Finite Group Databasebeta

← C2×C3×C5⋊#10587721 #30655573

Subgroups — C2×C3×C5⋊#10587721 #30655573

Order 16380.

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×C3×C5×#63271C211yesmaximal, characteristic, complemented
2C3×C5⋊#10587721C211yesmaximal, characteristic, direct product factor
3C3×C5⋊#10587721C211yesmaximal, characteristic, direct product factor
4C2×C3×C13×D35C311yesmaximal, characteristic, complemented
5C2×C5⋊#10587721C313yesmaximal, direct product factor
6C3×C5×#63271C2211yescharacteristic, complemented, Hall
7C2×C3×#10587721—51nomaximal, Hall
8C5⋊#10587721C2×C313yesdirect product factor
9C5⋊#10587721C2×C313yesdirect product factor
10C3×C13×D35C2×C311yescharacteristic, complemented
11C3×C13×D35C2×C311yescharacteristic, complemented
12C2×C5×#63271C2×C313yescomplemented
13C2×C3×C5×C7×C13C2×C311yescharacteristic, complemented
14C2×C3×D5×C13⋊C3—71nomaximal, Hall
15C2×C13×D35C3211yescharacteristic, complemented, Hall
16C2×C3×#63271D511yescharacteristic, complemented
17C3×#10587721—51no—
18C3×#10587721—51no—
19C3×C5×C7×C13C22×C311yescharacteristic, complemented
20C5×#63271C22×C313yescomplemented
21C2×C3×C5⋊AGL(1,𝔽7)—131nomaximal, Hall
22C2×C3×C5×C13⋊C3—71no—
23C3×D5×C13⋊C3—71no—
24C3×D5×C13⋊C3—71no—
25C2×C3×C13×D7—51no—
26C2×#10587721—53no—
27C2×C5×C7×C13C2×C3211yescharacteristic, complemented
28C13×D35C2×C3211yescharacteristic, complemented
29C13×D35C2×C3211yescharacteristic, complemented
30C3×#63271C2×D511yescharacteristic, complemented, Hall
31C2×C3×D5×C13—211no—
32C2×D5×C13⋊C3—73no—
33C2×C3×C5×ΓL(1,𝔽8)—131no—
34C3×C5⋊AGL(1,𝔽7)—131no—
35C3×C5⋊AGL(1,𝔽7)—131no—
36C3×C5×C13⋊C3—71noHall
37C2×C3×C7×C13C3×D511yescharacteristic, complemented
38C3×C13×D7—51no—
39C3×C13×D7—51no—
40C2×#63271C3×D513yescomplemented
41#10587721—53no—
42#10587721—53no—
43C22×C3×C13⋊C3—351noHall
44C5×C7×C13C22×C3211yescharacteristic, complemented, Hall
45C2×C3×D35C13⋊C311yescharacteristic, complemented
46C2×C5⋊AGL(1,𝔽7)—133no—
47C2×C3×C5×C13AGL(1,𝔽7)11yescharacteristic, complemented
48C3×D5×C13—211no—
49C3×D5×C13—211no—
50C2×C5×C13⋊C3—73no—
51D5×C13⋊C3—73no—
52D5×C13⋊C3—73no—
53C2×C13×D7—51noHall
54C3×C5×ΓL(1,𝔽8)—131noHall
55C3×C7×C13C2×C3×D511yescharacteristic, complemented
56#63271C2×C3×D513yescomplemented
57C2×D5×C13—211noHall
58C2×C3×AGL(1,𝔽7)—651noHall
59C2×C3×C13⋊C3—71no—
60C2×C3×C13⋊C3—351no—
61C2×C3×C13⋊C3—351no—
62C2×C3×C5×C7C2×C13⋊C311yescharacteristic, complemented
63C3×D35C2×C13⋊C311yescharacteristic, complemented
64C3×D35C2×C13⋊C311yescharacteristic, complemented
65C2×C5×ΓL(1,𝔽8)—133no—
66C5⋊AGL(1,𝔽7)—133no—
67C5⋊AGL(1,𝔽7)—133no—
68C3×C5×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
69C5×C13⋊C3—73no—
70C2×C7×C13C32×D511yescharacteristic, complemented
71C13×D7—51no—
72C13×D7—51no—
73C2×C32×D5—911noHall
74C22×C3×C13—1051no—
75C22×C13⋊C3—353no—
76C2×D35C3×C13⋊C311yescharacteristic, complemented, Hall
77C2×C5×C13C3×AGL(1,𝔽7)11yescharacteristic, complemented
78D5×C13—211no—
79D5×C13—211no—
80C2×C3×ΓL(1,𝔽8)—131no—
81C3×AGL(1,𝔽7)—651no—
82C3×AGL(1,𝔽7)—651no—
83C3×C13⋊C3—71noHall
84C3×C5×C7C22×C13⋊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—51no—
91C2×AGL(1,𝔽7)—653no—
92C2×C3×C13—351no—
93C2×C3×C13—1051no—
94C2×C3×C13C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
95C2×C13⋊C3—73no—
96C2×C13⋊C3—353no—
97C2×C13⋊C3—353no—
98C2×C5×C7C2×C3×C13⋊C311yescharacteristic, complemented
99D35C2×C3×C13⋊C311yescharacteristic, complemented
100D35C2×C3×C13⋊C311yescharacteristic, complemented
101C5×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
102C3×ΓL(1,𝔽8)—131noHall
103C2×C3×D5—211no—
104C2×C3×D5—913no—
105C22×C13—1051noHall
106C32×C5—911noHall
107C2×C3×C7D5×C13⋊C311yescharacteristic, complemented
