Finite Group Databasebeta

← C2×C17×#509 #14526418

Subgroups — C2×C17×#509 #14526418

Order 2176.

400 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C2×C17×#135C211yesmaximal, characteristic, complemented
2C22×D8×C17C211yesmaximal, characteristic, complemented
3C17×#509C218yesmaximal, direct product factor
4C2×C17×SD32C212yesmaximal, complemented
5C2×C17×D16C212yesmaximal, complemented
6C2×C17×M32C211yesmaximal, characteristic, complemented
7C2×#41×C17—21no—
8C22×D4×C17—21no—
9C17×#135—22no—
10C17×#135C2214yescomplemented
11C2×Q16×C17C2211yescharacteristic, complemented
12C2×SD16×C17—21no—
13C2×D8×C17—21no—
14C2×D8×C17—22no—
15C2×D8×C17C2212yes—
16C2×D8×C17C2214yescomplemented
17C2×D8×C17C2211yescharacteristic, complemented
18C22×C8×C17C2211yescharacteristic
19C17×SD32C2218yescomplemented
20C17×D16C2218yescomplemented
21C17×M32C2214yescomplemented
22C2×C16×C17C2212yescomplemented
23C24×C17—41no—
24#41×C17—24no—
25#41×C17—42no—
26C2×Q8×C17—21no—
27C2×D4×C17—22no—
28C2×D4×C17—24no—
29C2×D4×C17—21no—
30C2×D4×C17—41no—
31C2×D4×C17—42no—
32C2×D4×C17—41no—
33C22×C4×C17—41no—
34C22×C4×C17D411yescharacteristic
35Q16×C17—21no—
36Q16×C17C2312yescomplemented
37SD16×C17—24no—
38D8×C17—22no—
39D8×C17—24no—
40D8×C17—21no—
41D8×C17C2314yes—
42D8×C17C2312yescomplemented
43C2×C8×C17D411yescharacteristic
44C2×C8×C17D412yes—
45C2×C8×C17C2311yescharacteristic
46C2×C8×C17C2312yes—
47C16×C17C2314yescomplemented
48C23×C17—44no—
49C23×C17—41no—
50C23×C17—42no—
51C23×C17—84no—
52C23×C17D811yescharacteristic
53Q8×C17—22no—
54Q8×C17—41no—
55D4×C17—24no—
56D4×C17—22no—
57D4×C17—44no—
58D4×C17—44no—
59D4×C17—42no—
60D4×C17—41no—
61C2×C4×C17—41no—
62C2×C4×C17—44no—
63C2×C4×C17D812yes—
64C2×C4×C17D811yescharacteristic
65C2×C4×C17C2×D412yes—
66C2×C4×C17C2×D411yescharacteristic
67C8×C17C2×D411yescharacteristic
68C8×C17C2×D412yes—
69C8×C17C2411yescharacteristic
70C2×#509C1711yesmaximal, characteristic, direct product factor, Sylow
71C22×C17—22no—
72C22×C17—42no—
73C22×C17—44no—
74C22×C17—88no—
75C22×C17—82no—
76C22×C17—84no—
77C22×C17C2×D812yes—
78C22×C17C2×D811yescharacteristic
79C4×C17—42no—
80C4×C17C2×D812yes—
81C4×C17C2×D811yescharacteristic
82C4×C17C22×D411yescharacteristic
83C2×#135C2×C1711yescharacteristic, complemented
84C22×D8C2×C1711yescharacteristic, complemented
85#509C2×C1718yesdirect product factor
86C2×SD32C2×C1712yescomplemented
87C2×D16C2×C1712yescomplemented
88C2×M32C2×C1711yescharacteristic, complemented
89C2×C17—22no—
90C2×C17—82no—
91C2×C17—84no—
92C2×C17#50912yesdirect product factor
93C2×C17C22×D811yescharacteristic
94C2×#41—21no—
95C22×D4—21no—
96#135—22no—
97#135C22×C1714yescomplemented
98C2×Q16C22×C1711yescharacteristic, complemented
99C2×SD16—21no—
100C2×D8—21no—
101C2×D8—22no—
102C2×D8C22×C1712yes—
103C2×D8C22×C1714yescomplemented
104C2×D8C22×C1711yescharacteristic, complemented
