Finite Group Databasebeta

← #17023040

Subgroups — #17023040

Order 4352.

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

normal subgroups only

This group has 704 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#14539909C212yesmaximal
2#14539729C212yesmaximal, complemented
3C2×#11763207C211yesmaximal, characteristic, complemented
4#14528446C214yesmaximal, complemented
5#14528419C214yesmaximal, complemented
6#14527076C211yesmaximal, characteristic, complemented
7C17×#3131C211yesmaximal, characteristic
8C2×#10587555C2212yes—
9#11763207—22no—
10#11763207C2214yes—
11#11763205C2214yes—
12#11763188C2214yes—
13#11763161C2214yes—
14#11763155C2214yes—
15#11763125—24no—
16#11763099—24no—
17#11763019—24no—
18#11762964—24no—
19#11762862C2212yes—
20C2×#10587488—22no—
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⋊Q8C2311yescharacteristic
33C17×C4⋊Q8C2312yes—
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×#63231—28no—
44C2×C4⋊Dic17—22no—
45C2×C4⋊Dic17—22no—
46#10587555—22no—
47#10587555—22no—
48#10587555—22no—
49#10587555C2314yes—
50C17×#103—24no—
51C17×#102—24no—
52C4×C8×C17D412yes—
53C4×C8×C17C2312yes—
54#10587503—28no—
55C22⋊#3622D414yes—
56#10587488—28no—
57#10587481—28no—
58#10587479—28no—
59C4⋊#3622D414yes—
60C4×#3622C2314yes—
61#41×C17—44no—
62C2×Q8×C17—22no—
63C2×Q8×C17—21no—
64C2×D4×C17—21no—
65C2×D4×C17—22no—
66C2×D4×C17—41no—
67C22×C4×C17—41no—
68C22×C4×C17C2×D412yes—
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—42no—
71C2×Dic34—24no—
72C2×Dic34—42no—
73C2×Dic34—84no—
74C2×C8×C17—24no—
75C2×C8×C17C2×D414yes—
76C4⋊C4×C17—22no—
77C4⋊C4×C17—22no—
78C4⋊C4×C17—22no—
79C4⋊C4×C17—42no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—42no—
82C42×C17—41no—
83C42×C17C2×D412yes—
84C42×C17C2×D411yescharacteristic
85C42×C17C2411yescharacteristic
86#63231—88no—
87C4⋊Dic17—24no—
88C4⋊Dic17—24no—
89C4⋊Dic17—42no—
90C4⋊Dic17—42no—
91C2×#3622C2×D418yes—
92#19703—171nomaximal, Sylow
93C23×C17—41no—
94C23×C17#12711yescharacteristic
95Q8×C17—42no—
96Q8×C17—44no—
97D4×C17—42no—
98D4×C17—44no—
99D4×C17—82no—
100C2×C4×C17—24no—
101C2×C4×C17—42no—
102C2×C4×C17—41no—
103C2×C4×C17—82no—
104C2×C4×C17#12712yes—
105C2×C4×C17#12711yescharacteristic
106C2×C4×C17#13714yes—
107C2×C4×C17C22×D412yes—
108C2×C4×C17C22×D411yescharacteristic
109C2×Dic17—44no—
110C2×Dic17—84no—
111Dic34—48no—
112Dic34—84no—
113C8×C17—24no—
114#3622—28no—
115#3227—172no—
116#3209—172no—
117C2×#498—171no—
118#3131D1711yescharacteristic
119#1720—174no—
120#1716—174no—
121#1452—171no—
122C22×C17—22no—
123C22×C17—81no—
124C22×C17—82no—
125C22×C17#49712yes—
126C22×C17C2×#12711yescharacteristic
127C4×C17—22no—
128C4×C17—81no—
129C4×C17—82no—
130C4×C17#49712yes—
131C4×C17C2×#13712yes—
132C4×C17#57714yes—
133Dic17—84no—
134#533C2×D1711yescharacteristic
135#532C2×D1711yescharacteristic
136C2×C4⋊Q8—172no—
137#498—174no—
138#498—174no—
