Finite Group Databasebeta

← AGL(1,ℤ/8)×D397 #38763306

Subgroups — AGL(1,ℤ/8)×D397 #38763306

Order 25408.

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×D397C211yesmaximal, characteristic, complemented
2C2×D4×D397C211yesmaximal, characteristic, complemented
3#27163124C211yesmaximal, characteristic, complemented
4#27163123C212yesmaximal, complemented
5#27163122C211yesmaximal, characteristic, complemented
6#27163121C211yesmaximal, characteristic, complemented
7#27163120C212yesmaximal, complemented
8SD16×D397C212yesmaximal, complemented
9D8×D397C212yesmaximal, complemented
10M16×D397C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C397C211yesmaximal, characteristic, complemented
12C23×D397—21no—
13#20263732C2211yescharacteristic, complemented
14#20263731—21no—
15#20263731C2211yescharacteristic, complemented
16#20263730—21no—
17#20263730C2211yescharacteristic, complemented
18Q8×D397C2211yescharacteristic, complemented
19C2×D1588C2211yescharacteristic, complemented
20C2×C22⋊D397—21no—
21D4×D397—21no—
22D4×D397—21no—
23D4×D397—21no—
24D4×D397C2212yes—
25D4×D397C2212yescomplemented
26D4×D397C2211yescharacteristic, complemented
27C2×C4×D397—21no—
28C2×C4×D397C2211yescharacteristic
29#20263720C2212yescomplemented
30#20263719C2212yescomplemented
31#20263718C2212yescomplemented
32D3176C2212yescomplemented
33#20263716C2212yescomplemented
34#20263716C2212yescomplemented
35#20263715C2211yescharacteristic, complemented
36#20263714C2212yescomplemented
37C8×D397C2212yescomplemented
38#41×C397C2211yescharacteristic, complemented
39C2×D4×C397C2211yescharacteristic, complemented
40SD16×C397C2212yescomplemented
41D8×C397C2212yescomplemented
42M16×C397C2211yescharacteristic, complemented
43C22×D397—21no—
44C22×D397—21no—
45C22×D397—22no—
46C22×D397—22no—
47C22×D397—44no—
48C22×D397D411yescharacteristic
49Dic794—21no—
50Dic794C2311yescharacteristic, complemented
51D1588—21no—
52D1588—21no—
53D1588C2312yes—
54D1588C2311yescharacteristic, complemented
55C22⋊D397—21no—
56C22⋊D397—21no—
57C22⋊D397—22no—
58C22⋊D397—22no—
59C22⋊D397—21no—
60C2×Dic397—21no—
61C2×Dic397D411yescharacteristic
62C4×D397—21no—
63C4×D397—21no—
64C4×D397—21no—
65C4×D397D411yescharacteristic
66C4×D397D411yescharacteristic
67C4×D397C2311yescharacteristic
68C4×D397C2311yescharacteristic
69#15691787C2312yescomplemented
70C23×C397—21no—
71Q8×C397C2311yescharacteristic, complemented
72D4×C397—21no—
73D4×C397—21no—
74D4×C397C2312yes—
75D4×C397C2311yescharacteristic, complemented
76C2×C4×C397—21no—
77C2×C4×C397C2311yescharacteristic
78C8×C397C2312yescomplemented
79C22×C397—21no—
80C22×C397—22no—
81C22×C397—42no—
82C22×C397C2×D411yescharacteristic
83C2×D397—22no—
84C2×D397—21no—
85C2×D397—22no—
86C2×D397—44no—
87C2×D397—42no—
88C2×D397—42no—
89C2×D397—44no—
90C2×D397C2×D411yescharacteristic
91C2×D397C2×D411yescharacteristic
92C4×C397—21no—
93C4×C397C2×D411yescharacteristic
94C4×C397C2411yescharacteristic
95Dic397—21no—
96Dic397C2×D411yescharacteristic
97Dic397C2×D411yescharacteristic
98C2×C397—21no—
99C2×C397—42no—
100C2×C397—41no—
101C2×C397C22×D411yescharacteristic
102D397—21no—
103D397—41no—
104D397—42no—
105D397AGL(1,ℤ/8)12yesdirect product factor
106C397C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—3971nomaximal, Sylow
108C2×#41—3971no—
109C22×D4—3971no—
110AGL(1,ℤ/8)—3971no—
111AGL(1,ℤ/8)—3971no—
112AGL(1,ℤ/8)—3972no—
113AGL(1,ℤ/8)—3972no—
114AGL(1,ℤ/8)—3971no—
115AGL(1,ℤ/8)D39711yescharacteristic, direct product factor
116C2×SD16—3972no—
117C2×D8—3972no—
118C2×M16—3971no—
