Finite Group Databasebeta

← #14538367

Subgroups — #14538367

Order 2176.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#10587605C211yesmaximal, characteristic, complemented
2C22×C8×D17C211yesmaximal, characteristic, complemented
3#11763322C212yesmaximal, complemented
4#11763047C211yesmaximal, characteristic, complemented
5#11763046C211yesmaximal, characteristic, complemented
6#11763045C211yesmaximal, characteristic, complemented
7#11763041C212yesmaximal, complemented
8#11763040C212yesmaximal, complemented
9#11762846C211yesmaximal, characteristic, complemented
10#11762804C212yesmaximal, complemented
11C17×#467C211yesmaximal, characteristic, complemented
12C2×#63256C2212yes—
13C22×C4×D17C2211yescharacteristic
14C2×Q16×C17C2211yescharacteristic, complemented
15C22×C8×C17C2211yescharacteristic
16C17×C22⋊Q8C2212yes—
17#10587632C2212yes—
18C2×#63230C2212yescomplemented
19#10587616C2212yes—
20C22×#3622C2211yescharacteristic
21#10587605—24no—
22C2×D136C2211yescharacteristic, complemented
23C2×C8×D17—24no—
24C2×C8×D17C2211yescharacteristic
25C2×C8×D17C2211yescharacteristic
26C2×C8×D17C2211yescharacteristic
27C2×C8×D17C2211yescharacteristic
28#10587581C2212yes—
29#10587580—22no—
30#10587579—22no—
31#10587578C2212yes—
32#10587568—22no—
33#10587567—22no—
34C17×#107C2211yescharacteristic, complemented
35C17×#103C2212yescomplemented
36#10587506C2212yescomplemented
37#10587492C2212yescomplemented
38#10587489C2211yescharacteristic, complemented
39#10587480C2212yescomplemented
40#10587479C2212yescomplemented
41C23×D17D411yescharacteristic
42C2×Q8×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C2×C22⋊D17—22no—
45C22×Dic17D411yescharacteristic
46#63256—24no—
47#63256—24no—
48C2×D68—22no—
49C2×D68—22no—
50C2×D68C2312yes—
51C2×C4×D17—22no—
52C2×C4×D17—22no—
53C2×C4×D17—22no—
54C2×C4×D17D411yescharacteristic
55C2×C4×D17D411yescharacteristic
56C2×C4×D17C2311yescharacteristic
57C2×C4×D17C2311yescharacteristic
58Q16×C17—22no—
59C2×C8×C17—22no—
60C2×C8×C17C2311yescharacteristic
61C2×C8×C17C2311yescharacteristic
62C4⋊C4×C17—22no—
63C4⋊C4×C17C2312yes—
64C22⋊C4×C17—22no—
65#63230—24no—
66#63227—22no—
67#63227—22no—
68#63227—22no—
69#63227—22no—
70C4⋊Dic17C2312yes—
71#63225—22no—
72C4×Dic17—22no—
73C2×#3622—22no—
74C2×#3622C2311yescharacteristic
75C2×#3622C2311yescharacteristic
76D136—22no—
77C8×D17—22no—
78C8×D17—22no—
79C8×D17—22no—
80C8×D17D414yescomplemented
81C23×C17C2×D411yescharacteristic
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17C2×D411yescharacteristic
87C22×D17C2×D411yescharacteristic
88Q8×C17—22no—
89Q8×C17—22no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17C2×D411yescharacteristic
95C2×C4×C17C2411yescharacteristic
96C22⋊D17—44no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17C2×D411yescharacteristic
100C2×Dic17C2×D411yescharacteristic
101D68—22no—
102D68—22no—
103D68—44no—
104D68—44no—
105C4×D17—22no—
106C4×D17—21no—
107C4×D17—21no—
108C4×D17—22no—
109C4×D17—44no—
110C4×D17C2×D412yes—
111C4×D17#4112yes—
112C8×C17—21no—
113C8×C17C2×D412yes—
114#3622—21no—
115#3622C2×D412yes—
116#3050—171nomaximal, Sylow
117C22×C17—21no—
118C22×C17—21no—
119C22×C17#13512yescomplemented
120C22×C17C22×D411yescharacteristic
121C2×D17—28no—
122C2×D17—21no—
123C2×D17—21no—
124C2×D17—21no—
125C2×D17—21no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17#12112yes—
130C2×D17#13512yescomplemented
131C4×C17—21no—
132C4×C17—42no—
133C4×C17—42no—
134C4×C17C22×D411yescharacteristic
135C4×C17C2×#4111yescharacteristic
136Dic17—21no—
137Dic17—42no—
138Dic17#12112yes—
139C2×#135—171no—
140C23×C8—171no—
141#525—172no—
142#467—171no—
143#467D1711yescharacteristic, complemented
144#466—171no—
145#466—171no—
146#465—172no—
147#465—172no—
148#427—171no—
149#416—172no—
