Finite Group Databasebeta

← C3×#10601798 #14291407

Subgroups — C3×#10601798 #14291407

Order 1920.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#459C211yesmaximal, characteristic, complemented
2C3×#65362C211yesmaximal, characteristic, complemented
3C3×#65304C211yesmaximal, characteristic, complemented
4C3×D4×#168C211yesmaximal, characteristic, complemented
5C3×#64764C211yesmaximal, characteristic, complemented
6C3×#64705C211yesmaximal, characteristic, complemented
7C3×#64699C211yesmaximal, characteristic, complemented
8#10601798C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×D8—21no—
10C3×C5×#127C2211yescharacteristic
11C3×C4×C5×D4C2211yescharacteristic
12C2×C3×#686—21no—
13C2×C3×#686—21no—
14C2×C3×#686—21no—
15C22×C3×#168—21no—
16C3×#4020C2211yescharacteristic
17C3×C5×C4⋊C8C2211yescharacteristic
18C3×C5×#102—21no—
19C3×C22⋊#168—21no—
20C3×#3941—21no—
21C3×C4⋊#168C2211yescharacteristic
22C3×C4⋊#168C2211yescharacteristic
23C3×C4×#168C2211yescharacteristic
24C3×#1667—51nomaximal, Hall
25C5×#459C2×C311yescharacteristic, complemented
26#65362C2×C311yescharacteristic, complemented
27#65304C2×C311yescharacteristic, complemented
28D4×#168C2×C311yescharacteristic, complemented
29#64764C2×C311yescharacteristic, complemented
30#64705C2×C311yescharacteristic, complemented
31#64699C2×C311yescharacteristic, complemented
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4D411yescharacteristic
36C22×C3×C4×C5—21no—
37C2×C3×D20—21no—
38C2×C3×D20—21no—
39C2×C3×D20—21no—
40C3×C5×D8—42no—
41C2×C3×C5×C8—21no—
42C3×C5×C4⋊C4D411yescharacteristic
43C3×C5×C22⋊C4—21no—
44C3×C42×C5C2311yescharacteristic
45C3×#686—42no—
46C3×#686—42no—
47C3×#686—42no—
48C2×C3×#168—21no—
49C2×C3×#168—21no—
50C2×C3×#168—42no—
51C2×C3×#168D411yescharacteristic
52C2×C3×#168D411yescharacteristic
53C3×#493—51no—
54C3×#459—51no—
55C3×#459D511yescharacteristic, complemented
56C3×C8×D4—51no—
57C3×#335—51no—
58C3×#331—51no—
59C3×#331—51no—
60C2×C5×D8—21no—
61C5×#127C22×C311yescharacteristic
62C4×C5×D4C22×C311yescharacteristic
63C2×#686—21no—
64C2×#686—21no—
65C2×#686—21no—
66C22×#168—21no—
67#4020C22×C311yescharacteristic
68C5×C4⋊C8C22×C311yescharacteristic
69C5×#102—21no—
70C22⋊#168—21no—
71#3941—21no—
72C4⋊#168C22×C311yescharacteristic
73C4⋊#168C22×C311yescharacteristic
74C4×#168C22×C311yescharacteristic
75#1667—51noSylow
76C23×C3×C5—21no—
77C23×C3×C5—41no—
78C22×C3×D5—41no—
79C3×C5×D4—21no—
80C3×C5×D4—41no—
81C3×C5×D4—41no—
82C3×C5×D4—41no—
83C3×C5×D4—41no—
84C3×C5×D4—81no—
85C3×C5×D4D812yescomplemented
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5#4111yescharacteristic
92C3×D20—41no—
93C3×D20—41no—
94C3×D20—41no—
95C3×D20—41no—
96C3×D20—81no—
97C3×C5×C8—41no—
98C3×#168—21no—
99C3×#168—41no—
100C3×#168—41no—
101C3×#168D812yescomplemented
102C2×C3×D8—21no—
103C2×C3×D8—101no—
104C2×C3×D8—101no—
105C2×C3×D8—101no—
106C22×C3×C8—101no—
107C3×#127—51no—
108C3×#127C2×D511yescharacteristic
109C3×C4×D4C2×D511yescharacteristic
110C3×C4⋊C8—51no—
111C3×C4⋊C8—51no—
112C3×C4⋊C8C2×D511yescharacteristic
113C3×#102—21no—
114C3×#102—101no—
115C3×C22⋊C8—101no—
116C3×C4×C8—51no—
117C2×C5×D4—21no—
