Finite Group Databasebeta

← #17023004

Subgroups — #17023004

Order 4352.

744 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 179 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#14539925C212yesmaximal, complemented
2#14539902C212yesmaximal
3C2×#11763189C211yesmaximal, characteristic, complemented
4#14528463C214yesmaximal, complemented
5#14528443C214yesmaximal, complemented
6#14527076C211yesmaximal, characteristic, complemented
7C17×#3131C211yesmaximal, characteristic
8C22×Dic68—22no—
9C2×#10587555C2212yes—
10#11763212C2214yes—
11#11763205C2214yes—
12#11763189—22no—
13#11763189C2214yes—
14#11763159C2214yes—
15#11763155C2214yes—
16#11763137—24no—
17#11763120—24no—
18#11763094—24no—
19#11763089—24no—
20#11762862C2212yes—
21#11762531C2214yes—
22C17×#533C2211yescharacteristic
23C17×#532C2211yescharacteristic
24C17×#488—22no—
25C17×#487C2212yes—
26C17×#486C2212yes—
27C17×#416—22no—
28C2×C4×C8×C17C2211yescharacteristic
29C2×#41×C17—21no—
30C22×Dic34—22no—
31C22×C8×C17D412yes—
32C17×C4⋊Q8C2312yes—
33C17×C4⋊Q8C2311yescharacteristic
34C17×#127C2311yescharacteristic
35C17×#125—21no—
36C17×#124—21no—
37C17×C22⋊Q8—22no—
38C17×#121—22no—
39C4×Q8×C17—22no—
40C4×D4×C17—22no—
41C17×#117—21no—
42C2×C42×C17C2311yescharacteristic
43C2×C4⋊Dic17—22no—
44C2×C4⋊Dic17—22no—
45C2×Dic68—28no—
46C2×Dic68—48no—
47#10587555—22no—
48#10587555—22no—
49#10587555—22no—
50#10587555C2314yes—
51C17×#103—24no—
52C17×#102—24no—
53C4×C8×C17D412yes—
54C4×C8×C17C2312yes—
55#10587506—28no—
56#10587503—28no—
57C22⋊#3622D414yes—
58#10587479—28no—
59#10587478—28no—
60C4⋊#3622D414yes—
61C4×#3622C2314yes—
62#41×C17—44no—
63C2×Q8×C17—22no—
64C2×Q8×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17—22no—
67C2×D4×C17—41no—
68C22×C4×C17—41no—
69C22×C4×C17C2×D412yes—
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—42no—
72C2×Dic34—24no—
73C2×Dic34—48no—
74C2×Dic34—42no—
75C2×C8×C17—24no—
76C2×C8×C17C2×D414yes—
77C4⋊C4×C17—22no—
78C4⋊C4×C17—22no—
79C4⋊C4×C17—22no—
80C4⋊C4×C17—42no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—42no—
83C42×C17—41no—
84C42×C17C2×D412yes—
85C42×C17C2×D411yescharacteristic
86C42×C17C2411yescharacteristic
87C4⋊Dic17—24no—
88C4⋊Dic17—24no—
89C4⋊Dic17—42no—
90C4⋊Dic17—42no—
91C2×#3622C2×D418yes—
92Dic68—416no—
93#19699—171nomaximal, Sylow
94C23×C17—41no—
95C23×C17#12711yescharacteristic
96Q8×C17—42no—
97Q8×C17—44no—
98D4×C17—42no—
99D4×C17—44no—
100D4×C17—82no—
101C2×C4×C17—24no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17—82no—
105C2×C4×C17#12712yes—
106C2×C4×C17#12711yescharacteristic
107C2×C4×C17AGL(1,ℤ/8)14yes—
108C2×C4×C17C22×D412yes—
109C2×C4×C17C22×D411yescharacteristic
110C2×Dic17—44no—
111C2×Dic17—84no—
112Dic34—48no—
113Dic34—84no—
114C8×C17—24no—
115#3622—28no—
116#3229—172no—
117#3226—172no—
118C2×#494—171no—
119#3131D1711yescharacteristic
120#1723—174no—
121#1720—174no—
122#1452—171no—
123C22×C17—22no—
124C22×C17—81no—
125C22×C17—82no—
126C22×C17#49612yes—
127C22×C17C2×#12711yescharacteristic
128C4×C17—22no—
129C4×C17—81no—
130C4×C17—82no—
