Finite Group Databasebeta

← C3×#10602728 #14289158

Subgroups — C3×#10602728 #14289158

Order 1920.

420 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#557C211yesmaximal, characteristic, complemented
2C3×#65377C211yesmaximal, characteristic
3C3×#65316C211yesmaximal, characteristic
4C3×#65316C211yesmaximal, characteristic
5C3×#64767C211yesmaximal, characteristic, complemented
6C3×Q8⋊#168C211yesmaximal, characteristic, complemented
7C3×Q8⋊#168C211yesmaximal, characteristic, complemented
8#10602728C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×C4⋊Q8—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×#125—22no—
12C3×C5×#125—21no—
13C3×C4×C5×Q8—21no—
14C3×C4×C5×Q8—21no—
15C3×C4×C5×Q8C2211yescharacteristic
16C3×C4×C5×Q8C2211yescharacteristic
17C2×C3×#688—21no—
18C2×C3×#688—21no—
19C3×#4020C2211yescharacteristic
20C3×#3941—21no—
21C3×#3941—21no—
22C3×#3941—22no—
23C3×C4⋊#168C2211yescharacteristic
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1626—51nomaximal, Hall
27C5×#557C2×C311yescharacteristic, complemented
28#65377C2×C311yescharacteristic
29#65316C2×C311yescharacteristic
30#65316C2×C311yescharacteristic
31#64767C2×C311yescharacteristic, complemented
32Q8⋊#168C2×C311yescharacteristic, complemented
33Q8⋊#168C2×C311yescharacteristic, complemented
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C3×C5×Q8D411yescharacteristic
37C2×C3×D20—21no—
38C2×C3×D20—21no—
39C2×C3×D20—21no—
40C3×C5×C4⋊C4—22no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C4⋊C4—22no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4D411yescharacteristic
47C3×C5×C4⋊C4D411yescharacteristic
48C3×C5×C4⋊C4D412yes—
49C3×C42×C5—22no—
50C3×C42×C5—21no—
51C3×C42×C5—21no—
52C3×C42×C5C2311yescharacteristic
53C3×#688—42no—
54C3×#688—42no—
55C2×C3×#168—21no—
56C2×C3×#168—21no—
57C2×C3×#168—21no—
58C3×#557D511yescharacteristic, complemented
59C3×#486—51no—
60C3×#460—51no—
61C3×#460—51no—
62C3×#332—51no—
63C3×Q8⋊C8—51no—
64C3×Q8⋊C8—51no—
65C5×C4⋊Q8—21no—
66C5×C4⋊Q8C22×C311yescharacteristic
67C5×#125—22no—
68C5×#125—21no—
69C4×C5×Q8—21no—
70C4×C5×Q8—21no—
71C4×C5×Q8C22×C311yescharacteristic
72C4×C5×Q8C22×C311yescharacteristic
73C2×#688—21no—
74C2×#688—21no—
75#4020C22×C311yescharacteristic
76#3941—21no—
77#3941—21no—
78#3941—22no—
79C4⋊#168C22×C311yescharacteristic
80C4⋊#168C22×C311yescharacteristic
81C4×#168C22×C311yescharacteristic
82#1626—51noSylow
83C22×C3×D5—41no—
84C3×C5×Q8—21no—
85C3×C5×Q8—21no—
86C3×C5×Q8—21no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×Q8SD1612yescomplemented
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5C2×D411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C3×D20—42no—
99C3×D20—41no—
100C3×D20—41no—
101C3×D20—41no—
102C3×D20—41no—
103C3×#168—41no—
104C3×#168—41no—
105C3×#168—41no—
106C2×C3×SD16—101no—
107C2×C3×SD16—101no—
108C3×C4⋊Q8—21no—
109C3×C4⋊Q8C2×D511yescharacteristic
110C3×#127—51no—
111C3×#125—22no—
112C3×#125—21no—
113C3×C4×Q8—21no—
114C3×C4×Q8—21no—
115C3×C4×Q8C2×D511yescharacteristic
116C3×C4×Q8C2×D511yescharacteristic