139#498—342no—
140#494—174no—
141#488—22no—
142#487—174no—
143#487C2×D1712yes—
144#486—174no—
145#486C2×D1712yes—
146#481—344no—
147#476—344no—
148#462—344no—
149#451—344no—
150#431—172no—
151C2×#106—342no—
152#416—22no—
153C2×C4×C8C2×D1711yescharacteristic
154#322—174no—
155C2×C17—22no—
156C2×C17—161no—
157C2×C17C2×#49711yescharacteristic
158C2×C17#315412yes—
159C2×#41—21no—
160C22×Q8—342no—
161C2×Q16—348no—
162C22×C8C22⋊D1712yes—
163C4⋊Q8—174no—
164C4⋊Q8—342no—
165C4⋊Q8—342no—
166C4⋊Q8—342no—
167C4⋊Q8C22×D1711yescharacteristic
168C4⋊Q8C22×D1712yes—
169#127C22×D1711yescharacteristic
170#125—21no—
171#124—21no—
172C22⋊Q8—22no—
173#121—22no—
174C4×Q8—22no—
175C4×D4—22no—
176#117—21no—
177C2×C4⋊C4—342no—
178C2×C4⋊C4—342no—
179C2×C42C22×D1711yescharacteristic
180#106—348no—
181#106—348no—
182C4⋊C8—174no—
183#103—24no—
184#103—348no—
185#102—24no—
186#102—348no—
187C22⋊C8—174no—
188C4×C8—174no—
189C4×C8C22⋊D1712yes—
190C4×C8C22×D1712yes—
191C17#1970311yescharacteristic, complemented, Sylow
192#41—44no—
193C2×Q8—22no—
194C2×Q8—21no—
195C2×Q8—344no—
196C2×Q8—682no—
197C2×Q8—1364no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—41no—
201C22×C4—41no—
202C22×C4—682no—
203C22×C4D4×D1712yes—
204C22×C4C2×C22⋊D1711yescharacteristic
205Q16—1368no—
206C2×C8—24no—
207C2×C8—178no—
208C2×C8C2×C22⋊D1714yes—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—42no—
213C4⋊C4—344no—
214C4⋊C4—344no—
215C4⋊C4—682no—
216C4⋊C4—682no—
217C22⋊C4—41no—
218C22⋊C4—42no—
219C42—41no—
220C42D4×D1712yes—
221C42C2×C22⋊D1711yescharacteristic
222C42C23×D1711yescharacteristic
223C23—41no—
224C23#1058762611yescharacteristic
225Q8—42no—
226Q8—44no—
227Q8—688no—
228Q8—1364no—
229D4—42no—
230D4—44no—
231D4—82no—
232C2×C4—24no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—82no—
236C2×C4—684no—
237C2×C4—1364no—
238C2×C4#1058762612yes—
239C2×C4#1058762611yescharacteristic
240C2×C4#1058763714yes—
241C2×C4C2×D4×D1712yes—
242C2×C4C22×C22⋊D1711yescharacteristic
243C8—24no—
244C8—348no—
245C22—22no—
246C22—81no—
247C22—82no—
248C22#1176320412yes—
249C22C2×#1058762611yescharacteristic
250C4—22no—
251C4—81no—
252C4—82no—
253C4—1364no—
254C4#1176320412yes—
255C4C2×#1058763712yes—
256C4#1176361614yes—
257C2—22no—
258C2—161no—
259C2C2×#1176320411yescharacteristic
260C2#1453926112yes—
subgroupG/H
centerC22C2×#10587626
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3131C2
socleC22×C17C2×#127
prime radicalC17×#532C22
hypercenter#3131D17
hypocenterC17#19703
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
2. #14539729C2semidirect
3. C2×#11763207C2semidirect
4. #14528446C2semidirect
5. #14528419C2semidirect
6. #14527076C2semidirect
191. C17#19703semidirect

Maximal subgroups

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

subgroupindex
#145399092
#145397292
C2×#117632072
#145284462
#145284192
#145270762
C17×#31312
#1970317