Finite Group Databasebeta

← C3×#10604488 #14285299

Subgroups — C3×#10604488 #14285299

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65930C211yesmaximal, characteristic
2C3×#65758C211yesmaximal, characteristic, complemented
3C3×#65757C211yesmaximal, characteristic, complemented
4C3×#64925C212yesmaximal, complemented
5C3×#64873C212yesmaximal, complemented
6#10604488C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#711—21no—
8C2×C3×#704—21no—
9C3×C5×#121C411yescharacteristic
10C3×#4086C2211yescharacteristic
11C3×#4085—21no—
12C3×D4×Dic5C411yescharacteristic
13C3×#4076C411yescharacteristic
14C2×C3×C4×Dic5C2211yescharacteristic
15C3×#4039C411yescharacteristic
16C3×#4034C2211yescharacteristic
17C3×#4031—21no—
18C3×#4027—21no—
19C3×C22⋊#170—21no—
20C3×#4004C2212yes—
21C3×#4004C2212yes—
22C3×#4002—21no—
23C3×C4⋊#170—21no—
24C3×#1557—51nomaximal, Hall
25#65930C2×C311yescharacteristic
26#65758C2×C311yescharacteristic, complemented
27#65757C2×C311yescharacteristic, complemented
28#64925C2×C312yescomplemented
29#64873C2×C312yescomplemented
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4C2×C411yescharacteristic
32C22×C3×C4×C5C2×C411yescharacteristic
33C2×C3×C22⋊D5—21no—
34C2×C3×C22⋊D5—21no—
35C2×C3×C22⋊D5—21no—
36C22×C3×Dic5—21no—
37C22×C3×Dic5D411yescharacteristic
38C22×C3×Dic5D411yescharacteristic
39C3×#711—44no—
40C2×C3×D20C2×C411yescharacteristic
41C2×C3×C4×D5—21no—
42C2×C3×Dic10—21no—
43C3×#704—42no—
44C2×C3×#170—22no—
45C2×C3×#170—22no—
46C3×C5×C4⋊C4C2×C411yescharacteristic
47C3×C5×C22⋊C4—21no—
48C3×C22⋊Dic5—21no—
49C3×#685—21no—
50C3×#685—21no—
51C3×C4⋊Dic5C2×C411yescharacteristic
52C3×#683—21no—
53C3×C4×Dic5C2×C411yescharacteristic
54C3×C4×Dic5D411yescharacteristic
55C3×C4×Dic5D411yescharacteristic
56C3×C4×Dic5C2311yescharacteristic
57C3×#532—51no—
58C3×#432—51no—
59C3×#423—51no—
60C3×#332—52no—
61C3×#331—52no—
62C2×#711—21no—
63C2×#704—21no—
64C5×#121C3×C411yescharacteristic
65#4086C22×C311yescharacteristic
66#4085—21no—
67D4×Dic5C3×C411yescharacteristic
68#4076C3×C411yescharacteristic
69C2×C4×Dic5C22×C311yescharacteristic
70#4039C3×C411yescharacteristic
71#4034C22×C311yescharacteristic
72#4031—21no—
73#4027—21no—
74C22⋊#170—21no—
75#4004C22×C312yes—
76#4004C22×C312yes—
77#4002—21no—
78C4⋊#170—21no—
79#1557—51noSylow
80C23×C3×C5—21no—
81C23×C3×C5C22⋊C411yescharacteristic
82C22×C3×D5—21no—
83C3×C5×D4—41no—
84C3×C5×D4—42no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22×C411yescharacteristic
89C3×C22⋊D5—42no—
90C3×C22⋊D5—42no—
91C3×C22⋊D5—41no—
92C3×C22⋊D5—41no—
93C2×C3×Dic5—21no—
94C2×C3×Dic5—22no—
95C2×C3×Dic5—21no—
96C2×C3×Dic5—21no—
97C2×C3×Dic5—42no—
98C2×C3×Dic5C22⋊C411yescharacteristic
99C2×C3×Dic5C22⋊C411yescharacteristic
100C2×C3×Dic5C2×D411yescharacteristic
101C2×C3×Dic5C2×D411yescharacteristic
102C3×D20—41no—
103C3×C4×D5—42no—
104C3×Dic10—42no—
105C3×#170—42no—
106C3×#170—42no—
107C2×C3×#41—101no—
108C2×C3×M16—101no—
109C3×C4⋊Q8—51no—
110C3×#127—51no—
111C3×#124—101no—
112C3×#121—51no—
113C3×#121—101no—
114C3×#121AGL(1,𝔽5)11yescharacteristic
115C3×C4×D4—51no—
116C3×C4×D4—51no—
117C3×C4×D4—101no—
118C2×C3×C42—51no—
119C3×C4⋊C8—52no—
120C3×C4⋊C8—52no—
121C3×C4⋊C8—101no—
122C3×C22⋊C8—101no—
123C3×#97—101no—
124C2×C5×D4—21no—
125C2×C5×D4C2×C3×C411yescharacteristic
