Finite Group Databasebeta

← #17010625

Subgroups — #17010625

Order 4352.

918 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 195 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#14539164C211yesmaximal, characteristic
2#14538914C212yesmaximal
3#14528005C218yesmaximal, complemented
4#14527510C211yesmaximal, characteristic
5#14527231C212yesmaximal
6C2×Q82×C17C211yesmaximal, characteristic
7#11763366C2211yescharacteristic
8#11763177—22no—
9#11763159C2214yes—
10#11763139—22no—
11#11763094C2214yes—
12#11763019C2214yes—
13#11762965—22no—
14#11762914—24no—
15#11762866C2212yes—
16#11762818—22no—
17C4×#63223C2211yescharacteristic
18#11762558C2214yes—
19Q8⋊#3622C2218yes—
20Q82×C17—24no—
21Q82×C17—22no—
22Q82×C17C2214yes—
23C2×C17×C4⋊Q8—22no—
24C2×C17×C4⋊Q8—22no—
25C2×C17×C4⋊Q8C2211yescharacteristic
26C2×C4×Q8×C17—22no—
27C2×C4×Q8×C17C2212yes—
28C22×Q8×C17—22no—
29C22×Q8×C17D412yes—
30C22×Q8×C17D412yes—
31C17×C4⋊Q8—28no—
32C17×C4⋊Q8—28no—
33C17×C4⋊Q8—28no—
34C17×C4⋊Q8—28no—
35C17×C4⋊Q8—22no—
36C17×C4⋊Q8D412yes—
37C17×C4⋊Q8C2312yes—
38C4×Q8×C17—28no—
39C4×Q8×C17—28no—
40C4×Q8×C17—22no—
41C4×Q8×C17—22no—
42C4×Q8×C17D414yes—
43C4×Q8×C17C2314yes—
44C2×C4⋊C4×C17—24no—
45C2×C4⋊C4×C17—22no—
46C2×C4⋊C4×C17—22no—
47C2×C4⋊C4×C17D412yes—
48C2×C42×C17—22no—
49C2×C42×C17—22no—
50C2×C42×C17C2311yescharacteristic
51C2×#63231—24no—
52#10587612—21no—
53#10587610—22no—
54C2×#63223—21no—
55#10587556—21no—
56#10587555C2312yes—
57C4×Dic34—22no—
58#10587506—24no—
59#10587506—28no—
60C22⋊#3622—22no—
61#10587481—24no—
62#10587479—24no—
63C4⋊#3622C2314yes—
64#10587474—21no—
65#10587474—22no—
66C4×#3622C2312yes—
67C2×Q8×C17—28no—
68C2×Q8×C17—28no—
69C2×Q8×C17—22no—
70C2×Q8×C17—22no—
71C2×Q8×C17—44no—
72C2×Q8×C17—84no—
73C2×Q8×C17C2×D414yes—
74C2×Q8×C17C2×D414yes—
75C22×C4×C17—22no—
76C22×C4×C17—22no—
77C22×C4×C17—22no—
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D412yes—
80C2×Dic34—22no—
81C4⋊C4×C17—216no—
82C4⋊C4×C17—24no—
83C4⋊C4×C17—22no—
84C4⋊C4×C17—24no—
85C4⋊C4×C17—24no—
86C4⋊C4×C17—24no—
87C4⋊C4×C17C2×D414yes—
88C42×C17—28no—
89C42×C17—24no—
90C42×C17—24no—
91C42×C17C2×D412yes—
92C42×C17C2×D411yescharacteristic
93C42×C17C2411yescharacteristic
94C22⋊Dic17—41no—
95#63231—48no—
96C4⋊Dic17—22no—
97C4⋊Dic17—22no—
98#63225—42no—
99C4×Dic17—41no—
100#63223—42no—
101C2×#3622—22no—
102C2×#3622—24no—
103#18551—171nomaximal, Sylow
104C23×C17#12011yescharacteristic
105Q8×C17—28no—
106Q8×C17—24no—
107Q8×C17—44no—
108Q8×C17—82no—
109Q8×C17—88no—
110Q8×C17#13718yes—
111C2×C4×C17—28no—
112C2×C4×C17—24no—
113C2×C4×C17—22no—
114C2×C4×C17—24no—
115C2×C4×C17—24no—
116C2×C4×C17—24no—
117C2×C4×C17—44no—
118C2×C4×C17—44no—
119C2×C4×C17#12012yes—
120C2×C4×C17#12011yescharacteristic
121C2×C4×C17C22×D411yescharacteristic
122C2×C4×C17C22×D412yes—
123C2×Dic17—42no—
124Dic34—44no—
125#3622—42no—
126#3622—44no—
127C2×Q82D1711yescharacteristic
128#3139—171no—
129#3113—172no—
130#1645—178no—
131#1558—171no—
132#1491—172no—
133C22×C17—22no—
134C22×C17#45512yes—
