Finite Group Databasebeta

← C3×#10600496 #14287820

Subgroups — C3×#10600496 #14287820

Order 1920.

620 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 106 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×#65844C211yesmaximal, characteristic, complemented
2C3×C5×#394C211yesmaximal, characteristic, complemented
3C3×#65506C212yesmaximal, complemented
4C3×#64748C211yesmaximal, characteristic, complemented
5C3×#64671C212yesmaximal, complemented
6#10600496C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×D4C2211yescharacteristic
8C2×C3×#709—21no—
9C2×C3×C5×C4⋊C4C2211yescharacteristic
10C2×C3×C5×C22⋊C4—21no—
11C2×C3×C5×C22⋊C4—21no—
12C2×C3×#686—22no—
13C3×#4076—21no—
14C3×#4076C2212yes—
15C3×#4043—22no—
16C3×#4041—22no—
17C3×C5×#95C2211yescharacteristic
18C3×#3964—22no—
19C3×C22⋊#168C2212yes—
20C3×#1598—51nomaximal, Hall
21#65844C2×C311yescharacteristic, complemented
22C5×#394C2×C311yescharacteristic, complemented
23#65506C2×C312yescomplemented
24#64748C2×C311yescharacteristic, complemented
25#64671C2×C312yescomplemented
26C24×C3×C5—21no—
27C2×C3×C5×D4—22no—
28C2×C3×C5×D4—42no—
29C22×C3×C4×C5—21no—
30C22×C3×C4×C5D411yescharacteristic
31C22×C3×C4×C5D411yescharacteristic
32C22×C3×C4×C5C2311yescharacteristic
33C2×C3×C22⋊D5—22no—
34C3×#709—42no—
35C3×#709—42no—
36C2×C3×D20—21no—
37C2×C3×D20—21no—
38C2×C3×D20D412yes—
39C2×C3×C4×D5—22no—
40C2×C3×Dic10—21no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C22⋊C4—42no—
43C3×C5×C22⋊C4—42no—
44C3×#686—44no—
45C3×#685—22no—
46C3×#685—22no—
47C3×C4⋊Dic5—21no—
48C3×C4⋊Dic5D412yes—
49C2×C3×#168—22no—
50C3×#537—51no—
51C3×#447—52no—
52C3×#394D511yescharacteristic, complemented
53C3×#327—52no—
54C3×#324—51no—
55C22×C5×D4C22×C311yescharacteristic
56C2×#709—21no—
57C2×C5×C4⋊C4C22×C311yescharacteristic
58C2×C5×C22⋊C4—21no—
59C2×C5×C22⋊C4—21no—
60C2×#686—22no—
61#4076—21no—
62#4076C22×C312yes—
63#4043—22no—
64#4041—22no—
65C5×#95C22×C311yescharacteristic
66#3964—22no—
67C22⋊#168C22×C312yes—
68#1598—51noSylow
69C23×C3×C5—22no—
70C23×C3×C5—41no—
71C23×C3×C5—41no—
72C23×C3×C5—41no—
73C23×C3×C5—41no—
74C23×C3×C5—41no—
75C23×C3×C5—41no—
76C23×C3×C5C2×D411yescharacteristic
77C22×C3×D5—22no—
78C3×C5×D4—24no—
79C3×C5×D4—42no—
80C2×C3×C4×C5—21no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—41no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5D812yescomplemented
89C2×C3×C4×C5C2×D412yes—
90C3×C22⋊D5—42no—
91C3×C22⋊D5—42no—
92C2×C3×Dic5—22no—
93C3×D20—42no—
94C3×D20—42no—
95C3×D20—81no—
96C3×C4×D5—44no—
97C3×Dic10—42no—
98C3×#168—42no—
99C2×C3×#41—101no—
100C22×C3×D4C2×D511yescharacteristic
101C2×C3×D8—102no—
102C3×C22⋊Q8—102no—
103C3×#121—52no—
104C3×#121—102no—
105C3×#121—101no—
106C2×C3×C4⋊C4C2×D511yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C3×#102—102no—
110C3×C22⋊C8—52no—
111C3×#95C2×D511yescharacteristic
112C24×C5—21no—
113C2×C5×D4—22no—
114C2×C5×D4—42no—
115C22×C4×C5—21no—
116C22×C4×C5C3×D411yescharacteristic
117C22×C4×C5C3×D411yescharacteristic
118C22×C4×C5C23×C311yescharacteristic
119C2×C22⋊D5—22no—
120#709—42no—
121#709—42no—
