Finite Group Databasebeta

← AGL(1,ℤ/8)×D419 #41001145

Subgroups — AGL(1,ℤ/8)×D419 #41001145

Order 26816.

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×D419C211yesmaximal, characteristic, complemented
2C2×D4×D419C211yesmaximal, characteristic, complemented
3#27211746C211yesmaximal, characteristic, complemented
4#27211745C212yesmaximal, complemented
5#27211744C211yesmaximal, characteristic, complemented
6#27211743C211yesmaximal, characteristic, complemented
7#27211742C212yesmaximal, complemented
8SD16×D419C212yesmaximal, complemented
9D8×D419C212yesmaximal, complemented
10M16×D419C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C419C211yesmaximal, characteristic, complemented
12C23×D419—21no—
13#20269412C2211yescharacteristic, complemented
14#20269411—21no—
15#20269411C2211yescharacteristic, complemented
16#20269410—21no—
17#20269410C2211yescharacteristic, complemented
18Q8×D419C2211yescharacteristic, complemented
19C2×D1676C2211yescharacteristic, complemented
20C2×C22⋊D419—21no—
21D4×D419—21no—
22D4×D419—21no—
23D4×D419—21no—
24D4×D419C2212yes—
25D4×D419C2212yescomplemented
26D4×D419C2211yescharacteristic, complemented
27C2×C4×D419—21no—
28C2×C4×D419C2211yescharacteristic
29#20269400C2212yescomplemented
30#20269399C2212yescomplemented
31#20269398C2212yescomplemented
32D3352C2212yescomplemented
33#20269396C2212yescomplemented
34#20269396C2212yescomplemented
35#20269395C2211yescharacteristic, complemented
36#20269394C2212yescomplemented
37C8×D419C2212yescomplemented
38#41×C419C2211yescharacteristic, complemented
39C2×D4×C419C2211yescharacteristic, complemented
40SD16×C419C2212yescomplemented
41D8×C419C2212yescomplemented
42M16×C419C2211yescharacteristic, complemented
43C22×D419—21no—
44C22×D419—21no—
45C22×D419—22no—
46C22×D419—22no—
47C22×D419—44no—
48C22×D419D411yescharacteristic
49Dic838—21no—
50Dic838C2311yescharacteristic, complemented
51D1676—21no—
52D1676—21no—
53D1676C2312yes—
54D1676C2311yescharacteristic, complemented
55C22⋊D419—21no—
56C22⋊D419—21no—
57C22⋊D419—22no—
58C22⋊D419—22no—
59C22⋊D419—21no—
60C2×Dic419—21no—
61C2×Dic419D411yescharacteristic
62C4×D419—21no—
63C4×D419—21no—
64C4×D419—21no—
65C4×D419D411yescharacteristic
66C4×D419D411yescharacteristic
67C4×D419C2311yescharacteristic
68C4×D419C2311yescharacteristic
69C419⋊C8C2312yescomplemented
70C23×C419—21no—
71Q8×C419C2311yescharacteristic, complemented
72D4×C419—21no—
73D4×C419—21no—
74D4×C419C2312yes—
75D4×C419C2311yescharacteristic, complemented
76C2×C4×C419—21no—
77C2×C4×C419C2311yescharacteristic
78C8×C419C2312yescomplemented
79C22×C419—21no—
80C22×C419—22no—
81C22×C419—42no—
82C22×C419C2×D411yescharacteristic
83C2×D419—22no—
84C2×D419—21no—
85C2×D419—22no—
86C2×D419—44no—
87C2×D419—42no—
88C2×D419—42no—
89C2×D419—44no—
90C2×D419C2×D411yescharacteristic
91C2×D419C2×D411yescharacteristic
92C4×C419—21no—
93C4×C419C2×D411yescharacteristic
94C4×C419C2411yescharacteristic
95Dic419—21no—
96Dic419C2×D411yescharacteristic
97Dic419C2×D411yescharacteristic
98C2×C419—21no—
99C2×C419—42no—
100C2×C419—41no—
101C2×C419C22×D411yescharacteristic
102D419—21no—
103D419—41no—
104D419—42no—
105D419AGL(1,ℤ/8)12yesdirect product factor
106C419C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—4191nomaximal, Sylow
108C2×#41—4191no—
109C22×D4—4191no—
110AGL(1,ℤ/8)—4191no—
111AGL(1,ℤ/8)—4191no—
112AGL(1,ℤ/8)—4192no—
113AGL(1,ℤ/8)—4192no—
114AGL(1,ℤ/8)—4191no—
115AGL(1,ℤ/8)D41911yescharacteristic, direct product factor
116C2×SD16—4192no—
117C2×D8—4192no—
118C2×M16—4191no—