150C2×C17—22no—
151C2×C17C2×#12111yescharacteristic
152C2×C17C2×#13511yescharacteristic
153C2×C17C2×#13511yescharacteristic
154D17—22no—
155D17—22no—
156D17—82no—
157C2×#41—172no—
158C23×C4—171no—
159#135—344no—
160C2×Q16C2×D1711yescharacteristic, complemented
161C2×SD16—172no—
162C2×D8—171no—
163C22×C8—171no—
164C22×C8—171no—
165C22×C8—171no—
166C22×C8—171no—
167C22×C8—171no—
168C22×C8—344no—
169C22×C8C2×D1711yescharacteristic
170#123—342no—
171C22⋊Q8—172no—
172C22⋊Q8C2×D1712yes—
173#121—172no—
174#121—172no—
175#120—342no—
176C4×D4—342no—
177C4×D4—342no—
178#117—172no—
179#107—171no—
180#107C2×D1711yescharacteristic, complemented
181#106—172no—
182#103—172no—
183#103C2×D1712yescomplemented
184#102—172no—
185#102—172no—
186C17#305011yescharacteristic, complemented, Sylow
187C24—171no—
188#41—344no—
189#41—344no—
190C2×Q8C22×D1712yes—
191C2×D4—172no—
192C2×D4—342no—
193C2×D4—342no—
194C2×D4—342no—
195C22×C4—171no—
196C22×C4—171no—
197C22×C4—171no—
198C22×C4—171no—
199C22×C4—171no—
200C22×C4—342no—
201C22×C4—342no—
202C22×C4—342no—
203C22×C4C22×D1711yescharacteristic
204Q16—22no—
205SD16—344no—
206D8—342no—
207C2×C8—22no—
208C2×C8—171no—
209C2×C8—174no—
210C2×C8—171no—
211C2×C8—342no—
212C2×C8—342no—
213C2×C8—342no—
214C2×C8—342no—
215C2×C8C22×D1711yescharacteristic
216C2×C8C22×D1711yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—172no—
219C4⋊C4—342no—
220C4⋊C4C22×D1712yes—
221C22⋊C4—22no—
222C22⋊C4—342no—
223C22⋊C4—342no—
224C22⋊C4—342no—
225C22⋊C4—342no—
226C42—342no—
227C23—171no—
228C23—171no—
229C23—342no—
230C23—342no—
231C23—342no—
232C23—342no—
233C23D4×D1711yescharacteristic
234Q8—22no—
235Q8—22no—
236D4—342no—
237D4—342no—
238D4—684no—
239D4—684no—
240D4—684no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—171no—
246C2×C4—172no—
247C2×C4—172no—
248C2×C4—171no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—341no—
253C2×C4—341no—
254C2×C4—342no—
255C2×C4—684no—
256C2×C4D4×D1711yescharacteristic
257C2×C4C23×D1711yescharacteristic
258C8—21no—
259C8—172no—
260C8—341no—
261C8D4×D1712yes—
262C22—21no—
263C22—21no—
264C22—172no—
265C22—172no—
266C22—348no—
267C22—341no—
268C22—341no—
269C22—341no—
270C22—341no—
271C22—682no—
272C22—682no—
273C22—682no—
274C22#1058760512yescomplemented
275C22C2×D4×D1711yescharacteristic
276C4—21no—
277C4—42no—
278C4—42no—
279C4—172no—
280C4—341no—
281C4—682no—
282C4C2×D4×D1711yescharacteristic
283C4#41×D1711yescharacteristic
284C2—22no—
285C2—342no—
286C2—342no—
287C2—1362no—
288C2#121×D1711yescharacteristic
289C2#135×D1711yescharacteristic
290C2C2×#1058760511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#467C2
socleC22×C17C22×D4
hypercenter#467D17
hypocenterC17#3050
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#10587605C2semidirect
2. C22×C8×D17C2semidirect
3. #11763322C2semidirect
4. #11763047C2semidirect
5. #11763046C2semidirect
6. #11763045C2semidirect
7. #11763041C2semidirect
8. #11763040C2semidirect
9. #11762846C2semidirect
10. #11762804C2semidirect
11. C17×#467C2semidirect
14. C2×Q16×C17C22semidirect
18. C2×#63230C22semidirect
22. C2×D136C22semidirect
34. C17×#107C22semidirect
35. C17×#103C22semidirect
36. #10587506C22semidirect
37. #10587492C22semidirect
38. #10587489C22semidirect
39. #10587480C22semidirect
40. #10587479C22semidirect
80. C8×D17D4semidirect
119. C22×C17#135semidirect
130. C2×D17#135semidirect
143. #467D17semidirect
160. C2×Q16C2×D17semidirect
180. #107C2×D17semidirect
183. #103C2×D17semidirect
186. C17#3050semidirect
274. C22#10587605semidirect

Maximal subgroups

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

subgroupindex
C2×#105876052
C22×C8×D172
#117633222
#117630472
#117630462
#117630452
#117630412
#117630402
#117628462
#117628042
C17×#4672
#305017