Finite Group Databasebeta

← C22⋊D5×AGL(1,𝔽7) #13132435

Subgroups — C22⋊D5×AGL(1,𝔽7) #13132435

Order 1680.

216 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C22×C5×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2C2×D5×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3(C22⋊D35)⋊C3C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×C22⋊D5C211yesmaximal, characteristic, complemented
5#63306⋊C3C211yesmaximal, characteristic, complemented
6#63305⋊C3C211yesmaximal, characteristic, complemented
7Dic5×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
8D7×C22⋊D5C311yesmaximal, characteristic, complemented, Hall
9C22×C5×ΓL(1,𝔽8)C2211yescharacteristic, complemented
10C2×C5⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
11C2×D5×ΓL(1,𝔽8)C2211yescharacteristic, complemented
12C2×C5×AGL(1,𝔽7)—22no—
13C2×C5×AGL(1,𝔽7)C2211yescharacteristic
14C2×C5×AGL(1,𝔽7)C2211yescharacteristic, complemented
15D5×AGL(1,𝔽7)—22no—
16ΓL(1,𝔽8)⋊Dic5C2211yescharacteristic, complemented
17Dic5×ΓL(1,𝔽8)C2211yescharacteristic, complemented
18D4×AGL(1,𝔽7)—51nomaximal, Hall
19C22×C5×D7C2×C311yescharacteristic, complemented
20C2×D5×D7C2×C311yescharacteristic, complemented
21C22⋊D35C2×C311yescharacteristic, complemented
22C7×C22⋊D5C2×C311yescharacteristic, complemented
23#63306C2×C311yescharacteristic, complemented
24#63305C2×C311yescharacteristic, complemented
25D7×Dic5C2×C311yescharacteristic, complemented
26C2×C3×C22⋊D5—71nomaximal, Hall
27C2×C5×ΓL(1,𝔽8)—21no—
28C2×C5×ΓL(1,𝔽8)C2311yescharacteristic
29C5⋊AGL(1,𝔽7)—21no—
30D5×ΓL(1,𝔽8)—21no—
31C5×AGL(1,𝔽7)—21no—
32C5×AGL(1,𝔽7)D412yescomplemented
33C22×AGL(1,𝔽7)—51no—
34C22×AGL(1,𝔽7)D511yescharacteristic, complemented
35D4×ΓL(1,𝔽8)—51no—
36(C22⋊D7)⋊C3—51no—
37(C22⋊D7)⋊C3—51no—
38C4⋊AGL(1,𝔽7)—51no—
39C4×AGL(1,𝔽7)—51no—
40C22×C5×C7C22×C311yescharacteristic, complemented
41C2×D35C22×C311yescharacteristic, complemented
42C2×C7×D5C22×C311yescharacteristic, complemented
43C2×C5×D7—22no—
44C2×C5×D7C22×C311yescharacteristic
45C2×C5×D7C22×C311yescharacteristic, complemented
46D5×D7—22no—
47Dic35C22×C311yescharacteristic, complemented
48C7×Dic5C22×C311yescharacteristic, complemented
49C23×C3×C5—71no—
50C22×C3×D5—71no—
51C3×C22⋊D5—71no—
52C3×C22⋊D5—71no—
53C3×C22⋊D5—71no—
54C3×C22⋊D5—71no—
55C2×C3×Dic5—71no—
56D4×D7—51noHall
57C5×ΓL(1,𝔽8)C2×D411yescharacteristic, complemented, Hall
58C22×ΓL(1,𝔽8)—51no—
59C22×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
60C2×AGL(1,𝔽7)—22no—
61C2×AGL(1,𝔽7)—51no—
62C2×AGL(1,𝔽7)—102no—
63C2×AGL(1,𝔽7)C2×D511yescharacteristic
64C2×AGL(1,𝔽7)C2×D511yescharacteristic, complemented
65C4×ΓL(1,𝔽8)—51no—
66ΓL(1,𝔽8)⋊C4—51no—
67C2×C22⋊D5—71noHall
68C2×C5×C7—21no—
69C2×C5×C7C23×C311yescharacteristic
70D35—21no—
71C5×D7—21no—
72C5×D7C3×D412yescomplemented
73C7×D5—21no—
74C22×C3×C5—71no—
75C22×C3×C5—71no—
76C22×C3×C5—71no—
77C22×C3×C5—142no—
78C2×C3×D5—71no—
79C2×C3×D5—71no—
80C2×C3×D5—142no—
81C3×Dic5—71no—
82C3×Dic5—71no—
83C22×D7—51no—
84C22×D7C3×D511yescharacteristic, complemented
85C7×D4—51no—
86C22⋊D7—51no—
87C22⋊D7—51no—
88D28—51no—
89C4×D7—51no—
90C2×C3×D4—351noHall
91C2×ΓL(1,𝔽8)—21no—
92C2×ΓL(1,𝔽8)—101no—
93C2×ΓL(1,𝔽8)C22×D511yescharacteristic
94AGL(1,𝔽7)—21no—
95AGL(1,𝔽7)—101no—
96AGL(1,𝔽7)C22⋊D512yesdirect product factor
97C23×C5—71no—
98C22×D5—71no—
99C22⋊D5—71no—
100C22⋊D5—71no—
101C22⋊D5—71no—
102C22⋊D5AGL(1,𝔽7)11yescharacteristic, direct product factor
103C2×Dic5—71no—
104C5×C7C2×C3×D411yescharacteristic, complemented, Hall
105C2×C3×C5—72no—
106C2×C3×C5—71no—
