Finite Group Databasebeta

← C22×#17984866 #31821135

Subgroups — C22×#17984866 #31821135

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×#14555784C211yesmaximal, characteristic, complemented
2C22×D4×D139C211yesmaximal, characteristic, complemented
3C2×#17984866C2124yesmaximal, direct product factor
4C22×#14555774C211yesmaximal, characteristic, complemented
5C22×#14555773C211yesmaximal, characteristic, complemented
6C22×#14555770C211yesmaximal, characteristic, complemented
7C22×#14555768C211yesmaximal, characteristic, complemented
8C22×D8×C139C211yesmaximal, characteristic, complemented
9C24×D139—21no—
10C2×#14555784—26no—
11C2×#14555784C22112yescomplemented
12C22×Dic278C2211yescharacteristic, complemented
13C22×D556C2211yescharacteristic
14C22×C22⋊D139—21no—
15C22×C22⋊D139—21no—
16C2×D4×D139—26no—
17C2×D4×D139C22112yescomplemented
18C23×Dic139—21no—
19C22×C4×D139C2211yescharacteristic
20#17984866C22164yesdirect product factor
21C2×#14555774C22112yescomplemented
22C2×#14555773C22112yescomplemented
23C2×#14555770C22112yescomplemented
24C2×#14555768C22112yescomplemented
25C22×C139⋊C8C2211yescharacteristic, complemented
26C22×D4×C139C2211yescharacteristic
27C22×D4×C139C2211yescharacteristic, complemented
28C2×D8×C139C22112yescomplemented
29C22×C8×C139C2211yescharacteristic, complemented
30C23×D139—212no—
31C23×D139—21no—
32C23×D139—48no—
33C23×D139D411yescharacteristic
34#14555784—224no—
35#14555784C23116yescomplemented
36C2×Dic278—23no—
37C2×Dic278C2316yescomplemented
38C2×D556—23no—
39C2×D556C2316yes—
40C2×C22⋊D139—212no—
41C2×C22⋊D139—212no—
42D4×D139—224no—
43D4×D139C23116yescomplemented
44C22×Dic139—212no—
45C22×Dic139—21no—
46C22×Dic139D411yescharacteristic
47C2×C4×D139D416yes—
48C2×C4×D139C2316yes—
49#14555774C23116yescomplemented
50#14555773C23116yescomplemented
51#14555770C23116yescomplemented
52#14555768C23116yescomplemented
53C2×C139⋊C8C2316yescomplemented
54C24×C139—21no—
55C24×C139—21no—
56C2×D4×C139—23no—
57C2×D4×C139—23no—
58C2×D4×C139C2316yes—
59C2×D4×C139C2316yescomplemented
60C22×C4×C139C2311yescharacteristic
61D8×C139C23116yescomplemented
62C2×C8×C139C2316yescomplemented
63C22×D139—24no—
64C22×D139—26no—
65C22×D139—216no—
66C22×D139—448no—
67C22×D139—44no—
68C22×D139C2×D416yes—
69Dic278—26no—
70Dic278C2414yescomplemented
71D556—26no—
72D556C2414yes—
73C22⋊D139—216no—
74C22⋊D139—216no—
75C2×Dic139—26no—
76C2×Dic139—216no—
77C2×Dic139C2×D416yes—
78C4×D139C2×D4112yes—
79C4×D139C2414yes—
80C139⋊C8C2414yescomplemented
81C23×C139—26no—
82C23×C139—26no—
83C23×C139—44no—
84C23×C139—44no—
85C23×C139C2×D411yescharacteristic
86D4×C139—26no—
87D4×C139—26no—
88D4×C139C2414yes—
89D4×C139C2414yescomplemented
90C2×C4×C139C2×D413yes—
91C2×C4×C139C2413yes—
92C8×C139C2414yescomplemented
93C22×C139—24no—
94C22×C139—24no—
95C22×C139—412no—
96C22×C139—412no—
97C22×C139AGL(1,ℤ/8)14yescomplemented
98C22×C139C22×D413yes—
99C2×D139—212no—
100C2×D139—24no—
101C2×D139—432no—
102C2×D139—412no—
103C2×D139C22×D414yes—
104C4×C139C22×D413yes—
105C4×C139C2511yescharacteristic
106Dic139—24no—
107Dic139C22×D414yes—
108C2×C139—44no—
109C2×C139—44no—
110C2×C139C2×AGL(1,ℤ/8)16yescomplemented
111C2×C139C23×D411yescharacteristic
112D139—24no—
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×SD16—1391no—
121C22×D8—1391no—
122C22×D8D13911yescharacteristic, complemented
123C22×M16—1391no—
124C25—2781no—
125C2×#41—2786no—
126C2×#41—13912no—
127C22×Q8—1391no—
128C22×D4—2781no—
129C22×D4—2786no—
130C22×D4—2781no—
131C22×D4—13912no—
132C22×D4—1391no—