119C24—7941no—
120#41—7941no—
121#41—7941no—
122#41—3971no—
123#41—3971no—
124#41—3971no—
125#41C2×D39711yescharacteristic, complemented
126C2×Q8—3971no—
127C2×D4—7941no—
128C2×D4—7941no—
129C2×D4—7941no—
130C2×D4—7941no—
131C2×D4—3972no—
132C2×D4—3972no—
133C2×D4—3971no—
134C2×D4—3971no—
135C2×D4C2×D39711yescharacteristic, complemented
136C22×C4—7941no—
137C22×C4—3971no—
138SD16—3972no—
139SD16—3972no—
140SD16—3972no—
141SD16C2×D39712yescomplemented
142D8—3972no—
143D8—3972no—
144D8—3972no—
145D8C2×D39712yescomplemented
146M16—3971no—
147M16—3972no—
148M16C2×D39711yescharacteristic, complemented
149C2×C8—3972no—
150C23—21no—
151C23—7941no—
152C23—7942no—
153C23—7942no—
154C23—7941no—
155C23—15884no—
156C23—3971no—
157Q8—7941no—
158Q8—3971no—
159Q8C22×D39711yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—7941no—
163D4—7941no—
164D4—7941no—
165D4—7942no—
166D4—7942no—
167D4—7941no—
168D4—7941no—
169D4—3972no—
170D4—3971no—
171D4C22×D39712yes—
172D4C22×D39711yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—7941no—
175C2×C4—7941no—
176C2×C4—7941no—
177C2×C4—7941no—
178C2×C4—3971no—
179C2×C4—3971no—
180C2×C4—3971no—
181C2×C4—3971no—
182C2×C4—3971no—
183C2×C4C22×D39711yescharacteristic
184C8—3972no—
185C8C22×D39712yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—7942no—
190C22—7941no—
191C22—7942no—
192C22—15882no—
193C22—15884no—
194C22—15884no—
195C22—15882no—
196C22—3971no—
197C22—3971no—
198C22D4×D39711yescharacteristic
199C4—21no—
200C4—7941no—
201C4—3971no—
202C4—3971no—
203C4D4×D39711yescharacteristic
204C4C23×D39711yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—7941no—
209C2—15882no—
210C2—15881no—
211C2—3972no—
212C2C2×D4×D39711yescharacteristic
subgroupG/H
centerC2C2×D4×D397
commutator, cosocleC4×C397C24
FrattiniC4C23×D397
FittingAGL(1,ℤ/8)×C397C2
socleC2×C397C22×D4
hypercenterAGL(1,ℤ/8)D397
hypocenterC397C2×AGL(1,ℤ/8)
LCSG ▷ C4×C397 ▷ C2×C397 ▷ C397
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D397C2semidirect
2. C2×D4×D397C2semidirect
3. #27163124C2semidirect
4. #27163123C2semidirect
5. #27163122C2semidirect
6. #27163121C2semidirect
7. #27163120C2semidirect
8. SD16×D397C2semidirect
9. D8×D397C2semidirect
10. M16×D397C2semidirect
11. AGL(1,ℤ/8)×C397C2semidirect
13. #20263732C22semidirect
15. #20263731C22semidirect
17. #20263730C22semidirect
18. Q8×D397C22semidirect
19. C2×D1588C22semidirect
25. D4×D397C22semidirect
26. D4×D397C22semidirect
29. #20263720C22semidirect
30. #20263719C22semidirect
31. #20263718C22semidirect
32. D3176C22semidirect
33. #20263716C22semidirect
34. #20263716C22semidirect
35. #20263715C22semidirect
36. #20263714C22semidirect
37. C8×D397C22semidirect
38. #41×C397C22semidirect
39. C2×D4×C397C22semidirect
40. SD16×C397C22semidirect
41. D8×C397C22semidirect
42. M16×C397C22semidirect
50. Dic794C23semidirect
54. D1588C23semidirect
69. #15691787C23semidirect
71. Q8×C397C23semidirect
75. D4×C397C23semidirect
78. C8×C397C23semidirect
105. D397AGL(1,ℤ/8)direct (N × H)
106. C397C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D397direct (N × H)
125. #41C2×D397semidirect
135. C2×D4C2×D397semidirect
141. SD16C2×D397semidirect
145. D8C2×D397semidirect
148. M16C2×D397semidirect
159. Q8C22×D397semidirect
172. D4C22×D397semidirect
185. C8C22×D397semidirect

Maximal subgroups

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

subgroupindex
#41×D3972
C2×D4×D3972
#271631242
#271631232
#271631222
#271631212
#271631202
SD16×D3972
D8×D3972
M16×D3972
AGL(1,ℤ/8)×C3972
C2×AGL(1,ℤ/8)397