122C2×D20—21no—
123C2×D20—21no—
124C2×D20C3×D412yes—
125C2×C4×D5—22no—
126C2×Dic10—21no—
127C5×C4⋊C4—22no—
128C5×C22⋊C4—42no—
129C5×C22⋊C4—42no—
130#686—44no—
131#685—22no—
132#685—22no—
133C4⋊Dic5—21no—
134C4⋊Dic5C3×D412yes—
135C2×#168—22no—
136#537—51no—
137#447—52no—
138#394C3×D511yescharacteristic, complemented
139#327—52no—
140#324—51no—
141C22×C3×C5—21no—
142C22×C3×C5—21no—
143C22×C3×C5—42no—
144C22×C3×C5—42no—
145C22×C3×C5—42no—
146C22×C3×C5—81no—
147C22×C3×C5—81no—
148C22×C3×C5—81no—
149C22×C3×C5—81no—
150C22×C3×C5#12011yescharacteristic
151C22×C3×C5C2×D811yescharacteristic
152C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
153C2×C3×D5—42no—
154C2×C3×D5—42no—
155C2×C3×D5—42no—
156C3×C4×C5—22no—
157C3×C4×C5—22no—
158C3×C4×C5—41no—
159C3×C4×C5—81no—
160C3×Dic5—42no—
161C24×C3—21no—
162C3×#41—202no—
163C3×#41—202no—
164C2×C3×Q8—101no—
165C2×C3×D4—22no—
166C2×C3×D4—42no—
167C2×C3×D4—52no—
168C2×C3×D4—101no—
169C2×C3×D4—101no—
170C2×C3×D4—102no—
171C22×C3×C4—21no—
172C22×C3×C4—102no—
173C22×C3×C4C22⋊D511yescharacteristic
174C22×C3×C4C22⋊D511yescharacteristic
175C22×C3×C4C22×D511yescharacteristic
176C3×D8—204no—
177C2×C3×C8—102no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—52no—
180C3×C4⋊C4—101no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—102no—
184C3×C22⋊C4—102no—
185C23×C5—22no—
186C23×C5—41no—
187C23×C5—41no—
188C23×C5—41no—
189C23×C5—41no—
190C23×C5—41no—
191C23×C5—41no—
192C23×C5C2×C3×D411yescharacteristic
193C22×D5—22no—
194C5×D4—24no—
195C5×D4—42no—
196C2×C4×C5—21no—
197C2×C4×C5—21no—
198C2×C4×C5—21no—
199C2×C4×C5—41no—
200C2×C4×C5—41no—
201C2×C4×C5—41no—
202C2×C4×C5—41no—
203C2×C4×C5—41no—
204C2×C4×C5C3×D812yescomplemented
205C2×C4×C5C2×C3×D412yes—
206C22⋊D5—42no—
207C22⋊D5—42no—
208C2×Dic5—22no—
209D20—42no—
210D20—42no—
211D20—81no—
212C4×D5—44no—
213Dic10—42no—
214#168—42no—
215C2×#41—101no—
216C22×D4C2×C3×D511yescharacteristic
217C2×D8—102no—
218C22⋊Q8—102no—
219#121—52no—
220#121—102no—
221#121—101no—
222C2×C4⋊C4C2×C3×D511yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225#102—102no—
226C22⋊C8—52no—
227#95C2×C3×D511yescharacteristic
228C2×C3×C5—21no—
229C2×C3×C5—21no—
230C2×C3×C5—82no—
231C2×C3×C5#44711yescharacteristic
232C2×C3×C5#45411yescharacteristic
233C2×C3×C5B(4,𝔽2)11yescharacteristic
234C3×D5—82no—
235C23×C3—22no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—102no—
243C23×C3C2×C22⋊D511yescharacteristic
244C3×Q8—202no—
245C3×D4—24no—
246C3×D4—42no—
247C3×D4—202no—
248C3×D4—202no—
249C3×D4—202no—
250C3×D4—202no—
251C3×D4—401no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—102no—
261C2×C3×C4—204no—
262C2×C3×C4#68612yescomplemented
263C2×C3×C4D4×D512yes—
264C3×C8—202no—
265C22×C5—21no—
266C22×C5—21no—
267C22×C5—42no—
268C22×C5—42no—
269C22×C5—42no—
270C22×C5—81no—
271C22×C5—81no—
272C22×C5—81no—
273C22×C5—81no—
274C22×C5C3×#12011yescharacteristic
275C22×C5C2×C3×D811yescharacteristic
276C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
277C2×D5—42no—
278C2×D5—42no—
279C2×D5—42no—
280C4×C5—22no—
281C4×C5—22no—
282C4×C5—41no—
