Finite Group Databasebeta

← C22×#17984868 #31821137

Subgroups — C22×#17984868 #31821137

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
2C2×#17984868C2124yesmaximal, direct product factor
3C22×#14555773C212yesmaximal, complemented
4C22×#14555770C212yesmaximal, complemented
5C22×#14555769C211yesmaximal, characteristic, complemented
6C23×D4×C139C211yesmaximal, characteristic, complemented
7C2×#14555785—26no—
8C2×#14555785C22112yescomplemented
9C22×Dic278C2211yescharacteristic, complemented
10C22×D556C2211yescharacteristic, complemented
11C22×C22⋊D139—21no—
12C22×C4×D139—21no—
13#17984868C22164yesdirect product factor
14C2×#14555773C22124yescomplemented
15C2×#14555770C22124yescomplemented
16C2×#14555769C22112yescomplemented
17C22×C139⋊C8C2212yescomplemented
18C25×C139—21no—
19C22×D4×C139—21no—
20C22×D4×C139—26no—
21C22×D4×C139C2212yes—
22C22×D4×C139C22112yescomplemented
23C23×C4×C139C2211yescharacteristic
24C23×D139—21no—
25#14555785—224no—
26#14555785C23116yescomplemented
27C2×Dic278—23no—
28C2×Dic278C2316yescomplemented
29C2×D556—23no—
30C2×D556C2316yescomplemented
31C2×C22⋊D139—212no—
32C22×Dic139—21no—
33C2×C4×D139—212no—
34#14555773C23132yescomplemented
35#14555770C23132yescomplemented
36#14555769C23116yescomplemented
37C2×C139⋊C8C23112yescomplemented
38C24×C139—22no—
39C24×C139—212no—
40C24×C139—48no—
41C24×C139D411yescharacteristic
42C2×D4×C139—26no—
43C2×D4×C139—212no—
44C2×D4×C139—224no—
45C2×D4×C139C23112yes—
46C2×D4×C139C23116yescomplemented
47C22×C4×C139D411yescharacteristic
48C22×C4×C139D416yes—
49C22×C4×C139C2311yescharacteristic
50C22×C4×C139C2316yes—
51C22×D139—26no—
52C22×D139—44no—
53Dic278—26no—
54Dic278C2414yescomplemented
55D556—26no—
56D556C2414yescomplemented
57C22⋊D139—216no—
58C2×Dic139—26no—
59C4×D139—216no—
60C139⋊C8C2418yescomplemented
61C23×C139—24no—
62C23×C139—212no—
63C23×C139—216no—
64C23×C139—448no—
65C23×C139—48no—
66C23×C139C2×D416yes—
67C23×C139C2×D411yescharacteristic
68D4×C139—212no—
69D4×C139—216no—
70D4×C139C2418yes—
71C2×C4×C139C2×D416yes—
72C2×C4×C139C2×D413yes—
73C2×C4×C139C2×D4112yes—
74C2×C4×C139C2413yes—
75C2×C4×C139C2414yes—
76C22×C139—212no—
77C22×C139—28no—
78C22×C139—432no—
79C22×C139—424no—
80C22×C139AGL(1,ℤ/8)14yescomplemented
81C22×C139C22×D414yes—
82C22×C139C22×D413yes—
83C2×D139—24no—
84C2×D139—412no—
85C4×C139C22×D414yes—
86C4×C139C22×D413yes—
87C4×C139C2511yescharacteristic
88Dic139—24no—
89C2×C139—24no—
90C2×C139—48no—
91C2×C139C2×AGL(1,ℤ/8)16yescomplemented
92C2×C139C23×D411yescharacteristic
93D139—44no—
94C139C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1391nomaximal, Sylow
96C22×#41—1391no—
97C23×D4D13911yescharacteristic, complemented
98C2×AGL(1,ℤ/8)—13924no—
99C22×SD16—1392no—
100C22×D8—1392no—
101C22×M16—1391no—
102C25—21no—
103C2×#41—2786no—
104C2×#41—13912no—
105C22×Q8—1391no—
106C22×D4—26no—
107C22×D4—21no—
108C22×D4—2781no—
109C22×D4—1391no—
110C22×D4C2×D13912yes—
