Finite Group Databasebeta

← C5×D5×ΓL(1,𝔽8)×D11 #37528521

Subgroups — C5×D5×ΓL(1,𝔽8)×D11 #37528521

Order 23100.

192 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C5×ΓL(1,𝔽8)×D55C211yesmaximal, characteristic, complemented
2C5×D5×C11×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C52×ΓL(1,𝔽8)×D11C211yesmaximal, characteristic, complemented
4C5×C7×D5×D11C311yesmaximal, characteristic, complemented, Hall
5C52×C11×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
6C2×C5×ΓL(1,𝔽8)×D11—51nomaximal
7D5×ΓL(1,𝔽8)×D11C511yesmaximal, characteristic, direct product factor
8C5×C7×D55C2×C311yescharacteristic, complemented
9C5×C7×D5×C11C2×C311yescharacteristic, complemented
10C52×C7×D11C2×C311yescharacteristic, complemented
11C3×C5×D5×D11—71nomaximal, Hall
12ΓL(1,𝔽8)×D55C2×C511yescharacteristic, complemented
13D5×C11×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented
14C5×ΓL(1,𝔽8)×D11—22no—
15C5×ΓL(1,𝔽8)×D11—51no—
16C5×ΓL(1,𝔽8)×D11D511yescharacteristic, direct product factor
17C5×ΓL(1,𝔽8)×D11C2×C511yescharacteristic, complemented
18C2×C5×C11×ΓL(1,𝔽8)—51no—
19C2×C5×D5×ΓL(1,𝔽8)—111nomaximal, Hall
20C52×C7×C11C22×C311yescharacteristic, complemented, Hall
21C3×C5×D55—71no—
22C3×C5×D5×C11—71no—
23C3×C52×D11—71no—
24C2×C5×C7×D11—51no—
25C7×D5×D11C3×C511yescharacteristic, complemented
26C5×C11×ΓL(1,𝔽8)—22no—
27C5×C11×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
28C5×C11×ΓL(1,𝔽8)C22×C511yescharacteristic, complemented
29C5×D5×D11ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
30C2×C52×ΓL(1,𝔽8)—111no—
31C5×D5×ΓL(1,𝔽8)—111no—
32C5×D5×ΓL(1,𝔽8)D1111yescharacteristic, direct product factor
33C2×ΓL(1,𝔽8)×D11—51noHall
34C3×C52×C11—71noHall
35C2×C5×C7×C11—51no—
36C7×D55C2×C3×C511yescharacteristic, complemented
37C7×D5×C11C2×C3×C511yescharacteristic, complemented
38C5×C7×D11—22no—
39C5×C7×D11—51no—
40C5×C7×D11C3×D511yescharacteristic, complemented
41C5×C7×D11C2×C3×C511yescharacteristic, complemented
42C2×C5×C7×D5—111noHall
43C2×C3×C5×D11—351no—
44C3×D5×D11—71no—
45C5×D55C2×ΓL(1,𝔽8)11yescharacteristic, complemented
46C5×D5×C11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
47C52×D11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
48C52×ΓL(1,𝔽8)C2×D1111yescharacteristic, complemented, Hall
49C2×C11×ΓL(1,𝔽8)—51no—
50ΓL(1,𝔽8)×D11—51no—
51ΓL(1,𝔽8)×D11C5×D511yescharacteristic, direct product factor
52C22×C5×ΓL(1,𝔽8)—551no—
53C2×D5×ΓL(1,𝔽8)—111no—
54C5×C7×C11—22no—
55C5×C7×C11C2×C3×D511yescharacteristic, complemented
56C5×C7×C11C22×C3×C511yescharacteristic, complemented
57C2×C52×C7—111no—
58C5×C7×D5—111no—
59C5×C7×D5C3×D1111yescharacteristic, complemented
60C2×C3×C5×C11—351no—
61C3×D55—71no—
62C3×D5×C11—71no—
63C3×C5×D11—71no—
64C3×C5×D11—71no—
65C3×C5×D11—142no—
66C3×C5×D11—351no—