105C22×C8C22×C1711yescharacteristic
106SD32C22×C1718yescomplemented
107D16C22×C1718yescomplemented
108M32C22×C1714yescomplemented
109C2×C16C22×C1712yescomplemented
110C17C2×#50911yescharacteristic, direct product factor, Sylow
111C24—41no—
112#41—24no—
113#41—42no—
114C2×Q8—21no—
115C2×D4—22no—
116C2×D4—24no—
117C2×D4—21no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—42no—
121C22×C4—41no—
122C22×C4D4×C1711yescharacteristic
123Q16—21no—
124Q16C23×C1712yescomplemented
125SD16—24no—
126D8—22no—
127D8—24no—
128D8—21no—
129D8C23×C1714yes—
130D8C23×C1712yescomplemented
131C2×C8D4×C1711yescharacteristic
132C2×C8D4×C1712yes—
133C2×C8C23×C1711yescharacteristic
134C2×C8C23×C1712yes—
135C16C23×C1714yescomplemented
136C23—41no—
137C23—44no—
138C23—42no—
139C23—84no—
140C23D8×C1711yescharacteristic
141Q8—22no—
142Q8—41no—
143D4—24no—
144D4—22no—
145D4—44no—
146D4—44no—
147D4—42no—
148D4—41no—
149C2×C4—41no—
150C2×C4—44no—
151C2×C4D8×C1711yescharacteristic
152C2×C4D8×C1712yes—
153C2×C4C2×D4×C1711yescharacteristic
154C2×C4C2×D4×C1712yes—
155C8C2×D4×C1711yescharacteristic
156C8C2×D4×C1712yes—
157C8C24×C1711yescharacteristic
158C22—22no—
159C22—42no—
160C22—44no—
161C22—88no—
162C22—84no—
163C22—82no—
164C22C2×D8×C1712yes—
165C22C2×D8×C1711yescharacteristic
166C4—42no—
167C4C2×D8×C1712yes—
168C4C2×D8×C1711yescharacteristic
169C4C22×D4×C1711yescharacteristic
170C2—22no—
171C2—84no—
172C2—82no—
173C2C17×#50912yesdirect product factor
174C2C22×D8×C1711yescharacteristic
subgroupG/H
center, socleC22×C17C2×D8
commutator, Frattini, cosocleC8C24×C17
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C22×C4×C17 ◁ C22×C8×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C17×#135C2semidirect
2. C22×D8×C17C2semidirect
3. C17×#509C2direct (N × H)
4. C2×C17×SD32C2semidirect
5. C2×C17×D16C2semidirect
6. C2×C17×M32C2semidirect
10. C17×#135C22semidirect
11. C2×Q16×C17C22semidirect
16. C2×D8×C17C22semidirect
17. C2×D8×C17C22semidirect
19. C17×SD32C22semidirect
20. C17×D16C22semidirect
21. C17×M32C22semidirect
22. C2×C16×C17C22semidirect
36. Q16×C17C23semidirect
42. D8×C17C23semidirect
47. C16×C17C23semidirect
70. C2×#509C17direct (N × H)
83. C2×#135C2×C17semidirect
84. C22×D8C2×C17semidirect
85. #509C2×C17direct (N × H)
86. C2×SD32C2×C17semidirect
87. C2×D16C2×C17semidirect
88. C2×M32C2×C17semidirect
92. C2×C17#509direct (N × H)
97. #135C22×C17semidirect
98. C2×Q16C22×C17semidirect
103. C2×D8C22×C17semidirect
104. C2×D8C22×C17semidirect
106. SD32C22×C17semidirect
107. D16C22×C17semidirect
108. M32C22×C17semidirect
109. C2×C16C22×C17semidirect
110. C17C2×#509direct (N × H)
124. Q16C23×C17semidirect
130. D8C23×C17semidirect
135. C16C23×C17semidirect
173. C2C17×#509direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C17×#1352
C22×D8×C172
C17×#5092
C2×C17×SD322
C2×C17×D162
C2×C17×M322
C2×#50917