Finite Group Databasebeta

← #16949516

Subgroups — #16949516

Order 4352.

512 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 125 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#14531497C211yesmaximal, characteristic
2C2×#11762606C211yesmaximal, characteristic
3#101×Dic17C212yesmaximal, complemented
4#14526910C212yesmaximal, complemented
5C2×C17×#355C211yesmaximal, characteristic
6C2×Q8×Dic17C2211yescharacteristic
7C2×D4×Dic17C411yescharacteristic
8C2×#10587507C2212yes—
9M16×Dic17—22no—
10#11762716C2211yescharacteristic
11#11762712C411yescharacteristic
12#11762702—22no—
13#11762684—22no—
14#11762679C411yescharacteristic
15#11762655C411yescharacteristic
16#11762606C414yescomplemented
17#11762577—22no—
18C2×C17×#124C2211yescharacteristic
19C2×C17×#101C2212yes—
20C17×#355C414yescomplemented
21C22×Q8×C17C2311yescharacteristic
22C22×D4×C17C2×C411yescharacteristic
23C23×Dic17—21no—
24C2×M16×C17—22no—
25C17×#124—22no—
26C17×#124C2×C412yes—
27C17×#124C2×C412yes—
28C2×C22⋊C4×C17—22no—
29C2×C42×C17C2×C411yescharacteristic
30C2×C22⋊Dic17—21no—
31C2×C22⋊Dic17—22no—
32Q8×Dic17—42no—
33D4×Dic17—42no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17D412yes—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D412yes—
41C2×#63223—22no—
42C17×#101—22no—
43C17×#101C2×C414yescomplemented
44#10587507—22no—
45#10587507C2×C414yescomplemented
46#10587502—22no—
47#10587502—21no—
48#10587502—21no—
49#10587486—42no—
50C8×Dic17—42no—
51C24×C17—21no—
52C2×Q8×C17—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17C22×C411yescharacteristic
55C2×Q8×C17C22×C411yescharacteristic
56C2×Q8×C17#4112yes—
57C2×D4×C17—21no—
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C2×D4×C17C4212yes—
61C22×C4×C17—21no—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D412yes—
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17—42no—
67C22×Dic17—82no—
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C22⋊C412yes—
70C22×Dic17C22⋊C411yescharacteristic
71M16×C17—24no—
72M16×C17—42no—
73C2×C8×C17—42no—
74C22⋊C4×C17—24no—
75C22⋊C4×C17—24no—
76C42×C17—21no—
77C42×C17C4212yes—
78C22⋊Dic17—42no—
79C22⋊Dic17—82no—
80C4⋊Dic17—41no—
81C4⋊Dic17—42no—
82C4⋊Dic17—41no—
83C4⋊Dic17—82no—
84C4×Dic17—22no—
85C4×Dic17—22no—
86C4×Dic17—22no—
87C4×Dic17—41no—
88#63223—24no—
89#63223—42no—
90C2×#3622—42no—
91#12632—171nomaximal, Sylow
92C23×C17—22no—
93C23×C17—42no—
94C23×C17—42no—
95C23×C17—41no—
96C23×C17—41no—
97C23×C17C2×C22⋊C411yescharacteristic
98Q8×C17—41no—
99Q8×C17—41no—
100Q8×C17—42no—
101D4×C17—42no—
102D4×C17—42no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—22no—
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17C2×C4211yescharacteristic
111C2×C4×C17#11711yescharacteristic
112C2×C4×C17C4×D412yes—
113C2×C4×C17C4×D412yes—
114C2×Dic17—21no—
115C2×Dic17—22no—
116C2×Dic17—41no—
117C2×Dic17—42no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—42no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—84no—
125C2×Dic17#9912yes—
126C8×C17—44no—
127#3622—44no—
128#2326—171no—
129C2×#355—171no—
130C2×#355D1711yescharacteristic
131C4×#101—172no—
132#1394—172no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—44no—
136C22×C17—42no—
137C22×C17—82no—
138C22×C17—81no—
139C22×C17—81no—
140C22×C17C4×C22⋊C411yescharacteristic
141C22×C17C2×#9911yescharacteristic
142C22×C17#41011yescharacteristic
143C4×C17—22no—
144C4×C17—22no—
145C4×C17—21no—
146C4×C17—21no—
147C4×C17—42no—
148Dic17—22no—
149Dic17—42no—
150Dic17—41no—
151Dic17—81no—
