Finite Group Databasebeta

← #17023035

Subgroups — #17023035

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#14539912C212yesmaximal
2#14539732C212yesmaximal, complemented
3#14539107C211yesmaximal, characteristic
4#14528442C214yesmaximal, complemented
5#14528416C214yesmaximal, complemented
6#14527076C211yesmaximal, characteristic, complemented
7C2×C17×#498C211yesmaximal, characteristic, complemented
8#11763366C2211yescharacteristic
9#11763359C2211yescharacteristic
10#11763205C2214yes—
11#11763188C2214yes—
12#11763164—22no—
13#11763160C2212yes—
14#11763159C2214yes—
15#11763157C2214yes—
16#11763156C2212yes—
17#11763127—24no—
18#11763096—24no—
19#11763016—24no—
20#11762965—24no—
21#11762862C2212yes—
22#11762806—22no—
23#11762531C2214yes—
24C2×C17×C4⋊Q8C2212yes—
25C17×#498—22no—
26C17×#498C2214yes—
27C2×C17×#106—22no—
28C2×C4×C8×C17C2211yescharacteristic
29C22×Q8×C17—22no—
30C2×#63252—21no—
31C22×C8×C17D412yes—
32C17×C4⋊Q8—22no—
33C17×C4⋊Q8—22no—
34C17×C4⋊Q8—22no—
35C17×C4⋊Q8C2314yes—
36C2×C4⋊C4×C17—22no—
37C2×C4⋊C4×C17—22no—
38C2×C42×C17C2311yescharacteristic
39C2×#63231—28no—
40#10587616—22no—
41#10587612—21no—
42#10587610—22no—
43#10587561—21no—
44#10587560C2311yescharacteristic
45C4×D68—22no—
46#10587556—21no—
47#10587555C2311yescharacteristic
48#10587555C2312yes—
49C4×Dic34—22no—
50C17×#106—28no—
51C4×C8×C17D412yes—
52C4×C8×C17C2312yes—
53#10587506—28no—
54C22⋊#3622D414yes—
55#10587492—24no—
56#10587487—24no—
57#10587480—28no—
58#10587479—28no—
59C4⋊#3622D414yes—
60C4×#3622C2314yes—
61C2×Q8×C17—24no—
62C2×Q8×C17—42no—
63C2×Q8×C17—84no—
64C22×C4×C17—42no—
65C22×C4×C17C2×D412yes—
66C22×C4×C17C2×D411yescharacteristic
67C2×C22⋊D17—41no—
68#63252—44no—
69C2×D68—21no—
70C2×D68—22no—
71C2×C4×D17—41no—
72C2×Dic34—21no—
73C2×Dic34—22no—
74C2×C8×C17—24no—
75C2×C8×C17C2×D414yes—
76C4⋊C4×C17—24no—
77C4⋊C4×C17—24no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—42no—
80C42×C17C2×D412yes—
81C42×C17C2×D411yescharacteristic
82C42×C17C2411yescharacteristic
83C22⋊Dic17—41no—
84#63231—88no—
85#63227—42no—
86C4⋊Dic17—22no—
87C4⋊Dic17—22no—
88C4⋊Dic17—22no—
89#63225—42no—
90C4×Dic17—41no—
91C2×#3622C2×D418yes—
92#19703—171nomaximal, Sylow
93C23×C17#12711yescharacteristic
94C22×D17—41no—
95Q8×C17—48no—
96Q8×C17—84no—
97C2×C4×C17—24no—
98C2×C4×C17—44no—
99C2×C4×C17—84no—
100C2×C4×C17#12712yes—
101C2×C4×C17#12711yescharacteristic
102C2×C4×C17#13714yes—
103C2×C4×C17C22×D412yes—
104C2×C4×C17C22×D411yescharacteristic
105C22⋊D17—82no—
106C2×Dic17—42no—
107C2×Dic17—41no—
108D68—42no—
109D68—44no—
110C4×D17—82no—
111Dic34—42no—
112Dic34—44no—
113C8×C17—24no—
114#3622—28no—
115#3227—172no—
116#3209—172no—
117C2×#498D1711yescharacteristic, complemented
118#3131—171no—
119#1720—174no—
120#1716—174no—
121#1452—171no—
122C22×C17—22no—
123C22×C17#49712yes—
124C22×C17C2×#12711yescharacteristic
125C2×D17—81no—
126C2×D17—82no—
127C4×C17—22no—
128C4×C17—84no—
129C4×C17#49712yes—
130C4×C17C2×#13712yes—