67C2×C7×D11—51noHall
68C2×C3×C5×D5—771noHall
69C52×C11C22×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
70C11×ΓL(1,𝔽8)C2×C5×D511yescharacteristic, complemented, Hall
71C2×C5×D11—51no—
72D5×D11C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
73C2×C5×ΓL(1,𝔽8)—51no—
74C2×C5×ΓL(1,𝔽8)—111no—
75C2×C5×ΓL(1,𝔽8)—111no—
76C2×C5×ΓL(1,𝔽8)—222no—
77C2×C5×ΓL(1,𝔽8)—551no—
78D5×ΓL(1,𝔽8)—111no—
79D5×ΓL(1,𝔽8)C5×D1111yescharacteristic, direct product factor
80C52×C7C2×C3×D1111yescharacteristic, complemented, Hall
81C3×C5×C11—71no—
82C3×C5×C11—71no—
83C3×C5×C11—142no—
84C2×C7×C11—51no—
85C7×D11—51no—
86C7×D11C3×C5×D511yescharacteristic, complemented
87C2×C3×C52—771no—
88C3×C5×D5—71no—
89C3×C5×D5—771no—
90C22×C5×C7—551no—
91C2×C7×D5—111no—
92C2×C3×D11—351noHall
93C2×C5×C11—51no—
94D55C2×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
95C5×D11—22no—
96C5×D11—51no—
97C5×D11D5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
98C5×D11C2×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
99D5×C11C2×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
100C5×ΓL(1,𝔽8)—22no—
101C5×ΓL(1,𝔽8)D5×D1111yescharacteristic, direct product factor
102C5×ΓL(1,𝔽8)C2×C5×D1111yescharacteristic, complemented
103C2×C5×D5—111noHall
104C22×ΓL(1,𝔽8)—551noHall
105C7×C11C2×C3×C5×D511yescharacteristic, complemented, Hall
106C3×C52—71noHall
107C2×C5×C7—51no—
108C2×C5×C7—111no—
109C2×C5×C7—111no—
110C2×C5×C7—222no—
111C2×C5×C7—551no—
112C7×D5—111no—
113C7×D5C3×C5×D1111yescharacteristic, complemented
114C2×C3×C11—351no—
115C3×D11—71no—
116C3×D11—351no—
117C22×C3×C5—3851no—
118C2×C3×D5—771no—
119C5×C11—22no—
120C5×C11C2×D5×ΓL(1,𝔽8)11yescharacteristic, complemented
121C5×C11C22×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
122C2×C52—111no—
123C5×D5—111no—
124C5×D5ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
125C2×D11—51noHall
126C2×ΓL(1,𝔽8)—51no—
127C2×ΓL(1,𝔽8)—111no—
128C2×ΓL(1,𝔽8)—551no—
129C5×C7—22no—
130C5×C7C3×D5×D1111yescharacteristic, complemented
131C5×C7C2×C3×C5×D1111yescharacteristic, complemented
132C3×C11—71noHall
133C2×C3×C5—351no—
134C2×C3×C5—771no—
135C2×C3×C5—771no—
136C2×C3×C5—3851no—
137C2×C3×C5—1542no—
138C3×D5—71no—
139C3×D5—771no—
140C22×C7—551noHall
141C52C2×ΓL(1,𝔽8)×D1111yescharacteristic, complemented, Sylow
142C2×C11—51no—
143D11—51no—
144D11C5×D5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
145ΓL(1,𝔽8)C5×D5×D1111yescharacteristic, direct product factor, Hall
146C22×C5—551no—
147C2×D5—111no—
148C3×C5—71no—
149C3×C5—71no—
150C3×C5—142no—
151C2×C7—51no—
152C2×C7—111no—
153C2×C7—551no—
154C22×C3—3851noHall
155C11C2×C5×D5×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
156C2×C5—51no—
157C2×C5—111no—
158C2×C5—111no—
159C2×C5—222no—
160C2×C5—551no—
161D5—111no—
