Finite Group Databasebeta

← #16949513

Subgroups — #16949513

Order 4352.

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

normal subgroups only

This group has 512 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#14531498C211yesmaximal, characteristic
2C2×#11762605C212yesmaximal
3#101×Dic17C212yesmaximal, complemented
4#14526911C211yesmaximal, characteristic, complemented
5C17×#1394C211yesmaximal, characteristic, complemented
6C2×#10587633C2211yescharacteristic
7C2×Q8×Dic17C2211yescharacteristic
8C2×#10587579C411yescharacteristic
9C2×#10587507C2212yes—
10M16×Dic17—22no—
11#11762711C411yescharacteristic
12#11762698—22no—
13#11762683—22no—
14#11762678C411yescharacteristic
15#11762655C411yescharacteristic
16#11762605C418yescomplemented
17#11762577—22no—
18C2×C17×#101C2212yes—
19C17×#391C2211yescharacteristic
20C22×Q8×C17C2311yescharacteristic
21C22×D68C2×C411yescharacteristic
22C22×C4×D17—21no—
23C2×M16×C17—22no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C42×C17C2×C411yescharacteristic
26#10587633—21no—
27#10587633—21no—
28#10587633C2×C414yes—
29Q8×Dic17—42no—
30C2×#63227—22no—
31C2×#63227—21no—
32C2×#63227—22no—
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×C4⋊Dic17D412yes—
35C2×#63225—22no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D412yes—
39C2×#63223—22no—
40#10587579—42no—
41C17×#101—22no—
42C17×#101C2×C414yescomplemented
43C17×#95—22no—
44#10587507—22no—
45#10587507C2×C414yescomplemented
46#10587502—21no—
47#10587502—21no—
48#10587486—42no—
49C8×Dic17—42no—
50C23×D17—21no—
51C2×Q8×C17—41no—
52C2×Q8×C17—41no—
53C2×Q8×C17C22×C412yes—
54C2×Q8×C17#4111yescharacteristic
55C2×Q8×C17#4111yescharacteristic
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17C22×C411yescharacteristic
59C22×C4×C17C2×D412yes—
60C22×Dic17—21no—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C22⋊C412yes—
63C22×Dic17C22⋊C411yescharacteristic
64C2×D68—21no—
65C2×D68—41no—
66C2×D68—41no—
67C2×D68C4212yes—
68C2×C4×D17—42no—
69C2×C4×D17—82no—
70M16×C17—24no—
71M16×C17—42no—
72C2×C8×C17—42no—
73C4⋊C4×C17—41no—
74C42×C17—41no—
75#63227—24no—
76#63227—24no—
77#63227—42no—
78#63227—82no—
79C4⋊Dic17—41no—
80C4⋊Dic17—42no—
81#63225—82no—
82C4×Dic17—21no—
83C4×Dic17—22no—
84C4×Dic17—22no—
85C4×Dic17—22no—
86C4×Dic17C4212yes—
87#63223—24no—
88#63223—42no—
89C2×#3622—42no—
90#12632—171nomaximal, Sylow
91C23×C17C2×C22⋊C411yescharacteristic
92C22×D17—22no—
93C22×D17—42no—
94C22×D17—42no—
95C22×D17—41no—
96C22×D17—41no—
97Q8×C17—42no—
98Q8×C17—41no—
99Q8×C17—41no—
100C2×C4×C17—21no—
101C2×C4×C17—22no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—42no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17C2×C4211yescharacteristic
110C2×C4×C17#11711yescharacteristic
111C2×C4×C17C4×D412yes—
112C2×C4×C17C4×D412yes—
113C2×Dic17—22no—
114C2×Dic17—21no—
115C2×Dic17—22no—
116C2×Dic17—41no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—42no—
121C2×Dic17#9912yes—
122D68—42no—
123D68—42no—
124C4×D17—84no—
125C8×C17—44no—
126#3622—44no—
127#2326—171no—
128C2×#355—172no—
129C4×#101—172no—
130#1394—171no—
131#1394D1711yescharacteristic, complemented
132C22×C17—21no—
133C22×C17—21no—
134C22×C17C4×C22⋊C411yescharacteristic
135C22×C17C2×#9911yescharacteristic
136C22×C17#41011yescharacteristic
137C2×D17—44no—
138C2×D17—42no—
139C2×D17—82no—
140C2×D17—81no—
141C2×D17—81no—
142C4×C17—22no—
143C4×C17—22no—
144C4×C17—21no—
145C4×C17—21no—
146C4×C17—81no—
147C4×C17—81no—
