Finite Group Databasebeta

← AGL(1,ℤ/8)×D199 #27164802

Subgroups — AGL(1,ℤ/8)×D199 #27164802

Order 12736.

298 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 101 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
1#41×D199C211yesmaximal, characteristic, complemented
2C2×D4×D199C211yesmaximal, characteristic, complemented
3#20263983C211yesmaximal, characteristic, complemented
4#20263982C212yesmaximal, complemented
5#20263981C211yesmaximal, characteristic, complemented
6#20263980C211yesmaximal, characteristic, complemented
7#20263979C212yesmaximal, complemented
8SD16×D199C212yesmaximal, complemented
9D8×D199C212yesmaximal, complemented
10M16×D199C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C199C211yesmaximal, characteristic, complemented
12C23×D199—21no—
13#15691887C2211yescharacteristic, complemented
14#15691886—21no—
15#15691886C2211yescharacteristic, complemented
16#15691885—21no—
17#15691885C2211yescharacteristic, complemented
18Q8×D199C2211yescharacteristic, complemented
19C2×D796C2211yescharacteristic, complemented
20C2×C22⋊D199—21no—
21D4×D199—21no—
22D4×D199—21no—
23D4×D199—21no—
24D4×D199C2212yes—
25D4×D199C2212yescomplemented
26D4×D199C2211yescharacteristic, complemented
27C2×C4×D199—21no—
28C2×C4×D199C2211yescharacteristic
29#15691875C2212yescomplemented
30#15691874C2212yescomplemented
31#15691873C2212yescomplemented
32D1592C2212yescomplemented
33#15691871C2212yescomplemented
34#15691871C2212yescomplemented
35#15691870C2211yescharacteristic, complemented
36#15691869C2212yescomplemented
37C8×D199C2212yescomplemented
38#41×C199C2211yescharacteristic, complemented
39C2×D4×C199C2211yescharacteristic, complemented
40SD16×C199C2212yescomplemented
41D8×C199C2212yescomplemented
42M16×C199C2211yescharacteristic, complemented
43C22×D199—21no—
44C22×D199—21no—
45C22×D199—22no—
46C22×D199—22no—
47C22×D199—44no—
48C22×D199D411yescharacteristic
49Dic398—21no—
50Dic398C2311yescharacteristic, complemented
51D796—21no—
52D796—21no—
53D796C2312yes—
54D796C2311yescharacteristic, complemented
55C22⋊D199—21no—
56C22⋊D199—21no—
57C22⋊D199—22no—
58C22⋊D199—22no—
59C22⋊D199—21no—
60C2×Dic199—21no—
61C2×Dic199D411yescharacteristic
62C4×D199—21no—
63C4×D199—21no—
64C4×D199—21no—
65C4×D199D411yescharacteristic
66C4×D199D411yescharacteristic
67C4×D199C2311yescharacteristic
68C4×D199C2311yescharacteristic
69C199⋊C8C2312yescomplemented
70C23×C199—21no—
71Q8×C199C2311yescharacteristic, complemented
72D4×C199—21no—
73D4×C199—21no—
74D4×C199C2312yes—
75D4×C199C2311yescharacteristic, complemented
76C2×C4×C199—21no—
77C2×C4×C199C2311yescharacteristic
78C8×C199C2312yescomplemented
79C22×C199—21no—
80C22×C199—22no—
81C22×C199—42no—
82C22×C199C2×D411yescharacteristic
83C2×D199—22no—
84C2×D199—21no—
85C2×D199—22no—
86C2×D199—44no—
87C2×D199—42no—
88C2×D199—42no—
89C2×D199—44no—
90C2×D199C2×D411yescharacteristic
91C2×D199C2×D411yescharacteristic
92C4×C199—21no—
93C4×C199C2×D411yescharacteristic
94C4×C199C2411yescharacteristic
95Dic199—21no—
96Dic199C2×D411yescharacteristic
97Dic199C2×D411yescharacteristic
98C2×C199—21no—
99C2×C199—42no—
100C2×C199—41no—
101C2×C199C22×D411yescharacteristic
102D199—21no—
103D199—41no—
104D199—42no—
105D199AGL(1,ℤ/8)12yesdirect product factor
106C199C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—1991nomaximal, Sylow
108C2×#41—1991no—
109C22×D4—1991no—
110AGL(1,ℤ/8)—1991no—
111AGL(1,ℤ/8)—1991no—
112AGL(1,ℤ/8)—1992no—
113AGL(1,ℤ/8)—1992no—
114AGL(1,ℤ/8)—1991no—
115AGL(1,ℤ/8)D19911yescharacteristic, direct product factor
