Finite Group Databasebeta

← ΓL(1,𝔽8)×D11×D35 #44568708

Subgroups — ΓL(1,𝔽8)×D11×D35 #44568708

Order 32340.

168 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 48 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
1ΓL(1,𝔽8)×D385C211yesmaximal, characteristic, complemented
2C11×ΓL(1,𝔽8)×D35C211yesmaximal, characteristic, complemented
3C5×C7×ΓL(1,𝔽8)×D11C211yesmaximal, characteristic, complemented
4C7×D11×D35C311yesmaximal, characteristic, complemented, Hall
5C5×C7×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6D7×ΓL(1,𝔽8)×D11—51nomaximal, Hall
7C7×D385C2×C311yescharacteristic, complemented
8C7×C11×D35C2×C311yescharacteristic, complemented
9C5×C72×D11C2×C311yescharacteristic, complemented
10C3×D11×D35—71nomaximal
11D5×ΓL(1,𝔽8)×D11—71nomaximal
12ΓL(1,𝔽8)×D77—51no—
13C11×D7×ΓL(1,𝔽8)—51no—
14C7×ΓL(1,𝔽8)×D11D511yescharacteristic, complemented
15C2×ΓL(1,𝔽8)×D35—111nomaximal, Hall
16C5×C72×C11C22×C311yescharacteristic, complemented, Hall
17ΓL(1,𝔽8)×D55—71no—
18D5×C11×ΓL(1,𝔽8)—71no—
19C5×ΓL(1,𝔽8)×D11D711yescharacteristic, complemented
20C3×D385—71no—
21C3×C11×D35—71no—
22C3×C5×C7×D11—71no—
23C7×D7×D11—51noHall
24C7×C11×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented, Hall
25D11×D35ΓL(1,𝔽8)11yescharacteristic, direct product factor
26C7×D5×D11—71no—
27C2×C5×C7×ΓL(1,𝔽8)—111no—
28ΓL(1,𝔽8)×D35—111no—
29ΓL(1,𝔽8)×D35D1111yescharacteristic, direct product factor
30C3×C5×C7×C11—71no—
31C5×C11×ΓL(1,𝔽8)C2×D711yescharacteristic, complemented
32C7×D77—51no—
33C7×C11×D7—51no—
34C72×D11C3×D511yescharacteristic, complemented
35C2×C7×D35—111noHall
36C3×D7×D11—351no—
37C2×ΓL(1,𝔽8)×D11—351no—
38D385C2×ΓL(1,𝔽8)11yescharacteristic, complemented
39C11×D35C2×ΓL(1,𝔽8)11yescharacteristic, complemented
40C7×D55—71no—
41C7×D5×C11—71no—
42C5×C7×D11—61no—
43C5×C7×D11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
44C5×C7×D11C3×D711yescharacteristic, complemented
45C5×C7×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
46C3×D5×D11—491noHall
47C2×D7×ΓL(1,𝔽8)—551noHall
48C72×C11C2×C3×D511yescharacteristic, complemented, Hall
49C2×C5×C72—111no—
50C7×D35—111no—
51C7×D35C3×D1111yescharacteristic, complemented
52C3×D77—351no—
53C3×C11×D7—351no—
54C3×C7×D11—71no—
55C2×C11×ΓL(1,𝔽8)—351no—
56ΓL(1,𝔽8)×D11—351no—
57ΓL(1,𝔽8)×D11D3511yescharacteristic, direct product factor
58C2×C3×D35—771no—
59C2×D5×ΓL(1,𝔽8)—771no—
60C5×C7×C11—61no—
61C5×C7×C11C22×ΓL(1,𝔽8)11yescharacteristic, complemented
62C5×C7×C11C2×C3×D711yescharacteristic, complemented
63C3×D55—491no—
64C3×D5×C11—491no—
65C3×C5×D11—71no—
66C2×C7×D11—351no—
67D7×D11—51no—
68C2×C7×ΓL(1,𝔽8)—111no—
69D7×ΓL(1,𝔽8)—51no—
70D7×ΓL(1,𝔽8)—551no—
71C5×C72C2×C3×D1111yescharacteristic, complemented, Hall
72C3×C7×C11—71no—
73C11×ΓL(1,𝔽8)C2×D3511yescharacteristic, complemented
74D5×D11—71noHall
75C2×C3×C5×C7—771no—
76C3×D35—71no—
77C3×D35—771no—
78C2×C5×ΓL(1,𝔽8)—111no—
79D5×ΓL(1,𝔽8)—71no—
80D5×ΓL(1,𝔽8)—771no—
81C2×C7×D7—551noHall
82C3×C5×C11—71noHall
83C2×C7×C11—351no—
84D77—51no—
85C7×D11—61no—
86C7×D11—351no—
87C7×D11D5×ΓL(1,𝔽8)11yescharacteristic, complemented
88C7×D11C3×D3511yescharacteristic, complemented
89C11×D7—51no—
90C7×ΓL(1,𝔽8)D5×D1111yescharacteristic, complemented, Hall
91C2×D35—111no—
92C2×C7×D5—771no—
93C2×C3×D11—2451noHall
94D55—71no—
95C5×D11D7×ΓL(1,𝔽8)11yescharacteristic, complemented
96D5×C11—71no—
97C3×C5×C7—71no—
98C5×ΓL(1,𝔽8)D7×D1111yescharacteristic, complemented
99C2×C72—111no—
100C7×D7—51no—
101C7×D7—551no—
102C2×C3×D7—3851no—
103C22×ΓL(1,𝔽8)—3851no—
104C7×C11—61no—
105C7×C11C2×D5×ΓL(1,𝔽8)11yescharacteristic, complemented
106C7×C11C2×C3×D3511yescharacteristic, complemented
107C2×C5×C7—111no—
108C2×C5×C7—111no—
109C2×C5×C7—661no—
110D35—111no—
111D35ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
112C7×D5—71no—
113C7×D5—771no—
114C2×C3×C11—2451no—
115C3×D11—71no—
116C3×D11—2451no—
117C2×C3×D5—5391noHall
118C5×C11C2×D7×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
119C72C3×D5×D1111yescharacteristic, complemented, Sylow
120C2×D11—351noHall
121C2×C3×C7—771no—
122C3×D7—351no—
123C3×D7—3851no—
124C2×ΓL(1,𝔽8)—111no—
125C2×ΓL(1,𝔽8)—351no—
126C2×ΓL(1,𝔽8)—3851no—
127C5×C7—61no—
128C5×C7C2×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
129C5×C7C3×D7×D1111yescharacteristic, complemented
130C3×C11—71noHall
131C2×C3×C5—771no—
132C3×D5—5391no—
133C3×D5—491no—
134C22×C7—3851no—
135C2×D7—551no—
136C2×C11—351no—
137D11—351no—
138D11ΓL(1,𝔽8)×D3511yescharacteristic, direct product factor
139C3×C7—71no—
140ΓL(1,𝔽8)D11×D3511yescharacteristic, direct product factor
141C2×D5—771noHall
142C3×C5—71noHall
143C2×C7—111no—
144C2×C7—111no—
145C2×C7—351no—
146C2×C7—661no—
147C2×C7—3851no—
148D7—51no—
149D7—551no—
150C22×C3—26951noHall
151C11C2×ΓL(1,𝔽8)×D3511yescharacteristic, complemented, Sylow
152C2×C5—111no—
153D5—71no—
154D5—771no—
155C7—61no—
156C7D5×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
157C7C3×D11×D3511yescharacteristic, complemented
158C2×C3—771no—
159C2×C3—26951no—
160C2×C3—2451no—
161C5D7×ΓL(1,𝔽8)×D1111yescharacteristic, complemented, Sylow
162C22—3851noSylow
163C3—71noSylow
164C2—111no—
165C2—351no—
166C2—3851no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C72×C11C22×C3
LCSG ▷ C5×C72×C11
UCSC1

