Finite Group Databasebeta

← C22×#17984869 #31821138

Subgroups — C22×#17984869 #31821138

Order 17792.

1720 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 525 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
1C22×#14555786C211yesmaximal, characteristic, complemented
2C22×D4×D139C211yesmaximal, characteristic, complemented
3C2×#17984869C2124yesmaximal, direct product factor
4C22×#14555772C211yesmaximal, characteristic, complemented
5C22×D1112C211yesmaximal, characteristic, complemented
6C22×#14555770C211yesmaximal, characteristic, complemented
7C22×#14555768C211yesmaximal, characteristic, complemented
8C22×SD16×C139C211yesmaximal, characteristic, complemented
9C24×D139—21no—
10C2×#14555786—26no—
11C2×#14555786C22112yescomplemented
12C22×D556—21no—
13C22×D556C2211yescharacteristic
14C22×D556C2211yescharacteristic, complemented
15C22×C22⋊D139—21no—
16C2×D4×D139—26no—
17C2×D4×D139C22112yescomplemented
18C22×C4×D139—21no—
19C22×C4×D139C2211yescharacteristic
20#17984869C22164yesdirect product factor
21C2×#14555772C22112yescomplemented
22C2×D1112C22112yescomplemented
23C2×#14555770C22112yescomplemented
24C2×#14555768C22112yescomplemented
25C22×C139⋊C8C2211yescharacteristic, complemented
26C22×Q8×C139C2211yescharacteristic, complemented
27C22×D4×C139C2211yescharacteristic
28C2×SD16×C139C22112yescomplemented
29C22×C8×C139C2211yescharacteristic, complemented
30C23×D139—212no—
31C23×D139—21no—
32C23×D139—21no—
33C23×D139—48no—
34C23×D139D411yescharacteristic
35#14555786—224no—
36#14555786C23116yescomplemented
37C2×D556—212no—
38C2×D556—23no—
39C2×D556—23no—
40C2×D556C2316yes—
41C2×D556C2316yescomplemented
42C2×C22⋊D139—212no—
43D4×D139—224no—
44D4×D139C23116yescomplemented
45C22×Dic139D411yescharacteristic
46C2×C4×D139—212no—
47C2×C4×D139D416yes—
48C2×C4×D139C2316yes—
49#14555772C23116yescomplemented
50D1112C23116yescomplemented
51#14555770C23116yescomplemented
52#14555768C23116yescomplemented
53C2×C139⋊C8C2316yescomplemented
54C24×C139—21no—
55C2×Q8×C139—23no—
56C2×Q8×C139C2316yescomplemented
57C2×D4×C139—23no—
58C2×D4×C139C2316yes—
59C22×C4×C139—21no—
60C22×C4×C139C2311yescharacteristic
61SD16×C139C23116yescomplemented
62C2×C8×C139C2316yescomplemented
63C22×D139—24no—
64C22×D139—26no—
65C22×D139—26no—
66C22×D139—216no—
67C22×D139—448no—
68C22×D139—44no—
69C22×D139—44no—
70C22×D139C2×D416yes—
71D556—216no—
72D556—26no—
73D556—26no—
74D556C2414yes—
75D556C2414yescomplemented
76C22⋊D139—216no—
77C2×Dic139C2×D416yes—
78C4×D139—216no—
79C4×D139C2×D4112yes—
80C4×D139C2414yes—
81C139⋊C8C2414yescomplemented
82C23×C139—26no—
83C23×C139—44no—
84C23×C139C2×D411yescharacteristic
85Q8×C139—26no—
86Q8×C139C2414yescomplemented
87D4×C139—26no—
88D4×C139C2414yes—
89C2×C4×C139—26no—
90C2×C4×C139C2×D413yes—
91C2×C4×C139C2413yes—
92C8×C139C2414yescomplemented
93C22×C139—24no—
94C22×C139—412no—
95C22×C139AGL(1,ℤ/8)14yescomplemented
96C22×C139C22×D413yes—
97C2×D139—212no—
98C2×D139—24no—
99C2×D139—24no—
100C2×D139—432no—
101C2×D139—412no—
102C2×D139—412no—
103C2×D139C22×D414yes—
104C4×C139—24no—
105C4×C139C22×D413yes—
106C4×C139C2511yescharacteristic
107Dic139C22×D414yes—
108C2×C139—44no—
109C2×C139C2×AGL(1,ℤ/8)16yescomplemented
110C2×C139C23×D411yescharacteristic
111D139—24no—
112D139—44no—
113D139—44no—
114C139C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
115C22×AGL(1,ℤ/8)—1391nomaximal, Sylow
116C22×#41—1391no—
117C23×D4—1391no—
118C2×AGL(1,ℤ/8)—13924no—
119C22×SD16—1391no—
120C22×SD16D13911yescharacteristic, complemented
121C22×D8—1391no—
122C22×D8—1391no—
123C22×M16—1391no—
124C25—2781no—
125C2×#41—2786no—
126C2×#41—13912no—
127C22×Q8C2×D13911yescharacteristic, complemented
128C22×D4—2781no—
129C22×D4—2786no—
