Finite Group Databasebeta

← C22×#17984870 #31821139

Subgroups — C22×#17984870 #31821139

Order 17792.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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×#14555785C211yesmaximal, characteristic, complemented
2C23×D556C211yesmaximal, characteristic, complemented
3C2×#17984870C2124yesmaximal, direct product factor
4C22×#14555774C212yesmaximal, complemented
5C22×D1112C212yesmaximal, complemented
6C22×M16×C139C211yesmaximal, characteristic, complemented
7C24×D139—21no—
8C2×#14555785—26no—
9C2×#14555785C22112yescomplemented
10C22×Dic278C2211yescharacteristic, complemented
11C22×D556—21no—
12C22×D556—26no—
13C22×D556C2212yes—
14C22×D556C22112yescomplemented
15C22×D556C2211yescharacteristic, complemented
16C22×C22⋊D139—21no—
17C22×C4×D139—21no—
18#17984870C22164yesdirect product factor
19C2×#14555774C22124yescomplemented
20C2×D1112C22124yescomplemented
21C23×C4×C139C2211yescharacteristic
22C2×M16×C139C22112yescomplemented
23C22×C8×C139C2212yescomplemented
24C23×D139—21no—
25C23×D139—22no—
26C23×D139—212no—
27C23×D139—48no—
28#14555785—224no—
29#14555785C23116yescomplemented
30C2×Dic278—23no—
31C2×Dic278C2316yescomplemented
32C2×D556—26no—
33C2×D556—212no—
34C2×D556—224no—
35C2×D556—23no—
36C2×D556C23112yes—
37C2×D556C23116yescomplemented
38C2×D556C2316yescomplemented
39C2×C22⋊D139—212no—
40C22×Dic139—21no—
41C2×C4×D139—212no—
42#14555774C23132yescomplemented
43D1112C23132yescomplemented
44C24×C139D411yescharacteristic
45C22×C4×C139D411yescharacteristic
46C22×C4×C139D416yes—
47C22×C4×C139C2311yescharacteristic
48C22×C4×C139C2316yes—
49M16×C139C23116yescomplemented
50C2×C8×C139C23112yescomplemented
51C22×D139—26no—
52C22×D139—212no—
53C22×D139—216no—
54C22×D139—448no—
55C22×D139—44no—
56C22×D139—48no—
57Dic278—26no—
58Dic278C2414yescomplemented
59D556—212no—
60D556—216no—
61D556—26no—
62D556C2418yes—
63D556C2414yescomplemented
64C22⋊D139—216no—
65C2×Dic139—26no—
66C4×D139—216no—
67C23×C139—24no—
68C23×C139C2×D416yes—
69C23×C139C2×D411yescharacteristic
70C2×C4×C139C2×D416yes—
71C2×C4×C139C2×D413yes—
72C2×C4×C139C2×D4112yes—
73C2×C4×C139C2413yes—
74C2×C4×C139C2414yes—
75C8×C139C2418yescomplemented
76C22×C139—212no—
77C22×C139AGL(1,ℤ/8)14yescomplemented
78C22×C139C22×D414yes—
79C22×C139C22×D413yes—
80C2×D139—24no—
81C2×D139—28no—
82C2×D139—432no—
83C2×D139—412no—
84C2×D139—424no—
85C4×C139C22×D414yes—
86C4×C139C22×D413yes—
87C4×C139C2511yescharacteristic
88Dic139—24no—
89C2×C139—24no—
90C2×C139C2×AGL(1,ℤ/8)16yescomplemented
91C2×C139C23×D411yescharacteristic
92D139—44no—
93D139—48no—
94C139C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1391nomaximal, Sylow
96C22×#41—1391no—
97C23×D4—1391no—
98C2×AGL(1,ℤ/8)—13924no—
99C22×SD16—1392no—
100C22×D8—1392no—
101C22×M16D13911yescharacteristic, complemented
102C25—2781no—
103C2×#41—2786no—
104C2×#41—13912no—
105C22×Q8—1391no—
106C22×D4—2781no—
107C22×D4—2786no—
108C22×D4—2781no—
109C22×D4—1391no—
110C22×D4—1392no—
