Finite Group Databasebeta

← #14530737

Subgroups — #14530737

Order 2176.

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

normal subgroups only

This group has 216 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
2#11763252C212yesmaximal, complemented
3#11763067C211yesmaximal, characteristic, complemented
4#11762620C212yesmaximal, complemented
5C17×#349C211yesmaximal, characteristic, complemented
6C2×#63252—21no—
7C2×M16×C17C2211yescharacteristic
8#10587610—21no—
9#10587595—22no—
10#10587594—22no—
11C2×Dic68C2211yescharacteristic, complemented
12#10587590—21no—
13#10587590C2212yes—
14C17×M32C2212yescomplemented
15#10587488C2211yescharacteristic, complemented
16#10587487—21no—
17Dic136—22no—
18#10587471—22no—
19C22×C4×C17D411yescharacteristic
20C2×C22⋊D17—41no—
21#63252—22no—
22#63252—22no—
23#63252—41no—
24#63252—41no—
25C2×D68—21no—
26C2×C4×D17—41no—
27C2×Dic34—21no—
28C2×Dic34—21no—
29M16×C17—22no—
30C2×C8×C17D411yescharacteristic
31C2×C8×C17C2311yescharacteristic
32C22⋊Dic17—41no—
33C4⋊Dic17—21no—
34#63225—41no—
35Dic68—21no—
36Dic68—22no—
37Dic68D412yes—
38D136—21no—
39D136D412yes—
40#63219—22no—
41#63219—22no—
42C16×C17—22no—
43C23×C17D811yescharacteristic
44C22×D17—41no—
45C2×C4×C17—21no—
46C2×C4×C17—21no—
47C2×C4×C17D811yescharacteristic
48C2×C4×C17C2×D411yescharacteristic
49C22⋊D17—42no—
50C22⋊D17—42no—
51C2×Dic17—41no—
52C2×Dic17—41no—
53C2×Dic17—41no—
54D68—22no—
55D68—41no—
56C4×D17—42no—
57C4×D17—42no—
58Dic34—22no—
59Dic34—41no—
60Dic34—41no—
61Dic34—41no—
62C8×C17—21no—
63C8×C17C2×D412yes—
64#2191—171nomaximal, Sylow
65C22×C17—21no—
66C22×C17—41no—
67C22×C17C2×D811yescharacteristic
68C2×D17—42no—
69C2×D17—82no—
70C4×C17—21no—
71C4×C17#12011yescharacteristic
72C4×C17AGL(1,ℤ/8)11yescharacteristic
73Dic17—42no—
74Dic17—81no—
75Dic17—81no—
76#575—171no—
77#510—172no—
78#470—171no—
79#360—172no—
80#349D1711yescharacteristic, complemented
81C2×C17—21no—
82C2×C17—41no—
83C2×C17#44711yescharacteristic
84D17—82no—
85C2×#41—341no—
86#137—342no—
87AGL(1,ℤ/8)—342no—
88#135—172no—
89#135—341no—
90C2×Q16—171no—
91C2×M16C2×D1711yescharacteristic
92C22⋊Q8—341no—
93Q32—342no—
94SD32—342no—
95M32C2×D1712yescomplemented
96#106—171no—
97#103—341no—
98C17#219111yescharacteristic, complemented, Sylow
99#41—342no—
100#41—342no—
101#41—681no—
102#41—681no—
103C2×Q8—341no—
104C2×Q8—341no—
105C2×D4—341no—
106C2×D4—681no—
107C22×C4—681no—
108C22×C4D6811yescharacteristic
109Q16—172no—
110Q16—341no—
111Q16—342no—
112SD16—342no—
113SD16—342no—
114D8—172no—
115D8—341no—
116M16—22no—
117C2×C8D6811yescharacteristic
118C2×C8C22×D1711yescharacteristic
119C4⋊C4—341no—
120C4⋊C4—681no—
121C22⋊C4—681no—
122C16—22no—
123C23—681no—
124C23D13611yescharacteristic
125Q8—342no—
126Q8—681no—
127Q8—681no—
128Q8—681no—
129D4—342no—
130D4—681no—
131D4—682no—
132D4—682no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—682no—
136C2×C4—681no—
137C2×C4—682no—
138C2×C4—681no—
139C2×C4—681no—
140C2×C4D13611yescharacteristic
141C2×C4C2×D6811yescharacteristic
142C8—21no—
143C8D4×D1712yes—
144C22—21no—
145C22—41no—
146C22—682no—
147C22—1362no—
148C22C2×D13611yescharacteristic
149C4—21no—
150C4—682no—
151C4—1361no—
152C4—1361no—
153C4#1058756811yescharacteristic
154C4#1058759411yescharacteristic
155C2—21no—
156C2—41no—
157C2—1362no—
158C2#1176294511yescharacteristic
subgroupG/H
centerC2#11762945
commutator, cosocleC2×C8×C17C23
FrattiniC2×C8C22×D17
FittingC17×#349C2
socleC2×C17#447
prime radical#11763609C2
hypercenter#349D17
hypocenterC17#2191
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
2. #11763252C2semidirect
3. #11763067C2semidirect
4. #11762620C2semidirect
5. C17×#349C2semidirect
11. C2×Dic68C22semidirect
14. C17×M32C22semidirect
15. #10587488C22semidirect
80. #349D17semidirect
95. M32C2×D17semidirect
98. C17#2191semidirect

Maximal subgroups

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

subgroupindex
#117636092
#117632522
#117630672
#117626202
C17×#3492
#219117