135C22×C17C2×#12011yescharacteristic
136C4×C17—28no—
137C4×C17—22no—
138C4×C17—21no—
139C4×C17—21no—
140C4×C17—44no—
141C4×C17—44no—
142C4×C17C2×#13714yes—
143Dic17—82no—
144Q82—22no—
145Q82—24no—
146Q82C2×D1714yes—
147#533—171no—
148C2×C4⋊Q8—22no—
149C2×C4⋊Q8—22no—
150C2×C4⋊Q8C2×D1711yescharacteristic
151C2×C4×Q8—22no—
152C2×C4×Q8C2×D1712yes—
153#491—342no—
154#487—174no—
155#483—342no—
156#475—174no—
157#462—174no—
158#451—342no—
159#441—344no—
160#432—172no—
161#419—342no—
162C4×M16—171no—
163#333—174no—
164Q8⋊C8—178no—
165C2×C17—22no—
166C2×C17#300412yes—
167C2×C17C2×#45511yescharacteristic
168C22×Q8—22no—
169C22×Q8C22⋊D1712yes—
170C22×Q8C22⋊D1712yes—
171C2×Q16—344no—
172C2×M16—341no—
173C4⋊Q8—28no—
174C4⋊Q8—28no—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—22no—
178C4⋊Q8—172no—
179C4⋊Q8C22⋊D1712yes—
180C4⋊Q8C22×D1712yes—
181#125—341no—
182C22⋊Q8—342no—
183C4×Q8—28no—
184C4×Q8—28no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—342no—
188C4×Q8C22⋊D1714yes—
189C4×Q8C22×D1714yes—
190#117—341no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4C22⋊D1712yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42C22×D1711yescharacteristic
198#106—344no—
199C4⋊C8—174no—
200#103—348no—
201#103—344no—
202#103—344no—
203C22⋊C8—342no—
204#97—341no—
205#97—342no—
206C4×C8—172no—
207C17#1855111yescharacteristic, complemented, Sylow
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—44no—
213C2×Q8—84no—
214C2×Q8—342no—
215C2×Q8C2×C22⋊D1714yes—
216C2×Q8C2×C22⋊D1714yes—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4C2×C22⋊D1712yes—
221C22×C4C2×C22⋊D1711yescharacteristic
222Q16—688no—
223M16—682no—
224C2×C8—342no—
225C2×C8—344no—
226C4⋊C4—216no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—342no—
233C4⋊C4—342no—
234C4⋊C4—682no—
235C4⋊C4C2×C22⋊D1714yes—
236C22⋊C4—681no—
237C42—28no—
238C42—24no—
239C42—24no—
240C42—681no—
241C42C2×C22⋊D1711yescharacteristic
242C42C2×C22⋊D1712yes—
243C42C23×D1711yescharacteristic
244C23#1058763911yescharacteristic
245Q8—28no—
246Q8—24no—
247Q8—44no—
248Q8—82no—
249Q8—88no—
250Q8—684no—
251Q8#1058762818yes—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—682no—
261C2×C4#1058763912yes—
262C2×C4#1058763911yescharacteristic
263C2×C4C22×C22⋊D1711yescharacteristic
264C2×C4C22×C22⋊D1712yes—
265C8—682no—
266C8—684no—
267C22—22no—
268C22#1176298712yes—
269C22C2×#1058763911yescharacteristic
270C4—28no—
271C4—22no—
272C4—21no—
273C4—21no—
274C4—44no—
275C4—44no—
276C4—1362no—
277C4C2×#1058762814yes—
278C2—22no—
279C2#1453801812yes—
280C2C2×#1176298711yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×Q82×C17C2
socleC22×C17C2×#120
prime radicalC23×C17#120
hypercenterC2×Q82D17
hypocenterC17#18551
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #14528005C2semidirect
207. C17#18551semidirect

Maximal subgroups

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

subgroupindex
#145391642
#145389142
#145280052
#145275102
#145272312
C2×Q82×C172
#1855117