Finite Group Databasebeta

← #16945860

Subgroups — #16945860

Order 4352.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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#14530596C212yesmaximal, complemented
2#14526825C212yesmaximal, complemented
3#14526823C212yesmaximal, complemented
4C17×#3152C211yesmaximal, characteristic
5#11762630C2214yes—
6#11762591—22no—
7#11762587—22no—
8C2×C17×#137—22no—
9C17×#532C411yescharacteristic
10C2×C17×C4⋊Q8C411yescharacteristic
11C17×#497—22no—
12C17×#494C411yescharacteristic
13C17×#494C2211yescharacteristic
14C17×#492C411yescharacteristic
15C17×#492C2211yescharacteristic
16C4×M16×C17C2211yescharacteristic
17C2×#41×C17—21no—
18C22×Q8×C17—21no—
19C17×#137—48no—
20C2×Q16×C17—22no—
21C2×Q16×C17—22no—
22C2×SD16×C17—22no—
23C2×SD16×C17—22no—
24C2×M16×C17D412yes—
25C17×C4⋊Q8—21no—
26C17×C4⋊Q8—41no—
27C17×C4⋊Q8C2×C411yescharacteristic
28C17×C4⋊Q8C2×C411yescharacteristic
29C17×C4⋊Q8C2×C411yescharacteristic
30C17×#127C2×C411yescharacteristic
31C17×#124—21no—
32C17×#121—41no—
33C4×D4×C17—41no—
34C2×C4⋊C4×C17—41no—
35C2×C42×C17C2×C411yescharacteristic
36C17×#97D412yes—
37C4×C8×C17C2×C411yescharacteristic
38C4×C8×C17C2311yescharacteristic
39#10587483—24no—
40#41×C17—44no—
41C2×Q8×C17—21no—
42C2×Q8×C17—21no—
43C2×Q8×C17—21no—
44C2×Q8×C17—44no—
45C2×Q8×C17—41no—
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C22×C4×C17—21no—
50C22×C4×C17—41no—
51C22×C4×C17—41no—
52C22×C4×C17C22⋊C411yescharacteristic
53Q16×C17—44no—
54Q16×C17—44no—
55SD16×C17—44no—
56SD16×C17—44no—
57M16×C17—24no—
58C2×C8×C17C22⋊C412yes—
59C2×C8×C17C2×D412yes—
60C4⋊C4×C17—41no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—42no—
64C22⋊C4×C17—42no—
65C42×C17—21no—
66C42×C17C22⋊C411yescharacteristic
67C42×C17C22×C411yescharacteristic
68#63214—44no—
69#11993—171nomaximal, Sylow
70C23×C17—41no—
71C23×C17#10011yescharacteristic
72Q8×C17—42no—
73Q8×C17—42no—
74Q8×C17—42no—
75Q8×C17—82no—
76D4×C17—42no—
77D4×C17—82no—
78D4×C17—82no—
79C2×C4×C17—21no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—42no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
85C2×C4×C17—41no—
86C2×C4×C17—82no—
87C2×C4×C17—82no—
88C2×C4×C17#10011yescharacteristic
89C2×C4×C17C2×C22⋊C411yescharacteristic
90C8×C17—22no—
91C8×C17AGL(1,ℤ/8)14yes—
92#3152D1711yescharacteristic
93#2165—172no—
94#1359—172no—
95#1358—172no—
96C22×C17—21no—
97C22×C17—41no—
98C22×C17—82no—
99C22×C17—81no—
100C22×C17C2×#10011yescharacteristic
101C4×C17—22no—
102C4×C17—21no—
103C4×C17—41no—
104C4×C17—81no—
105C4×C17—81no—
106C4×C17—81no—
107C4×C17#41812yes—
108C2×#137—22no—
109#532Dic1711yescharacteristic
110C2×C4⋊Q8Dic1711yescharacteristic
111#497—22no—
112#494Dic1711yescharacteristic
113#494C2×D1711yescharacteristic
114#492Dic1711yescharacteristic
115#492C2×D1711yescharacteristic
116C4×M16C2×D1711yescharacteristic
117#364—174no—
118#349—342no—
119#347—342no—
120C2×C17—21no—
121C2×C17—41no—
122C2×C17—161no—
123C2×C17#155511yescharacteristic
124C2×#41—21no—
125C22×Q8—21no—
126#137—48no—
127C2×Q16—22no—
128C2×Q16—22no—
129C2×SD16—22no—
130C2×SD16—22no—
131C2×M16C22⋊D1712yes—
132C4⋊Q8—21no—
133C4⋊Q8—41no—
134C4⋊Q8C2×Dic1711yescharacteristic
135C4⋊Q8C2×Dic1711yescharacteristic
136C4⋊Q8C2×Dic1711yescharacteristic
137#127C2×Dic1711yescharacteristic
138#124—21no—
139#121—41no—
140C4×D4—41no—
141C2×C4⋊C4—41no—
142C2×C42C2×Dic1711yescharacteristic
143M32—344no—
144#97C22⋊D1712yes—
145C4×C8C2×Dic1711yescharacteristic
146C4×C8C22×D1711yescharacteristic
147C17#1199311yescharacteristic, complemented, Sylow
148#41—44no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—44no—
153C2×Q8—41no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C22⋊Dic1711yescharacteristic
161Q16—44no—
162Q16—44no—
163SD16—44no—
164SD16—44no—
165M16—24no—
166C2×C8C22⋊Dic1712yes—
167C2×C8C2×C22⋊D1712yes—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C22⋊C4—42no—
173C42—21no—
174C42C22⋊Dic1711yescharacteristic
175C42C22×Dic1711yescharacteristic
176C16—684no—
177C23—41no—
178C23#1058750411yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—42no—
182Q8—82no—
183D4—42no—
184D4—82no—
185D4—82no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4#1058750411yescharacteristic
196C2×C4C2×C22⋊Dic1711yescharacteristic
197C8—22no—
198C8#1058761814yes—
199C22—21no—
200C22—41no—
201C22—82no—
202C22—81no—
203C22C2×#1058750411yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—81no—
210C4#1176281412yes—
211C2—21no—
212C2—41no—
213C2—161no—
214C2#1452749811yescharacteristic
subgroupG/H
centerC2#14527498
commutatorC42×C17C22×C4
FrattiniC4×C8C22×D17
FittingC17×#3152C2
socleC2×C17#1555
prime radicalC17×#532C4
cosocleC4×C8×C17C23
hypercenter#3152D17
hypocenterC17#11993
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. #14530596C2semidirect
2. #14526825C2semidirect
3. #14526823C2semidirect
147. C17#11993semidirect

Maximal subgroups

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

subgroupindex
#145305962
#145268252
#145268232
C17×#31522
#1199317