111C22×D4—13912no—
112C23×C4—2781no—
113C23×C4C2×D13911yescharacteristic
114AGL(1,ℤ/8)—13964no—
115C2×SD16—13924no—
116C2×D8—13924no—
117C2×M16C2×D139112yescomplemented
118C22×C8C2×D13912yescomplemented
119C24—27812no—
120C24—2781no—
121C24—2782no—
122C24—5568no—
123C24D55611yescharacteristic
124#41—27824no—
125#41—13916no—
126C2×Q8—2783no—
127C2×Q8—1396no—
128C2×D4—27812no—
129C2×D4—27812no—
130C2×D4—2786no—
131C2×D4—27824no—
132C2×D4—2783no—
133C2×D4—1396no—
134C2×D4—13912no—
135C2×D4—13916no—
136C22×C4—2781no—
137C22×C4—27812no—
138C22×C4D55616yes—
139C22×C4D55611yescharacteristic
140C22×C4C22×D13916yes—
141C22×C4C22×D13911yescharacteristic
142SD16—13932no—
143D8—13932no—
144M16C22×D139116yescomplemented
145C2×C8C22×D139112yescomplemented
146C23—24no—
147C23—2786no—
148C23—27816no—
149C23—27812no—
150C23—55648no—
151C23—5568no—
152C23—5564no—
153C23C2×D55616yes—
154C23C2×D55611yescharacteristic
155Q8—2786no—
156Q8—1394no—
157D4—27816no—
158D4—27816no—
159D4—27812no—
160D4—2786no—
161D4—1394no—
162D4—1398no—
163C2×C4—2786no—
164C2×C4—27816no—
165C2×C4C2×D55616yes—
166C2×C4C2×D55613yes—
167C2×C4C2×D556112yes—
168C2×C4C23×D13913yes—
169C2×C4C23×D13914yes—
170C8C23×D13918yescomplemented
171C22—212no—
172C22—2784no—
173C22—2788no—
174C22—55632no—
175C22—55624no—
176C22—55612no—
177C22#1798487014yesdirect product factor
178C22C22×D55614yes—
179C22C22×D55613yes—
180C4—2784no—
181C4C22×D55614yes—
182C4C22×D55613yes—
183C4C24×D13911yescharacteristic
184C2—24no—
185C2—5568no—
186C2—5564no—
187C2C2×#1798487016yesdirect product factor
188C2C23×D55611yescharacteristic
subgroupG/H
centerC23C2×D556
commutator, cosocleC4×C139C25
FrattiniC4C24×D139
FittingC22×M16×C139C2
socleC23×C139C2×D4
hypercenterC22×M16D139
hypocenterC139C22×AGL(1,ℤ/8)
LCSG ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#14555785C2semidirect
2. C23×D556C2semidirect
3. C2×#17984870C2direct (N × H)
4. C22×#14555774C2semidirect
5. C22×D1112C2semidirect
6. C22×M16×C139C2semidirect
9. C2×#14555785C22semidirect
10. C22×Dic278C22semidirect
14. C22×D556C22semidirect
15. C22×D556C22semidirect
18. #17984870C22direct (N × H)
19. C2×#14555774C22semidirect
20. C2×D1112C22semidirect
22. C2×M16×C139C22semidirect
23. C22×C8×C139C22semidirect
29. #14555785C23semidirect
31. C2×Dic278C23semidirect
37. C2×D556C23semidirect
38. C2×D556C23semidirect
42. #14555774C23semidirect
43. D1112C23semidirect
49. M16×C139C23semidirect
50. C2×C8×C139C23semidirect
58. Dic278C24semidirect
63. D556C24semidirect
75. C8×C139C24semidirect
77. C22×C139AGL(1,ℤ/8)semidirect
90. C2×C139C2×AGL(1,ℤ/8)semidirect
94. C139C22×AGL(1,ℤ/8)semidirect
101. C22×M16D139semidirect
117. C2×M16C2×D139semidirect
118. C22×C8C2×D139semidirect
144. M16C22×D139semidirect
145. C2×C8C22×D139semidirect
170. C8C23×D139semidirect
177. C22#17984870direct (N × H)
187. C2C2×#17984870direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145557852
C23×D5562
C2×#179848702
C22×#145557742
C22×D11122
C22×M16×C1392
C22×AGL(1,ℤ/8)139