Finite Group Databasebeta

← AGL(1,ℤ/8)×D521 #45688216

Subgroups — AGL(1,ℤ/8)×D521 #45688216

Order 33344.

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×D521C211yesmaximal, characteristic, complemented
2C2×D4×D521C211yesmaximal, characteristic, complemented
3#30659736C211yesmaximal, characteristic, complemented
4#30659735C212yesmaximal, complemented
5#30659734C211yesmaximal, characteristic, complemented
6#30659733C211yesmaximal, characteristic, complemented
7#30659732C212yesmaximal, complemented
8SD16×D521C212yesmaximal, complemented
9D8×D521C212yesmaximal, complemented
10M16×D521C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C521C211yesmaximal, characteristic, complemented
12C23×D521—21no—
13#22552365C2211yescharacteristic, complemented
14#22552364—21no—
15#22552364C2211yescharacteristic, complemented
16#22552363—21no—
17#22552363C2211yescharacteristic, complemented
18Q8×D521C2211yescharacteristic, complemented
19C2×D2084C2211yescharacteristic, complemented
20C2×C22⋊D521—21no—
21D4×D521—21no—
22D4×D521—21no—
23D4×D521—21no—
24D4×D521C2212yes—
25D4×D521C2212yescomplemented
26D4×D521C2211yescharacteristic, complemented
27C2×C4×D521—21no—
28C2×C4×D521C2211yescharacteristic
29#22552353C2212yescomplemented
30#22552352C2212yescomplemented
31#22552351C2212yescomplemented
32D4168C2212yescomplemented
33#22552349C2212yescomplemented
34#22552349C2212yescomplemented
35#22552348C2211yescharacteristic, complemented
36#22552347C2212yescomplemented
37C8×D521C2212yescomplemented
38#41×C521C2211yescharacteristic, complemented
39C2×D4×C521C2211yescharacteristic, complemented
40SD16×C521C2212yescomplemented
41D8×C521C2212yescomplemented
42M16×C521C2211yescharacteristic, complemented
43C22×D521—21no—
44C22×D521—21no—
45C22×D521—22no—
46C22×D521—22no—
47C22×D521—44no—
48C22×D521D411yescharacteristic
49Dic1042—21no—
50Dic1042C2311yescharacteristic, complemented
51D2084—21no—
52D2084—21no—
53D2084C2312yes—
54D2084C2311yescharacteristic, complemented
55C22⋊D521—21no—
56C22⋊D521—21no—
57C22⋊D521—22no—
58C22⋊D521—22no—
59C22⋊D521—21no—
60C2×Dic521—21no—
61C2×Dic521D411yescharacteristic
62C4×D521—21no—
63C4×D521—21no—
64C4×D521—21no—
65C4×D521D411yescharacteristic
66C4×D521D411yescharacteristic
67C4×D521C2311yescharacteristic
68C4×D521C2311yescharacteristic
69#16861960C2312yescomplemented
70C23×C521—21no—
71Q8×C521C2311yescharacteristic, complemented
72D4×C521—21no—
73D4×C521—21no—
74D4×C521C2312yes—
75D4×C521C2311yescharacteristic, complemented
76C2×C4×C521—21no—
77C2×C4×C521C2311yescharacteristic
78C8×C521C2312yescomplemented
79C22×C521—21no—
80C22×C521—22no—
81C22×C521—42no—
82C22×C521C2×D411yescharacteristic
83C2×D521—22no—
84C2×D521—21no—
85C2×D521—22no—
86C2×D521—44no—
87C2×D521—42no—
88C2×D521—42no—
89C2×D521—44no—
90C2×D521C2×D411yescharacteristic
91C2×D521C2×D411yescharacteristic
92C4×C521—21no—
93C4×C521C2×D411yescharacteristic
94C4×C521C2411yescharacteristic
95Dic521—21no—
96Dic521C2×D411yescharacteristic
97Dic521C2×D411yescharacteristic
98C2×C521—21no—
99C2×C521—42no—
100C2×C521—41no—
101C2×C521C22×D411yescharacteristic
102D521—21no—
103D521—41no—
104D521—42no—
105D521AGL(1,ℤ/8)12yesdirect product factor
106C521C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—5211nomaximal, Sylow
108C2×#41—5211no—
109C22×D4—5211no—
110AGL(1,ℤ/8)—5211no—
111AGL(1,ℤ/8)—5211no—
112AGL(1,ℤ/8)—5212no—
113AGL(1,ℤ/8)—5212no—
114AGL(1,ℤ/8)—5211no—
115AGL(1,ℤ/8)D52111yescharacteristic, direct product factor
116C2×SD16—5212no—
117C2×D8—5212no—
118C2×M16—5211no—
119C24—10421no—