148Dic17—22no—
149Dic17—42no—
150Dic17—41no—
151Dic17—42no—
152C2×#124—171no—
153C2×C4×Q8—171no—
154C2×C4×D4—171no—
155C2×#101—172no—
156C2×#101C2×D1712yes—
157C4×M16—342no—
158#391C2×D1711yescharacteristic
159#390—171no—
160#388—342no—
161#385—342no—
162#384—171no—
163#376—171no—
164#355—178no—
165#343—342no—
166C2×C17—21no—
167C2×C17—21no—
168C2×C17C4×#9911yescharacteristic
169C2×C17#213311yescharacteristic
170C2×C17#213411yescharacteristic
171D17—82no—
172C22×Q8C22×D1711yescharacteristic
173C22×D4—171no—
174C23×C4—341no—
175C2×M16—22no—
176C2×M16—342no—
177#124—174no—
178#124—341no—
179#124—341no—
180C4×Q8—682no—
181C4×D4—682no—
182C2×C4⋊C4—171no—
183C2×C4⋊C4—172no—
184C2×C4⋊C4—342no—
185C2×C4⋊C4C4×D1711yescharacteristic
186C2×C22⋊C4—342no—
187C2×C22⋊C4—341no—
188C2×C22⋊C4—342no—
189C2×C42—171no—
190C2×C42—171no—
191C2×C42—172no—
192C2×C42C4×D1711yescharacteristic
193#101—22no—
194#101—174no—
195#101—342no—
196#101C4×D1714yescomplemented
197#97—682no—
198C4×C8—682no—
199#95—22no—
200#95—341no—
201#95—341no—
202C17#1263211yescharacteristic, complemented, Sylow
203C24—341no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8C2×C4×D1712yes—
207C2×Q8#6325411yescharacteristic
208C2×Q8#6325611yescharacteristic
209C2×D4—172no—
210C2×D4—341no—
211C2×D4—681no—
212C2×D4—681no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—171no—
216C22×C4—171no—
217C22×C4—172no—
218C22×C4—341no—
219C22×C4—682no—
220C22×C4—1362no—
221C22×C4C2×C4×D1711yescharacteristic
222C22×C4D4×D1712yes—
223M16—24no—
224M16—42no—
225M16—344no—
226M16—682no—
227C2×C8—42no—
228C2×C8—682no—
229C4⋊C4—41no—
230C4⋊C4—681no—
231C4⋊C4—682no—
232C4⋊C4—1362no—
233C22⋊C4—344no—
234C22⋊C4—344no—
235C22⋊C4—682no—
236C22⋊C4—1362no—
237C42—41no—
238C42—172no—
239C42—342no—
240C42—342no—
241C42—342no—
242C42—341no—
243C23—342no—
244C23—682no—
245C23—682no—
246C23—681no—
247C23—681no—
248C23C22⋊C4×D1711yescharacteristic
249Q8—42no—
250Q8—41no—
251Q8—41no—
252D4—682no—
253D4—682no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—172no—
264C2×C4—341no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—681no—
268C2×C4—681no—
269C2×C4—681no—
270C2×C4—682no—
271C2×C4—681no—
272C2×C4—1364no—
273C2×C4C42×D1711yescharacteristic
274C2×C4#1058756712yes—
275C2×C4#1058757811yescharacteristic
276C2×C4#1058757912yes—
277C8—44no—
278C8—684no—
279C22—21no—
280C22—21no—
281C22—684no—
282C22—682no—
283C22—1362no—
284C22—1361no—
285C22—1361no—
286C22#1176265611yescharacteristic
287C22#99×D1711yescharacteristic
288C22#1176278111yescharacteristic
289C4—22no—
290C4—22no—
291C4—21no—
292C4—21no—
293C4—81no—
294C4—81no—
295C4—342no—
296C4—682no—
297C4—681no—
298C4—682no—
299C2—21no—
300C2—21no—
301C2—1362no—
302C2#1452857111yescharacteristic
303C2#1453047211yescharacteristic
304C2#1453047411yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC2×C4×C17C2×C42
FrattiniC22×Q8C22×D17
FittingC17×#1394C2
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC22×Q8×C17C23
hypercenter#1394D17
hypocenterC17#12632
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #101×Dic17C2semidirect
4. #14526911C2semidirect
5. C17×#1394C2semidirect
16. #11762605C4semidirect
42. C17×#101C2×C4semidirect
45. #10587507C2×C4semidirect
131. #1394D17semidirect
196. #101C4×D17semidirect
202. C17#12632semidirect

Maximal subgroups

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

subgroupindex
#145314982
C2×#117626052
#101×Dic172
#145269112
C17×#13942
#1263217