Finite Group Databasebeta

← AGL(1,ℤ/8)×D353 #36378571

Subgroups — AGL(1,ℤ/8)×D353 #36378571

Order 22592.

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×D353C211yesmaximal, characteristic, complemented
2C2×D4×D353C211yesmaximal, characteristic, complemented
3#26015020C211yesmaximal, characteristic, complemented
4#26015019C212yesmaximal, complemented
5#26015018C211yesmaximal, characteristic, complemented
6#26015017C211yesmaximal, characteristic, complemented
7#26015016C212yesmaximal, complemented
8SD16×D353C212yesmaximal, complemented
9D8×D353C212yesmaximal, complemented
10M16×D353C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C353C211yesmaximal, characteristic, complemented
12C23×D353—21no—
13#19150738C2211yescharacteristic, complemented
14#19150737—21no—
15#19150737C2211yescharacteristic, complemented
16#19150736—21no—
17#19150736C2211yescharacteristic, complemented
18Q8×D353C2211yescharacteristic, complemented
19C2×D1412C2211yescharacteristic, complemented
20C2×C22⋊D353—21no—
21D4×D353—21no—
22D4×D353—21no—
23D4×D353—21no—
24D4×D353C2212yes—
25D4×D353C2212yescomplemented
26D4×D353C2211yescharacteristic, complemented
27C2×C4×D353—21no—
28C2×C4×D353C2211yescharacteristic
29#19150726C2212yescomplemented
30#19150725C2212yescomplemented
31#19150724C2212yescomplemented
32D2824C2212yescomplemented
33#19150722C2212yescomplemented
34#19150722C2212yescomplemented
35#19150721C2211yescharacteristic, complemented
36#19150720C2212yescomplemented
37C8×D353C2212yescomplemented
38#41×C353C2211yescharacteristic, complemented
39C2×D4×C353C2211yescharacteristic, complemented
40SD16×C353C2212yescomplemented
41D8×C353C2212yescomplemented
42M16×C353C2211yescharacteristic, complemented
43C22×D353—21no—
44C22×D353—21no—
45C22×D353—22no—
46C22×D353—22no—
47C22×D353—44no—
48C22×D353D411yescharacteristic
49Dic706—21no—
50Dic706C2311yescharacteristic, complemented
51D1412—21no—
52D1412—21no—
53D1412C2312yes—
54D1412C2311yescharacteristic, complemented
55C22⋊D353—21no—
56C22⋊D353—21no—
57C22⋊D353—22no—
58C22⋊D353—22no—
59C22⋊D353—21no—
60C2×Dic353—21no—
61C2×Dic353D411yescharacteristic
62C4×D353—21no—
63C4×D353—21no—
64C4×D353—21no—
65C4×D353D411yescharacteristic
66C4×D353D411yescharacteristic
67C4×D353C2311yescharacteristic
68C4×D353C2311yescharacteristic
69#15667725C2312yescomplemented
70C23×C353—21no—
71Q8×C353C2311yescharacteristic, complemented
72D4×C353—21no—
73D4×C353—21no—
74D4×C353C2312yes—
75D4×C353C2311yescharacteristic, complemented
76C2×C4×C353—21no—
77C2×C4×C353C2311yescharacteristic
78C8×C353C2312yescomplemented
79C22×C353—21no—
80C22×C353—22no—
81C22×C353—42no—
82C22×C353C2×D411yescharacteristic
83C2×D353—22no—
84C2×D353—21no—
85C2×D353—22no—
86C2×D353—44no—
87C2×D353—42no—
88C2×D353—42no—
89C2×D353—44no—
90C2×D353C2×D411yescharacteristic
91C2×D353C2×D411yescharacteristic
92C4×C353—21no—
93C4×C353C2×D411yescharacteristic
94C4×C353C2411yescharacteristic
95Dic353—21no—
96Dic353C2×D411yescharacteristic
97Dic353C2×D411yescharacteristic
98C2×C353—21no—
99C2×C353—42no—
100C2×C353—41no—
101C2×C353C22×D411yescharacteristic
102D353—21no—
103D353—41no—
104D353—42no—
105D353AGL(1,ℤ/8)12yesdirect product factor
106C353C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—3531nomaximal, Sylow
108C2×#41—3531no—
109C22×D4—3531no—
110AGL(1,ℤ/8)—3531no—
111AGL(1,ℤ/8)—3531no—
112AGL(1,ℤ/8)—3532no—
113AGL(1,ℤ/8)—3532no—
114AGL(1,ℤ/8)—3531no—
115AGL(1,ℤ/8)D35311yescharacteristic, direct product factor
116C2×SD16—3532no—
117C2×D8—3532no—
118C2×M16—3531no—
