Finite Group Databasebeta

← D55×C3⋊AGL(1,𝔽7) #28357607

Subgroups — D55×C3⋊AGL(1,𝔽7) #28357607

Order 13860.

176 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 42 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⋊(C11⋊AGL(1,𝔽7)))C211yesmaximal, characteristic, complemented
2C5×C11×C3⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C3×ΓL(1,𝔽8)×D55C211yesmaximal, characteristic, complemented
4D21×D55C311yesmaximal, characteristic, complemented
5AGL(1,𝔽7)×D55—31nomaximal
6C3×C5×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7D11×C3⋊AGL(1,𝔽7)—51nomaximal, Hall
8C5⋊(C11⋊AGL(1,𝔽7))—31no—
9ΓL(1,𝔽8)×D55—21no—
10ΓL(1,𝔽8)×D55D311yescharacteristic, complemented
11C5×C11×AGL(1,𝔽7)—31no—
12D1155C2×C311yescharacteristic, complemented
13C5×C11×D21C2×C311yescharacteristic, complemented
14C3×C7×D55C2×C311yescharacteristic, complemented
15C3×D3×D55—71nomaximal, Hall
16D7×D55—31noHall
17C3⋊(C11⋊AGL(1,𝔽7))—51no—
18C11×C3⋊AGL(1,𝔽7)D511yescharacteristic, complemented
19C3×ΓL(1,𝔽8)×D11—51no—
20D5×C3⋊AGL(1,𝔽7)—111nomaximal, Hall
21C3×C5×C7×C11C22×C311yescharacteristic, complemented
22C5×C11×ΓL(1,𝔽8)—21no—
23C5×C11×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
24C3×D165—71no—
25C3×C5×D3×C11—71no—
26C32×D55—71no—
27D11×D21—51no—
28D11×AGL(1,𝔽7)—151no—
29D385—31no—
30C7×D55C3×D311yescharacteristic, complemented
31C5×C11×D7—31no—
32C3×C11×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented, Hall
33C2×C3×D55—211no—
34D3×D55—71no—
35C3⋊(C5⋊AGL(1,𝔽7))—111no—
36C5×C3⋊AGL(1,𝔽7)D1111yescharacteristic, complemented
37C3×D5×ΓL(1,𝔽8)—111no—
38C32×C5×C11—71noHall
39D231—51no—
40C11×D21C3×D511yescharacteristic, complemented
41C3×C7×D11—51no—
42C11⋊AGL(1,𝔽7)—151no—
43C11×AGL(1,𝔽7)—31no—
44ΓL(1,𝔽8)×D11—51no—
45ΓL(1,𝔽8)×D11—101no—
46D5×D21—111no—
47D5×AGL(1,𝔽7)—331no—
48C3×D3×D11—351noHall
49C5×C7×C11C2×C3×D311yescharacteristic, complemented, Hall
50C2×C3×C5×C11—211no—
51D165—71no—
52C5×D3×C11—71no—
53C3×D55—141no—
54C3×D55—141no—
55C3×D55—211no—
56C3×D55AGL(1,𝔽7)11yescharacteristic, complemented
57C3×C5×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
58D7×D11—151noHall
59C2×C3⋊AGL(1,𝔽7)—551noHall
60C3×C7×C11C2×C3×D511yescharacteristic, complemented
61C11×ΓL(1,𝔽8)—21no—
62C11×ΓL(1,𝔽8)D3×D511yescharacteristic, complemented
63C2×D55—211noHall
64D105—111no—
65C5×D21C3×D1111yescharacteristic, complemented
66C3×C7×D5—111no—
67C5⋊AGL(1,𝔽7)—331no—
68D5×ΓL(1,𝔽8)—111no—
69D5×ΓL(1,𝔽8)—221no—
70C5×AGL(1,𝔽7)—31no—
71C3×D33—351no—
72C3×D3×C11—71no—
73C32×D11—701no—
74C3×D3×D5—771noHall
75C3×C5×C11—141no—
76C3×C5×C11—141no—
77C3×C5×C11C2×AGL(1,𝔽7)11yescharacteristic, complemented
78D77—151no—
79C7×D11—51no—
80C11×D7—31no—
81D5×D7—331noHall
82C2×C3×D11—1051no—
83D3×D11—351no—
84C2×C3×ΓL(1,𝔽8)—551no—
85C3⋊AGL(1,𝔽7)—551no—
86C3⋊AGL(1,𝔽7)D5511yescharacteristic, direct product factor
87C2×C5×C11—211no—
88D55—211no—
89D55C3⋊AGL(1,𝔽7)11yescharacteristic, direct product factor
90C3×C5×C7C2×C3×D1111yescharacteristic, complemented
91C5×ΓL(1,𝔽8)—21no—
92C5×ΓL(1,𝔽8)D3×D1111yescharacteristic, complemented
93C32×C11—141noHall
94C3×D15—771no—
95C3×C5×D3—71no—
96C32×D5—771no—
97C2×D21—551no—
98C2×AGL(1,𝔽7)—1651no—
99C7×C11C3×D3×D511yescharacteristic, complemented, Hall
100D35—331no—
101C5×D7—31no—
102C7×D5—111no—
103C2×C3×C11—211no—
104D33—351no—