120#41—5211no—
121#41—5211no—
122#41—5211no—
123#41—10421no—
124#41—10421no—
125#41C2×D52111yescharacteristic, complemented
126C2×Q8—5211no—
127C2×D4—5212no—
128C2×D4—5212no—
129C2×D4—5211no—
130C2×D4—5211no—
131C2×D4—10421no—
132C2×D4—10421no—
133C2×D4—10421no—
134C2×D4—10421no—
135C2×D4C2×D52111yescharacteristic, complemented
136C22×C4—5211no—
137C22×C4—10421no—
138SD16—5212no—
139SD16—5212no—
140SD16—5212no—
141SD16C2×D52112yescomplemented
142D8—5212no—
143D8—5212no—
144D8—5212no—
145D8C2×D52112yescomplemented
146M16—5211no—
147M16—5212no—
148M16C2×D52111yescharacteristic, complemented
149C2×C8—5212no—
150C23—21no—
151C23—5211no—
152C23—10421no—
153C23—10422no—
154C23—10422no—
155C23—10421no—
156C23—20844no—
157Q8—5211no—
158Q8—10421no—
159Q8C22×D52111yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—5212no—
163D4—5211no—
164D4—10421no—
165D4—10422no—
166D4—10422no—
167D4—10421no—
168D4—10421no—
169D4—10421no—
170D4—10421no—
171D4C22×D52112yes—
172D4C22×D52111yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—5211no—
175C2×C4—5211no—
176C2×C4—5211no—
177C2×C4—5211no—
178C2×C4—5211no—
179C2×C4—10421no—
180C2×C4—10421no—
181C2×C4—10421no—
182C2×C4—10421no—
183C2×C4C22×D52111yescharacteristic
184C8—5212no—
185C8C22×D52112yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—5211no—
190C22—5211no—
191C22—10422no—
192C22—10421no—
193C22—10422no—
194C22—20842no—
195C22—20844no—
196C22—20842no—
197C22—20844no—
198C22D4×D52111yescharacteristic
199C4—21no—
200C4—5211no—
201C4—5211no—
202C4—10421no—
203C4D4×D52111yescharacteristic
204C4C23×D52111yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—5212no—
209C2—10421no—
210C2—20842no—
211C2—20841no—
212C2C2×D4×D52111yescharacteristic
subgroupG/H
centerC2C2×D4×D521
commutator, cosocleC4×C521C24
FrattiniC4C23×D521
FittingAGL(1,ℤ/8)×C521C2
socleC2×C521C22×D4
hypercenterAGL(1,ℤ/8)D521
hypocenterC521C2×AGL(1,ℤ/8)
LCSG ▷ C4×C521 ▷ C2×C521 ▷ C521
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D521C2semidirect
2. C2×D4×D521C2semidirect
3. #30659736C2semidirect
4. #30659735C2semidirect
5. #30659734C2semidirect
6. #30659733C2semidirect
7. #30659732C2semidirect
8. SD16×D521C2semidirect
9. D8×D521C2semidirect
10. M16×D521C2semidirect
11. AGL(1,ℤ/8)×C521C2semidirect
13. #22552365C22semidirect
15. #22552364C22semidirect
17. #22552363C22semidirect
18. Q8×D521C22semidirect
19. C2×D2084C22semidirect
25. D4×D521C22semidirect
26. D4×D521C22semidirect
29. #22552353C22semidirect
30. #22552352C22semidirect
31. #22552351C22semidirect
32. D4168C22semidirect
33. #22552349C22semidirect
34. #22552349C22semidirect
35. #22552348C22semidirect
36. #22552347C22semidirect
37. C8×D521C22semidirect
38. #41×C521C22semidirect
39. C2×D4×C521C22semidirect
40. SD16×C521C22semidirect
41. D8×C521C22semidirect
42. M16×C521C22semidirect
50. Dic1042C23semidirect
54. D2084C23semidirect
69. #16861960C23semidirect
71. Q8×C521C23semidirect
75. D4×C521C23semidirect
78. C8×C521C23semidirect
105. D521AGL(1,ℤ/8)direct (N × H)
106. C521C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D521direct (N × H)
125. #41C2×D521semidirect
135. C2×D4C2×D521semidirect
141. SD16C2×D521semidirect
145. D8C2×D521semidirect
148. M16C2×D521semidirect
159. Q8C22×D521semidirect
172. D4C22×D521semidirect
185. C8C22×D521semidirect

Maximal subgroups

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

subgroupindex
#41×D5212
C2×D4×D5212
#306597362
#306597352
#306597342
#306597332
#306597322
SD16×D5212
D8×D5212
M16×D5212
AGL(1,ℤ/8)×C5212
C2×AGL(1,ℤ/8)521