162D5C5×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
163C7C3×C5×D5×D1111yescharacteristic, complemented, Sylow
164C2×C3—351no—
165C2×C3—771no—
166C2×C3—3851no—
167C5—22no—
168C5D5×ΓL(1,𝔽8)×D1111yescharacteristic, direct product factor
169C5C2×C5×ΓL(1,𝔽8)×D1111yescharacteristic, complemented
170C22—551noSylow
171C3—71noSylow
172C2—51no—
173C2—111no—
174C2—551no—
subgroupG/H
center, hypercenterC5D5×ΓL(1,𝔽8)×D11
commutator, cosocle, hypocenterC5×C7×C11C22×C3×C5
FrattiniC1G
Fitting, socleC52×C7×C11C22×C3
LCSG ▷ C5×C7×C11
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
1. C5×ΓL(1,𝔽8)×D55C2semidirect
2. C5×D5×C11×ΓL(1,𝔽8)C2semidirect
3. C52×ΓL(1,𝔽8)×D11C2semidirect
4. C5×C7×D5×D11C3semidirect
5. C52×C11×ΓL(1,𝔽8)C22semidirect
7. D5×ΓL(1,𝔽8)×D11C5direct (N × H)
8. C5×C7×D55C2×C3semidirect
9. C5×C7×D5×C11C2×C3semidirect
10. C52×C7×D11C2×C3semidirect
12. ΓL(1,𝔽8)×D55C2×C5semidirect
13. D5×C11×ΓL(1,𝔽8)C2×C5semidirect
16. C5×ΓL(1,𝔽8)×D11D5direct (N × H)
17. C5×ΓL(1,𝔽8)×D11C2×C5semidirect
20. C52×C7×C11C22×C3semidirect
25. C7×D5×D11C3×C5semidirect
27. C5×C11×ΓL(1,𝔽8)C2×D5semidirect
28. C5×C11×ΓL(1,𝔽8)C22×C5semidirect
29. C5×D5×D11ΓL(1,𝔽8)direct (N × H)
32. C5×D5×ΓL(1,𝔽8)D11direct (N × H)
36. C7×D55C2×C3×C5semidirect
37. C7×D5×C11C2×C3×C5semidirect
40. C5×C7×D11C3×D5semidirect
41. C5×C7×D11C2×C3×C5semidirect
45. C5×D55C2×ΓL(1,𝔽8)semidirect
46. C5×D5×C11C2×ΓL(1,𝔽8)semidirect
47. C52×D11C2×ΓL(1,𝔽8)semidirect
48. C52×ΓL(1,𝔽8)C2×D11semidirect
51. ΓL(1,𝔽8)×D11C5×D5direct (N × H)
55. C5×C7×C11C2×C3×D5semidirect
56. C5×C7×C11C22×C3×C5semidirect
59. C5×C7×D5C3×D11semidirect
69. C52×C11C22×ΓL(1,𝔽8)semidirect
70. C11×ΓL(1,𝔽8)C2×C5×D5semidirect
72. D5×D11C5×ΓL(1,𝔽8)direct (N × H)
79. D5×ΓL(1,𝔽8)C5×D11direct (N × H)
80. C52×C7C2×C3×D11semidirect
86. C7×D11C3×C5×D5semidirect
94. D55C2×C5×ΓL(1,𝔽8)semidirect
97. C5×D11D5×ΓL(1,𝔽8)direct (N × H)
98. C5×D11C2×C5×ΓL(1,𝔽8)semidirect
99. D5×C11C2×C5×ΓL(1,𝔽8)semidirect
101. C5×ΓL(1,𝔽8)D5×D11direct (N × H)
102. C5×ΓL(1,𝔽8)C2×C5×D11semidirect
105. C7×C11C2×C3×C5×D5semidirect
113. C7×D5C3×C5×D11semidirect
120. C5×C11C2×D5×ΓL(1,𝔽8)semidirect
121. C5×C11C22×C5×ΓL(1,𝔽8)semidirect
124. C5×D5ΓL(1,𝔽8)×D11direct (N × H)
130. C5×C7C3×D5×D11semidirect
131. C5×C7C2×C3×C5×D11semidirect
141. C52C2×ΓL(1,𝔽8)×D11semidirect
144. D11C5×D5×ΓL(1,𝔽8)direct (N × H)
145. ΓL(1,𝔽8)C5×D5×D11direct (N × H)
155. C11C2×C5×D5×ΓL(1,𝔽8)semidirect
162. D5C5×ΓL(1,𝔽8)×D11direct (N × H)
163. C7C3×C5×D5×D11semidirect
168. C5D5×ΓL(1,𝔽8)×D11direct (N × H)
169. C5C2×C5×ΓL(1,𝔽8)×D11semidirect

Maximal subgroups

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

subgroupindex
C5×ΓL(1,𝔽8)×D552
C5×D5×C11×ΓL(1,𝔽8)2
C52×ΓL(1,𝔽8)×D112
C5×C7×D5×D113
C2×C5×ΓL(1,𝔽8)×D115
D5×ΓL(1,𝔽8)×D115
C3×C5×D5×D117
C2×C5×D5×ΓL(1,𝔽8)11