Finite Group Databasebeta

← D33⋊(C11⋊AGL(1,𝔽7)) #44479107

Subgroups — D33⋊(C11⋊AGL(1,𝔽7)) #44479107

Order 30492.

296 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 40 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
1C11×ΓL(1,𝔽8)×D33C211yesmaximal, characteristic, complemented
2C11×C3⋊(C11⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
3C3×C11⋊(C11⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
4D33⋊D77C311yesmaximal, characteristic, complemented
5D11×C11⋊AGL(1,𝔽7)—31nomaximal
6C3×C112×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
7C11×D231C2×C311yescharacteristic, complemented
8C7×C11×D33C2×C311yescharacteristic, complemented
9C3×C7⋊(C11⋊D11)C2×C311yescharacteristic, complemented
10C11×ΓL(1,𝔽8)×D11—31no—
11C11⋊(C11⋊AGL(1,𝔽7))—21no—
12C11⋊(C11⋊AGL(1,𝔽7))D311yescharacteristic, complemented
13C11×C11⋊AGL(1,𝔽7)—31no—
14C3×D33⋊D11—71nomaximal, Hall
15D11×D77—31noHall
16D33⋊AGL(1,𝔽7)—111nomaximal
17D3×C11⋊AGL(1,𝔽7)—111nomaximal
18C112×ΓL(1,𝔽8)—21no—
19C112×ΓL(1,𝔽8)C2×D311yescharacteristic, complemented
20C3×C7×C112C22×C311yescharacteristic, complemented
21C3×C11×D33—71no—
22C3×C11×D33—141no—
23C32×C11⋊D11—71no—
24C7⋊(C11⋊D11)C3×D311yescharacteristic, complemented
25C11×D77—31no—
26C7×C11×D11—31no—
27D33⋊D11—71no—
28C3×D112—211no—
29C3⋊(C11⋊AGL(1,𝔽7))D1111yescharacteristic, complemented
30C11×C3⋊AGL(1,𝔽7)—111no—
31ΓL(1,𝔽8)×D33D1111yescharacteristic, complemented
32D3×C11×ΓL(1,𝔽8)—111no—
33C3×C11⋊AGL(1,𝔽7)—111no—
34C3×C11⋊AGL(1,𝔽7)—111no—
35C3×C11⋊AGL(1,𝔽7)—222no—
36C3×C11⋊AGL(1,𝔽7)—223no—
37C32×C112—141noHall
38D21⋊D11—111no—
39D3×D77—111no—
40C2×C11⋊AGL(1,𝔽7)—331no—
41D11×AGL(1,𝔽7)—331no—
42C7×C112C2×C3×D311yescharacteristic, complemented, Hall
43C11×D33—71no—
44C11×D33AGL(1,𝔽7)11yescharacteristic, complemented
45C3×C11⋊D11—71no—
46C3×C11⋊D11—71no—
47C3×C11⋊D11—141no—
48C3×C11×D11—211no—
49C3×C11×D11—421no—
50C3×C11×ΓL(1,𝔽8)—22no—
51C3×C11×ΓL(1,𝔽8)—23no—
52C3×C11×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented
53C3×C11×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented
54D112—211noHall
55D231C3×D1111yescharacteristic, complemented
56C11×D21—111no—
57C7×D33C3×D1111yescharacteristic, complemented
58D3×C7×C11—111no—
59C3×D77—111no—
60C3×D77—111no—
61C3×D77—222no—
62C3×D77—223no—
63C2×C11×ΓL(1,𝔽8)—331no—
64C11⋊AGL(1,𝔽7)—31no—
65C11⋊AGL(1,𝔽7)—111no—
66C11⋊AGL(1,𝔽7)—111no—
67C11⋊AGL(1,𝔽7)—221no—
68C11⋊AGL(1,𝔽7)—221no—
69C11⋊AGL(1,𝔽7)—222no—
70C11⋊AGL(1,𝔽7)—223no—
71C11⋊AGL(1,𝔽7)—224no—
72C11⋊AGL(1,𝔽7)—226no—
73C11×AGL(1,𝔽7)—331no—
74ΓL(1,𝔽8)×D11—31no—
75C3×D3×D11—771no—
76C3×D3×D11—771no—
77C3×C112—141no—
78C3×C112—141no—
79C3×C112C2×AGL(1,𝔽7)11yescharacteristic, complemented
80C2×D77—331no—
81D7×D11—331no—
82D3×AGL(1,𝔽7)—1211noHall
83C11⋊D11—71no—
84C11×D11—31no—
85C11×D11—211no—
86C3×C7×C11—22no—
87C3×C7×C11—23no—
88C3×C7×C11C2×C3×D1111yescharacteristic, complemented
