Finite Group Databasebeta

← C3×D5×#10587723 #30655366

Subgroups — C3×D5×#10587723 #30655366

Order 16380.

240 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 60 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×D5×#63271C211yesmaximal, characteristic, complemented
2C3×C5⋊#10587723C211yesmaximal, characteristic, complemented
3C3×C5×#10587723C211yesmaximal, characteristic, complemented
4C3×D5×D91C311yesmaximal, characteristic, complemented
5D5×#10587723C313yesmaximal, direct product factor
6C3×C5×#63271C2211yescharacteristic, complemented, Hall
7C2×C3×#10587723—51nomaximal, Hall
8C5⋊#10587723C2×C313yescomplemented
9D5×#63271C2×C313yescomplemented
10C5×#10587723C2×C313yescomplemented
11C3×D455C2×C311yescharacteristic, complemented
12C3×C7×D5×C13C2×C311yescharacteristic, complemented
13C3×C5×D91C2×C311yescharacteristic, complemented
14C3×D5×D13⋊C3—71nomaximal, Hall
15D5×D91C3211yescharacteristic, complemented, Hall
16C2×C3×#63271—51no—
17C3×#10587723—51no—
18C3×#10587723D511yescharacteristic, direct product factor
19C3×C5×C7×C13C22×C311yescharacteristic, complemented
20C5×#63271C22×C313yescomplemented
21C3×D5×AGL(1,𝔽7)—131nomaximal, Hall
22C3×C5⋊(D13⋊C3)—71no—
23C3×D5×C13⋊C3—71no—
24C3×C5×D13⋊C3—71no—
25C2×C3×D91—51no—
26C2×#10587723—53no—
27D455C2×C3211yescharacteristic, complemented
28C7×D5×C13C2×C3211yescharacteristic, complemented
29C5×D91C2×C3211yescharacteristic, complemented
30C3×#63271C2×D511yescharacteristic, complemented, Hall
31C3×D5×D13—211no—
32D5×D13⋊C3—73no—
33C3×C5⋊AGL(1,𝔽7)—131no—
34C3×D5×ΓL(1,𝔽8)—131no—
35C3×C5×AGL(1,𝔽7)—131no—
36C3×C5×C13⋊C3—71noHall
37C2×C3×C7×C13—51no—
38C3×D91—51no—
39C3×D91C3×D511yescharacteristic, complemented
40C2×#63271—53no—
41#10587723—53no—
42#10587723C3×D513yesdirect product factor
43C2×C3×D13⋊C3—351noHall
44C5×C7×C13C22×C3211yescharacteristic, complemented, Hall
45C3×D5×D7—391no—
46D5×AGL(1,𝔽7)—133no—
47C3×D65—211no—
48C3×D5×C13AGL(1,𝔽7)11yescharacteristic, complemented
49C3×C5×D13—211no—
50C5⋊(D13⋊C3)—73no—
51D5×C13⋊C3—73no—
52C5×D13⋊C3—73no—
53C2×D91—51noHall
54C3×C5×ΓL(1,𝔽8)—131noHall
55C3×C7×C13C2×C3×D511yescharacteristic, complemented
56#63271C2×C3×D513yescomplemented
57D5×D13—211noHall
58C2×C3×AGL(1,𝔽7)—651noHall
59C2×C3×C13⋊C3—351no—
60C3×D13⋊C3—71no—
61C3×D13⋊C3—351no—
62C3×D35—131no—
63C3×C7×D5D13⋊C311yescharacteristic, complemented
64C3×C5×D7—391no—
65C5⋊AGL(1,𝔽7)—133no—
66D5×ΓL(1,𝔽8)—133no—
67C5×AGL(1,𝔽7)—133no—
68C3×C5×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
69C5×C13⋊C3—73no—
70C2×C7×C13—51no—
71D91—51no—
72D91C32×D511yescharacteristic, complemented
73C2×C32×D5—911noHall
74C2×C3×D13—1051no—
75C2×D13⋊C3—353no—
76D5×D7—391noHall
77D65—211no—
78C5×D13—211no—
79D5×C13C3×AGL(1,𝔽7)11yescharacteristic, complemented
80C2×C3×ΓL(1,𝔽8)—651no—
81C3×AGL(1,𝔽7)—131no—
82C3×AGL(1,𝔽7)—651no—
83C3×C13⋊C3—71noHall
84C3×C5×C7C2×D13⋊C311yescharacteristic, complemented
85C5×ΓL(1,𝔽8)—133no—
86C7×C13C2×C32×D511yescharacteristic, complemented, Hall
87C2×C32×C5—911no—
88C32×D5—911no—
89C32×D5—911no—
90C2×C3×D7—651no—
91C2×AGL(1,𝔽7)—653no—
92C2×C3×C13—51no—
93C3×D13—71no—
94C3×D13—1051no—
95C2×C13⋊C3—353no—
96D13⋊C3—73no—
97D13⋊C3—353no—
98D35—131no—
99C5×D7—391no—
