
171C3×C22⋊C4—102no—
172C3×C22⋊C4—202no—
173C3×C42—101no—
174C3×C42D2011yescharacteristic, complemented
175C23×C5C2×C3×D412yes—
176C22×D5—22no—
177C22×D5—22no—
178C22×D5—42no—
179C22×D5—82no—
180C5×Q8—41no—
181C5×D4—41no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—42no—
185C2×C4×C5C2×C3×D411yescharacteristic
186C22⋊D5—42no—
187C22⋊D5—41no—
188C22⋊D5—82no—
189C2×Dic5—22no—
190C2×Dic5—22no—
191D20—21no—
192D20—81no—
193D20—81no—
194D20—84no—
195C4×D5—41no—
196Dic10—21no—
197Dic10—81no—
198#168—41no—
199H(5,𝔽2)—51no—
200#137—101no—
201#124—102no—
202#124—101no—
203#124—101no—
204#124C2×C3×D511yescharacteristic
205#120—52no—
206#120—102no—
207#120—102no—
208#104—101no—
209#100—51no—
210#99—102no—
211#99C2×C3×D512yes—
212C2×C3×C5—21no—
213C2×C3×C5—42no—
214C2×C3×C5B(4,𝔽2)11yescharacteristic
215C3×D5—82no—
216C3×D5—81no—
217C23×C3—102no—
218C23×C3—102no—
219C23×C3—202no—
220C23×C3—402no—
221C23×C3D4×D512yes—
222C3×Q8—41no—
223C3×Q8—101no—
224C3×Q8—401no—
225C3×D4—41no—
226C3×D4—101no—
227C3×D4—202no—
228C3×D4—201no—
229C3×D4—401no—
230C3×D4—401no—
231C3×D4—404no—
232C3×D4—402no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—42no—
236C2×C3×C4—102no—
237C2×C3×C4—102no—
238C2×C3×C4—201no—
239C2×C3×C4C2×D2011yescharacteristic
240C3×C8—201no—
241C22×C5—22no—
242C22×C5—42no—
243C22×C5C3×#12011yescharacteristic
244C2×D5—42no—
245C2×D5—42no—
246C2×D5—44no—
247C2×D5—41no—
248C2×D5—42no—
249C2×D5—82no—
250C2×D5—82no—
251C4×C5—21no—
252C4×C5—41no—
253C4×C5—41no—
254C4×C5—82no—
255Dic5—42no—
256Dic5—41no—
257C24—52no—
258#41—101no—
259#41—201no—
260C2×Q8—101no—
261C2×Q8C3×D2011yescharacteristic, complemented
262C2×D4—101no—
263C2×D4—101no—
264C2×D4—102no—
265C2×D4—102no—
266C2×D4—202no—
267C2×D4—201no—
268C2×D4C22×C3×D511yescharacteristic
269Q16—201no—
270SD16—201no—
271M16—101no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—52no—
275C22⋊C4—102no—
276C22⋊C4—102no—
277C22⋊C4—102no—
278C22⋊C4—202no—
279C42—101no—
280C42C3×D2011yescharacteristic, complemented
281C3×C5#219911yescharacteristic, complemented, Hall
282C22×C3—22no—
283C22×C3—42no—
284C22×C3—202no—
285C22×C3—202no—
286C22×C3—204no—
287C22×C3—201no—
288C22×C3—202no—
289C22×C3—402no—
290C22×C3—402no—
291C22×C3#402811yescharacteristic
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—82no—
296C3×C4—202no—
297C3×C4—201no—
298C2×C5—21no—
299C2×C5—42no—
300C2×C5C3×B(4,𝔽2)11yescharacteristic
301D5—82no—
302D5—81no—
303C23—102no—
304C23—102no—
305C23—202no—
306C23—402no—
307C23C3×D4×D512yes—
308Q8—41no—
309Q8—101no—
310Q8—401no—
311D4—41no—
312D4—101no—
313D4—202no—
314D4—201no—
315D4—401no—
316D4—401no—
317D4—404no—
318D4—402no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—102no—
323C2×C4—102no—
324C2×C4—201no—
325C2×C4C2×C3×D2011yescharacteristic
326C8—201no—
327C2×C3—21no—
328C2×C3—42no—
329C2×C3—402no—
330C2×C3—401no—
331C2×C3#6503011yescharacteristic
332C5C3×#219911yescharacteristic, complemented, Sylow
333C22—22no—
334C22—42no—
335C22—202no—
336C22—202no—
337C22—204no—
338C22—201no—
339C22—202no—
340C22—402no—
341C22—402no—
342C22C3×#402811yescharacteristic
343C4—21no—
344C4—41no—
345C4—41no—
346C4—82no—
347C4—202no—
348C4—201no—
349C3#1059994711yescharacteristic, direct product factor, Sylow
350C2—21no—
351C2—42no—
352C2—402no—
353C2—401no—
354C2C3×#6503011yescharacteristic
subgroupG/H
centerC2×C3#65030
commutator, cosocleC2×C5×D4C23×C3
FrattiniC2×D4C22×C3×D5
FittingC3×C5×#354C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC3×#354D5
hypocenterC5C3×#2199
LCSG ▷ C2×C5×D4 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#354

Complements

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

NH ≅ G/Ntype
1. C3×#66017C2semidirect
2. C3×#65350C2semidirect
3. C3×#65030C2semidirect
4. C3×C5×#354C2semidirect
5. C3×#64691C2semidirect
6. #10599947C3direct (N × H)
21. #66017C2×C3semidirect
22. #65350C2×C3semidirect
23. #65030C2×C3semidirect
24. C5×#354C2×C3semidirect
25. #64691C2×C3semidirect
26. C23×C3×D5D4semidirect
27. C2×C3×C5×Q8D4semidirect
40. C3×C42×C5D4semidirect
41. C3×C22⋊Dic5D4semidirect
53. C3×#354D5semidirect
106. C23×D5C3×D4semidirect
107. C2×C5×Q8C3×D4semidirect
120. C42×C5C3×D4semidirect
121. C22⋊Dic5C3×D4semidirect
133. #354C3×D5semidirect
155. C2×C3×Q8D20semidirect
174. C3×C42D20semidirect
261. C2×Q8C3×D20semidirect
280. C42C3×D20semidirect
281. C3×C5#2199semidirect
332. C5C3×#2199semidirect
349. C3#10599947direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660172
C3×#653502
C3×#650302
C3×C5×#3542
C3×#646912
#105999473
C3×#21995