283C4×C5—81no—
284Dic5—42no—
285C24—21no—
286#41—202no—
287#41—202no—
288C2×Q8—101no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—52no—
292C2×D4—101no—
293C2×D4—101no—
294C2×D4—102no—
295C22×C4—21no—
296C22×C4—102no—
297C22×C4C3×C22⋊D511yescharacteristic
298C22×C4C3×C22⋊D511yescharacteristic
299C22×C4C22×C3×D511yescharacteristic
300D8—204no—
301C2×C8—102no—
302C4⋊C4—22no—
303C4⋊C4—52no—
304C4⋊C4—101no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—102no—
308C22⋊C4—102no—
309C3×C5#159811yescharacteristic, complemented, Hall
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3—42no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—202no—
320C22×C3—202no—
321C22×C3—202no—
322C22×C3C2×#68611yescharacteristic
323C22×C3#408311yescharacteristic
324C22×C3#409511yescharacteristic
325C3×C4—22no—
326C3×C4—22no—
327C3×C4—41no—
328C3×C4—81no—
329C3×C4—202no—
330C2×C5—21no—
331C2×C5—21no—
332C2×C5—82no—
333C2×C5C3×#44711yescharacteristic
334C2×C5C3×#45411yescharacteristic
335C2×C5C3×B(4,𝔽2)11yescharacteristic
336D5—82no—
337C23—22no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—102no—
345C23C2×C3×C22⋊D511yescharacteristic
346Q8—202no—
347D4—24no—
348D4—42no—
349D4—202no—
350D4—202no—
351D4—202no—
352D4—202no—
353D4—401no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—102no—
363C2×C4—204no—
364C2×C4C3×#68612yescomplemented
365C2×C4C3×D4×D512yes—
366C8—202no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—82no—
370C2×C3—402no—
371C2×C3#6532111yescharacteristic
372C2×C3#6532511yescharacteristic
373C2×C3#6535111yescharacteristic
374C5C3×#159811yescharacteristic, complemented, Sylow
375C22—21no—
376C22—21no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—202no—
385C22—202no—
386C22—202no—
387C22C2×C3×#68611yescharacteristic
388C22C3×#408311yescharacteristic
389C22C3×#409511yescharacteristic
390C4—22no—
391C4—22no—
392C4—41no—
393C4—81no—
394C4—202no—
395C3#1060049611yescharacteristic, direct product factor, Sylow
396C2—21no—
397C2—21no—
398C2—82no—
399C2—402no—
400C2C3×#6532111yescharacteristic
401C2C3×#6532511yescharacteristic
402C2C3×#6535111yescharacteristic
subgroupG/H
centerC22×C3#4083
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#394C2
socleC22×C3×C5#120
hypercenterC3×#394D5
hypocenterC5C3×#1598
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#394

Complements

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

NH ≅ G/Ntype
1. C3×#65844C2semidirect
2. C3×C5×#394C2semidirect
3. C3×#65506C2semidirect
4. C3×#64748C2semidirect
5. C3×#64671C2semidirect
6. #10600496C3direct (N × H)
21. #65844C2×C3semidirect
22. C5×#394C2×C3semidirect
23. #65506C2×C3semidirect
24. #64748C2×C3semidirect
25. #64671C2×C3semidirect
52. C3×#394D5semidirect
88. C2×C3×C4×C5D8semidirect
138. #394C3×D5semidirect
204. C2×C4×C5C3×D8semidirect
262. C2×C3×C4#686semidirect
309. C3×C5#1598semidirect
364. C2×C4C3×#686semidirect
374. C5C3×#1598semidirect
395. C3#10600496direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658442
C3×C5×#3942
C3×#655062
C3×#647482
C3×#646712
#106004963
C3×#15985