Finite Group Databasebeta

← C3×#66170 #11755847

Subgroups — C3×#66170 #11755847

Order 960.

516 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 214 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×C5×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#4150C211yesmaximal, characteristic, complemented
3C3×#4097C213yesmaximal, complemented
4C3×#4096C213yesmaximal, complemented
5C3×#4094C211yesmaximal, characteristic, complemented
6C2×C3×#687C213yesmaximal, complemented
7C3×#4078C213yesmaximal, complemented
8#66170C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×#41—21no—
10C3×C5×#41C2213yes—
11C3×C5×#41C2211yescharacteristic
12C2×C3×C5×D4—23no—
13C2×C3×C5×D4C2213yes—
14C3×Q8×D5—21no—
15C3×#711—23no—
16C3×#709C2213yescomplemented
17C2×C3×Dic10C2213yescomplemented
18C3×#689C2213yescomplemented
19C3×#688C2211yescharacteristic, complemented
20C3×#687C2216yescomplemented
21C3×#687C2213yescomplemented
22C3×#686C2213yescomplemented
23C3×#681C2213yescomplemented
24C2×C3×#168C2213yescomplemented
25C3×#577—51nomaximal, Hall
26C5×H(5,𝔽2)C2×C311yescharacteristic, complemented
27#4150C2×C311yescharacteristic, complemented
28#4097C2×C313yescomplemented
29#4096C2×C313yescomplemented
30#4094C2×C311yescharacteristic, complemented
31C2×#687C2×C313yescomplemented
32#4078C2×C313yescomplemented
33C23×C3×C5—23no—
34C3×C5×Q8D411yescharacteristic
35C3×C5×Q8C2311yescharacteristic
36C3×C5×D4—23no—
37C3×C5×D4—23no—
38C3×C5×D4D413yes—
39C3×C5×D4C2313yes—
40C2×C3×C4×C5—23no—
41C2×C3×C4×C5C2313yes—
42C3×C22⋊D5—23no—
43C2×C3×Dic5—23no—
44C3×D20C2311yescharacteristic
45C3×C4×D5—23no—
46C3×Dic10—23no—
47C3×Dic10C2313yes—
48C3×#168C2311yescharacteristic
49C3×#168C2313yes—
50C3×2−1+4—51no—
51C3×H(5,𝔽2)D511yescharacteristic, complemented
52C3×#137—53no—
53C3×AGL(1,ℤ/8)—53no—
54C3×#135—53no—
55C2×C3×SD16—53no—
56C3×#131—51no—
57C5×#41—21no—
58C5×#41C22×C313yes—
59C5×#41C22×C311yescharacteristic
60C2×C5×D4—23no—
61C2×C5×D4C22×C313yes—
62Q8×D5—21no—
63#711—23no—
64#709C22×C313yescomplemented
65C2×Dic10C22×C313yescomplemented
66#689C22×C313yescomplemented
67#688C22×C311yescharacteristic, complemented
68#687C22×C313yescomplemented
69#687C22×C316yescomplemented
70#686C22×C313yescomplemented
71#681C22×C313yescomplemented
72C2×#168C22×C313yescomplemented
73#577—51noSylow
74C22×C3×C5—23no—
75C22×C3×C5—83no—
76C22×C3×C5C2×D413yes—
77C2×C3×D5—21no—
78C3×C4×C5—21no—
79C3×C4×C5C2×D413yes—
80C3×C4×C5C2411yescharacteristic
81C3×Dic5—23no—
82C3×#41—21no—
83C3×#41—53no—
84C3×#41—103no—
85C3×#41C2×D513yes—
86C3×#41C2×D511yescharacteristic
87C2×C3×Q8—53no—
88C2×C3×Q8—101no—
89C2×C3×D4—23no—
90C2×C3×D4C2×D513yes—
91C3×Q16—53no—
92C3×SD16—51no—
93C3×SD16—53no—
94C3×SD16—56no—
95C3×D8—53no—
96C3×M16—53no—
97C2×C3×C8—53no—
98C23×C5—23no—
99C5×Q8C3×D411yescharacteristic
100C5×Q8C23×C311yescharacteristic
101C5×D4—23no—
102C5×D4—23no—
103C5×D4C3×D413yes—
104C5×D4C23×C313yes—
105C2×C4×C5—23no—
106C2×C4×C5C23×C313yes—
107C22⋊D5—23no—
108C2×Dic5—23no—
109D20C23×C311yescharacteristic
110C4×D5—23no—
111Dic10—23no—
112Dic10C23×C313yes—
113#168C23×C311yescharacteristic
114#168C23×C313yes—
1152−1+4—51no—
116H(5,𝔽2)C3×D511yescharacteristic, complemented
117#137—53no—
118AGL(1,ℤ/8)—53no—
119#135—53no—