89C3×C7×C11C2×C3×D1111yescharacteristic, complemented
90C11×ΓL(1,𝔽8)—21no—
91C11×ΓL(1,𝔽8)—24no—
92C11×ΓL(1,𝔽8)—26no—
93C11×ΓL(1,𝔽8)—21no—
94C11×ΓL(1,𝔽8)—22no—
95C11×ΓL(1,𝔽8)—23no—
96C11×ΓL(1,𝔽8)D3×D1111yescharacteristic, complemented
97C11×ΓL(1,𝔽8)D3×D1111yescharacteristic, complemented
98C3×D33—71no—
99C3×D33—141no—
100C3×D3×C11—771no—
101C3×D3×C11—1541no—
102C32×D11—771no—
103C32×D11—771no—
104C32×D11—1542no—
105C32×D11—1543no—
106C2×C7×C11—331no—
107D77—31no—
108D77—111no—
109D77—111no—
110D77—222no—
111D77—223no—
112C7×D11—31no—
113C11×D7—331no—
114C2×C3×D11—2311no—
115C2×C3×D11—2311no—
116D3×D11—771no—
117D3×D11—771no—
118C3⋊AGL(1,𝔽7)—111no—
119D3×ΓL(1,𝔽8)—111no—
120C3×AGL(1,𝔽7)—1211no—
121C112D3×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
122C32×C11—142no—
123C32×C11—141no—
124C32×C11—141no—
125C32×C11—283no—
126D3×D7—1211no—
127C2×AGL(1,𝔽7)—3631no—
128C7×C11—22no—
129C7×C11—23no—
130C7×C11C3×D3×D1111yescharacteristic, complemented
131C7×C11C3×D3×D1111yescharacteristic, complemented
132C2×C3×C11—4621no—
133C2×C3×C11—2311no—
134D33—71no—
135D33C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
136C3×D11—211no—
137C3×D11—421no—
138C3×D11—771no—
139C3×D11—771no—
140C3×D11—771no—
141C3×D11—771no—
142C3×D11—1542no—
143C3×D11—1543no—
144C3×D11—1542no—
145C3×D11—1543no—
146C3×D11—1541no—
147C3×D11—1541no—
148C3×D11—1544no—
149C3×D11—1546no—
150D3×C11—111no—
151D3×C11—771no—
152C3×ΓL(1,𝔽8)D11211yescharacteristic, complemented, Hall
153C2×D11—2311no—
154C2×D11—2311no—
155D21—111no—
156C3×D7—1211no—
157D3×C7—111no—
158C2×ΓL(1,𝔽8)—331no—
159AGL(1,𝔽7)—331no—
160AGL(1,𝔽7)—1211no—
161AGL(1,𝔽7)—2421no—
162C2×C3×D3—8471noHall
163C3×C11—22no—
164C3×C11—23no—
165C3×C11—141no—
166C3×C11—141no—
167C3×C11—141no—
168C3×C11—282no—
169C3×C11—283no—
170C3×C11—284no—
171C3×C11—286no—
172C3×C11—281no—
173C3×C11D11×AGL(1,𝔽7)11yescharacteristic, complemented
174C3×C11C2×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
175C2×D7—3631noHall
176C2×C11—331no—
177C2×C11—2311no—
178D11—31no—
179D11—211no—
180D11—771no—
181D11—771no—
182D11—1542no—
183D11—1543no—
184C3×C7C3×D11211yescharacteristic, complemented
185ΓL(1,𝔽8)—21no—
186ΓL(1,𝔽8)D33⋊D1111yescharacteristic, complemented
187C2×C32—8471no—
188C3×D3—771no—
189C3×D3—771no—
190C2×C7—331no—
191D7—331no—
192D7—1211no—
193C22×C3—25411no—
194C2×D3—8471no—
195C11—22no—
196C11—23no—
197C11D3×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
198C11D33⋊AGL(1,𝔽7)11yescharacteristic, complemented
199C32—71noSylow
200C7C3×D33⋊D1111yescharacteristic, complemented, Sylow
201C2×C3—8471no—
202C2×C3—8471no—
203C2×C3—16941no—
204C2×C3—4621no—
205C2×C3—2311no—
206D3—111no—
207D3—771no—
208C22—25411noSylow
209C3—141no—
210C3—281no—
211C3D11×C11⋊AGL(1,𝔽7)11yescharacteristic, complemented
212C2—331no—
213C2—8471no—
214C2—2311no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C112C22×C3
LCSG ▷ C3×C7×C112
UCSC1

