Finite Group Databasebeta

← #14530732

Subgroups — #14530732

Order 2176.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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
1#11763609C211yesmaximal, characteristic, complemented
2#11763247C212yesmaximal, complemented
3#11763062C211yesmaximal, characteristic, complemented
4#11762620C212yesmaximal, complemented
5C17×#349C211yesmaximal, characteristic, complemented
6C2×#63252—21no—
7C2×M16×C17C2211yescharacteristic
8#10587616—21no—
9#10587595—22no—
10#10587594—22no—
11#10587590—21no—
12#10587590C2212yes—
13C2×D136C2211yescharacteristic, complemented
14C17×M32C2212yescomplemented
15#10587492—21no—
16#10587488C2211yescharacteristic, complemented
17#10587471—22no—
18D272—22no—
19C22×C4×C17D411yescharacteristic
20C2×C22⋊D17—41no—
21C2×C22⋊D17—41no—
22#63252—22no—
23#63252—22no—
24#63252—41no—
25#63252—41no—
26C2×D68—21no—
27C2×D68—21no—
28C2×C4×D17—41no—
29C2×Dic34—21no—
30M16×C17—22no—
31C2×C8×C17D411yescharacteristic
32C2×C8×C17C2311yescharacteristic
33#63227—41no—
34C4⋊Dic17—21no—
35Dic68—21no—
36Dic68D412yescomplemented
37D136—21no—
38D136—22no—
39D136D412yescomplemented
40#63219—22no—
41#63219—22no—
42C16×C17—22no—
43C23×C17D811yescharacteristic
44C22×D17—41no—
45C22×D17—41no—
46C2×C4×C17—21no—
47C2×C4×C17—21no—
48C2×C4×C17D811yescharacteristic
49C2×C4×C17C2×D411yescharacteristic
50C22⋊D17—42no—
51C22⋊D17—42no—
52C22⋊D17—82no—
53C2×Dic17—41no—
54C2×Dic17—41no—
55D68—22no—
56D68—41no—
57D68—41no—
58D68—41no—
59C4×D17—42no—
60C4×D17—42no—
61Dic34—22no—
62Dic34—41no—
63C8×C17—21no—
64C8×C17C2×D412yes—
65#2190—171nomaximal, Sylow
66C22×C17—21no—
67C22×C17—41no—
68C22×C17C2×D811yescharacteristic
69C2×D17—42no—
70C2×D17—82no—
71C2×D17—81no—
72C2×D17—81no—
73C2×D17—81no—
74C4×C17—21no—
75C4×C17#12011yescharacteristic
76C4×C17AGL(1,ℤ/8)11yescharacteristic
77Dic17—42no—
78Dic17—81no—
79#575—171no—
80#509—172no—
81#469—171no—
82#360—172no—
83#349D1711yescharacteristic, complemented
84C2×C17—21no—
85C2×C17—41no—
86C2×C17#44711yescharacteristic
87D17—82no—
88D17—161no—
89C2×#41—341no—
90#137—342no—
91AGL(1,ℤ/8)—342no—
92#135—172no—
93#135—341no—
94C2×D8—171no—
95C2×M16C2×D1711yescharacteristic
96#121—341no—
97SD32—342no—
98D16—342no—
99M32C2×D1712yescomplemented
100#106—171no—
101#102—341no—
102C17#219011yescharacteristic, complemented, Sylow
103#41—342no—
104#41—342no—
105#41—681no—
106#41—681no—
107C2×Q8—341no—
108C2×D4—341no—
109C2×D4—341no—
110C2×D4—681no—
111C2×D4—681no—
112C22×C4—681no—
113C22×C4D6811yescharacteristic
114Q16—172no—
115Q16—341no—
116SD16—342no—
117SD16—342no—
118D8—172no—
119D8—341no—
120D8—342no—
121M16—22no—
122C2×C8D6811yescharacteristic
123C2×C8C22×D1711yescharacteristic
124C4⋊C4—341no—
125C22⋊C4—681no—
126C16—22no—
127C23—681no—
128C23—681no—
129C23D13611yescharacteristic
130Q8—342no—
131Q8—681no—
132D4—342no—
133D4—681no—
134D4—682no—
135D4—682no—
136D4—681no—
137D4—681no—
138D4—1362no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—682no—
142C2×C4—681no—
143C2×C4—682no—
144C2×C4—681no—
145C2×C4D13611yescharacteristic
146C2×C4C2×D6811yescharacteristic
147C8—21no—
148C8D4×D1712yes—
149C22—21no—
150C22—41no—
151C22—682no—
152C22—1361no—
153C22—1361no—
154C22—1362no—
155C22—1361no—
156C22C2×D13611yescharacteristic
157C4—21no—
158C4—682no—
159C4—1361no—
160C4#1058756811yescharacteristic
161C4#1058759411yescharacteristic
162C2—21no—
163C2—41no—
164C2—2721no—
165C2—1362no—
166C2#1176294511yescharacteristic
subgroupG/H
centerC2#11762945
commutator, cosocleC2×C8×C17C23
FrattiniC2×C8C22×D17
FittingC17×#349C2
socleC2×C17#447
hypercenter#349D17
hypocenterC17#2190
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
1. #11763609C2semidirect
2. #11763247C2semidirect
3. #11763062C2semidirect
4. #11762620C2semidirect
5. C17×#349C2semidirect
13. C2×D136C22semidirect
14. C17×M32C22semidirect
16. #10587488C22semidirect
36. Dic68D4semidirect
39. D136D4semidirect
83. #349D17semidirect
99. M32C2×D17semidirect
102. C17#2190semidirect

Maximal subgroups

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

subgroupindex
#117636092
#117632472
#117630622
#117626202
C17×#3492
#219017