116C2×SD16—1992no—
117C2×D8—1992no—
118C2×M16—1991no—
119C24—3981no—
120#41—3981no—
121#41—3981no—
122#41—1991no—
123#41—1991no—
124#41—1991no—
125#41C2×D19911yescharacteristic, complemented
126C2×Q8—1991no—
127C2×D4—3981no—
128C2×D4—3981no—
129C2×D4—3981no—
130C2×D4—3981no—
131C2×D4—1992no—
132C2×D4—1992no—
133C2×D4—1991no—
134C2×D4—1991no—
135C2×D4C2×D19911yescharacteristic, complemented
136C22×C4—3981no—
137C22×C4—1991no—
138SD16—1992no—
139SD16—1992no—
140SD16—1992no—
141SD16C2×D19912yescomplemented
142D8—1992no—
143D8—1992no—
144D8—1992no—
145D8C2×D19912yescomplemented
146M16—1991no—
147M16—1992no—
148M16C2×D19911yescharacteristic, complemented
149C2×C8—1992no—
150C23—21no—
151C23—7964no—
152C23—3981no—
153C23—3982no—
154C23—3982no—
155C23—3981no—
156C23—1991no—
157Q8—3981no—
158Q8—1991no—
159Q8C22×D19911yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—3981no—
163D4—3981no—
164D4—3981no—
165D4—3982no—
166D4—3982no—
167D4—3981no—
168D4—3981no—
169D4—1991no—
170D4—1992no—
171D4C22×D19912yes—
172D4C22×D19911yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—3981no—
175C2×C4—3981no—
176C2×C4—3981no—
177C2×C4—3981no—
178C2×C4—1991no—
179C2×C4—1991no—
180C2×C4—1991no—
181C2×C4—1991no—
182C2×C4—1991no—
183C2×C4C22×D19911yescharacteristic
184C8—1992no—
185C8C22×D19912yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—7962no—
190C22—7964no—
191C22—7964no—
192C22—7962no—
193C22—3982no—
194C22—3981no—
195C22—3982no—
196C22—1991no—
197C22—1991no—
198C22D4×D19911yescharacteristic
199C4—21no—
200C4—3981no—
201C4—1991no—
202C4—1991no—
203C4D4×D19911yescharacteristic
204C4C23×D19911yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—7962no—
209C2—7961no—
210C2—3981no—
211C2—1992no—
212C2C2×D4×D19911yescharacteristic
subgroupG/H
centerC2C2×D4×D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
FittingAGL(1,ℤ/8)×C199C2
socleC2×C199C22×D4
hypercenterAGL(1,ℤ/8)D199
hypocenterC199C2×AGL(1,ℤ/8)
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D199C2semidirect
2. C2×D4×D199C2semidirect
3. #20263983C2semidirect
4. #20263982C2semidirect
5. #20263981C2semidirect
6. #20263980C2semidirect
7. #20263979C2semidirect
8. SD16×D199C2semidirect
9. D8×D199C2semidirect
10. M16×D199C2semidirect
11. AGL(1,ℤ/8)×C199C2semidirect
13. #15691887C22semidirect
15. #15691886C22semidirect
17. #15691885C22semidirect
18. Q8×D199C22semidirect
19. C2×D796C22semidirect
25. D4×D199C22semidirect
26. D4×D199C22semidirect
29. #15691875C22semidirect
30. #15691874C22semidirect
31. #15691873C22semidirect
32. D1592C22semidirect
33. #15691871C22semidirect
34. #15691871C22semidirect
35. #15691870C22semidirect
36. #15691869C22semidirect
37. C8×D199C22semidirect
38. #41×C199C22semidirect
39. C2×D4×C199C22semidirect
40. SD16×C199C22semidirect
41. D8×C199C22semidirect
42. M16×C199C22semidirect
50. Dic398C23semidirect
54. D796C23semidirect
69. C199⋊C8C23semidirect
71. Q8×C199C23semidirect
75. D4×C199C23semidirect
78. C8×C199C23semidirect
105. D199AGL(1,ℤ/8)direct (N × H)
106. C199C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D199direct (N × H)
125. #41C2×D199semidirect
135. C2×D4C2×D199semidirect
141. SD16C2×D199semidirect
145. D8C2×D199semidirect
148. M16C2×D199semidirect
159. Q8C22×D199semidirect
172. D4C22×D199semidirect
185. C8C22×D199semidirect

Maximal subgroups

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

subgroupindex
#41×D1992
C2×D4×D1992
#202639832
#202639822
#202639812
#202639802
#202639792
SD16×D1992
D8×D1992
M16×D1992
AGL(1,ℤ/8)×C1992
C2×AGL(1,ℤ/8)199