Finite Group Databasebeta

← #16931784

Subgroups — #16931784

Order 4352.

526 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 103 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#14534164C211yesmaximal, characteristic
2#14527303C212yesmaximal
3#14527114C211yesmaximal, characteristic
4#14526670C212yesmaximal
5C17×#2313C211yesmaximal, characteristic
6C2×#10587610C2211yescharacteristic
7#11762773C2212yes—
8#11762739C412yescomplemented
9#11762729C412yescomplemented
10#11762704—22no—
11#11762682—21no—
12#11762682—21no—
13C4×C22⋊Dic17C2211yescharacteristic
14#11762545—24no—
15#11762535—22no—
16C22×C4⋊C4×C17C2211yescharacteristic
17C17×#387—24no—
18C17×#381—21no—
19C17×#380—21no—
20C4×C22⋊C4×C17C2212yes—
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C22×Dic34C2×C411yescharacteristic
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17C2×C412yes—
29C2×C4⋊C4×C17D412yes—
30C2×C22⋊C4×C17—22no—
31C2×C42×C17—22no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35#10587610—22no—
36#10587610—22no—
37C2×#63225—22no—
38C2×#63225—22no—
39C2×#63225—21no—
40C2×C4×Dic17—21no—
41C2×#63223—22no—
42C17×#95—22no—
43C17×#95—24no—
44C17×#95D414yes—
45#10587502—22no—
46#10587502—22no—
47#10587502C2×C412yes—
48C22⋊#3622—22no—
49C22⋊#3622—22no—
50C24×C17C2×D411yescharacteristic
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—24no—
56C22×C4×C17—42no—
57C22×C4×C17—81no—
58C22×C4×C17—81no—
59C22×C4×C17C22⋊C412yes—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C22×C411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D412yes—
64C22×C4×C17#4112yes—
65C22×Dic17—22no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C2×Dic34—22no—
69C2×Dic34—81no—
70C4⋊C4×C17—24no—
71C4⋊C4×C17—24no—
72C4⋊C4×C17—82no—
73C22⋊C4×C17—44no—
74C42×C17—82no—
75C22⋊Dic17—24no—
76C22⋊Dic17—41no—
77C22⋊Dic17—41no—
78C4⋊Dic17—22no—
79#63225—44no—
80#63225—42no—
81#63225—82no—
82C4×Dic17—81no—
83#63223—82no—
84C2×#3622—42no—
85#9412—171nomaximal, Sylow
86C23×C17—22no—
87C23×C17—22no—
88C23×C17—41no—
89C23×C17C2×C22⋊C411yescharacteristic
90C23×C17#12011yescharacteristic
91C23×C17#12311yescharacteristic
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—42no—
97C2×C4×C17—41no—
98C2×C4×C17—41no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—44no—
105C2×C4×C17—44no—
106C2×C4×C17—44no—
107C2×C4×C17—81no—
108C2×C4×C17C4×D412yes—
109C2×C4×C17#12112yes—
110C2×Dic17—24no—
111C2×Dic17—42no—
112C2×Dic17—42no—
113C2×Dic17—42no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—42no—
117Dic34—82no—
118#3622—82no—
119#2583—171no—
120#2313D1711yescharacteristic
121#1508—172no—
122#1463—171no—
123#1297—172no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—44no—
132C22×C17#38611yescharacteristic
133C22×C17#42112yes—
134C22×C17#45612yes—
135C4×C17—22no—
136C4×C17—22no—
137C4×C17—42no—
138C4×C17—42no—
139C4×C17—84no—
140Dic17—44no—
141Dic17—81no—
142Dic17—81no—
143C2×C22⋊Q8—171no—
144C22×C4⋊C4C2×D1711yescharacteristic
145#407—172no—
146#398—172no—
147#395—172no—
148#389—342no—
149#387—24no—
150#385—341no—
151#385—341no—
152#381—21no—
153#380—21no—
154C4×C22⋊C4—171no—
155C4×C22⋊C4C2×D1712yes—
156#328—344no—
157#324—342no—
158C2×C17—22no—
159C2×C17—22no—
160C2×C17—41no—
161C2×C17#188611yescharacteristic
162C2×C17#189711yescharacteristic
163C2×C17#191311yescharacteristic
164C22×Q8—171no—
165C23×C4C22⋊D1711yescharacteristic
166C23×C4C22⋊D1711yescharacteristic
167C23×C4C22×D1711yescharacteristic
168C2×M16—342no—
169C22⋊Q8—342no—
170C22⋊Q8—342no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—171no—
175C2×C4⋊C4—341no—
176C2×C4⋊C4—342no—
177C2×C4⋊C4—342no—
178C2×C4⋊C4C4×D1712yes—
179C2×C4⋊C4D6812yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—341no—
182C2×C22⋊C4—341no—
183C2×C42—22no—
184C2×C42—341no—
185C22⋊C8—342no—
186C22⋊C8—342no—
187#95—22no—
188#95—24no—
189#95—172no—
190#95—342no—
191#95—342no—
192#95C22⋊D1714yes—
193C17#941211yescharacteristic, complemented, Sylow
194C24C2×C22⋊D1711yescharacteristic
195C2×Q8—342no—
196C2×Q8—1361no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—24no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—342no—
206C22×C4—341no—
207C22×C4—341no—
208C22×C4#6322712yes—
209C22×C4#6322712yes—
210C22×C4C2×C4×D1711yescharacteristic
211C22×C4C2×D6811yescharacteristic
212C22×C4#6325212yes—
213C22×C4C2×C22⋊D1712yes—
214M16—1362no—
215C2×C8—682no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—82no—
219C4⋊C4—342no—
220C4⋊C4—684no—
221C4⋊C4—682no—
222C4⋊C4—1362no—
223C22⋊C4—44no—
224C22⋊C4—344no—
225C22⋊C4—681no—
226C22⋊C4—681no—
227C42—82no—
228C42—1361no—
229C23—22no—
230C23—22no—
231C23—41no—
232C23C2×#6322711yescharacteristic
233C23#1058761511yescharacteristic
234C23#1058763911yescharacteristic
235Q8—1362no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—81no—
252C2×C4—344no—
253C2×C4—682no—
254C2×C4—682no—
255C2×C4—682no—
256C2×C4—681no—
257C2×C4—681no—
258C2×C4—682no—
259C2×C4C4×C22⋊D1712yes—
260C2×C4#1058761612yes—
261C8—1362no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1176268811yescharacteristic
271C22#1176282712yes—
272C22#1176299212yes—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—42no—
277C4—84no—
278C4—684no—
279C4—1361no—
280C4—1361no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#1452917711yescharacteristic
285C2#1452923211yescharacteristic
286C2#1452932611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2313C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#2313D17
hypocenterC17#9412
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
8. #11762739C4semidirect
9. #11762729C4semidirect
193. C17#9412semidirect

Maximal subgroups

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

subgroupindex
#145341642
#145273032
#145271142
#145266702
C17×#23132
#941217