Finite Group Databasebeta

← C3×#10604132 #14315860

Subgroups — C3×#10604132 #14315860

Order 1920.

308 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 88 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×#65741C211yesmaximal, characteristic, complemented
2C3×#65731C211yesmaximal, characteristic, complemented
3C3×#65203C211yesmaximal, characteristic
4C3×#64907C211yesmaximal, characteristic, complemented
5C3×#64906C211yesmaximal, characteristic, complemented
6C3×#64847C211yesmaximal, characteristic, complemented
7C3×#64845C211yesmaximal, characteristic, complemented
8#10604132C311yesmaximal, characteristic, direct product factor, Hall
9C3×#4133—21no—
10C3×D4×AGL(1,𝔽5)—21no—
11C3×#4065C2211yescharacteristic
12C3×D5×D8C2211yescharacteristic
13C3×#4010—21no—
14C3×#4007—21no—
15C3×#3995C2211yescharacteristic
16C3×#3993C2211yescharacteristic
17C3×#3992C2211yescharacteristic
18C3×#3990C2211yescharacteristic
19C3×C5×SD32C411yescharacteristic, complemented
20C3×#3960C411yescharacteristic, complemented
21C3×#3958C411yescharacteristic, complemented
22C3×D80C411yescharacteristic, complemented
23C3×#3930C2211yescharacteristic
24C3×AGL(1,ℤ/16)—51nomaximal, Hall
25#65741C2×C311yescharacteristic, complemented
26#65731C2×C311yescharacteristic, complemented
27#65203C2×C311yescharacteristic
28#64907C2×C311yescharacteristic, complemented
29#64906C2×C311yescharacteristic, complemented
30#64847C2×C311yescharacteristic, complemented
31#64845C2×C311yescharacteristic, complemented
32C22×C3×AGL(1,𝔽5)—41no—
33C3×#713—21no—
34C3×D4×D5—21no—
35C3×C22⋊AGL(1,𝔽5)—41no—
36C3×C4⋊AGL(1,𝔽5)—21no—
37C3×C4×AGL(1,𝔽5)D411yescharacteristic
38C3×#700—21no—
39C3×#700—41no—
40C3×D5⋊C8—41no—
41C3×D5⋊C8D411yescharacteristic
42C3×C5×Q16C2×C411yescharacteristic, complemented
43C3×C5×D8C2×C411yescharacteristic
44C3×#688—21no—
45C3×#686—21no—
46C3×D40C2×C411yescharacteristic
47C3×D40C2×C411yescharacteristic, complemented
48C3×C8×D5C2311yescharacteristic
49C3×#674—21no—
50C3×C5×C16C2×C411yescharacteristic, complemented
51C3×#672C2×C411yescharacteristic, complemented
52C3×#509—51no—
53C3×#444—51no—
54C3×#442—51no—
55C3×#365—51no—
56C3×#361—51no—
57C3×#360—51no—
58C3×#347—51no—
59#4133—21no—
60D4×AGL(1,𝔽5)—21no—
61#4065C22×C311yescharacteristic
62D5×D8C22×C311yescharacteristic
63#4010—21no—
64#4007—21no—
65#3995C22×C311yescharacteristic
66#3993C22×C311yescharacteristic
67#3992C22×C311yescharacteristic
68#3990C22×C311yescharacteristic
69C5×SD32C3×C411yescharacteristic, complemented
70#3960C3×C411yescharacteristic, complemented
71#3958C3×C411yescharacteristic, complemented
72D80C3×C411yescharacteristic, complemented
73#3930C22×C311yescharacteristic
74AGL(1,ℤ/16)—51noSylow
75C22×C3×D5—41no—
76C2×C3×AGL(1,𝔽5)—21no—
77C2×C3×AGL(1,𝔽5)—41no—
78C2×C3×AGL(1,𝔽5)—82no—
79C3×C5×Q8—21no—
80C3×C5×D4—21no—
81C3×C22⋊D5—41no—
82C3×D20—21no—
83C3×D20—21no—
84C3×D20—41no—
85C3×C4×D5—41no—
86C3×C4×D5C2×D411yescharacteristic
87C3×#170—21no—
88C3×#170—41no—
89C3×C5×C8C22×C411yescharacteristic
90C3×#168#4111yescharacteristic
91C3×#135—51no—
92C2×C3×D8—51no—
93C3×#131—101no—
94C3×C4×D4—101no—
95C3×SD32—51no—
96C3×SD32AGL(1,𝔽5)11yescharacteristic, complemented
97C3×D16—51no—
98C3×D16—51no—
99C3×M32—51no—
100C3×M32—51no—
101C3×#108—51no—
102C3×#106—51no—
103C3×#104—101no—
104C3×#102—101no—
105C3×#97—51no—
106C22×AGL(1,𝔽5)—41no—
107#713—21no—
108D4×D5—21no—
109C22⋊AGL(1,𝔽5)—41no—
110C4⋊AGL(1,𝔽5)—21no—
111C4×AGL(1,𝔽5)C3×D411yescharacteristic
112#700—21no—
113#700—41no—
114D5⋊C8—41no—
115D5⋊C8C3×D411yescharacteristic
116C5×Q16C2×C3×C411yescharacteristic, complemented
117C5×D8C2×C3×C411yescharacteristic
118#688—21no—
119#686—21no—
120D40C2×C3×C411yescharacteristic
121D40C2×C3×C411yescharacteristic, complemented
122C8×D5C23×C311yescharacteristic
123#674—21no—
124C5×C16C2×C3×C411yescharacteristic, complemented