126C22×C4×C5C2×C3×C411yescharacteristic
127C2×C22⋊D5—21no—
128C2×C22⋊D5—21no—
129C2×C22⋊D5—21no—
130C22×Dic5—21no—
131C22×Dic5C3×D411yescharacteristic
132C22×Dic5C3×D411yescharacteristic
133#711—44no—
134C2×D20C2×C3×C411yescharacteristic
135C2×C4×D5—21no—
136C2×Dic10—21no—
137#704—42no—
138C2×#170—22no—
139C2×#170—22no—
140C5×C4⋊C4C2×C3×C411yescharacteristic
141C5×C22⋊C4—21no—
142C22⋊Dic5—21no—
143#685—21no—
144#685—21no—
145C4⋊Dic5C2×C3×C411yescharacteristic
146#683—21no—
147C4×Dic5C2×C3×C411yescharacteristic
148C4×Dic5C3×D411yescharacteristic
149C4×Dic5C3×D411yescharacteristic
150C4×Dic5C23×C311yescharacteristic
151#532—51no—
152#432—51no—
153#423—51no—
154#332—52no—
155#331—52no—
156C22×C3×C5—21no—
157C22×C3×C5—21no—
158C22×C3×C5—21no—
159C22×C3×C5—41no—
160C22×C3×C5—41no—
161C22×C3×C5—41no—
162C22×C3×C5C2×C22⋊C411yescharacteristic
163C2×C3×D5—41no—
164C2×C3×D5—41no—
165C2×C3×D5—41no—
166C3×C4×C5—21no—
167C3×C4×C5—21no—
168C3×C4×C5—41no—
169C3×Dic5—21no—
170C3×Dic5—21no—
171C3×Dic5—41no—
172C3×Dic5AGL(1,ℤ/8)11yescharacteristic
173C3×Dic5AGL(1,ℤ/8)12yes—
174C3×Dic5#13711yescharacteristic
175C3×#41—204no—
176C2×C3×Q8—101no—
177C2×C3×D4—21no—
178C2×C3×D4—51no—
179C2×C3×D4—101no—
180C2×C3×D4—101no—
181C2×C3×D4—101no—
182C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
183C22×C3×C4—51no—
184C22×C3×C4—51no—
185C22×C3×C4—101no—
186C22×C3×C4—101no—
187C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
188C3×M16—202no—
189C2×C3×C8—102no—
190C2×C3×C8—102no—
191C3×C4⋊C4—51no—
192C3×C4⋊C4—101no—
193C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
194C3×C22⋊C4—21no—
195C3×C22⋊C4—101no—
196C3×C22⋊C4—101no—
197C3×C22⋊C4—101no—
198C3×C42—51no—
199C3×C42—51no—
200C3×C42—51no—
201C3×C42—51no—
202C23×C5—21no—
203C23×C5C3×C22⋊C411yescharacteristic
204C22×D5—21no—
205C5×D4—41no—
206C5×D4—42no—
207C2×C4×C5—21no—
208C2×C4×C5—41no—
209C2×C4×C5C3×C22⋊C411yescharacteristic
210C2×C4×C5C22×C3×C411yescharacteristic
211C22⋊D5—42no—
212C22⋊D5—42no—
213C22⋊D5—41no—
214C22⋊D5—41no—
215C2×Dic5—21no—
216C2×Dic5—22no—
217C2×Dic5—21no—
218C2×Dic5—21no—
219C2×Dic5—42no—
220C2×Dic5C3×C22⋊C411yescharacteristic
221C2×Dic5C3×C22⋊C411yescharacteristic
222C2×Dic5C2×C3×D411yescharacteristic
223C2×Dic5C2×C3×D411yescharacteristic
224D20—41no—
225C4×D5—42no—
226Dic10—42no—
227#170—42no—
228#170—42no—
229C2×#41—101no—
230C2×M16—101no—
231C4⋊Q8—51no—
232#127—51no—
233#124—101no—
234#121—51no—
235#121—101no—
236#121C3×AGL(1,𝔽5)11yescharacteristic
237C4×D4—51no—
238C4×D4—51no—
239C4×D4—101no—
240C2×C42—51no—
241C4⋊C8—52no—
242C4⋊C8—52no—
243C4⋊C8—101no—
244C22⋊C8—101no—
245#97—101no—
246C2×C3×C5—41no—
247C2×C3×C5—81no—
248C2×C3×C5#41311yescharacteristic
249C2×C3×C5#41711yescharacteristic
250C2×C3×C5#41811yescharacteristic
251C3×D5—81no—
252C23×C3—21no—
253C23×C3—101no—
254C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
255C3×Q8—202no—
256C3×D4—41no—
257C3×D4—42no—
258C3×D4—201no—
259C3×D4—201no—
260C3×D4—201no—
261C3×D4—202no—
262C3×D4—202no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—51no—
266C2×C3×C4—51no—
267C2×C3×C4—51no—
268C2×C3×C4—51no—
269C2×C3×C4—101no—
270C2×C3×C4—101no—
271C2×C3×C4—102no—
272C2×C3×C4—101no—
273C2×C3×C4—202no—
274C2×C3×C4—202no—
275C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