130C22×D4—2781no—
131C22×D4—13912no—
132C22×D4—1391no—
133C22×D4—1391no—
134C22×D4C2×D13911yescharacteristic
135C23×C4—2781no—
136C23×C4—1391no—
137AGL(1,ℤ/8)—13964no—
138C2×SD16—13912no—
139C2×SD16C2×D139112yescomplemented
140C2×D8—13912no—
141C2×D8—13912no—
142C2×M16—13912no—
143C22×C8—1391no—
144C22×C8C2×D13911yescharacteristic, complemented
145C24—21no—
146C24—27812no—
147C24—2781no—
148C24—2781no—
149C24—5568no—
150C24—1391no—
151#41—27824no—
152#41—13916no—
153C2×Q8—23no—
154C2×Q8C22×D13916yescomplemented
155C2×D4—23no—
156C2×D4—27812no—
157C2×D4—27824no—
158C2×D4—27812no—
159C2×D4—2783no—
160C2×D4—2783no—
161C2×D4—13916no—
162C2×D4—1396no—
163C2×D4—1396no—
164C2×D4C22×D13916yes—
165C22×C4—21no—
166C22×C4—27812no—
167C22×C4—1391no—
168C22×C4—1396no—
169C22×C4—1396no—
170C22×C4C22×D13911yescharacteristic
171SD16—13916no—
172SD16C22×D139116yescomplemented
173D8—13916no—
174D8—13916no—
175M16—13916no—
176C2×C8—1396no—
177C2×C8C22×D13916yescomplemented
178C23—26no—
179C23—44no—
180C23—27816no—
181C23—2784no—
182C23—2786no—
183C23—2786no—
184C23—55648no—
185C23—5564no—
186C23—5564no—
187C23—1396no—
188C23D4×D13911yescharacteristic
189Q8—26no—
190Q8C23×D13914yescomplemented
191D4—26no—
192D4—27816no—
193D4—27816no—
194D4—2786no—
195D4—2786no—
196D4—1394no—
197D4—1394no—
198D4C23×D13914yes—
199C2×C4—26no—
200C2×C4—27816no—
201C2×C4—1396no—
202C2×C4—13912no—
203C2×C4—1394no—
204C2×C4D4×D13913yes—
205C2×C4C23×D13913yes—
206C8—1394no—
207C8C23×D13914yescomplemented
208C22—24no—
209C22—412no—
210C22—27812no—
211C22—2784no—
212C22—2784no—
213C22—55632no—
214C22—55612no—
215C22—55612no—
216C22—1394no—
217C22#1798486914yesdirect product factor
218C22C2×D4×D13913yes—
219C4—24no—
220C4—1394no—
221C4C2×D4×D13913yes—
222C4C24×D13911yescharacteristic
223C2—44no—
224C2—2784no—
225C2—5564no—
226C2—5564no—
227C2C2×#1798486916yesdirect product factor
228C2C22×D4×D13911yescharacteristic
subgroupG/H
centerC23D4×D139
commutator, cosocleC4×C139C25
FrattiniC4C24×D139
FittingC22×SD16×C139C2
socleC23×C139C2×D4
hypercenterC22×SD16D139
hypocenterC139C22×AGL(1,ℤ/8)
LCSG ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14555786C2semidirect
2. C22×D4×D139C2semidirect
3. C2×#17984869C2direct (N × H)
4. C22×#14555772C2semidirect
5. C22×D1112C2semidirect
6. C22×#14555770C2semidirect
7. C22×#14555768C2semidirect
8. C22×SD16×C139C2semidirect
11. C2×#14555786C22semidirect
14. C22×D556C22semidirect
17. C2×D4×D139C22semidirect
20. #17984869C22direct (N × H)
21. C2×#14555772C22semidirect
22. C2×D1112C22semidirect
23. C2×#14555770C22semidirect
24. C2×#14555768C22semidirect
25. C22×C139⋊C8C22semidirect
26. C22×Q8×C139C22semidirect
28. C2×SD16×C139C22semidirect
29. C22×C8×C139C22semidirect
36. #14555786C23semidirect
41. C2×D556C23semidirect
44. D4×D139C23semidirect
49. #14555772C23semidirect
50. D1112C23semidirect
51. #14555770C23semidirect
52. #14555768C23semidirect
53. C2×C139⋊C8C23semidirect
56. C2×Q8×C139C23semidirect
61. SD16×C139C23semidirect
62. C2×C8×C139C23semidirect
75. D556C24semidirect
81. C139⋊C8C24semidirect
86. Q8×C139C24semidirect
92. C8×C139C24semidirect
95. C22×C139AGL(1,ℤ/8)semidirect
109. C2×C139C2×AGL(1,ℤ/8)semidirect
114. C139C22×AGL(1,ℤ/8)semidirect
120. C22×SD16D139semidirect
127. C22×Q8C2×D139semidirect
139. C2×SD16C2×D139semidirect
144. C22×C8C2×D139semidirect
154. C2×Q8C22×D139semidirect
172. SD16C22×D139semidirect
177. C2×C8C22×D139semidirect
190. Q8C23×D139semidirect
207. C8C23×D139semidirect
217. C22#17984869direct (N × H)
227. C2C2×#17984869direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145557862
C22×D4×D1392
C2×#179848692
C22×#145557722
C22×D11122
C22×#145557702
C22×#145557682
C22×SD16×C1392
C22×AGL(1,ℤ/8)139