117C3×C4⋊C8—51no—
118C3×C4⋊C8—51no—
119C3×#102—101no—
120C3×#102—101no—
121C3×#102—102no—
122C3×C4×C8—51no—
123C2×C5×Q8—21no—
124C2×C5×Q8C3×D411yescharacteristic
125C2×C5×Q8C3×D411yescharacteristic
126C2×D20—21no—
127C2×D20—21no—
128C2×D20—21no—
129C5×C4⋊C4—22no—
130C5×C4⋊C4—22no—
131C5×C4⋊C4—22no—
132C5×C4⋊C4—21no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4—21no—
135C5×C4⋊C4C3×D411yescharacteristic
136C5×C4⋊C4C3×D411yescharacteristic
137C5×C4⋊C4C3×D412yes—
138C42×C5—22no—
139C42×C5—21no—
140C42×C5—21no—
141C42×C5C23×C311yescharacteristic
142#688—42no—
143#688—42no—
144C2×#168—21no—
145C2×#168—21no—
146C2×#168—21no—
147#557C3×D511yescharacteristic, complemented
148#486—51no—
149#460—51no—
150#460—51no—
151#332—51no—
152Q8⋊C8—51no—
153Q8⋊C8—51no—
154C22×C3×C5#12011yescharacteristic
155C2×C3×D5—81no—
156C2×C3×D5—81no—
157C2×C3×D5—81no—
158C3×C4×C5—22no—
159C3×C4×C5—21no—
160C3×C4×C5—41no—
161C3×C4×C5—42no—
162C3×C4×C5—41no—
163C3×C4×C5—41no—
164C3×C4×C5C2×SD1611yescharacteristic
165C3×C4×C5#13511yescharacteristic
166C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
167C3×C4×C5#13711yescharacteristic
168C2×C3×Q8—21no—
169C2×C3×Q8C22⋊D511yescharacteristic
170C2×C3×Q8C22⋊D511yescharacteristic
171C2×C3×D4—101no—
172C2×C3×D4—101no—
173C2×C3×D4—101no—
174C3×SD16—202no—
175C3×SD16—202no—
176C2×C3×C8—101no—
177C2×C3×C8—101no—
178C2×C3×C8—101no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4C22⋊D511yescharacteristic
186C3×C4⋊C4C22⋊D511yescharacteristic
187C3×C4⋊C4C22⋊D512yes—
188C3×C42—22no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42C22×D511yescharacteristic
192C22×D5—41no—
193C5×Q8—21no—
194C5×Q8—21no—
195C5×Q8—21no—
196C5×Q8—41no—
197C5×Q8—41no—
198C5×Q8C3×SD1612yescomplemented
199C2×C4×C5—21no—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—22no—
204C2×C4×C5C2×C3×D411yescharacteristic
205C2×C4×C5C2×C3×D411yescharacteristic
206C2×C4×C5C2×C3×D411yescharacteristic
207D20—42no—
208D20—41no—
209D20—41no—
210D20—41no—
211D20—41no—
212#168—41no—
213#168—41no—
214#168—41no—
215C2×SD16—101no—
216C2×SD16—101no—
217C4⋊Q8—21no—
218C4⋊Q8C2×C3×D511yescharacteristic
219#127—51no—
220#125—22no—
221#125—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8C2×C3×D511yescharacteristic
225C4×Q8C2×C3×D511yescharacteristic
226C4⋊C8—51no—
227C4⋊C8—51no—
228#102—101no—
229#102—101no—
230#102—102no—
231C4×C8—51no—
232C2×C3×C5#44811yescharacteristic
233C2×C3×C5#44911yescharacteristic
234C2×C3×C5#45411yescharacteristic
235C3×D5—161no—
236C23×C3—201no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×Q8#68812yescomplemented
243C3×D4—202no—
244C3×D4—201no—
245C3×D4—201no—
246C3×D4—201no—
247C3×D4—201no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—22no—
253C2×C3×C4C2×C22⋊D511yescharacteristic
254C2×C3×C4C2×C22⋊D511yescharacteristic
255C2×C3×C4C2×C22⋊D511yescharacteristic
256C3×C8—201no—
257C3×C8—201no—
258C3×C8—201no—
259C22×C5C3×#12011yescharacteristic
260C2×D5—81no—
261C2×D5—81no—
262C2×D5—81no—
263C4×C5—22no—
264C4×C5—21no—
265C4×C5—41no—
