Finite Group Databasebeta

← #16950322

Subgroups — #16950322

Order 4352.

500 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 95 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#14531327C211yesmaximal, characteristic
2C2×#11762605C212yesmaximal
3#14528778C212yesmaximal, complemented
4#14526911C211yesmaximal, characteristic, complemented
5C17×#1394C211yesmaximal, characteristic, complemented
6C2×#10587633C2211yescharacteristic
7C2×Q8×Dic17C2211yescharacteristic
8C2×#10587579C411yescharacteristic
9#11762832—22no—
10C2×#10587507C2212yes—
11#11762711C411yescharacteristic
12#11762701—22no—
13#11762674C411yescharacteristic
14#11762656—22no—
15C42×Dic17C411yescharacteristic
16#11762605—24no—
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—22no—
27#10587633D412yes—
28#10587633Q812yes—
29Q8×Dic17—41no—
30Q8×Dic17—41no—
31C2×#63227—22no—
32C2×#63227—21no—
33C2×#63227—22no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17D412yes—
36C2×C4×Dic17—21no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D412yes—
41C2×#63223—22no—
42#10587579—41no—
43#10587579—41no—
44C17×#101—22no—
45C17×#101—22no—
46C17×#95—22no—
47#10587507—22no—
48#10587507—22no—
49#10587502—22no—
50#10587485—42no—
51#10587485—42no—
52C23×D17—21no—
53C2×Q8×C17—41no—
54C2×Q8×C17—41no—
55C2×Q8×C17C2×D411yescharacteristic
56C2×Q8×C17C2×Q811yescharacteristic
57C2×Q8×C17#4112yes—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17C22×C411yescharacteristic
61C22×C4×C17C2×D412yes—
62C22×Dic17—21no—
63C22×Dic17C22⋊C411yescharacteristic
64C22×Dic17C22⋊C412yes—
65C22×Dic17C22⋊C411yescharacteristic
66C2×D68—21no—
67C2×D68—41no—
68C2×D68—41no—
69C2×D68C4⋊C412yes—
70C2×C4×D17—44no—
71C2×C4×D17—41no—
72C2×C4×D17—41no—
73M16×C17—42no—
74M16×C17—42no—
75C2×C8×C17—42no—
76C4⋊C4×C17—41no—
77C42×C17—41no—
78#63227—24no—
79#63227—24no—
80#63227—42no—
81#63227—44no—
82C4⋊Dic17—41no—
83C4⋊Dic17—42no—
84C4×Dic17—21no—
85C4×Dic17—22no—
86C4×Dic17—22no—
87C4×Dic17—21no—
88C4×Dic17—21no—
89C4×Dic17—42no—
90C4×Dic17—42no—
91C4×Dic17C4⋊C412yes—
92#63223—42no—
93#63223—42no—
94C2×#3622—42no—
95#12801—171nomaximal, Sylow
96C23×C17C2×C22⋊C411yescharacteristic
97C22×D17—22no—
98C22×D17—42no—
99C22×D17—42no—
100C22×D17—41no—
101C22×D17—41no—
102Q8×C17—41no—
103Q8×C17—41no—
104Q8×C17—42no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17C2×C4⋊C411yescharacteristic
115C2×C4×C17#11711yescharacteristic
116C2×C4×C17#12112yes—
117C2×C4×C17C22⋊Q812yes—
118C2×Dic17—22no—
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—21no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—42no—
126C2×Dic17#9911yescharacteristic
127C2×Dic17#9911yescharacteristic
128D68—42no—
129D68—42no—
130C4×D17—82no—
131C4×D17—82no—
132C8×C17—82no—
133#3622—82no—
134C4×#124—171no—
135C2×#355—172no—
136#1801—172no—
137#1394—171no—
138#1394D1711yescharacteristic, complemented
139C22×C17—21no—
140C22×C17—21no—
141C22×C17#38011yescharacteristic
142C22×C17C2×#9911yescharacteristic
143C22×C17#41011yescharacteristic
144C2×D17—44no—
145C2×D17—42no—
146C2×D17—82no—
147C2×D17—81no—
148C2×D17—81no—
149C4×C17—22no—
150C4×C17—21no—
151C4×C17—21no—
152C4×C17—22no—
153C4×C17—42no—
154C4×C17—81no—
155Dic17—21no—
156Dic17—42no—
157Dic17—42no—
158Dic17—41no—
159Dic17#35512yescomplemented
160C2×#124—171no—
161C2×C4×Q8—171no—
162C2×C4×D4—171no—