118C2×C5×D4—21no—
119C2×C5×D4—21no—
120C2×C5×D4C3×D411yescharacteristic
121C22×C4×C5—21no—
122C2×D20—21no—
123C2×D20—21no—
124C2×D20—21no—
125C5×D8—42no—
126C2×C5×C8—21no—
127C5×C4⋊C4C3×D411yescharacteristic
128C5×C22⋊C4—21no—
129C42×C5C23×C311yescharacteristic
130#686—42no—
131#686—42no—
132#686—42no—
133C2×#168—21no—
134C2×#168—21no—
135C2×#168—42no—
136C2×#168C3×D411yescharacteristic
137C2×#168C3×D411yescharacteristic
138#493—51no—
139#459—51no—
140#459C3×D511yescharacteristic, complemented
141C8×D4—51no—
142#335—51no—
143#331—51no—
144#331—51no—
145C22×C3×C5—22no—
146C22×C3×C5—41no—
147C22×C3×C5—41no—
148C22×C3×C5—81no—
149C22×C3×C5—81no—
150C22×C3×C5—81no—
151C22×C3×C5#12111yescharacteristic
152C2×C3×D5—81no—
153C2×C3×D5—81no—
154C2×C3×D5—81no—
155C3×C4×C5—21no—
156C3×C4×C5—41no—
157C3×C4×C5C2×D811yescharacteristic
158C3×C4×C5C2×D811yescharacteristic
159C3×C4×C5#13511yescharacteristic
160C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—101no—
165C2×C3×D4—101no—
166C2×C3×D4—101no—
167C2×C3×D4C22⋊D511yescharacteristic
168C22×C3×C4—21no—
169C3×D8—42no—
170C3×D8—202no—
171C3×D8—202no—
172C3×D8—202no—
173C2×C3×C8—21no—
174C2×C3×C8—51no—
175C2×C3×C8—51no—
176C2×C3×C8—101no—
177C2×C3×C8—101no—
178C2×C3×C8—202no—
179C3×C4⋊C4C22⋊D511yescharacteristic
180C3×C22⋊C4—21no—
181C3×C42C22×D511yescharacteristic
182C23×C5—21no—
183C23×C5—41no—
184C22×D5—41no—
185C5×D4—21no—
186C5×D4—41no—
187C5×D4—41no—
188C5×D4—41no—
189C5×D4—41no—
190C5×D4—81no—
191C5×D4C3×D812yescomplemented
192C2×C4×C5—21no—
193C2×C4×C5—41no—
194C2×C4×C5—41no—
195C2×C4×C5C2×C3×D411yescharacteristic
196C2×C4×C5C2×C3×D411yescharacteristic
197C2×C4×C5C3×#4111yescharacteristic
198D20—41no—
199D20—41no—
200D20—41no—
201D20—41no—
202D20—81no—
203C5×C8—41no—
204#168—21no—
205#168—41no—
206#168—41no—
207#168C3×D812yescomplemented
208C2×D8—21no—
209C2×D8—101no—
210C2×D8—101no—
211C2×D8—101no—
212C22×C8—101no—
213#127—51no—
214#127C2×C3×D511yescharacteristic
215C4×D4C2×C3×D511yescharacteristic
216C4⋊C8—51no—
217C4⋊C8—51no—
218C4⋊C8C2×C3×D511yescharacteristic
219#102—21no—
220#102—101no—
221C22⋊C8—101no—
222C4×C8—51no—
223C2×C3×C5—42no—
224C2×C3×C5—161no—
225C2×C3×C5#45911yescharacteristic
226C2×C3×C5#46611yescharacteristic
227C2×C3×C5#47211yescharacteristic
228C3×D5—161no—
229C23×C3—21no—
230C23×C3—41no—
231C23×C3—201no—
232C3×D4—21no—
233C3×D4—41no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—81no—
238C3×D4—201no—
239C3×D4—201no—
240C3×D4—201no—
241C3×D4—201no—
242C3×D4—401no—
243C3×D4#68612yescomplemented
244C2×C3×C4—21no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4D4×D511yescharacteristic
248C2×C3×C4#71111yescharacteristic
249C2×C3×C4C2×C22⋊D511yescharacteristic
250C3×C8—41no—
251C3×C8—52no—
252C3×C8—101no—
253C3×C8—201no—
254C3×C8—201no—
255C22×C5—22no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—81no—
259C22×C5—81no—
260C22×C5—81no—
261C22×C5C3×#12111yescharacteristic
262C2×D5—81no—
263C2×D5—81no—
264C2×D5—81no—
265C4×C5—21no—
266C4×C5—41no—
267C4×C5C2×C3×D811yescharacteristic
268C4×C5C2×C3×D811yescharacteristic
269C4×C5C3×#13511yescharacteristic