111C22×D4C2×D139112yescomplemented
112C23×C4—2781no—
113C23×C4C2×D13911yescharacteristic
114AGL(1,ℤ/8)—13964no—
115C2×SD16—13924no—
116C2×D8—13924no—
117C2×M16—13912no—
118C22×C8—1392no—
119C24—212no—
120C24—22no—
121C24—48no—
122C24—2781no—
123C24C22⋊D13911yescharacteristic
124#41—27824no—
125#41—13916no—
126C2×Q8—2783no—
127C2×Q8—1396no—
128C2×D4—26no—
129C2×D4—212no—
130C2×D4—224no—
131C2×D4—27812no—
132C2×D4—2783no—
133C2×D4—1396no—
134C2×D4C22×D139112yes—
135C2×D4C22×D139116yescomplemented
136C22×C4—2781no—
137C22×C4—27812no—
138C22×C4C22⋊D13916yes—
139C22×C4C22⋊D13911yescharacteristic
140C22×C4C22×D13916yes—
141C22×C4C22×D13911yescharacteristic
142SD16—13932no—
143D8—13932no—
144M16—13916no—
145C2×C8—13912no—
146C23—24no—
147C23—212no—
148C23—216no—
149C23—448no—
150C23—48no—
151C23—2786no—
152C23—5564no—
153C23C2×C22⋊D13916yes—
154C23C2×C22⋊D13911yescharacteristic
155Q8—2786no—
156Q8—1394no—
157D4—212no—
158D4—216no—
159D4—27816no—
160D4—2786no—
161D4—1394no—
162D4C23×D13918yes—
163C2×C4—2786no—
164C2×C4—27816no—
165C2×C4C2×C22⋊D13916yes—
166C2×C4C2×C22⋊D13913yes—
167C2×C4C2×C22⋊D139112yes—
168C2×C4C23×D13913yes—
169C2×C4C23×D13914yes—
170C8—1398no—
171C22—212no—
172C22—28no—
173C22—432no—
174C22—424no—
175C22—2784no—
176C22—55612no—
177C22#1798486814yesdirect product factor
178C22C22×C22⋊D13914yes—
179C22C22×C22⋊D13913yes—
180C4—2784no—
181C4C22×C22⋊D13914yes—
182C4C22×C22⋊D13913yes—
183C4C24×D13911yescharacteristic
184C2—24no—
185C2—48no—
186C2—5564no—
187C2C2×#1798486816yesdirect product factor
188C2C23×C22⋊D13911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D139
commutator, cosocleC4×C139C25
FrattiniC4C24×D139
FittingC23×D4×C139C2
socleC23×C139C2×D4
hypercenterC23×D4D139
hypocenterC139C22×AGL(1,ℤ/8)
LCSG ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14555785C2semidirect
2. C2×#17984868C2direct (N × H)
3. C22×#14555773C2semidirect
4. C22×#14555770C2semidirect
5. C22×#14555769C2semidirect
6. C23×D4×C139C2semidirect
8. C2×#14555785C22semidirect
9. C22×Dic278C22semidirect
10. C22×D556C22semidirect
13. #17984868C22direct (N × H)
14. C2×#14555773C22semidirect
15. C2×#14555770C22semidirect
16. C2×#14555769C22semidirect
17. C22×C139⋊C8C22semidirect
22. C22×D4×C139C22semidirect
26. #14555785C23semidirect
28. C2×Dic278C23semidirect
30. C2×D556C23semidirect
34. #14555773C23semidirect
35. #14555770C23semidirect
36. #14555769C23semidirect
37. C2×C139⋊C8C23semidirect
46. C2×D4×C139C23semidirect
54. Dic278C24semidirect
56. D556C24semidirect
60. C139⋊C8C24semidirect
80. C22×C139AGL(1,ℤ/8)semidirect
91. C2×C139C2×AGL(1,ℤ/8)semidirect
94. C139C22×AGL(1,ℤ/8)semidirect
97. C23×D4D139semidirect
111. C22×D4C2×D139semidirect
135. C2×D4C22×D139semidirect
177. C22#17984868direct (N × H)
187. C2C2×#17984868direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145557852
C2×#179848682
C22×#145557732
C22×#145557702
C22×#145557692
C23×D4×C1392
C22×AGL(1,ℤ/8)139