107C2×C3×C5—141no—
108C2×C3×C5—141no—
109C3×D5—141no—
110C3×D5—141no—
111C22×C7—51no—
112C22×C7C2×C3×D511yescharacteristic, complemented
113C2×D7—22no—
114C2×D7—51no—
115C2×D7—102no—
116C2×D7C2×C3×D511yescharacteristic
117C2×D7C2×C3×D511yescharacteristic, complemented
118C4×C7—51no—
119Dic7—51no—
120C23×C3—71no—
121C23×C3—351no—
122C3×D4—351no—
123C3×D4—351no—
124C3×D4—351no—
125C3×D4—351no—
126C2×C3×C4—351no—
127ΓL(1,𝔽8)C2×C22⋊D511yescharacteristic, complemented, Hall
128C22×C5—71no—
129C22×C5—71no—
130C22×C5—142no—
131C22×C5C2×AGL(1,𝔽7)11yescharacteristic, complemented
132C2×D5—71no—
133C2×D5—142no—
134C2×D5C2×AGL(1,𝔽7)11yescharacteristic, complemented
135Dic5—71no—
136Dic5C2×AGL(1,𝔽7)11yescharacteristic, complemented
137C2×D4—351noSylow
138C3×C5—71noHall
139C2×C7—21no—
140C2×C7—101no—
141C2×C7C22×C3×D511yescharacteristic
142D7—21no—
143D7—101no—
144D7C3×C22⋊D512yescomplemented
145C22×C3—71no—
146C22×C3—71no—
147C22×C3—71no—
148C22×C3—142no—
149C22×C3—351no—
150C22×C3—351no—
151C22×C3—702no—
152C3×C4—351no—
153C3×C4—351no—
154C2×C5—21no—
155C2×C5—72no—
156C2×C5—141no—
157C2×C5C22×AGL(1,𝔽7)11yescharacteristic
158D5—21no—
159D5—141no—
160C23—71no—
161C23—351no—
162D4—51no—
163D4—351no—
164D4—351no—
165D4—351no—
166C2×C4—351no—
167C7C2×C3×C22⋊D511yescharacteristic, complemented, Sylow
168C2×C3—72no—
169C2×C3—71no—
170C2×C3—141no—
171C2×C3—141no—
172C2×C3—701no—
173C2×C3—701no—
174C5D4×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
175C22—51no—
176C22—71no—
177C22—71no—
178C22—142no—
179C22—351no—
180C22—702no—
181C22D5×AGL(1,𝔽7)11yescharacteristic, complemented
182C4—51no—
183C4—351no—
184C3—71noSylow
185C2—21no—
186C2—72no—
187C2—101no—
188C2—141no—
189C2—701no—
190C2C2×D5×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D5×AGL(1,𝔽7)
commutator, socle, cosocleC2×C5×C7C23×C3
FittingC22×C5×C7C22×C3
hypercenterC22D5×AGL(1,𝔽7)
hypocenterC5×C7C2×C3×D4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C5×AGL(1,𝔽7)C2semidirect
2. C2×D5×AGL(1,𝔽7)C2semidirect
3. (C22⋊D35)⋊C3C2semidirect
4. ΓL(1,𝔽8)×C22⋊D5C2semidirect
5. #63306⋊C3C2semidirect
6. #63305⋊C3C2semidirect
7. Dic5×AGL(1,𝔽7)C2semidirect
8. D7×C22⋊D5C3semidirect
9. C22×C5×ΓL(1,𝔽8)C22semidirect
10. C2×C5⋊AGL(1,𝔽7)C22semidirect
11. C2×D5×ΓL(1,𝔽8)C22semidirect
14. C2×C5×AGL(1,𝔽7)C22semidirect
16. ΓL(1,𝔽8)⋊Dic5C22semidirect
17. Dic5×ΓL(1,𝔽8)C22semidirect
19. C22×C5×D7C2×C3semidirect
20. C2×D5×D7C2×C3semidirect
21. C22⋊D35C2×C3semidirect
22. C7×C22⋊D5C2×C3semidirect
23. #63306C2×C3semidirect
24. #63305C2×C3semidirect
25. D7×Dic5C2×C3semidirect
32. C5×AGL(1,𝔽7)D4semidirect
34. C22×AGL(1,𝔽7)D5semidirect
40. C22×C5×C7C22×C3semidirect
41. C2×D35C22×C3semidirect
42. C2×C7×D5C22×C3semidirect
45. C2×C5×D7C22×C3semidirect
47. Dic35C22×C3semidirect
48. C7×Dic5C22×C3semidirect
57. C5×ΓL(1,𝔽8)C2×D4semidirect
59. C22×ΓL(1,𝔽8)C2×D5semidirect
64. C2×AGL(1,𝔽7)C2×D5semidirect
72. C5×D7C3×D4semidirect
84. C22×D7C3×D5semidirect
96. AGL(1,𝔽7)C22⋊D5direct (N × H)
102. C22⋊D5AGL(1,𝔽7)direct (N × H)
104. C5×C7C2×C3×D4semidirect
112. C22×C7C2×C3×D5semidirect
117. C2×D7C2×C3×D5semidirect
127. ΓL(1,𝔽8)C2×C22⋊D5semidirect
131. C22×C5C2×AGL(1,𝔽7)semidirect
134. C2×D5C2×AGL(1,𝔽7)semidirect
136. Dic5C2×AGL(1,𝔽7)semidirect
144. D7C3×C22⋊D5semidirect
167. C7C2×C3×C22⋊D5semidirect
174. C5D4×AGL(1,𝔽7)semidirect
181. C22D5×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C22×C5×AGL(1,𝔽7)2
C2×D5×AGL(1,𝔽7)2
(C22⋊D35)⋊C32
ΓL(1,𝔽8)×C22⋊D52
#63306⋊C32
#63305⋊C32
Dic5×AGL(1,𝔽7)2
D7×C22⋊D53
D4×AGL(1,𝔽7)5
C2×C3×C22⋊D57