131C4×C17#49612yes—
132C4×C17C2×AGL(1,ℤ/8)12yes—
133C4×C17#57814yes—
134Dic17—84no—
135C22×Q16—342no—
136#533C2×D1711yescharacteristic
137#532C2×D1711yescharacteristic
138C2×C4⋊Q8—172no—
139#500—174no—
140#498—174no—
141#494—174no—
142#494—342no—
143#488—22no—
144#487—174no—
145#487C2×D1712yes—
146#486—174no—
147#486C2×D1712yes—
148#483—344no—
149#480—344no—
150#475—344no—
151#474—344no—
152#431—172no—
153#416—22no—
154C2×C4×C8C2×D1711yescharacteristic
155#322—174no—
156C2×C17—22no—
157C2×C17—161no—
158C2×C17C2×#49611yescharacteristic
159C2×C17#315612yes—
160C2×#41—21no—
161C22×Q8—342no—
162C2×Q16—348no—
163C2×Q16—688no—
164C22×C8C22⋊D1712yes—
165C4⋊Q8—174no—
166C4⋊Q8—342no—
167C4⋊Q8—342no—
168C4⋊Q8—342no—
169C4⋊Q8C22×D1711yescharacteristic
170C4⋊Q8C22×D1712yes—
171#127C22×D1711yescharacteristic
172#125—21no—
173#124—21no—
174C22⋊Q8—22no—
175#121—22no—
176C4×Q8—22no—
177C4×D4—22no—
178#117—21no—
179C2×C4⋊C4—342no—
180C2×C4⋊C4—342no—
181C2×C42C22×D1711yescharacteristic
182#107—348no—
183#106—348no—
184C4⋊C8—174no—
185#103—24no—
186#103—348no—
187#102—24no—
188#102—348no—
189C22⋊C8—174no—
190C4×C8—174no—
191C4×C8C22⋊D1712yes—
192C4×C8C22×D1712yes—
193C17#1969911yescharacteristic, complemented, Sylow
194#41—44no—
195C2×Q8—22no—
196C2×Q8—21no—
197C2×Q8—344no—
198C2×Q8—682no—
199C2×Q8—688no—
200C2×D4—22no—
201C2×D4—21no—
202C2×D4—41no—
203C22×C4—41no—
204C22×C4—682no—
205C22×C4D4×D1712yes—
206C22×C4C2×C22⋊D1711yescharacteristic
207Q16—6816no—
208C2×C8—24no—
209C2×C8—178no—
210C2×C8C2×C22⋊D1714yes—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—344no—
216C4⋊C4—344no—
217C4⋊C4—682no—
218C4⋊C4—682no—
219C22⋊C4—41no—
220C22⋊C4—42no—
221C42—41no—
222C42D4×D1712yes—
223C42C2×C22⋊D1711yescharacteristic
224C42C23×D1711yescharacteristic
225C23—41no—
226C23#1058762611yescharacteristic
227Q8—42no—
228Q8—44no—
229Q8—688no—
230Q8—1364no—
231D4—42no—
232D4—44no—
233D4—82no—
234C2×C4—24no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—82no—
238C2×C4—684no—
239C2×C4—1364no—
240C2×C4#1058762612yes—
241C2×C4#1058762611yescharacteristic
242C2×C4#1058763514yes—
243C2×C4C2×D4×D1712yes—
244C2×C4C22×C22⋊D1711yescharacteristic
245C8—24no—
246C8—348no—
247C22—22no—
248C22—82no—
249C22—81no—
250C22#1176319912yes—
251C22C2×#1058762611yescharacteristic
252C4—22no—
253C4—82no—
254C4—81no—
255C4—1364no—
256C4#1176319912yes—
257C4C2×#1058763512yes—
258C4#1176362314yes—
259C2—22no—
260C2—161no—
261C2C2×#1176319911yescharacteristic
262C2#1453927912yes—
subgroupG/H
centerC22C2×#10587626
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3131C2
socleC22×C17C2×#127
prime radicalC17×#532C22
hypercenter#3131D17
hypocenterC17#19699
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #14539925C2semidirect
3. C2×#11763189C2semidirect
4. #14528463C2semidirect
5. #14528443C2semidirect
6. #14527076C2semidirect
193. C17#19699semidirect

Maximal subgroups

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

subgroupindex
#145399252
#145399022
C2×#117631892
#145284632
#145284432
#145270762
C17×#31312
#1969917