Complements

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

NH ≅ G/Ntype
1. C11×ΓL(1,𝔽8)×D33C2semidirect
2. C11×C3⋊(C11⋊AGL(1,𝔽7))C2semidirect
3. C3×C11⋊(C11⋊AGL(1,𝔽7))C2semidirect
4. D33⋊D77C3semidirect
6. C3×C112×ΓL(1,𝔽8)C22semidirect
7. C11×D231C2×C3semidirect
8. C7×C11×D33C2×C3semidirect
9. C3×C7⋊(C11⋊D11)C2×C3semidirect
12. C11⋊(C11⋊AGL(1,𝔽7))D3semidirect
19. C112×ΓL(1,𝔽8)C2×D3semidirect
20. C3×C7×C112C22×C3semidirect
24. C7⋊(C11⋊D11)C3×D3semidirect
29. C3⋊(C11⋊AGL(1,𝔽7))D11semidirect
31. ΓL(1,𝔽8)×D33D11semidirect
42. C7×C112C2×C3×D3semidirect
44. C11×D33AGL(1,𝔽7)semidirect
52. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
53. C3×C11×ΓL(1,𝔽8)C2×D11semidirect
55. D231C3×D11semidirect
57. C7×D33C3×D11semidirect
79. C3×C112C2×AGL(1,𝔽7)semidirect
88. C3×C7×C11C2×C3×D11semidirect
89. C3×C7×C11C2×C3×D11semidirect
96. C11×ΓL(1,𝔽8)D3×D11semidirect
97. C11×ΓL(1,𝔽8)D3×D11semidirect
121. C112D3×AGL(1,𝔽7)semidirect
130. C7×C11C3×D3×D11semidirect
131. C7×C11C3×D3×D11semidirect
135. D33C11⋊AGL(1,𝔽7)semidirect
152. C3×ΓL(1,𝔽8)D112semidirect
173. C3×C11D11×AGL(1,𝔽7)semidirect
174. C3×C11C2×C11⋊AGL(1,𝔽7)semidirect
184. C3×C7C3×D112semidirect
186. ΓL(1,𝔽8)D33⋊D11semidirect
197. C11D3×C11⋊AGL(1,𝔽7)semidirect
198. C11D33⋊AGL(1,𝔽7)semidirect
200. C7C3×D33⋊D11semidirect
211. C3D11×C11⋊AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C11×ΓL(1,𝔽8)×D332
C11×C3⋊(C11⋊AGL(1,𝔽7))2
C3×C11⋊(C11⋊AGL(1,𝔽7))2
D33⋊D773
D11×C11⋊AGL(1,𝔽7)3
C3×D33⋊D117
D33⋊AGL(1,𝔽7)11
D3×C11⋊AGL(1,𝔽7)11