Finite Group Databasebeta

← AGL(1,ℤ/8)×D223 #28384364

Subgroups — AGL(1,ℤ/8)×D223 #28384364

Order 14272.

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×D223C211yesmaximal, characteristic, complemented
2C2×D4×D223C211yesmaximal, characteristic, complemented
3#20297332C211yesmaximal, characteristic, complemented
4#20297331C212yesmaximal, complemented
5#20297330C211yesmaximal, characteristic, complemented
6#20297329C211yesmaximal, characteristic, complemented
7#20297328C212yesmaximal, complemented
8SD16×D223C212yesmaximal, complemented
9D8×D223C212yesmaximal, complemented
10M16×D223C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C223C211yesmaximal, characteristic, complemented
12C23×D223—21no—
13#16813171C2211yescharacteristic, complemented
14#16813170—21no—
15#16813170C2211yescharacteristic, complemented
16#16813169—21no—
17#16813169C2211yescharacteristic, complemented
18Q8×D223C2211yescharacteristic, complemented
19C2×D892C2211yescharacteristic, complemented
20C2×C22⋊D223—21no—
21D4×D223—21no—
22D4×D223—21no—
23D4×D223—21no—
24D4×D223C2212yes—
25D4×D223C2212yescomplemented
26D4×D223C2211yescharacteristic, complemented
27C2×C4×D223—21no—
28C2×C4×D223C2211yescharacteristic
29#16813159C2212yescomplemented
30#16813158C2212yescomplemented
31#16813157C2212yescomplemented
32D1784C2212yescomplemented
33#16813155C2212yescomplemented
34#16813155C2212yescomplemented
35#16813154C2211yescharacteristic, complemented
36#16813153C2212yescomplemented
37C8×D223C2212yescomplemented
38#41×C223C2211yescharacteristic, complemented
39C2×D4×C223C2211yescharacteristic, complemented
40SD16×C223C2212yescomplemented
41D8×C223C2212yescomplemented
42M16×C223C2211yescharacteristic, complemented
43C22×D223—21no—
44C22×D223—21no—
45C22×D223—22no—
46C22×D223—22no—
47C22×D223—44no—
48C22×D223D411yescharacteristic
49Dic446—21no—
50Dic446C2311yescharacteristic, complemented
51D892—21no—
52D892—21no—
53D892C2312yes—
54D892C2311yescharacteristic, complemented
55C22⋊D223—21no—
56C22⋊D223—21no—
57C22⋊D223—22no—
58C22⋊D223—22no—
59C22⋊D223—21no—
60C2×Dic223—21no—
61C2×Dic223D411yescharacteristic
62C4×D223—21no—
63C4×D223—21no—
64C4×D223—21no—
65C4×D223D411yescharacteristic
66C4×D223D411yescharacteristic
67C4×D223C2311yescharacteristic
68C4×D223C2311yescharacteristic
69C223⋊C8C2312yescomplemented
70C23×C223—21no—
71Q8×C223C2311yescharacteristic, complemented
72D4×C223—21no—
73D4×C223—21no—
74D4×C223C2312yes—
75D4×C223C2311yescharacteristic, complemented
76C2×C4×C223—21no—
77C2×C4×C223C2311yescharacteristic
78C8×C223C2312yescomplemented
79C22×C223—21no—
80C22×C223—22no—
81C22×C223—42no—
82C22×C223C2×D411yescharacteristic
83C2×D223—22no—
84C2×D223—21no—
85C2×D223—22no—
86C2×D223—44no—
87C2×D223—42no—
88C2×D223—42no—
89C2×D223—44no—
90C2×D223C2×D411yescharacteristic
91C2×D223C2×D411yescharacteristic
92C4×C223—21no—
93C4×C223C2×D411yescharacteristic
94C4×C223C2411yescharacteristic
95Dic223—21no—
96Dic223C2×D411yescharacteristic
97Dic223C2×D411yescharacteristic
98C2×C223—21no—
99C2×C223—42no—
100C2×C223—41no—
101C2×C223C22×D411yescharacteristic
102D223—21no—
103D223—41no—
104D223—42no—
105D223AGL(1,ℤ/8)12yesdirect product factor
106C223C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—2231nomaximal, Sylow
108C2×#41—2231no—
109C22×D4—2231no—
110AGL(1,ℤ/8)—2231no—
111AGL(1,ℤ/8)—2231no—
112AGL(1,ℤ/8)—2232no—
113AGL(1,ℤ/8)—2232no—
114AGL(1,ℤ/8)—2231no—
115AGL(1,ℤ/8)D22311yescharacteristic, direct product factor
116C2×SD16—2232no—
117C2×D8—2232no—
