Finite Group Databasebeta

← #30655511

Subgroups — #30655511

Order 16380.

176 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 35 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⋊#10587716)C211yesmaximal, characteristic, complemented
2C5×C3⋊#10587722C211yesmaximal, characteristic, complemented
3C3×C5⋊#10587720C211yesmaximal, characteristic, complemented
4D35⋊D39C311yesmaximal, characteristic, complemented
5#10587722⋊D5—31nomaximal
6C3×C5×#63270C2211yescharacteristic, complemented, Hall
7#10587720⋊D3—51nomaximal, Hall
8C5⋊#10587720—21no—
9C5⋊#10587720D311yescharacteristic, complemented
10C5⋊#10587716—31no—
11C5×#10587722—31no—
12C7×D195C2×C311yescharacteristic, complemented
13C5×D273C2×C311yescharacteristic, complemented
14C3×C13×D35C2×C311yescharacteristic, complemented
15D5×C3⋊(D13⋊C3)—71nomaximal, Hall
16D35⋊D13—31noHall
17C3⋊#10587722D511yescharacteristic, complemented
18C3⋊#10587716—51no—
19C3×#10587720—51no—
20C3×C5×C7×C13C22×C311yescharacteristic, complemented
21C5×#63270—21no—
22C5×#63270C2×D311yescharacteristic, complemented
23D15⋊AGL(1,𝔽7)—131nomaximal, Hall
24C3⋊(C5⋊(D13⋊C3))—141no—
25C5×C3⋊(D13⋊C3)—71no—
26C3×D5×C13⋊C3—71no—
27D7×D39—51no—
28#10587716⋊C2—151no—
29C13×D35C3×D311yescharacteristic, complemented
30C7×D65—31no—
31C5×D91—31no—
32C3×#63270C2×D511yescharacteristic, complemented, Hall
33D5×D39—71no—
34D5×D13⋊C3—211no—
35ΓL(1,𝔽8)×D15—131no—
36C5×C3⋊AGL(1,𝔽7)—131no—
37C3×C5⋊AGL(1,𝔽7)—261no—
38C3×C5×C13⋊C3—141noHall
39D273C3×D511yescharacteristic, complemented
40C7×D39—51no—
41C3×C13×D7—51no—
42#10587722—31no—
43#10587720—51no—
44#10587720—101no—
45#10587716—151no—
46C2×C3⋊(D13⋊C3)—351noHall
47C5×C7×C13C2×C3×D311yescharacteristic, complemented, Hall
48D15⋊D7—131no—
49D5×AGL(1,𝔽7)—391no—
50D195AGL(1,𝔽7)11yescharacteristic, complemented
51C5×D39—71no—
52C3×D5×C13—71no—
53C5⋊(D13⋊C3)—421no—
54D5×C13⋊C3—71no—
55D5×C13⋊C3—141no—
56C5×D13⋊C3—211no—
57D7×D13—151noHall
58C3×C5×ΓL(1,𝔽8)—261noHall
59C3×C7×C13C2×C3×D511yescharacteristic, complemented
60#63270—21no—
61#63270D3×D511yescharacteristic, complemented
62D5×D13—211noHall
63D3×AGL(1,𝔽7)—651noHall
64C2×C3×C13⋊C3—351no—
65C3⋊(D13⋊C3)—71no—
66C3⋊(D13⋊C3)—701no—
67C7×D15—131no—
68C5×D21—131no—
69C3×D35D13⋊C311yescharacteristic, complemented
70C5⋊AGL(1,𝔽7)—261no—
71C5⋊AGL(1,𝔽7)—261no—
72D5×ΓL(1,𝔽8)—391no—
73C5×AGL(1,𝔽7)—391no—
74C3×C5×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
75C5×C13⋊C3—141no—
76C5×C13⋊C3—141no—
77D91—31no—
78C7×D13—151no—
79C13×D7—51no—
80C3×D3×D5—911noHall
81C2×D39—351no—
82C2×D13⋊C3—1051no—
83D5×D7—391noHall
84D65—31no—
85C5×D13—211no—
86D5×C13—71no—
87C3⋊AGL(1,𝔽7)—131no—
88D3×ΓL(1,𝔽8)—651no—
89C3×AGL(1,𝔽7)—651no—
90C3×C13⋊C3—141noHall
91C3×C5×C7C2×D13⋊C311yescharacteristic, complemented
92C5×ΓL(1,𝔽8)—261no—
93C5×ΓL(1,𝔽8)—261no—
94C7×C13C3×D3×D511yescharacteristic, complemented, Hall
95C3×D15—1821no—
96C3×C5×D3—911no—
97C32×D5—1821no—
98D3×D7—651no—
99C2×AGL(1,𝔽7)—1951no—
100C2×C3×C13—351no—
101D39—51no—
102D39—71no—