276C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
277C3×C8—202no—
278C3×C8—202no—
279C22×C5—21no—
280C22×C5—21no—
281C22×C5—21no—
282C22×C5—41no—
283C22×C5—41no—
284C22×C5—41no—
285C22×C5C2×C3×C22⋊C411yescharacteristic
286C2×D5—41no—
287C2×D5—41no—
288C2×D5—41no—
289C4×C5—21no—
290C4×C5—21no—
291C4×C5—41no—
292Dic5—21no—
293Dic5—21no—
294Dic5—41no—
295Dic5C3×AGL(1,ℤ/8)11yescharacteristic
296Dic5C3×AGL(1,ℤ/8)12yes—
297Dic5C3×#13711yescharacteristic
298#41—204no—
299C2×Q8—101no—
300C2×D4—21no—
301C2×D4—51no—
302C2×D4—101no—
303C2×D4—101no—
304C2×D4—101no—
305C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
306C22×C4—51no—
307C22×C4—51no—
308C22×C4—101no—
309C22×C4—101no—
310C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
311M16—202no—
312C2×C8—102no—
313C2×C8—102no—
314C4⋊C4—51no—
315C4⋊C4—101no—
316C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—101no—
319C22⋊C4—101no—
320C22⋊C4—101no—
321C42—51no—
322C42—51no—
323C42—51no—
324C42—51no—
325C3×C5#155711yescharacteristic, complemented, Hall
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—201no—
333C22×C3—201no—
334C22×C3—201no—
335C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—41no—
339C3×C4—51no—
340C3×C4—51no—
341C3×C4—52no—
342C3×C4—101no—
343C3×C4—101no—
344C3×C4—201no—
345C2×C5—41no—
346C2×C5—81no—
347C2×C5C3×#41311yescharacteristic
348C2×C5C3×#41711yescharacteristic
349C2×C5C3×#41811yescharacteristic
350D5—81no—
351C23—21no—
352C23—101no—
353C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
354Q8—202no—
355D4—41no—
356D4—42no—
357D4—201no—
358D4—202no—
359D4—201no—
360D4—201no—
361D4—202no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—51no—
365C2×C4—51no—
366C2×C4—51no—
367C2×C4—51no—
368C2×C4—101no—
369C2×C4—102no—
370C2×C4—101no—
371C2×C4—101no—
372C2×C4—202no—
373C2×C4—202no—
374C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
375C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
376C8—202no—
377C8—202no—
378C2×C3—41no—
379C2×C3—81no—
380C2×C3—401no—
381C2×C3#6576711yescharacteristic
382C2×C3#6577711yescharacteristic
383C2×C3#6579311yescharacteristic
384C5C3×#155711yescharacteristic, complemented, Sylow
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—201no—
392C22—201no—
393C22—201no—
394C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
395C4—21no—
396C4—21no—
397C4—41no—
398C4—52no—
399C4—51no—
400C4—51no—
401C4—101no—
402C4—101no—
403C4—201no—
404C3#1060448811yescharacteristic, direct product factor, Sylow
405C2—41no—
406C2—81no—
407C2—401no—
408C2C3×#6576711yescharacteristic
409C2C3×#6577711yescharacteristic
410C2C3×#6579311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#121C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65930C2
cosocleC4×Dic5C23×C3
hypercenterC3×#121AGL(1,𝔽5)
hypocenterC5C3×#1557
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. C3×#65758C2semidirect
3. C3×#65757C2semidirect
4. C3×#64925C2semidirect
5. C3×#64873C2semidirect
6. #10604488C3direct (N × H)
26. #65758C2×C3semidirect
27. #65757C2×C3semidirect
28. #64925C2×C3semidirect
29. #64873C2×C3semidirect
325. C3×C5#1557semidirect
384. C5C3×#1557semidirect
404. C3#10604488direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659302
C3×#657582
C3×#657572
C3×#649252
C3×#648732
#106044883
C3×#15575