119C24—8381no—
120#41—8381no—
121#41—8381no—
122#41—4191no—
123#41—4191no—
124#41—4191no—
125#41C2×D41911yescharacteristic, complemented
126C2×Q8—4191no—
127C2×D4—8381no—
128C2×D4—8381no—
129C2×D4—8381no—
130C2×D4—8381no—
131C2×D4—4192no—
132C2×D4—4192no—
133C2×D4—4191no—
134C2×D4—4191no—
135C2×D4C2×D41911yescharacteristic, complemented
136C22×C4—8381no—
137C22×C4—4191no—
138SD16—4192no—
139SD16—4192no—
140SD16—4192no—
141SD16C2×D41912yescomplemented
142D8—4192no—
143D8—4192no—
144D8—4192no—
145D8C2×D41912yescomplemented
146M16—4191no—
147M16—4192no—
148M16C2×D41911yescharacteristic, complemented
149C2×C8—4192no—
150C23—21no—
151C23—8381no—
152C23—8382no—
153C23—8382no—
154C23—8381no—
155C23—16764no—
156C23—4191no—
157Q8—8381no—
158Q8—4191no—
159Q8C22×D41911yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—8381no—
163D4—8381no—
164D4—8381no—
165D4—8382no—
166D4—8382no—
167D4—8381no—
168D4—8381no—
169D4—4191no—
170D4—4192no—
171D4C22×D41912yes—
172D4C22×D41911yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—8381no—
175C2×C4—8381no—
176C2×C4—8381no—
177C2×C4—8381no—
178C2×C4—4191no—
179C2×C4—4191no—
180C2×C4—4191no—
181C2×C4—4191no—
182C2×C4—4191no—
183C2×C4C22×D41911yescharacteristic
184C8—4192no—
185C8C22×D41912yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—8382no—
190C22—8381no—
191C22—8382no—
192C22—16762no—
193C22—16764no—
194C22—16764no—
195C22—16762no—
196C22—4191no—
197C22—4191no—
198C22D4×D41911yescharacteristic
199C4—21no—
200C4—8381no—
201C4—4191no—
202C4—4191no—
203C4D4×D41911yescharacteristic
204C4C23×D41911yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—8381no—
209C2—16762no—
210C2—16761no—
211C2—4192no—
212C2C2×D4×D41911yescharacteristic
subgroupG/H
centerC2C2×D4×D419
commutator, cosocleC4×C419C24
FrattiniC4C23×D419
FittingAGL(1,ℤ/8)×C419C2
socleC2×C419C22×D4
hypercenterAGL(1,ℤ/8)D419
hypocenterC419C2×AGL(1,ℤ/8)
LCSG ▷ C4×C419 ▷ C2×C419 ▷ C419
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D419C2semidirect
2. C2×D4×D419C2semidirect
3. #27211746C2semidirect
4. #27211745C2semidirect
5. #27211744C2semidirect
6. #27211743C2semidirect
7. #27211742C2semidirect
8. SD16×D419C2semidirect
9. D8×D419C2semidirect
10. M16×D419C2semidirect
11. AGL(1,ℤ/8)×C419C2semidirect
13. #20269412C22semidirect
15. #20269411C22semidirect
17. #20269410C22semidirect
18. Q8×D419C22semidirect
19. C2×D1676C22semidirect
25. D4×D419C22semidirect
26. D4×D419C22semidirect
29. #20269400C22semidirect
30. #20269399C22semidirect
31. #20269398C22semidirect
32. D3352C22semidirect
33. #20269396C22semidirect
34. #20269396C22semidirect
35. #20269395C22semidirect
36. #20269394C22semidirect
37. C8×D419C22semidirect
38. #41×C419C22semidirect
39. C2×D4×C419C22semidirect
40. SD16×C419C22semidirect
41. D8×C419C22semidirect
42. M16×C419C22semidirect
50. Dic838C23semidirect
54. D1676C23semidirect
69. C419⋊C8C23semidirect
71. Q8×C419C23semidirect
75. D4×C419C23semidirect
78. C8×C419C23semidirect
105. D419AGL(1,ℤ/8)direct (N × H)
106. C419C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D419direct (N × H)
125. #41C2×D419semidirect
135. C2×D4C2×D419semidirect
141. SD16C2×D419semidirect
145. D8C2×D419semidirect
148. M16C2×D419semidirect
159. Q8C22×D419semidirect
172. D4C22×D419semidirect
185. C8C22×D419semidirect

Maximal subgroups

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

subgroupindex
#41×D4192
C2×D4×D4192
#272117462
#272117452
#272117442
#272117432
#272117422
SD16×D4192
D8×D4192
M16×D4192
AGL(1,ℤ/8)×C4192
C2×AGL(1,ℤ/8)419