133C22×D4C2×D13911yescharacteristic
134C22×D4C2×D13911yescharacteristic, complemented
135C23×C4—2781no—
136C23×C4—1391no—
137AGL(1,ℤ/8)—13964no—
138C2×SD16—13912no—
139C2×SD16—13912no—
140C2×D8—13912no—
141C2×D8C2×D139112yescomplemented
142C2×M16—13912no—
143C22×C8—1391no—
144C22×C8C2×D13911yescharacteristic, complemented
145C24—21no—
146C24—21no—
147C24—27812no—
148C24—2781no—
149C24—5568no—
150C24—1391no—
151#41—27824no—
152#41—13916no—
153C2×Q8—2783no—
154C2×Q8—1396no—
155C2×D4—23no—
156C2×D4—23no—
157C2×D4—27812no—
158C2×D4—27824no—
159C2×D4—27812no—
160C2×D4—2783no—
161C2×D4—13916no—
162C2×D4—1396no—
163C2×D4C22×D13916yes—
164C2×D4C22×D13916yescomplemented
165C22×C4—2781no—
166C22×C4—27812no—
167C22×C4—1391no—
168C22×C4—1396no—
169C22×C4—1396no—
170C22×C4C22×D13911yescharacteristic
171SD16—13916no—
172SD16—13916no—
173D8—13916no—
174D8C22×D139116yescomplemented
175M16—13916no—
176C2×C8—1396no—
177C2×C8C22×D13916yescomplemented
178C23—26no—
179C23—26no—
180C23—44no—
181C23—44no—
182C23—27816no—
183C23—2784no—
184C23—2786no—
185C23—55648no—
186C23—5564no—
187C23—1396no—
188C23D4×D13911yescharacteristic
189Q8—2786no—
190Q8—1394no—
191D4—26no—
192D4—26no—
193D4—27816no—
194D4—27816no—
195D4—2786no—
196D4—1394no—
197D4C23×D13914yes—
198D4C23×D13914yescomplemented
199C2×C4—2786no—
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—24no—
210C22—412no—
211C22—412no—
212C22—27812no—
213C22—2784no—
214C22—55632no—
215C22—55612no—
216C22—1394no—
217C22#1798486614yesdirect product factor
218C22C2×D4×D13913yes—
219C4—2784no—
220C4—1394no—
221C4C2×D4×D13913yes—
222C4C24×D13911yescharacteristic
223C2—44no—
224C2—44no—
225C2—2784no—
226C2—5564no—
227C2C2×#1798486616yesdirect product factor
228C2C22×D4×D13911yescharacteristic
subgroupG/H
centerC23D4×D139
commutator, cosocleC4×C139C25
FrattiniC4C24×D139
FittingC22×D8×C139C2
socleC23×C139C2×D4
hypercenterC22×D8D139
hypocenterC139C22×AGL(1,ℤ/8)
LCSG ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#14555784C2semidirect
2. C22×D4×D139C2semidirect
3. C2×#17984866C2direct (N × H)
4. C22×#14555774C2semidirect
5. C22×#14555773C2semidirect
6. C22×#14555770C2semidirect
7. C22×#14555768C2semidirect
8. C22×D8×C139C2semidirect
11. C2×#14555784C22semidirect
12. C22×Dic278C22semidirect
17. C2×D4×D139C22semidirect
20. #17984866C22direct (N × H)
21. C2×#14555774C22semidirect
22. C2×#14555773C22semidirect
23. C2×#14555770C22semidirect
24. C2×#14555768C22semidirect
25. C22×C139⋊C8C22semidirect
27. C22×D4×C139C22semidirect
28. C2×D8×C139C22semidirect
29. C22×C8×C139C22semidirect
35. #14555784C23semidirect
37. C2×Dic278C23semidirect
43. D4×D139C23semidirect
49. #14555774C23semidirect
50. #14555773C23semidirect
51. #14555770C23semidirect
52. #14555768C23semidirect
53. C2×C139⋊C8C23semidirect
59. C2×D4×C139C23semidirect
61. D8×C139C23semidirect
62. C2×C8×C139C23semidirect
70. Dic278C24semidirect
80. C139⋊C8C24semidirect
89. D4×C139C24semidirect
92. C8×C139C24semidirect
97. C22×C139AGL(1,ℤ/8)semidirect
110. C2×C139C2×AGL(1,ℤ/8)semidirect
114. C139C22×AGL(1,ℤ/8)semidirect
122. C22×D8D139semidirect
134. C22×D4C2×D139semidirect
141. C2×D8C2×D139semidirect
144. C22×C8C2×D139semidirect
164. C2×D4C22×D139semidirect
174. D8C22×D139semidirect
177. C2×C8C22×D139semidirect
198. D4C23×D139semidirect
207. C8C23×D139semidirect
217. C22#17984866direct (N × H)
227. C2C2×#17984866direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145557842
C22×D4×D1392
C2×#179848662
C22×#145557742
C22×#145557732
C22×#145557702
C22×#145557682
C22×D8×C1392
C22×AGL(1,ℤ/8)139