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×D385C2semidirect
2. C11×ΓL(1,𝔽8)×D35C2semidirect
3. C5×C7×ΓL(1,𝔽8)×D11C2semidirect
4. C7×D11×D35C3semidirect
5. C5×C7×C11×ΓL(1,𝔽8)C22semidirect
7. C7×D385C2×C3semidirect
8. C7×C11×D35C2×C3semidirect
9. C5×C72×D11C2×C3semidirect
14. C7×ΓL(1,𝔽8)×D11D5semidirect
16. C5×C72×C11C22×C3semidirect
19. C5×ΓL(1,𝔽8)×D11D7semidirect
24. C7×C11×ΓL(1,𝔽8)C2×D5semidirect
25. D11×D35ΓL(1,𝔽8)direct (N × H)
29. ΓL(1,𝔽8)×D35D11direct (N × H)
31. C5×C11×ΓL(1,𝔽8)C2×D7semidirect
34. C72×D11C3×D5semidirect
38. D385C2×ΓL(1,𝔽8)semidirect
39. C11×D35C2×ΓL(1,𝔽8)semidirect
43. C5×C7×D11C2×ΓL(1,𝔽8)semidirect
44. C5×C7×D11C3×D7semidirect
45. C5×C7×ΓL(1,𝔽8)C2×D11semidirect
48. C72×C11C2×C3×D5semidirect
51. C7×D35C3×D11semidirect
57. ΓL(1,𝔽8)×D11D35direct (N × H)
61. C5×C7×C11C22×ΓL(1,𝔽8)semidirect
62. C5×C7×C11C2×C3×D7semidirect
71. C5×C72C2×C3×D11semidirect
73. C11×ΓL(1,𝔽8)C2×D35semidirect
87. C7×D11D5×ΓL(1,𝔽8)semidirect
88. C7×D11C3×D35semidirect
90. C7×ΓL(1,𝔽8)D5×D11semidirect
95. C5×D11D7×ΓL(1,𝔽8)semidirect
98. C5×ΓL(1,𝔽8)D7×D11semidirect
105. C7×C11C2×D5×ΓL(1,𝔽8)semidirect
106. C7×C11C2×C3×D35semidirect
111. D35ΓL(1,𝔽8)×D11direct (N × H)
118. C5×C11C2×D7×ΓL(1,𝔽8)semidirect
119. C72C3×D5×D11semidirect
128. C5×C7C2×ΓL(1,𝔽8)×D11semidirect
129. C5×C7C3×D7×D11semidirect
138. D11ΓL(1,𝔽8)×D35direct (N × H)
140. ΓL(1,𝔽8)D11×D35direct (N × H)
151. C11C2×ΓL(1,𝔽8)×D35semidirect
156. C7D5×ΓL(1,𝔽8)×D11semidirect
157. C7C3×D11×D35semidirect
161. C5D7×ΓL(1,𝔽8)×D11semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×D3852
C11×ΓL(1,𝔽8)×D352
C5×C7×ΓL(1,𝔽8)×D112
C7×D11×D353
D7×ΓL(1,𝔽8)×D115
C3×D11×D357
D5×ΓL(1,𝔽8)×D117
C2×ΓL(1,𝔽8)×D3511