131C4×C17#57714yes—
132Dic17—81no—
133Dic17—82no—
134#533—171no—
135#532—171no—
136C2×C4⋊Q8C2×D1712yes—
137#498—22no—
138#498—174no—
139#498C2×D1714yes—
140#494—174no—
141#488—342no—
142#487—174no—
143#487—172no—
144#486—174no—
145#486—172no—
146#481—344no—
147#476—344no—
148#462—344no—
149#451—344no—
150#431—172no—
151C2×#106—22no—
152#416—342no—
153C2×C4×C8C2×D1711yescharacteristic
154#322—174no—
155C2×C17—22no—
156C2×C17C2×#49711yescharacteristic
157C2×C17#315412yes—
158D17—161no—
159C2×#41—341no—
160C22×Q8—22no—
161C2×Q16—348no—
162C22×C8C22⋊D1712yes—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—171no—
167C4⋊Q8—172no—
168C4⋊Q8C22×D1714yes—
169#127—171no—
170#125—341no—
171#124—341no—
172C22⋊Q8—342no—
173#121—342no—
174C4×Q8—342no—
175C4×D4—342no—
176#117—341no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C42C22×D1711yescharacteristic
180#106—28no—
181#106—348no—
182C4⋊C8—174no—
183#103—348no—
184#103—344no—
185#102—348no—
186#102—344no—
187C22⋊C8—174no—
188C4×C8—174no—
189C4×C8C22⋊D1712yes—
190C4×C8C22×D1712yes—
191C17#1970311yescharacteristic, complemented, Sylow
192#41—684no—
193C2×Q8—24no—
194C2×Q8—42no—
195C2×Q8—84no—
196C2×Q8—342no—
197C2×Q8—341no—
198C2×D4—342no—
199C2×D4—341no—
200C2×D4—681no—
201C22×C4—42no—
202C22×C4—681no—
203C22×C4D4×D1712yes—
204C22×C4C2×C22⋊D1711yescharacteristic
205Q16—1368no—
206C2×C8—24no—
207C2×C8—178no—
208C2×C8C2×C22⋊D1714yes—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—342no—
214C4⋊C4—342no—
215C4⋊C4—342no—
216C4⋊C4—682no—
217C22⋊C4—681no—
218C22⋊C4—682no—
219C42—681no—
220C42D4×D1712yes—
221C42C2×C22⋊D1711yescharacteristic
222C42C23×D1711yescharacteristic
223C23—681no—
224C23#1058762611yescharacteristic
225Q8—48no—
226Q8—84no—
227Q8—682no—
228Q8—684no—
229D4—682no—
230D4—684no—
231D4—1362no—
232C2×C4—24no—
233C2×C4—44no—
234C2×C4—84no—
235C2×C4—682no—
236C2×C4—681no—
237C2×C4—1362no—
238C2×C4#1058762612yes—
239C2×C4#1058762611yescharacteristic
240C2×C4#1058762814yes—
241C2×C4C2×D4×D1712yes—
242C2×C4C22×C22⋊D1711yescharacteristic
243C8—24no—
244C8—348no—
245C22—22no—
246C22—1361no—
247C22—1362no—
248C22#1176320112yes—
249C22C2×#1058762611yescharacteristic
250C4—22no—
251C4—84no—
252C4—1361no—
253C4—1362no—
254C4#1176320112yes—
255C4C2×#1058762812yes—
256C4#1176361514yes—
257C2—22no—
258C2—2721no—
259C2C2×#1176320111yescharacteristic
260C2#1453925412yes—
subgroupG/H
centerC22C2×#10587626
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#498C2
socleC22×C17C2×#127
prime radical#11763359C22
hypercenterC2×#498D17
hypocenterC17#19703
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
2. #14539732C2semidirect
4. #14528442C2semidirect
5. #14528416C2semidirect
6. #14527076C2semidirect
7. C2×C17×#498C2semidirect
117. C2×#498D17semidirect
191. C17#19703semidirect

Maximal subgroups

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

subgroupindex
#145399122
#145397322
#145391072
#145284422
#145284162
#145270762
C2×C17×#4982
#1970317