105C3×D11—51no—
106C3×D11—701no—
107C3×D11—701no—
108C3×D11—1051no—
109D3×C11—71no—
110C3×ΓL(1,𝔽8)C2×D5511yescharacteristic, complemented, Hall
111C2×C3×D5—2311no—
112D3×D5—771no—
113C5×C11C2×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented, Hall
114C32×C5—71noHall
115C2×D11—1051noHall
116C2×C3×C7—551no—
117D21—551no—
118D21C3×D5511yescharacteristic, complemented
119C2×ΓL(1,𝔽8)—551no—
120C2×ΓL(1,𝔽8)—1101no—
121AGL(1,𝔽7)—31no—
122AGL(1,𝔽7)—1651no—
123C2×C3×D3—3851noHall
124C5×C7C3×D3×D1111yescharacteristic, complemented, Hall
125C3×C11—141no—
126C3×C11—141no—
127C3×C11D5×AGL(1,𝔽7)11yescharacteristic, complemented
128C2×C3×C5—211no—
129D15—771no—
130C3×D5—111no—
131C3×D5—1541no—
132C3×D5—1541no—
133C3×D5—2311no—
134C5×D3—71no—
135C2×D7—1651noHall
136C2×C11—211no—
137D11—51no—
138D11—1051no—
139C3×C7C2×C3×D5511yescharacteristic, complemented
140ΓL(1,𝔽8)—21no—
141ΓL(1,𝔽8)D3×D5511yescharacteristic, complemented
142C2×D5—2311noHall
143C2×C32—7701no—
144C3×D3—71no—
145C3×D3—3851no—
146C3×C5—141no—
147C3×C5—141no—
148C3×C5D11×AGL(1,𝔽7)11yescharacteristic, complemented
149C2×C7—551no—
150D7—31no—
151D7—1651no—
152C22×C3—11551no—
153C2×D3—3851no—
154C11D5×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
155C2×C5—211no—
156D5—111no—
157D5—2311no—
158C32—141noSylow
159C7C3×D3×D5511yescharacteristic, complemented, Sylow
160C2×C3—7701no—
161C2×C3—7701no—
162C2×C3—211no—
163C2×C3—551no—
164C2×C3—11551no—
165D3—71no—
166D3—3851no—
167C5D11×C3⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
168C22—11551noSylow
169C3—141no—
170C3—141no—
171C3AGL(1,𝔽7)×D5511yescharacteristic, complemented
172C2—211no—
173C2—551no—
174C2—11551no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C7×C11C22×C3
LCSG ▷ C3×C5×C7×C11
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3⋊(C5⋊(C11⋊AGL(1,𝔽7)))C2semidirect
2. C5×C11×C3⋊AGL(1,𝔽7)C2semidirect
3. C3×ΓL(1,𝔽8)×D55C2semidirect
4. D21×D55C3semidirect
6. C3×C5×C11×ΓL(1,𝔽8)C22semidirect
10. ΓL(1,𝔽8)×D55D3semidirect
12. D1155C2×C3semidirect
13. C5×C11×D21C2×C3semidirect
14. C3×C7×D55C2×C3semidirect
18. C11×C3⋊AGL(1,𝔽7)D5semidirect
21. C3×C5×C7×C11C22×C3semidirect
23. C5×C11×ΓL(1,𝔽8)C2×D3semidirect
30. C7×D55C3×D3semidirect
32. C3×C11×ΓL(1,𝔽8)C2×D5semidirect
36. C5×C3⋊AGL(1,𝔽7)D11semidirect
40. C11×D21C3×D5semidirect
49. C5×C7×C11C2×C3×D3semidirect
56. C3×D55AGL(1,𝔽7)semidirect
57. C3×C5×ΓL(1,𝔽8)C2×D11semidirect
60. C3×C7×C11C2×C3×D5semidirect
62. C11×ΓL(1,𝔽8)D3×D5semidirect
65. C5×D21C3×D11semidirect
77. C3×C5×C11C2×AGL(1,𝔽7)semidirect
86. C3⋊AGL(1,𝔽7)D55direct (N × H)
89. D55C3⋊AGL(1,𝔽7)direct (N × H)
90. C3×C5×C7C2×C3×D11semidirect
92. C5×ΓL(1,𝔽8)D3×D11semidirect
99. C7×C11C3×D3×D5semidirect
110. C3×ΓL(1,𝔽8)C2×D55semidirect
113. C5×C11C2×C3⋊AGL(1,𝔽7)semidirect
118. D21C3×D55semidirect
124. C5×C7C3×D3×D11semidirect
127. C3×C11D5×AGL(1,𝔽7)semidirect
139. C3×C7C2×C3×D55semidirect
141. ΓL(1,𝔽8)D3×D55semidirect
148. C3×C5D11×AGL(1,𝔽7)semidirect
154. C11D5×C3⋊AGL(1,𝔽7)semidirect
159. C7C3×D3×D55semidirect
167. C5D11×C3⋊AGL(1,𝔽7)semidirect
171. C3AGL(1,𝔽7)×D55semidirect

Maximal subgroups

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

subgroupindex
C3⋊(C5⋊(C11⋊AGL(1,𝔽7)))2
C5×C11×C3⋊AGL(1,𝔽7)2
C3×ΓL(1,𝔽8)×D552
D21×D553
AGL(1,𝔽7)×D553
D11×C3⋊AGL(1,𝔽7)5
C3×D3×D557
D5×C3⋊AGL(1,𝔽7)11