Finite Group Databasebeta

← #17960044

Subgroups — #17960044

Order 4352.

360 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 64 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#14545153C212yesmaximal, complemented
2#14543795C212yesmaximal, complemented
3#14543793C212yesmaximal, complemented
4#14539241C211yesmaximal, characteristic
5#11763732C2214yes—
6#11763705—22no—
7#11763701—22no—
8C2×#10587628—22no—
9#11763359C411yescharacteristic
10#11763201—22no—
11#11763188C411yescharacteristic
12#11763188C2211yescharacteristic
13#11763183C411yescharacteristic
14#11763183C2211yescharacteristic
15C4×#63223C2211yescharacteristic
16C2×C17×C4⋊Q8C411yescharacteristic
17C22×Q8×C17—21no—
18C2×#63252—21no—
19C17×C4⋊Q8—21no—
20C17×C4⋊Q8—41no—
21C17×C4⋊Q8C2×C411yescharacteristic
22C17×C4⋊Q8C2×C411yescharacteristic
23C2×C4⋊C4×C17—41no—
24C2×C42×C17C2×C411yescharacteristic
25C2×#63231—22no—
26C2×#63231—22no—
27#10587628—48no—
28C2×#63230—22no—
29C2×#63230—22no—
30#10587616—41no—
31C2×#63223D412yes—
32#10587561—21no—
33#10587560C2×C411yescharacteristic
34C4×D68—41no—
35#10587555C2×C411yescharacteristic
36#10587538—24no—
37#10587474D412yes—
38C4×#3622C2×C411yescharacteristic
39C4×#3622C2311yescharacteristic
40C2×Q8×C17—21no—
41C2×Q8×C17—21no—
42C2×Q8×C17—44no—
43C2×Q8×C17—41no—
44C22×C4×C17—21no—
45C22×C4×C17—41no—
46C22×C4×C17C22⋊C411yescharacteristic
47C2×C22⋊D17—41no—
48#63252—44no—
49C2×D68—21no—
50C2×D68—41no—
51C2×C4×D17—41no—
52C2×Dic34—21no—
53C4⋊C4×C17—41no—
54C4⋊C4×C17—41no—
55C4⋊C4×C17—42no—
56C42×C17—21no—
57C42×C17C22⋊C411yescharacteristic
58C42×C17C22×C411yescharacteristic
59#63231—44no—
60#63231—44no—
61#63230—44no—
62#63230—44no—
63#63227—42no—
64C4⋊Dic17—41no—
65#63223—24no—
66C2×#3622C22⋊C412yes—
67C2×#3622C2×D412yes—
68#63216—44no—
69#11993—171nomaximal, Sylow
70C23×C17#10011yescharacteristic
71C22×D17—41no—
72Q8×C17—42no—
73Q8×C17—42no—
74Q8×C17—82no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—42no—
79C2×C4×C17—41no—
80C2×C4×C17—41no—
81C2×C4×C17—82no—
82C2×C4×C17#10011yescharacteristic
83C2×C4×C17C2×C22⋊C411yescharacteristic
84C22⋊D17—82no—
85C2×Dic17—41no—
86D68—42no—
87D68—82no—
88C4×D17—82no—
89Dic34—42no—
90#3622—22no—
91#3622AGL(1,ℤ/8)14yes—
92#3152—171no—
93#2165—172no—
94#1359—172no—
95#1358—172no—
96C22×C17—21no—
97C22×C17—41no—
98C22×C17C2×#10011yescharacteristic
99C2×D17—82no—
100C2×D17—81no—
101C4×C17—22no—
102C4×C17—21no—
103C4×C17—41no—
104C4×C17—81no—
105C4×C17—81no—
106C4×C17#41812yes—
107Dic17—81no—
108C2×#137—342no—
109#532—171no—
110C2×C4⋊Q8#59511yescharacteristic
111#497—342no—
112#494—171no—
113#494—171no—
114#492—171no—
115#492—171no—
116C4×M16—171no—
117#364—174no—
118#349—342no—
119#347—342no—
120C2×C17—21no—
121C2×C17—41no—
122C2×C17#155511yescharacteristic
123D17—161no—
124C2×#41—341no—
125C22×Q8—21no—
126#137—688no—
127C2×Q16—342no—
128C2×Q16—342no—
129C2×SD16—342no—
130C2×SD16—342no—
131C2×M16—172no—
132C4⋊Q8—21no—
133C4⋊Q8—41no—
134C4⋊Q8—171no—
135C4⋊Q8C2×#59511yescharacteristic
136C4⋊Q8C2×#59511yescharacteristic
137#127—171no—
138#124—341no—
139#121—681no—
140C4×D4—681no—
141C2×C4⋊C4—41no—
142C2×C42C2×#59511yescharacteristic
143M32—344no—
144#97—172no—
145C4×C8—171no—
146C4×C8—171no—
147C17#1199311yescharacteristic, complemented, Sylow
148#41—684no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—44no—
152C2×Q8—41no—
153C2×Q8—341no—
154C2×D4—341no—
155C2×D4—681no—
156C2×D4—681no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—681no—
160C22×C4C22⋊#59511yescharacteristic
161Q16—684no—
162Q16—684no—
163SD16—684no—
164SD16—684no—
165M16—344no—
166C2×C8—172no—
167C2×C8—172no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—681no—
172C22⋊C4—682no—
173C42—21no—
174C42C22⋊#59511yescharacteristic
175C42C22×#59511yescharacteristic
176C16—684no—
177C23—681no—
178C23#1058755311yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—82no—
182Q8—682no—
183D4—682no—
184D4—1362no—
185D4—1362no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—681no—
194C2×C4—1362no—
195C2×C4#1058755311yescharacteristic
196C2×C4C2×C22⋊#59511yescharacteristic
197C8—174no—
198C8—342no—
199C22—21no—
200C22—41no—
201C22—1362no—
202C22—1361no—
203C22C2×#1058755311yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—1361no—
210C4#1176380112yes—
211C2—21no—
212C2—41no—
213C2—2721no—
214C2#1454431111yescharacteristic
subgroupG/H
centerC2#14544311
commutatorC42×C17C22×C4
FrattiniC42C22×#595
FittingC2×C17×C4⋊Q8C4
socleC2×C17#1555
prime radical#11763359C4
cosocleC4×#3622C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11993
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #14545153C2semidirect
2. #14543795C2semidirect
3. #14543793C2semidirect
147. C17#11993semidirect

Maximal subgroups

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

subgroupindex
#145451532
#145437952
#145437932
#145392412
#1199317