266C4×C5—42no—
267C4×C5—41no—
268C4×C5—41no—
269C4×C5C2×C3×SD1611yescharacteristic
270C4×C5C3×#13511yescharacteristic
271C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
272C4×C5C3×#13711yescharacteristic
273C2×Q8—21no—
274C2×Q8C3×C22⋊D511yescharacteristic
275C2×Q8C3×C22⋊D511yescharacteristic
276C2×D4—101no—
277C2×D4—101no—
278C2×D4—101no—
279SD16—202no—
280SD16—202no—
281C2×C8—101no—
282C2×C8—101no—
283C2×C8—101no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4C3×C22⋊D511yescharacteristic
291C4⋊C4C3×C22⋊D511yescharacteristic
292C4⋊C4C3×C22⋊D512yes—
293C42—22no—
294C42—21no—
295C42—21no—
296C42C22×C3×D511yescharacteristic
297C3×C5#162611yescharacteristic, complemented, Hall
298C22×C3—401no—
299C22×C3—401no—
300C22×C3—401no—
301C22×C3#409911yescharacteristic
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4—42no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4C2×#68811yescharacteristic
309C3×C4#408811yescharacteristic
310C3×C4#409511yescharacteristic
311C3×C4#409611yescharacteristic
312C2×C5C3×#44811yescharacteristic
313C2×C5C3×#44911yescharacteristic
314C2×C5C3×#45411yescharacteristic
315D5—161no—
316C23—201no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—41no—
321Q8—41no—
322Q8C3×#68812yescomplemented
323D4—202no—
324D4—201no—
325D4—201no—
326D4—201no—
327D4—201no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4C2×C3×C22⋊D511yescharacteristic
334C2×C4C2×C3×C22⋊D511yescharacteristic
335C2×C4C2×C3×C22⋊D511yescharacteristic
336C8—201no—
337C8—201no—
338C8—201no—
339C2×C3—801no—
340C2×C3#6551611yescharacteristic
341C2×C3#6552711yescharacteristic
342C2×C3#6553411yescharacteristic
343C5C3×#162611yescharacteristic, complemented, Sylow
344C22—401no—
345C22—401no—
346C22—401no—
347C22C3×#409911yescharacteristic
348C4—22no—
349C4—21no—
350C4—41no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4C2×C3×#68811yescharacteristic
355C4C3×#408811yescharacteristic
356C4C3×#409511yescharacteristic
357C4C3×#409611yescharacteristic
358C3#1060272811yescharacteristic, direct product factor, Sylow
359C2—801no—
360C2C3×#6551611yescharacteristic
361C2C3×#6552711yescharacteristic
362C2C3×#6553411yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#557C2
socleC22×C3×C5#120
prime radicalC3×#4020C22
hypercenterC3×#557D5
hypocenterC5C3×#1626
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#557

Complements

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

NH ≅ G/Ntype
1. C3×C5×#557C2semidirect
5. C3×#64767C2semidirect
6. C3×Q8⋊#168C2semidirect
7. C3×Q8⋊#168C2semidirect
8. #10602728C3direct (N × H)
27. C5×#557C2×C3semidirect
31. #64767C2×C3semidirect
32. Q8⋊#168C2×C3semidirect
33. Q8⋊#168C2×C3semidirect
58. C3×#557D5semidirect
89. C3×C5×Q8SD16semidirect
147. #557C3×D5semidirect
198. C5×Q8C3×SD16semidirect
242. C3×Q8#688semidirect
297. C3×C5#1626semidirect
322. Q8C3×#688semidirect
343. C5C3×#1626semidirect
358. C3#10602728direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5572
C3×#653772
C3×#653162
C3×#647672
C3×Q8⋊#1682
#106027283
C3×#16265