108C3×D7—51no—
109C3×D7—51no—
110C2×ΓL(1,𝔽8)—133no—
111AGL(1,𝔽7)—653no—
112AGL(1,𝔽7)—653no—
113C3×C13C2×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
114C13⋊C3—73no—
115C22×C32—4551noHall
116C5×C7C22×C3×C13⋊C311yescharacteristic, complemented, Hall
117C2×C3×C5—913no—
118C2×C3×C5#1058772111yescharacteristic, complemented
119C3×D5—211no—
120C3×D5—211no—
121C3×D5—913no—
122C3×D5—913no—
123C2×D7—51noHall
124C2×C13—351no—
125C2×C13—1051no—
126C2×C13C3×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
127C3×C7C2×D5×C13⋊C311yescharacteristic, complemented
128ΓL(1,𝔽8)—133no—
129C2×D5—211noHall
130C2×C32—911no—
131C2×C32—4551no—
132C2×C32—4551no—
133C3×C5—913no—
134C3×C5C2×#1058772111yescharacteristic, complemented
135C2×C7C3×D5×C13⋊C311yescharacteristic, complemented
136D7—51no—
137D7—51no—
138C13C2×C3×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
139C22×C3—351no—
140C22×C3—4553no—
141C2×C5C3×#1058772111yescharacteristic, complemented
142D5—211no—
143D5—211no—
144C32—911noSylow
145C7C2×C3×D5×C13⋊C311yescharacteristic, complemented, Sylow
146C2×C3—351no—
147C2×C3—913no—
148C2×C3—1051no—
149C2×C3—4553no—
150C2×C3—4553no—
151C2×C3C5⋊#1058772111yescharacteristic, direct product factor
152C5C2×C3×#1058772111yescharacteristic, complemented, Sylow
153C22—351noSylow
154C3—913no—
155C3C2×C5⋊#1058772111yescharacteristic, direct product factor
156C2—351no—
157C2—1051no—
158C2C3×C5⋊#1058772111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C5⋊#10587721
commutator, cosocle, hypocenterC5×C7×C13C22×C32
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C13C2×C3
LCSG ▷ C5×C7×C13
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#63271C2semidirect
2. C3×C5⋊#10587721C2direct (N × H)
3. C3×C5⋊#10587721C2direct (N × H)
4. C2×C3×C13×D35C3semidirect
5. C2×C5⋊#10587721C3direct (N × H)
6. C3×C5×#63271C22semidirect
8. C5⋊#10587721C2×C3direct (N × H)
9. C5⋊#10587721C2×C3direct (N × H)
10. C3×C13×D35C2×C3semidirect
11. C3×C13×D35C2×C3semidirect
12. C2×C5×#63271C2×C3semidirect
13. C2×C3×C5×C7×C13C2×C3semidirect
15. C2×C13×D35C32semidirect
16. C2×C3×#63271D5semidirect
19. C3×C5×C7×C13C22×C3semidirect
20. C5×#63271C22×C3semidirect
27. C2×C5×C7×C13C2×C32semidirect
28. C13×D35C2×C32semidirect
29. C13×D35C2×C32semidirect
30. C3×#63271C2×D5semidirect
37. C2×C3×C7×C13C3×D5semidirect
40. C2×#63271C3×D5semidirect
44. C5×C7×C13C22×C32semidirect
45. C2×C3×D35C13⋊C3semidirect
47. C2×C3×C5×C13AGL(1,𝔽7)semidirect
55. C3×C7×C13C2×C3×D5semidirect
56. #63271C2×C3×D5semidirect
62. C2×C3×C5×C7C2×C13⋊C3semidirect
63. C3×D35C2×C13⋊C3semidirect
64. C3×D35C2×C13⋊C3semidirect
68. C3×C5×C13C2×AGL(1,𝔽7)semidirect
70. C2×C7×C13C32×D5semidirect
76. C2×D35C3×C13⋊C3semidirect
77. C2×C5×C13C3×AGL(1,𝔽7)semidirect
84. C3×C5×C7C22×C13⋊C3semidirect
86. C7×C13C2×C32×D5semidirect
94. C2×C3×C13C5⋊AGL(1,𝔽7)semidirect
98. C2×C5×C7C2×C3×C13⋊C3semidirect
99. D35C2×C3×C13⋊C3semidirect
100. D35C2×C3×C13⋊C3semidirect
101. C5×C13C2×C3×AGL(1,𝔽7)semidirect
107. C2×C3×C7D5×C13⋊C3semidirect
113. C3×C13C2×C5⋊AGL(1,𝔽7)semidirect
116. C5×C7C22×C3×C13⋊C3semidirect
118. C2×C3×C5#10587721semidirect
126. C2×C13C3×C5⋊AGL(1,𝔽7)semidirect
127. C3×C7C2×D5×C13⋊C3semidirect
134. C3×C5C2×#10587721semidirect
135. C2×C7C3×D5×C13⋊C3semidirect
138. C13C2×C3×C5⋊AGL(1,𝔽7)semidirect
141. C2×C5C3×#10587721semidirect
145. C7C2×C3×D5×C13⋊C3semidirect
151. C2×C3C5⋊#10587721direct (N × H)
152. C5C2×C3×#10587721semidirect
155. C3C2×C5⋊#10587721direct (N × H)
158. C2C3×C5⋊#10587721direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#632712
C3×C5⋊#105877212
C2×C3×C13×D353
C2×C5⋊#105877213
C2×C3×#105877215
C2×C3×D5×C13⋊C37
C2×C3×C5⋊AGL(1,𝔽7)13