Finite Group Databasebeta

← C22×#17984867 #31821136

Subgroups — C22×#17984867 #31821136

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
1C23×D556C211yesmaximal, characteristic, complemented
2C2×#17984867C2124yesmaximal, direct product factor
3C22×#14555772C212yesmaximal, complemented
4C22×#14555770C212yesmaximal, complemented
5C22×#14555769C211yesmaximal, characteristic, complemented
6C22×#41×C139C211yesmaximal, characteristic, complemented
7C24×D139—21no—
8C22×D556—21no—
9C22×D556—26no—
10C22×D556C2212yes—
11C22×D556C22112yescomplemented
12#17984867C22164yesdirect product factor
13C2×#14555772C22124yescomplemented
14C2×#14555770C22124yescomplemented
15C2×#14555769C22112yescomplemented
16C22×C139⋊C8C2212yescomplemented
17C2×#41×C139—26no—
18C2×#41×C139C22112yescomplemented
19C22×Q8×C139C2211yescharacteristic, complemented
20C22×D4×C139—21no—
21C22×D4×C139C2211yescharacteristic, complemented
22C23×C4×C139—21no—
23C23×C4×C139C2211yescharacteristic
24C23×D139—22no—
25C23×D139—212no—
26C23×D139—48no—
27C2×D556—26no—
28C2×D556—212no—
29C2×D556—224no—
30C2×D556C23112yes—
31C2×D556C23116yescomplemented
32#14555772C23132yescomplemented
33#14555770C23132yescomplemented
34#14555769C23116yescomplemented
35C2×C139⋊C8C23112yescomplemented
36C24×C139—21no—
37C24×C139D411yescharacteristic
38#41×C139—224no—
39#41×C139C23116yescomplemented
40C2×Q8×C139—23no—
41C2×Q8×C139C2316yescomplemented
42C2×D4×C139—212no—
43C2×D4×C139—23no—
44C2×D4×C139C2316yescomplemented
45C22×C4×C139—21no—
46C22×C4×C139—212no—
47C22×C4×C139D411yescharacteristic
48C22×C4×C139D416yes—
49C22×C4×C139C2311yescharacteristic
50C22×C4×C139C2316yes—
51C22×D139—212no—
52C22×D139—216no—
53C22×D139—448no—
54C22×D139—48no—
55D556—212no—
56D556—216no—
57D556C2418yes—
58C139⋊C8C2418yescomplemented
59C23×C139—24no—
60C23×C139—26no—
61C23×C139—44no—
62C23×C139C2×D416yes—
63C23×C139C2×D411yescharacteristic
64Q8×C139—26no—
65Q8×C139C2414yescomplemented
66D4×C139—216no—
67D4×C139—26no—
68D4×C139C2414yescomplemented
69C2×C4×C139—26no—
70C2×C4×C139—216no—
71C2×C4×C139C2×D416yes—
72C2×C4×C139C2×D413yes—
73C2×C4×C139C2×D4112yes—
74C2×C4×C139C2413yes—
75C2×C4×C139C2414yes—
76C22×C139—212no—
77C22×C139—24no—
78C22×C139—412no—
79C22×C139AGL(1,ℤ/8)14yescomplemented
80C22×C139C22×D414yes—
81C22×C139C22×D413yes—
82C2×D139—28no—
83C2×D139—432no—
84C2×D139—424no—
85C4×C139—24no—
86C4×C139C22×D414yes—
87C4×C139C22×D413yes—
88C4×C139C2511yescharacteristic
89C2×C139—24no—
90C2×C139—44no—
91C2×C139C2×AGL(1,ℤ/8)16yescomplemented
92C2×C139C23×D411yescharacteristic
93D139—48no—
94C139C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1391nomaximal, Sylow
96C22×#41D13911yescharacteristic, complemented
97C23×D4—1391no—
98C2×AGL(1,ℤ/8)—13924no—
99C22×SD16—1392no—
100C22×D8—1392no—
101C22×M16—1391no—
102C25—2781no—
103C2×#41—26no—
104C2×#41C2×D139112yescomplemented
105C22×Q8C2×D13911yescharacteristic, complemented
106C22×D4—21no—
107C22×D4—2786no—
108C22×D4—2781no—
109C22×D4—1392no—
110C22×D4—13912no—
111C22×D4C2×D13911yescharacteristic, complemented
112C23×C4—21no—
113C23×C4C2×D13911yescharacteristic