125#672C2×C3×C411yescharacteristic, complemented
126#509—51no—
127#444—51no—
128#442—51no—
129#365—51no—
130#361—51no—
131#360—51no—
132#347—51no—
133C22×C3×C5—41no—
134C2×C3×D5—41no—
135C2×C3×D5—41no—
136C2×C3×D5—81no—
137C2×C3×D5—81no—
138C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
139C3×AGL(1,𝔽5)—41no—
140C3×AGL(1,𝔽5)—81no—
141C3×C4×C5—41no—
142C3×C4×C5C4×D411yescharacteristic
143C3×Dic5AGL(1,ℤ/8)11yescharacteristic
144C3×#41—101no—
145C2×C3×D4—101no—
146C22×C3×C4—201no—
147C3×Q16C2×AGL(1,𝔽5)11yescharacteristic, complemented
148C3×SD16—101no—
149C3×D8—51no—
150C3×D8—51no—
151C3×D8—101no—
152C3×D8C2×AGL(1,𝔽5)11yescharacteristic
153C3×M16—101no—
154C3×M16—201no—
155C2×C3×C8—51no—
156C2×C3×C8—51no—
157C2×C3×C8—201no—
158C3×C4⋊C4—101no—
159C3×C22⋊C4—201no—
160C3×C42—51no—
161C3×C16—51no—
162C3×C16—101no—
163C3×C16C2×AGL(1,𝔽5)11yescharacteristic, complemented
164C22×D5—41no—
165C2×AGL(1,𝔽5)—21no—
166C2×AGL(1,𝔽5)—41no—
167C2×AGL(1,𝔽5)—82no—
168C5×Q8—21no—
169C5×D4—21no—
170C22⋊D5—41no—
171D20—21no—
172D20—21no—
173D20—41no—
174C4×D5—41no—
175C4×D5C2×C3×D411yescharacteristic
176#170—21no—
177#170—41no—
178C5×C8C22×C3×C411yescharacteristic
179#168C3×#4111yescharacteristic
180#135—51no—
181C2×D8—51no—
182#131—101no—
183C4×D4—101no—
184SD32—51no—
185SD32C3×AGL(1,𝔽5)11yescharacteristic, complemented
186D16—51no—
187D16—51no—
188M32—51no—
189M32—51no—
190#108—51no—
191#106—51no—
192#104—101no—
193#102—101no—
194#97—51no—
195C2×C3×C5—81no—
196C2×C3×C5#44211yescharacteristic
197C3×D5—21no—
198C3×D5—81no—
199C3×D5—81no—
200C23×C3—201no—
201C3×Q8—21no—
202C3×D4—21no—
203C3×D4—101no—
204C3×D4—101no—
205C3×D4—201no—
206C3×D4—201no—
207C2×C3×C4—51no—
208C2×C3×C4—101no—
209C2×C3×C4—201no—
210C2×C3×C4—201no—
211C2×C3×C4—402no—
212C3×C8—51no—
213C3×C8—101no—
214C3×C8—201no—
215C3×C8C22×AGL(1,𝔽5)11yescharacteristic
216C22×C5—41no—
217C2×D5—41no—
218C2×D5—41no—
219C2×D5—81no—
220C2×D5—81no—
221C2×D5C3×AGL(1,ℤ/8)11yescharacteristic
222AGL(1,𝔽5)—41no—
223AGL(1,𝔽5)—81no—
224C4×C5—41no—
225C4×C5C3×C4×D411yescharacteristic
226Dic5C3×AGL(1,ℤ/8)11yescharacteristic
227#41—101no—
228C2×D4—101no—
229C22×C4—201no—
230Q16C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
231SD16—101no—
232D8—51no—
233D8—51no—
234D8—101no—
235D8C2×C3×AGL(1,𝔽5)11yescharacteristic
236M16—101no—
237M16—201no—
238C2×C8—51no—
239C2×C8—51no—
240C2×C8—201no—
241C4⋊C4—101no—
242C22⋊C4—201no—
243C42—51no—
244C16—51no—
245C16—101no—
246C16C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
247C3×C5AGL(1,ℤ/16)11yescharacteristic, complemented, Hall
248C22×C3—41no—
249C22×C3—51no—
250C22×C3—201no—
251C22×C3—201no—
252C22×C3—401no—
253C22×C3—401no—
254C3×C4—41no—
255C3×C4—51no—
256C3×C4—201no—
257C3×C4—401no—
258C3×C4D4×AGL(1,𝔽5)11yescharacteristic
259C2×C5—81no—
260C2×C5C3×#44211yescharacteristic
261D5—21no—
262D5—81no—
263D5—81no—
264C23—201no—
265Q8—21no—
266D4—21no—
267D4—101no—
268D4—101no—
269D4—201no—
270D4—201no—
271C2×C4—51no—
272C2×C4—101no—
273C2×C4—201no—
274C2×C4—201no—
275C2×C4—402no—
276C8—51no—
277C8—101no—
278C8—201no—
279C8C22×C3×AGL(1,𝔽5)11yescharacteristic
280C2×C3—81no—
281C2×C3—101no—
282C2×C3—401no—
283C2×C3—401no—
284C2×C3#6573111yescharacteristic
285C5C3×AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
286C22—41no—
287C22—51no—
288C22—201no—
289C22—201no—
290C22—401no—
291C22—401no—
292C4—41no—
293C4—51no—
294C4—201no—
295C4—401no—
296C4C3×D4×AGL(1,𝔽5)11yescharacteristic
297C3#1060413211yescharacteristic, direct product factor, Sylow
298C2—81no—
299C2—101no—
300C2—401no—
301C2—401no—
302C2C3×#6573111yescharacteristic
subgroupG/H
centerC2×C3#65731
commutatorC5×C8C22×C3×C4
FrattiniC8C22×C3×AGL(1,𝔽5)
FittingC3×C5×SD32C4
socleC2×C3×C5#442
prime radicalC3×#65203C2
cosocleC8×D5C23×C3
hypercenterC3×SD32AGL(1,𝔽5)
hypocenterC5C3×AGL(1,ℤ/16)
LCSG ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×SD32