118C2×M16—2231no—
119C24—4461no—
120#41—4461no—
121#41—4461no—
122#41—2231no—
123#41—2231no—
124#41—2231no—
125#41C2×D22311yescharacteristic, complemented
126C2×Q8—2231no—
127C2×D4—4461no—
128C2×D4—4461no—
129C2×D4—4461no—
130C2×D4—4461no—
131C2×D4—2232no—
132C2×D4—2232no—
133C2×D4—2231no—
134C2×D4—2231no—
135C2×D4C2×D22311yescharacteristic, complemented
136C22×C4—4461no—
137C22×C4—2231no—
138SD16—2232no—
139SD16—2232no—
140SD16—2232no—
141SD16C2×D22312yescomplemented
142D8—2232no—
143D8—2232no—
144D8—2232no—
145D8C2×D22312yescomplemented
146M16—2231no—
147M16—2232no—
148M16C2×D22311yescharacteristic, complemented
149C2×C8—2232no—
150C23—21no—
151C23—8924no—
152C23—4461no—
153C23—4462no—
154C23—4462no—
155C23—4461no—
156C23—2231no—
157Q8—4461no—
158Q8—2231no—
159Q8C22×D22311yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—4461no—
163D4—4461no—
164D4—4461no—
165D4—4462no—
166D4—4462no—
167D4—4461no—
168D4—4461no—
169D4—2231no—
170D4—2232no—
171D4C22×D22312yes—
172D4C22×D22311yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—4461no—
175C2×C4—4461no—
176C2×C4—4461no—
177C2×C4—4461no—
178C2×C4—2231no—
179C2×C4—2231no—
180C2×C4—2231no—
181C2×C4—2231no—
182C2×C4—2231no—
183C2×C4C22×D22311yescharacteristic
184C8—2232no—
185C8C22×D22312yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—8922no—
190C22—8924no—
191C22—8924no—
192C22—8922no—
193C22—4462no—
194C22—4461no—
195C22—4462no—
196C22—2231no—
197C22—2231no—
198C22D4×D22311yescharacteristic
199C4—21no—
200C4—4461no—
201C4—2231no—
202C4—2231no—
203C4D4×D22311yescharacteristic
204C4C23×D22311yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—8922no—
209C2—8921no—
210C2—4461no—
211C2—2232no—
212C2C2×D4×D22311yescharacteristic
subgroupG/H
centerC2C2×D4×D223
commutator, cosocleC4×C223C24
FrattiniC4C23×D223
FittingAGL(1,ℤ/8)×C223C2
socleC2×C223C22×D4
hypercenterAGL(1,ℤ/8)D223
hypocenterC223C2×AGL(1,ℤ/8)
LCSG ▷ C4×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D223C2semidirect
2. C2×D4×D223C2semidirect
3. #20297332C2semidirect
4. #20297331C2semidirect
5. #20297330C2semidirect
6. #20297329C2semidirect
7. #20297328C2semidirect
8. SD16×D223C2semidirect
9. D8×D223C2semidirect
10. M16×D223C2semidirect
11. AGL(1,ℤ/8)×C223C2semidirect
13. #16813171C22semidirect
15. #16813170C22semidirect
17. #16813169C22semidirect
18. Q8×D223C22semidirect
19. C2×D892C22semidirect
25. D4×D223C22semidirect
26. D4×D223C22semidirect
29. #16813159C22semidirect
30. #16813158C22semidirect
31. #16813157C22semidirect
32. D1784C22semidirect
33. #16813155C22semidirect
34. #16813155C22semidirect
35. #16813154C22semidirect
36. #16813153C22semidirect
37. C8×D223C22semidirect
38. #41×C223C22semidirect
39. C2×D4×C223C22semidirect
40. SD16×C223C22semidirect
41. D8×C223C22semidirect
42. M16×C223C22semidirect
50. Dic446C23semidirect
54. D892C23semidirect
69. C223⋊C8C23semidirect
71. Q8×C223C23semidirect
75. D4×C223C23semidirect
78. C8×C223C23semidirect
105. D223AGL(1,ℤ/8)direct (N × H)
106. C223C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D223direct (N × H)
125. #41C2×D223semidirect
135. C2×D4C2×D223semidirect
141. SD16C2×D223semidirect
145. D8C2×D223semidirect
148. M16C2×D223semidirect
159. Q8C22×D223semidirect
172. D4C22×D223semidirect
185. C8C22×D223semidirect

Maximal subgroups

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

subgroupindex
#41×D2232
C2×D4×D2232
#202973322
#202973312
#202973302
#202973292
#202973282
SD16×D2232
D8×D2232
M16×D2232
AGL(1,ℤ/8)×C2232
C2×AGL(1,ℤ/8)223