163#423—342no—
164C2×#101—172no—
165C2×#101C2×D1712yes—
166#391C2×D1711yescharacteristic
167#390—171no—
168#388—342no—
169#382—171no—
170C4×C22⋊C4—342no—
171C43—171no—
172#355—344no—
173#343—342no—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17#179911yescharacteristic
177C2×C17C2×#35511yescharacteristic
178C2×C17#213211yescharacteristic
179D17—82no—
180C22×Q8C22×D1711yescharacteristic
181C22×D4—171no—
182C23×C4—341no—
183C2×M16—22no—
184C2×M16—342no—
185#124—172no—
186#124—172no—
187#124—342no—
188C4×Q8—681no—
189C4×Q8—681no—
190C4×D4—681no—
191C4×D4—681no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—172no—
194C2×C4⋊C4C4×D1711yescharacteristic
195C2×C22⋊C4—342no—
196C2×C22⋊C4—341no—
197C2×C22⋊C4—342no—
198C2×C42—171no—
199C2×C42—171no—
200C2×C42—172no—
201C2×C42—341no—
202C2×C42—341no—
203C2×C42C4×D1711yescharacteristic
204C4⋊C8—682no—
205C4⋊C8—682no—
206#101—22no—
207#101—22no—
208#101—342no—
209#101—342no—
210#95—22no—
211#95—342no—
212C17#1280111yescharacteristic, complemented, Sylow
213C24—341no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8#6325212yes—
217C2×Q8D4×D1711yescharacteristic
218C2×Q8Q8×D1711yescharacteristic
219C2×D4—172no—
220C2×D4—341no—
221C2×D4—681no—
222C2×D4—681no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—171no—
226C22×C4—172no—
227C22×C4—171no—
228C22×C4—341no—
229C22×C4—681no—
230C22×C4—681no—
231C22×C4—684no—
232C22×C4C2×C4×D1711yescharacteristic
233C22×C4D4×D1712yes—
234M16—42no—
235M16—42no—
236M16—682no—
237M16—682no—
238C2×C8—42no—
239C2×C8—682no—
240C4⋊C4—41no—
241C4⋊C4—681no—
242C4⋊C4—682no—
243C22⋊C4—344no—
244C22⋊C4—344no—
245C22⋊C4—684no—
246C22⋊C4—682no—
247C42—41no—
248C42—172no—
249C42—342no—
250C42—342no—
251C42—341no—
252C42—341no—
253C42—341no—
254C42—682no—
255C42—682no—
256C23—342no—
257C23—682no—
258C23—682no—
259C23—681no—
260C23—681no—
261C23C22⋊C4×D1711yescharacteristic
262Q8—42no—
263Q8—41no—
264Q8—41no—
265D4—682no—
266D4—682no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—171no—
277C2×C4—171no—
278C2×C4—342no—
279C2×C4—341no—
280C2×C4—342no—
281C2×C4—342no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—682no—
286C2×C4—1362no—
287C2×C4—1362no—
288C2×C4#1058755811yescharacteristic
289C2×C4#1058757012yes—
290C2×C4C4⋊C4×D1711yescharacteristic
291C2×C4#1058758212yes—
292C8—82no—
293C8—1362no—
294C22—21no—
295C22—21no—
296C22—684no—
297C22—682no—
298C22—1362no—
299C22—1361no—
300C22—1361no—
301C22#1176266811yescharacteristic
302C22#99×D1711yescharacteristic
303C22#1176278111yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—42no—
309C4—81no—
310C4—172no—
311C4—341no—
312C4—682no—
313C4—681no—
314C4—682no—
315C2—21no—
316C2—21no—
317C2—1362no—
318C2#1452877111yescharacteristic
319C2D17×#35511yescharacteristic
320C2#1453046811yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#1394C2
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC22×Q8×C17C23
hypercenter#1394D17
hypocenterC17#12801
LCSG ▷ C22×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. #14528778C2semidirect
4. #14526911C2semidirect
5. C17×#1394C2semidirect
138. #1394D17semidirect
159. Dic17#355semidirect
212. C17#12801semidirect

Maximal subgroups

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

subgroupindex
#145313272
C2×#117626052
#145287782
#145269112
C17×#13942
#1280117