103C2×C13⋊C3—351no—
104C2×C13⋊C3—701no—
105D13⋊C3—211no—
106D13⋊C3—1051no—
107D35C3⋊(D13⋊C3)11yescharacteristic, complemented
108C5×D7—391no—
109C7×D5—391no—
110C5×C13D3×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
111C3×ΓL(1,𝔽8)—131noHall
112C2×C3×D5—2731no—
113D3×D5—911no—
114C2×D13—1051noHall
115C32×C5—3641noHall
116D21—131no—
117C3×D7—51no—
118D3×C7—651no—
119C2×ΓL(1,𝔽8)—1951no—
120AGL(1,𝔽7)—391no—
121AGL(1,𝔽7)—1301no—
122AGL(1,𝔽7)—1301no—
123C3×C13D5×AGL(1,𝔽7)11yescharacteristic, complemented
124C13⋊C3—71no—
125C13⋊C3—281no—
126C2×C3×D3—4551noHall
127C5×C7C2×C3⋊(D13⋊C3)11yescharacteristic, complemented, Hall
128C2×C3×C5—2731no—
129D15—131no—
130C3×D5—71no—
131C3×D5—5461no—
132C3×D5—1821no—
133C3×D5—1821no—
134C5×D3—911no—
135C2×D7—1951noHall
136C2×C13—351no—
137D13—151no—
138D13—211no—
139C3×C7D5×D13⋊C311yescharacteristic, complemented
140ΓL(1,𝔽8)—261no—
141ΓL(1,𝔽8)—261no—
142C2×D5—2731noHall
143C2×C32—9101no—
144C3×D3—911no—
145C3×D3—9101no—
146C3×C5—3641no—
147C3×C5—1821no—
148C3×C5#10587716⋊C211yescharacteristic, complemented
149C2×C7—1951no—
150D7—51no—
151D7—391no—
152C13D15⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
153C22×C3—13651no—
154C2×D3—4551no—
155C2×C5—2731no—
156D5—71no—
157D5—391no—
158C32—3641noSylow
159C7D5×C3⋊(D13⋊C3)11yescharacteristic, complemented, Sylow
160C2×C3—2731no—
161C2×C3—351no—
162C2×C3—9101no—
163C2×C3—9101no—
164C2×C3—27301no—
165D3—651no—
166D3—911no—
167C5#10587720⋊D311yescharacteristic, complemented, Sylow
168C22—13651noSylow
169C3—3641no—
170C3—3641no—
171C3#10587722⋊D511yescharacteristic, complemented
172C2—2731no—
173C2—351no—
174C2—1951no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C7×C13C22×C3
LCSG ▷ C3×C5×C7×C13
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3⋊(C5⋊#10587716)C2semidirect
2. C5×C3⋊#10587722C2semidirect
3. C3×C5⋊#10587720C2semidirect
4. D35⋊D39C3semidirect
6. C3×C5×#63270C22semidirect
9. C5⋊#10587720D3semidirect
12. C7×D195C2×C3semidirect
13. C5×D273C2×C3semidirect
14. C3×C13×D35C2×C3semidirect
17. C3⋊#10587722D5semidirect
20. C3×C5×C7×C13C22×C3semidirect
22. C5×#63270C2×D3semidirect
29. C13×D35C3×D3semidirect
32. C3×#63270C2×D5semidirect
39. D273C3×D5semidirect
47. C5×C7×C13C2×C3×D3semidirect
50. D195AGL(1,𝔽7)semidirect
59. C3×C7×C13C2×C3×D5semidirect
61. #63270D3×D5semidirect
69. C3×D35D13⋊C3semidirect
74. C3×C5×C13C2×AGL(1,𝔽7)semidirect
91. C3×C5×C7C2×D13⋊C3semidirect
94. C7×C13C3×D3×D5semidirect
107. D35C3⋊(D13⋊C3)semidirect
110. C5×C13D3×AGL(1,𝔽7)semidirect
123. C3×C13D5×AGL(1,𝔽7)semidirect
127. C5×C7C2×C3⋊(D13⋊C3)semidirect
139. C3×C7D5×D13⋊C3semidirect
148. C3×C5#10587716⋊C2semidirect
152. C13D15⋊AGL(1,𝔽7)semidirect
159. C7D5×C3⋊(D13⋊C3)semidirect
167. C5#10587720⋊D3semidirect
171. C3#10587722⋊D5semidirect

Maximal subgroups

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

subgroupindex
C3⋊(C5⋊#10587716)2
C5×C3⋊#105877222
C3×C5⋊#105877202
D35⋊D393
#10587722⋊D53
#10587720⋊D35
D5×C3⋊(D13⋊C3)7
D15⋊AGL(1,𝔽7)13