Finite Group Databasebeta

← #16927104

Subgroups — #16927104

Order 4352.

540 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 113 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#14531323C211yesmaximal, characteristic
2#14528638C212yesmaximal, complemented
3C2×#11762548C212yesmaximal
4#14526665C211yesmaximal, characteristic, complemented
5C17×#1295C211yesmaximal, characteristic, complemented
6C2×#10587561C2211yescharacteristic
7C2×#10587573C411yescharacteristic
8C2×#10587579C411yescharacteristic
9C2×#10587485—22no—
10C2×#10587474C2212yes—
11#11762714C411yescharacteristic
12#11762710C411yescharacteristic
13#11762698—22no—
14#11762656—22no—
15C42×Dic17C2211yescharacteristic
16#11762563—22no—
17#11762548—24no—
18C2×C17×#97C2212yes—
19C17×#382C2211yescharacteristic
20C22×D68C2×C411yescharacteristic
21C22×C4×D17—21no—
22C22×Dic34C2×C411yescharacteristic
23C22×C8×C17—22no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C4⋊C4×C17C2×C411yescharacteristic
26C2×C42×C17C2311yescharacteristic
27C2×#63227—22no—
28C2×#63227—21no—
29C2×#63227—22no—
30C2×#63225—21no—
31C2×C4×Dic17—21no—
32C2×C4×Dic17C2×C411yescharacteristic
33C2×C4×Dic17C2×C411yescharacteristic
34C2×C4×Dic17D412yes—
35C2×C4×Dic17D412yes—
36C22×#3622—22no—
37#10587579—42no—
38#10587573—42no—
39#10587561—22no—
40#10587561D412yes—
41#10587561Q812yes—
42C17×#97—22no—
43C17×#97—22no—
44C17×#95—22no—
45#10587502—22no—
46#10587485—28no—
47#10587474—22no—
48#10587474—22no—
49C23×D17—21no—
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17C22×C411yescharacteristic
53C22×C4×C17C2×D412yes—
54C22×Dic17—21no—
55C22×Dic17C22⋊C411yescharacteristic
56C22×Dic17C22⋊C411yescharacteristic
57C22×Dic17C22⋊C412yes—
58C2×D68—21no—
59C2×D68—41no—
60C2×D68—41no—
61C2×D68C4⋊C412yes—
62C2×C4×D17—44no—
63C2×C4×D17—42no—
64C2×Dic34—21no—
65C2×Dic34—41no—
66C2×Dic34—41no—
67C2×Dic34C4⋊C412yes—
68C2×C8×C17—24no—
69C2×C8×C17—42no—
70C2×C8×C17—42no—
71C4⋊C4×C17—41no—
72C4⋊C4×C17—41no—
73C42×C17C2×D411yescharacteristic
74C42×C17C2×Q811yescharacteristic
75C42×C17#4112yes—
76#63227—24no—
77#63227—24no—
78#63227—42no—
79#63227—44no—
80#63225—42no—
81C4×Dic17—22no—
82C4×Dic17—24no—
83C4×Dic17—41no—
84C4×Dic17—42no—
85C4×Dic17—42no—
86C2×#3622—24no—
87C2×#3622—42no—
88C2×#3622—42no—
89#8583—171nomaximal, Sylow
90C23×C17C2×C22⋊C411yescharacteristic
91C22×D17—22no—
92C22×D17—41no—
93C22×D17—41no—
94C22×D17—42no—
95C22×D17—42no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17C2×C4⋊C411yescharacteristic
105C2×C4×C17#11711yescharacteristic
106C2×C4×C17#12112yes—
107C2×C4×C17C22⋊Q812yes—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—21no—
113C2×Dic17—22no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17#10012yes—
117C2×Dic17#10414yes—
118D68—42no—
119D68—42no—
120C4×D17—84no—
121Dic34—42no—
122Dic34—42no—
123C8×C17—44no—
124#3622—44no—
125C4×#124—171no—
126#1767—172no—
127C2×#329—172no—
128#1295—171no—
129#1295D1711yescharacteristic, complemented
130C22×C17#38011yescharacteristic
131C22×C17C2×#10011yescharacteristic
132C22×C17#41311yescharacteristic
133C22×C17C2×#10412yes—
134C22×C17#42112yes—
135C2×D17—42no—
136C2×D17—44no—
137C2×D17—81no—
138C2×D17—82no—
139C2×D17—81no—
140C4×C17—21no—
141C4×C17—21no—
142C4×C17—22no—
143C4×C17—22no—
144C4×C17—81no—
145C4×C17—81no—