152Dic17—81no—
153C2×#124C2×D1711yescharacteristic
154C2×C4×Q8—171no—
155C2×C4×D4—171no—
156C2×#101—172no—
157C2×#101C2×D1712yes—
158C4×M16—342no—
159#391—171no—
160#390—171no—
161#388—342no—
162#385—342no—
163#384—171no—
164#376—171no—
165#355—174no—
166#355Dic1714yescomplemented
167#343—342no—
168C2×C17—21no—
169C2×C17—21no—
170C2×C17—82no—
171C2×C17C4×#9911yescharacteristic
172C2×C17#213311yescharacteristic
173C2×C17#213411yescharacteristic
174C22×Q8C22×D1711yescharacteristic
175C22×D4C4×D1711yescharacteristic
176C23×C4—341no—
177C2×M16—22no—
178C2×M16—342no—
179#124—22no—
180#124C4×D1712yes—
181#124C2×Dic1712yes—
182C4×Q8—682no—
183C4×D4—682no—
184C2×C4⋊C4—171no—
185C2×C4⋊C4—172no—
186C2×C4⋊C4—171no—
187C2×C4⋊C4—342no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—341no—
190C2×C22⋊C4—342no—
191C2×C42—171no—
192C2×C42—172no—
193C2×C42—171no—
194C2×C42C4×D1711yescharacteristic
195#101—22no—
196#101—174no—
197#101—342no—
198#101C2×Dic1714yescomplemented
199#97—682no—
200C4×C8—682no—
201#95—341no—
202#95—341no—
203#95—342no—
204C17#1263211yescharacteristic, complemented, Sylow
205C24—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8C2×C4×D1711yescharacteristic
209C2×Q8#6325412yes—
210C2×Q8C22×Dic1711yescharacteristic
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4C4×Dic1712yes—
215C22×C4—21no—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—172no—
219C22×C4—341no—
220C22×C4—341no—
221C22×C4—682no—
222C22×C4—1362no—
223C22×C4C2×C4×D1711yescharacteristic
224C22×C4D4×D1712yes—
225M16—24no—
226M16—42no—
227M16—344no—
228M16—682no—
229C2×C8—42no—
230C2×C8—682no—
231C4⋊C4—681no—
232C4⋊C4—682no—
233C4⋊C4—681no—
234C4⋊C4—1362no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—682no—
238C22⋊C4—1362no—
239C42—21no—
240C42—342no—
241C42—342no—
242C42—342no—
243C42—681no—
244C42C4×Dic1712yes—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23C22⋊C4×D1711yescharacteristic
251Q8—41no—
252Q8—42no—
253Q8—41no—
254D4—42no—
255D4—42no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—172no—
264C2×C4—341no—
265C2×C4—342no—
266C2×C4—681no—
267C2×C4—682no—
268C2×C4—681no—
269C2×C4—681no—
270C2×C4—682no—
271C2×C4—681no—
272C2×C4—681no—
273C2×C4—681no—
274C2×C4—1364no—
275C2×C4#1058756311yescharacteristic
276C2×C4#1058756712yes—
277C2×C4C2×C4×Dic1711yescharacteristic
278C2×C4D4×Dic1712yes—
279C8—44no—
280C8—684no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—42no—
285C22—82no—
286C22—81no—
287C22—81no—
288C22C22⋊C4×Dic1711yescharacteristic
289C22#99×D1711yescharacteristic
290C22#1176278111yescharacteristic
291C4—22no—
292C4—22no—
293C4—21no—
294C4—21no—
295C4—42no—
296C4—342no—
297C4—681no—
298C4—682no—
299C4—1361no—
300C4—1361no—
301C2—21no—
302C2—21no—
303C2—82no—
304C2#99×Dic1711yescharacteristic
305C2#1453047011yescharacteristic
306C2#1453047411yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC2×C4×C17C2×C42
FrattiniC22×Q8C22×D17
FittingC2×C17×#355C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC22×Q8×C17C23
hypercenterC2×#355D17
hypocenterC17#12632
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #101×Dic17C2semidirect
4. #14526910C2semidirect
16. #11762606C4semidirect
20. C17×#355C4semidirect
43. C17×#101C2×C4semidirect
45. #10587507C2×C4semidirect
166. #355Dic17semidirect
198. #101C2×Dic17semidirect
204. C17#12632semidirect

Maximal subgroups

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

subgroupindex
#145314972
C2×#117626062
#101×Dic172
#145269102
C2×C17×#3552
#1263217