Complements

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

NH ≅ G/Ntype
1. C3×#65741C2semidirect
2. C3×#65731C2semidirect
4. C3×#64907C2semidirect
5. C3×#64906C2semidirect
6. C3×#64847C2semidirect
7. C3×#64845C2semidirect
8. #10604132C3direct (N × H)
19. C3×C5×SD32C4semidirect
20. C3×#3960C4semidirect
21. C3×#3958C4semidirect
22. C3×D80C4semidirect
25. #65741C2×C3semidirect
26. #65731C2×C3semidirect
28. #64907C2×C3semidirect
29. #64906C2×C3semidirect
30. #64847C2×C3semidirect
31. #64845C2×C3semidirect
42. C3×C5×Q16C2×C4semidirect
47. C3×D40C2×C4semidirect
50. C3×C5×C16C2×C4semidirect
51. C3×#672C2×C4semidirect
69. C5×SD32C3×C4semidirect
70. #3960C3×C4semidirect
71. #3958C3×C4semidirect
72. D80C3×C4semidirect
96. C3×SD32AGL(1,𝔽5)semidirect
116. C5×Q16C2×C3×C4semidirect
121. D40C2×C3×C4semidirect
124. C5×C16C2×C3×C4semidirect
125. #672C2×C3×C4semidirect
147. C3×Q16C2×AGL(1,𝔽5)semidirect
163. C3×C16C2×AGL(1,𝔽5)semidirect
185. SD32C3×AGL(1,𝔽5)semidirect
230. Q16C2×C3×AGL(1,𝔽5)semidirect
246. C16C2×C3×AGL(1,𝔽5)semidirect
247. C3×C5AGL(1,ℤ/16)semidirect
285. C5C3×AGL(1,ℤ/16)semidirect
297. C3#10604132direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#657412
C3×#657312
C3×#652032
C3×#649072
C3×#649062
C3×#648472
C3×#648452
#106041323
C3×AGL(1,ℤ/16)5