119C24—7061no—
120#41—3531no—
121#41—3531no—
122#41—3531no—
123#41—7061no—
124#41—7061no—
125#41C2×D35311yescharacteristic, complemented
126C2×Q8—3531no—
127C2×D4—3532no—
128C2×D4—3532no—
129C2×D4—3531no—
130C2×D4—3531no—
131C2×D4—7061no—
132C2×D4—7061no—
133C2×D4—7061no—
134C2×D4—7061no—
135C2×D4C2×D35311yescharacteristic, complemented
136C22×C4—3531no—
137C22×C4—7061no—
138SD16—3532no—
139SD16—3532no—
140SD16—3532no—
141SD16C2×D35312yescomplemented
142D8—3532no—
143D8—3532no—
144D8—3532no—
145D8C2×D35312yescomplemented
146M16—3531no—
147M16—3532no—
148M16C2×D35311yescharacteristic, complemented
149C2×C8—3532no—
150C23—21no—
151C23—3531no—
152C23—14124no—
153C23—7061no—
154C23—7062no—
155C23—7062no—
156C23—7061no—
157Q8—3531no—
158Q8—7061no—
159Q8C22×D35311yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—3532no—
163D4—3531no—
164D4—7061no—
165D4—7062no—
166D4—7062no—
167D4—7061no—
168D4—7061no—
169D4—7061no—
170D4—7061no—
171D4C22×D35312yes—
172D4C22×D35311yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—3531no—
175C2×C4—3531no—
176C2×C4—3531no—
177C2×C4—3531no—
178C2×C4—3531no—
179C2×C4—7061no—
180C2×C4—7061no—
181C2×C4—7061no—
182C2×C4—7061no—
183C2×C4C22×D35311yescharacteristic
184C8—3532no—
185C8C22×D35312yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—3531no—
190C22—3531no—
191C22—14122no—
192C22—14124no—
193C22—14122no—
194C22—14124no—
195C22—7062no—
196C22—7061no—
197C22—7062no—
198C22D4×D35311yescharacteristic
199C4—21no—
200C4—3531no—
201C4—3531no—
202C4—7061no—
203C4D4×D35311yescharacteristic
204C4C23×D35311yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—3532no—
209C2—14122no—
210C2—14121no—
211C2—7061no—
212C2C2×D4×D35311yescharacteristic
subgroupG/H
centerC2C2×D4×D353
commutator, cosocleC4×C353C24
FrattiniC4C23×D353
FittingAGL(1,ℤ/8)×C353C2
socleC2×C353C22×D4
hypercenterAGL(1,ℤ/8)D353
hypocenterC353C2×AGL(1,ℤ/8)
LCSG ▷ C4×C353 ▷ C2×C353 ▷ C353
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D353C2semidirect
2. C2×D4×D353C2semidirect
3. #26015020C2semidirect
4. #26015019C2semidirect
5. #26015018C2semidirect
6. #26015017C2semidirect
7. #26015016C2semidirect
8. SD16×D353C2semidirect
9. D8×D353C2semidirect
10. M16×D353C2semidirect
11. AGL(1,ℤ/8)×C353C2semidirect
13. #19150738C22semidirect
15. #19150737C22semidirect
17. #19150736C22semidirect
18. Q8×D353C22semidirect
19. C2×D1412C22semidirect
25. D4×D353C22semidirect
26. D4×D353C22semidirect
29. #19150726C22semidirect
30. #19150725C22semidirect
31. #19150724C22semidirect
32. D2824C22semidirect
33. #19150722C22semidirect
34. #19150722C22semidirect
35. #19150721C22semidirect
36. #19150720C22semidirect
37. C8×D353C22semidirect
38. #41×C353C22semidirect
39. C2×D4×C353C22semidirect
40. SD16×C353C22semidirect
41. D8×C353C22semidirect
42. M16×C353C22semidirect
50. Dic706C23semidirect
54. D1412C23semidirect
69. #15667725C23semidirect
71. Q8×C353C23semidirect
75. D4×C353C23semidirect
78. C8×C353C23semidirect
105. D353AGL(1,ℤ/8)direct (N × H)
106. C353C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D353direct (N × H)
125. #41C2×D353semidirect
135. C2×D4C2×D353semidirect
141. SD16C2×D353semidirect
145. D8C2×D353semidirect
148. M16C2×D353semidirect
159. Q8C22×D353semidirect
172. D4C22×D353semidirect
185. C8C22×D353semidirect

Maximal subgroups

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

subgroupindex
#41×D3532
C2×D4×D3532
#260150202
#260150192
#260150182
#260150172
#260150162
SD16×D3532
D8×D3532
M16×D3532
AGL(1,ℤ/8)×C3532
C2×AGL(1,ℤ/8)353