270C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—101no—
275C2×D4—101no—
276C2×D4—101no—
277C2×D4C3×C22⋊D511yescharacteristic
278C22×C4—21no—
279D8—42no—
280D8—202no—
281D8—202no—
282D8—202no—
283C2×C8—21no—
284C2×C8—51no—
285C2×C8—51no—
286C2×C8—101no—
287C2×C8—101no—
288C2×C8—202no—
289C4⋊C4C3×C22⋊D511yescharacteristic
290C22⋊C4—21no—
291C42C22×C3×D511yescharacteristic
292C3×C5#166711yescharacteristic, complemented, Hall
293C22×C3—22no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3—401no—
300C22×C3—401no—
301C22×C3—401no—
302C22×C3#408511yescharacteristic
303C3×C4—21no—
304C3×C4—41no—
305C3×C4D5×D811yescharacteristic
306C3×C4#405711yescharacteristic
307C3×C4C2×#68611yescharacteristic
308C3×C4#409611yescharacteristic
309C2×C5—42no—
310C2×C5—161no—
311C2×C5C3×#45911yescharacteristic
312C2×C5C3×#46611yescharacteristic
313C2×C5C3×#47211yescharacteristic
314D5—161no—
315C23—21no—
316C23—41no—
317C23—201no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—81no—
324D4—201no—
325D4—201no—
326D4—201no—
327D4—201no—
328D4—401no—
329D4C3×#68612yescomplemented
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4C3×D4×D511yescharacteristic
334C2×C4C3×#71111yescharacteristic
335C2×C4C2×C3×C22⋊D511yescharacteristic
336C8—41no—
337C8—52no—
338C8—101no—
339C8—201no—
340C8—201no—
341C2×C3—42no—
342C2×C3—161no—
343C2×C3—801no—
344C2×C3#6532711yescharacteristic
345C2×C3#6543911yescharacteristic
346C2×C3#6548811yescharacteristic
347C5C3×#166711yescharacteristic, complemented, Sylow
348C22—22no—
349C22—41no—
350C22—41no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—401no—
355C22—401no—
356C22—401no—
357C22C3×#408511yescharacteristic
358C4—21no—
359C4—41no—
360C4C3×D5×D811yescharacteristic
361C4C3×#405711yescharacteristic
362C4C2×C3×#68611yescharacteristic
363C4C3×#409611yescharacteristic
364C3#1060179811yescharacteristic, direct product factor, Sylow
365C2—42no—
366C2—161no—
367C2—801no—
368C2C3×#6532711yescharacteristic
369C2C3×#6543911yescharacteristic
370C2C3×#6548811yescharacteristic
subgroupG/H
centerC22×C3#4085
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#459C2
socleC22×C3×C5#121
hypercenterC3×#459D5
hypocenterC5C3×#1667
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#459

Complements

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

NH ≅ G/Ntype
1. C3×C5×#459C2semidirect
2. C3×#65362C2semidirect
3. C3×#65304C2semidirect
4. C3×D4×#168C2semidirect
5. C3×#64764C2semidirect
6. C3×#64705C2semidirect
7. C3×#64699C2semidirect
8. #10601798C3direct (N × H)
25. C5×#459C2×C3semidirect
26. #65362C2×C3semidirect
27. #65304C2×C3semidirect
28. D4×#168C2×C3semidirect
29. #64764C2×C3semidirect
30. #64705C2×C3semidirect
31. #64699C2×C3semidirect
55. C3×#459D5semidirect
85. C3×C5×D4D8semidirect
101. C3×#168D8semidirect
140. #459C3×D5semidirect
191. C5×D4C3×D8semidirect
207. #168C3×D8semidirect
243. C3×D4#686semidirect
292. C3×C5#1667semidirect
329. D4C3×#686semidirect
347. C5C3×#1667semidirect
364. C3#10601798direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#4592
C3×#653622
C3×#653042
C3×D4×#1682
C3×#647642
C3×#647052
C3×#646992
#106017983
C3×#16675