120C2×SD16—53no—
121#131—51no—
122C2×C3×C5—23no—
123C2×C3×C5—43no—
124C2×C3×C5C22×D411yescharacteristic
125C3×D5—41no—
126C23×C3—23no—
127C3×Q8—53no—
128C3×Q8—103no—
129C3×Q8C22⋊D511yescharacteristic
130C3×Q8C22×D511yescharacteristic
131C3×D4—23no—
132C3×D4—23no—
133C3×D4—51no—
134C3×D4—103no—
135C3×D4C22⋊D513yes—
136C3×D4C22×D513yes—
137C2×C3×C4—23no—
138C2×C3×C4—103no—
139C2×C3×C4—103no—
140C2×C3×C4C22×D513yes—
141C3×C8—51no—
142C3×C8—53no—
143C22×C5—23no—
144C22×C5—83no—
145C22×C5C2×C3×D413yes—
146C2×D5—21no—
147C4×C5—21no—
148C4×C5C2×C3×D413yes—
149C4×C5C24×C311yescharacteristic
150Dic5—23no—
151#41—21no—
152#41—53no—
153#41—103no—
154#41C2×C3×D513yes—
155#41C2×C3×D511yescharacteristic
156C2×Q8—53no—
157C2×Q8—101no—
158C2×D4—23no—
159C2×D4C2×C3×D513yes—
160Q16—53no—
161SD16—51no—
162SD16—53no—
163SD16—56no—
164D8—53no—
165M16—53no—
166C2×C8—53no—
167C3×C5#57711yescharacteristic, complemented, Hall
168C22×C3—23no—
169C22×C3—83no—
170C22×C3—101no—
171C22×C3C2×C22⋊D513yes—
172C3×C4—21no—
173C3×C4—103no—
174C3×C4C2×C22⋊D513yes—
175C3×C4C23×D511yescharacteristic
176C2×C5—23no—
177C2×C5—43no—
178C2×C5C22×C3×D411yescharacteristic
179D5—41no—
180C23—23no—
181Q8—53no—
182Q8—103no—
183Q8C3×C22⋊D511yescharacteristic
184Q8C22×C3×D511yescharacteristic
185D4—23no—
186D4—23no—
187D4—51no—
188D4—103no—
189D4C3×C22⋊D513yes—
190D4C22×C3×D513yes—
191C2×C4—23no—
192C2×C4—103no—
193C2×C4—103no—
194C2×C4C22×C3×D513yes—
195C8—51no—
196C8—53no—
197C2×C3—23no—
198C2×C3—43no—
199C2×C3—201no—
200C2×C3C22×C22⋊D511yescharacteristic
201C5C3×#57711yescharacteristic, complemented, Sylow
202C22—23no—
203C22—83no—
204C22—101no—
205C22C2×C3×C22⋊D513yes—
206C4—21no—
207C4—103no—
208C4C2×C3×C22⋊D513yes—
209C4C23×C3×D511yescharacteristic
210C3#6617011yescharacteristic, direct product factor, Sylow
211C2—23no—
212C2—43no—
213C2—201no—
214C2C22×C3×C22⋊D511yescharacteristic
subgroupG/H
centerC2×C3C22×C22⋊D5
commutator, cosocleC4×C5C24×C3
FrattiniC4C23×C3×D5
FittingC3×C5×H(5,𝔽2)C2
socleC2×C3×C5C22×D4
hypercenterC3×H(5,𝔽2)D5
hypocenterC5C3×#577
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C3×C5×H(5,𝔽2)C2semidirect
2. C3×#4150C2semidirect
3. C3×#4097C2semidirect
4. C3×#4096C2semidirect
5. C3×#4094C2semidirect
6. C2×C3×#687C2semidirect
7. C3×#4078C2semidirect
8. #66170C3direct (N × H)
16. C3×#709C22semidirect
17. C2×C3×Dic10C22semidirect
18. C3×#689C22semidirect
19. C3×#688C22semidirect
20. C3×#687C22semidirect
21. C3×#687C22semidirect
22. C3×#686C22semidirect
23. C3×#681C22semidirect
24. C2×C3×#168C22semidirect
26. C5×H(5,𝔽2)C2×C3semidirect
27. #4150C2×C3semidirect
28. #4097C2×C3semidirect
29. #4096C2×C3semidirect
30. #4094C2×C3semidirect
31. C2×#687C2×C3semidirect
32. #4078C2×C3semidirect
51. C3×H(5,𝔽2)D5semidirect
64. #709C22×C3semidirect
65. C2×Dic10C22×C3semidirect
66. #689C22×C3semidirect
67. #688C22×C3semidirect
68. #687C22×C3semidirect
69. #687C22×C3semidirect
70. #686C22×C3semidirect
71. #681C22×C3semidirect
72. C2×#168C22×C3semidirect
116. H(5,𝔽2)C3×D5semidirect
167. C3×C5#577semidirect
201. C5C3×#577semidirect
210. C3#66170direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×H(5,𝔽2)2
C3×#41502
C3×#40972
C3×#40962
C3×#40942
C2×C3×#6872
C3×#40782
#661703
C3×#5775