146Dic17—24no—
147Dic17—22no—
148Dic17—42no—
149Dic17—41no—
150C2×#124—171no—
151C2×C4×Q8—171no—
152C2×C4×D4—171no—
153C2×C4⋊C8—342no—
154C2×#97—172no—
155C2×#97C2×D1712yes—
156#391—171no—
157#390—171no—
158#388—342no—
159#382C2×D1711yescharacteristic
160C4×C22⋊C4—342no—
161C43—171no—
162#336—342no—
163#329—344no—
164C2×C17#152511yescharacteristic
165C2×C17#153212yes—
166C2×C17#153311yescharacteristic
167C2×C17#180011yescharacteristic
168C2×C17#180612yes—
169D17—82no—
170C22×Q8—171no—
171C22×D4—171no—
172C23×C4—341no—
173C22×C8—22no—
174C22×C8—342no—
175#124—172no—
176#124—172no—
177#124—342no—
178C4×Q8—682no—
179C4×D4—682no—
180C2×C4⋊C4—341no—
181C2×C4⋊C4C4×D1711yescharacteristic
182C2×C4⋊C4C4×D1711yescharacteristic
183C2×C22⋊C4—342no—
184C2×C22⋊C4—341no—
185C2×C22⋊C4—342no—
186C2×C42—171no—
187C2×C42—172no—
188C2×C42—171no—
189C2×C42—172no—
190C2×C42—341no—
191C2×C42C22×D1711yescharacteristic
192C4⋊C8—348no—
193#97—22no—
194#97—22no—
195#97—342no—
196#97—342no—
197#95—22no—
198#95—342no—
199C17#858311yescharacteristic, complemented, Sylow
200C24—341no—
201C2×Q8—172no—
202C2×Q8—341no—
203C2×Q8—681no—
204C2×Q8—681no—
205C2×D4—172no—
206C2×D4—341no—
207C2×D4—681no—
208C2×D4—681no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—172no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—341no—
215C22×C4—682no—
216C22×C4—684no—
217C22×C4C2×C4×D1711yescharacteristic
218C22×C4D4×D1712yes—
219C2×C8—24no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—344no—
223C2×C8—682no—
224C2×C8—682no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—682no—
228C22⋊C4—344no—
229C22⋊C4—344no—
230C22⋊C4—684no—
231C22⋊C4—682no—
232C42—344no—
233C42—342no—
234C42—681no—
235C42—682no—
236C42—682no—
237C42#6325212yes—
238C42D4×D1711yescharacteristic
239C42Q8×D1711yescharacteristic
240C23—342no—
241C23—681no—
242C23—681no—
243C23—682no—
244C23—682no—
245C23C22⋊C4×D1711yescharacteristic
246Q8—682no—
247Q8—682no—
248D4—682no—
249D4—682no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—172no—
259C2×C4—174no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—341no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4—1364no—
269C2×C4#1058755811yescharacteristic
270C2×C4#1058757012yes—
271C2×C4C4⋊C4×D1711yescharacteristic
272C2×C4#1058758212yes—
273C8—44no—
274C8—684no—
275C22—682no—
276C22—684no—
277C22—1361no—
278C22—1362no—
279C22—1361no—
280C22#1176266811yescharacteristic
281C22#100×D1711yescharacteristic
282C22#1176279211yescharacteristic
283C22#104×D1712yes—
284C22#1176282412yes—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—344no—
292C4—342no—
293C4—682no—
294C4—681no—
295C2—1362no—
296C2#1452737411yescharacteristic
297C2#1452740312yes—
298C2#1452740611yescharacteristic
299C2#1452877511yescharacteristic
300C2#1452880312yes—
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1295C2
socleC23×C17C2×C22⋊C4
prime radicalC22×D68C2×C4
cosocleC2×C42×C17C23
hypercenter#1295D17
hypocenterC17#8583
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. #14528638C2semidirect
4. #14526665C2semidirect
5. C17×#1295C2semidirect
129. #1295D17semidirect
199. C17#8583semidirect

Maximal subgroups

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

subgroupindex
#145313232
#145286382
C2×#117625482
#145266652
C17×#12952
#858317