100C7×D5C3×D13⋊C311yescharacteristic, complemented
101C5×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
102C3×ΓL(1,𝔽8)—131noHall
103C2×C3×D5—2731no—
104C2×C3×D5—913no—
105C2×D13—1051noHall
106C32×C5—911noHall
107C2×C3×C7—51no—
108C3×D7—391no—
109C3×D7—1951no—
110C2×ΓL(1,𝔽8)—653no—
111AGL(1,𝔽7)—133no—
112AGL(1,𝔽7)—653no—
113C3×C13D5×AGL(1,𝔽7)11yescharacteristic, complemented
114C13⋊C3—73no—
115C22×C32—4551noHall
116C5×C7C2×C3×D13⋊C311yescharacteristic, complemented, Hall
117C2×C3×C5—2731no—
118C2×C3×C5—913no—
119C3×D5—2731no—
120C3×D5—913no—
121C3×D5—913no—
122C3×D5#1058772311yescharacteristic, direct product factor
123C2×D7—651noHall
124C2×C13—51no—
125D13—71no—
126D13—1051no—
127C3×C7D5×D13⋊C311yescharacteristic, complemented
128ΓL(1,𝔽8)—133no—
129C2×D5—2731noHall
130C2×C32—911no—
131C2×C32—4551no—
132C2×C32—4551no—
133C3×C5—913no—
134C3×C5C2×#1058772311yescharacteristic, complemented
135C2×C7—51no—
136D7—391no—
137D7—1951no—
138C13C3×D5×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
139C22×C3—13651no—
140C22×C3—4553no—
141C2×C5—2731no—
142D5—2731no—
143D5C3×#1058772311yescharacteristic, direct product factor
144C32—911noSylow
145C7C3×D5×D13⋊C311yescharacteristic, complemented, Sylow
146C2×C3—51no—
147C2×C3—2731no—
148C2×C3—13651no—
149C2×C3—913no—
150C2×C3—4553no—
151C2×C3—4553no—
152C5C2×C3×#1058772311yescharacteristic, complemented, Sylow
153C22—13651noSylow
154C3—913no—
155C3D5×#1058772311yescharacteristic, direct product factor
156C2—51no—
157C2—2731no—
158C2—13651no—
subgroupG/H
center, hypercenterC3D5×#10587723
commutator, cosocle, hypocenterC5×C7×C13C22×C32
FrattiniC1G
Fitting, socleC3×C5×C7×C13C22×C3
LCSG ▷ C5×C7×C13
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×D5×#63271C2semidirect
2. C3×C5⋊#10587723C2semidirect
3. C3×C5×#10587723C2semidirect
4. C3×D5×D91C3semidirect
5. D5×#10587723C3direct (N × H)
6. C3×C5×#63271C22semidirect
8. C5⋊#10587723C2×C3semidirect
9. D5×#63271C2×C3semidirect
10. C5×#10587723C2×C3semidirect
11. C3×D455C2×C3semidirect
12. C3×C7×D5×C13C2×C3semidirect
13. C3×C5×D91C2×C3semidirect
15. D5×D91C32semidirect
18. C3×#10587723D5direct (N × H)
19. C3×C5×C7×C13C22×C3semidirect
20. C5×#63271C22×C3semidirect
27. D455C2×C32semidirect
28. C7×D5×C13C2×C32semidirect
29. C5×D91C2×C32semidirect
30. C3×#63271C2×D5semidirect
39. C3×D91C3×D5semidirect
42. #10587723C3×D5direct (N × H)
44. C5×C7×C13C22×C32semidirect
48. C3×D5×C13AGL(1,𝔽7)semidirect
55. C3×C7×C13C2×C3×D5semidirect
56. #63271C2×C3×D5semidirect
63. C3×C7×D5D13⋊C3semidirect
68. C3×C5×C13C2×AGL(1,𝔽7)semidirect
72. D91C32×D5semidirect
79. D5×C13C3×AGL(1,𝔽7)semidirect
84. C3×C5×C7C2×D13⋊C3semidirect
86. C7×C13C2×C32×D5semidirect
100. C7×D5C3×D13⋊C3semidirect
101. C5×C13C2×C3×AGL(1,𝔽7)semidirect
113. C3×C13D5×AGL(1,𝔽7)semidirect
116. C5×C7C2×C3×D13⋊C3semidirect
122. C3×D5#10587723direct (N × H)
127. C3×C7D5×D13⋊C3semidirect
134. C3×C5C2×#10587723semidirect
138. C13C3×D5×AGL(1,𝔽7)semidirect
143. D5C3×#10587723direct (N × H)
145. C7C3×D5×D13⋊C3semidirect
152. C5C2×C3×#10587723semidirect
155. C3D5×#10587723direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×#632712
C3×C5⋊#105877232
C3×C5×#105877232
C3×D5×D913
D5×#105877233
C2×C3×#105877235
C3×D5×D13⋊C37
C3×D5×AGL(1,𝔽7)13