Finite Group Databasebeta

← #14528037

Subgroups — #14528037

Order 2176.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#11763533C211yesmaximal, characteristic
2#11763091C211yesmaximal, characteristic, complemented
3#11763011C211yesmaximal, characteristic
4#11762555C211yesmaximal, characteristic, complemented
5#11762541C211yesmaximal, characteristic, complemented
6#11762538C211yesmaximal, characteristic
7C17×#500C211yesmaximal, characteristic, complemented
8C17×C4⋊Q8C2211yescharacteristic
9C17×C4⋊Q8C2211yescharacteristic
10#10587633—21no—
11Q8×Dic17—21no—
12C2×#63230—21no—
13#10587584—22no—
14#10587583—21no—
15#10587578—21no—
16C4×D68C2211yescharacteristic
17C4×Dic34C2211yescharacteristic
18C17×#107—22no—
19C4×C8×C17C2211yescharacteristic
20#10587506—21no—
21#10587481—21no—
22#10587479—21no—
23C4⋊#3622C2211yescharacteristic
24C4⋊#3622C2211yescharacteristic
25C2×Q8×C17—21no—
26C2×Q8×C17—21no—
27C2×D68D411yescharacteristic
28C2×C4×D17—21no—
29C2×Dic34D411yescharacteristic
30C2×C8×C17—21no—
31C4⋊C4×C17—21no—
32C4⋊C4×C17—21no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17D412yes—
35C42×C17C2311yescharacteristic
36#63230—42no—
37#63227—22no—
38#63227—22no—
39#63227—21no—
40C4⋊Dic17—22no—
41C4⋊Dic17D411yescharacteristic
42C4⋊Dic17D411yescharacteristic
43#63225—21no—
44C4×Dic17—22no—
45C4×Dic17—21no—
46C2×#3622—21no—
47C2×#3622—21no—
48C22×D17—21no—
49Q8×C17—41no—
50Q8×C17—41no—
51Q8×C17—41no—
52Q8×C17—41no—
53C2×C4×C17—22no—
54C2×C4×C17—41no—
55C2×C4×C17C2×D411yescharacteristic
56C2×C4×C17C2×D411yescharacteristic
57C2×C4×C17C2×D411yescharacteristic
58C2×Dic17—21no—
59C2×Dic17—21no—
60C2×Dic17—21no—
61D68—21no—
62D68—21no—
63C4×D17—41no—
64C4×D17—41no—
65Dic34—21no—
66Dic34—21no—
67C8×C17—22no—
68#3622—41no—
69#3622—41no—
70#1650—171nomaximal, Sylow
71C22×C17#12011yescharacteristic
72C2×D17—41no—
73C2×D17—41no—
74C2×D17—41no—
75C4×C17—21no—
76C4×C17—42no—
77C4×C17—81no—
78C4×C17#13511yescharacteristic
79C4×C17#13511yescharacteristic
80C4×C17AGL(1,ℤ/8)11yescharacteristic
81C4×C17#13711yescharacteristic
82Dic17—41no—
83Dic17—41no—
84Dic17—41no—
85#556—171no—
86#500D1711yescharacteristic, complemented
87#475—171no—
88#461—171no—
89#332—171no—
90Q8⋊C8—171no—
91D4⋊C8—171no—
92C2×C17#44911yescharacteristic
93C2×C17#45211yescharacteristic
94C2×C17#45611yescharacteristic
95D17—81no—
96C2×SD16—341no—
97C4⋊Q8C2×D1711yescharacteristic
98C4⋊Q8C2×D1711yescharacteristic
99#126—342no—
100#124—341no—
101C22⋊Q8—341no—
102C4×Q8—171no—
103C4×Q8—341no—
104C4×D4—171no—
105#117—341no—
106#107—22no—
107#106—341no—
108C4⋊C8—171no—
109C4⋊C8—171no—
110#103—341no—
111#103—341no—
112C4×C8C2×D1711yescharacteristic
113C17#165011yescharacteristic, complemented, Sylow
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—171no—
117C2×D4—171no—
118C22×C4—341no—
119SD16—682no—
120C2×C8—21no—
121C2×C8—341no—
122C2×C8—341no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—171no—
127C4⋊C4—171no—
128C4⋊C4—342no—
129C4⋊C4—341no—
130C4⋊C4C22⋊D1712yes—
131C22⋊C4—342no—
132C22⋊C4—342no—
133C22⋊C4—341no—
134C42—342no—
135C42—341no—
136C42C22×D1711yescharacteristic
137C23—341no—
138Q8—41no—
139Q8—41no—
140Q8—41no—
141Q8—41no—
142Q8—341no—
143Q8—341no—
144D4—341no—
145D4—341no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—341no—
149C2×C4—341no—
150C2×C4—341no—
151C2×C4—681no—
152C2×C4—681no—
153C2×C4D4×D1711yescharacteristic
154C2×C4D4×D1711yescharacteristic
155C2×C4C2×C22⋊D1711yescharacteristic
156C8—22no—
157C8—681no—
158C8—681no—
159C22—681no—
160C22—681no—
161C22—681no—
162C22#1058762311yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—681no—
167C4—681no—
168C4—681no—
169C4#1058759711yescharacteristic
170C4#1058759811yescharacteristic
171C4#1058760411yescharacteristic
172C4#1058760511yescharacteristic
173C2—1361no—
174C2#1176295111yescharacteristic
175C2#1176297011yescharacteristic
176C2#1176299111yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#500C2
socleC22×C17#120
prime radicalC2×D68D4
hypercenter#500D17
hypocenterC17#1650
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #11763091C2semidirect
4. #11762555C2semidirect
5. #11762541C2semidirect
7. C17×#500C2semidirect
86. #500D17semidirect
113. C17#1650semidirect

Maximal subgroups

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

subgroupindex
#117635332
#117630912
#117630112
#117625552
#117625412
#117625382
C17×#5002
#165017