114AGL(1,ℤ/8)—13964no—
115C2×SD16—13924no—
116C2×D8—13924no—
117C2×M16—13912no—
118C22×C8—1392no—
119C24—21no—
120C24—27812no—
121C24—2782no—
122C24—5568no—
123C24C22⋊D13911yescharacteristic
124#41—224no—
125#41C22×D139116yescomplemented
126C2×Q8—23no—
127C2×Q8C22×D13916yescomplemented
128C2×D4—212no—
129C2×D4—23no—
130C2×D4—27812no—
131C2×D4—2786no—
132C2×D4—27824no—
133C2×D4—13912no—
134C2×D4—13916no—
135C2×D4C22×D13916yescomplemented
136C22×C4—21no—
137C22×C4—212no—
138C22×C4C22⋊D13916yes—
139C22×C4C22⋊D13911yescharacteristic
140C22×C4C22×D13916yes—
141C22×C4C22×D13911yescharacteristic
142SD16—13932no—
143D8—13932no—
144M16—13916no—
145C2×C8—13912no—
146C23—24no—
147C23—26no—
148C23—44no—
149C23—27816no—
150C23—27812no—
151C23—55648no—
152C23—5568no—
153C23C2×C22⋊D13916yes—
154C23C2×C22⋊D13911yescharacteristic
155Q8—26no—
156Q8C23×D13914yescomplemented
157D4—216no—
158D4—26no—
159D4—27816no—
160D4—27812no—
161D4—1398no—
162D4C23×D13914yescomplemented
163C2×C4—26no—
164C2×C4—216no—
165C2×C4C2×C22⋊D13916yes—
166C2×C4C2×C22⋊D13913yes—
167C2×C4C2×C22⋊D139112yes—
168C2×C4C23×D13913yes—
169C2×C4C23×D13914yes—
170C8—1398no—
171C22—212no—
172C22—24no—
173C22—412no—
174C22—2788no—
175C22—55632no—
176C22—55624no—
177C22#1798486714yesdirect product factor
178C22C22×C22⋊D13914yes—
179C22C22×C22⋊D13913yes—
180C4—24no—
181C4C22×C22⋊D13914yes—
182C4C22×C22⋊D13913yes—
183C4C24×D13911yescharacteristic
184C2—24no—
185C2—44no—
186C2—5568no—
187C2C2×#1798486716yesdirect product factor
188C2C23×C22⋊D13911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D139
commutator, cosocleC4×C139C25
FrattiniC4C24×D139
FittingC22×#41×C139C2
socleC23×C139C2×D4
hypercenterC22×#41D139
hypocenterC139C22×AGL(1,ℤ/8)
LCSG ▷ C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C23×D556C2semidirect
2. C2×#17984867C2direct (N × H)
3. C22×#14555772C2semidirect
4. C22×#14555770C2semidirect
5. C22×#14555769C2semidirect
6. C22×#41×C139C2semidirect
11. C22×D556C22semidirect
12. #17984867C22direct (N × H)
13. C2×#14555772C22semidirect
14. C2×#14555770C22semidirect
15. C2×#14555769C22semidirect
16. C22×C139⋊C8C22semidirect
18. C2×#41×C139C22semidirect
19. C22×Q8×C139C22semidirect
21. C22×D4×C139C22semidirect
31. C2×D556C23semidirect
32. #14555772C23semidirect
33. #14555770C23semidirect
34. #14555769C23semidirect
35. C2×C139⋊C8C23semidirect
39. #41×C139C23semidirect
41. C2×Q8×C139C23semidirect
44. C2×D4×C139C23semidirect
58. C139⋊C8C24semidirect
65. Q8×C139C24semidirect
68. D4×C139C24semidirect
79. C22×C139AGL(1,ℤ/8)semidirect
91. C2×C139C2×AGL(1,ℤ/8)semidirect
94. C139C22×AGL(1,ℤ/8)semidirect
96. C22×#41D139semidirect
104. C2×#41C2×D139semidirect
105. C22×Q8C2×D139semidirect
111. C22×D4C2×D139semidirect
125. #41C22×D139semidirect
127. C2×Q8C22×D139semidirect
135. C2×D4C22×D139semidirect
156. Q8C23×D139semidirect
162. D4C23×D139semidirect
177. C22#17984867direct (N × H)
187. C2C2×#17984867direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D5562
C2×#179848672
C22×#145557722
C22×#145557702
C22×#145557692
C22×#41×C1392
C22×AGL(1,ℤ/8)139