Finite Group Databasebeta

← #14538406

Subgroups — #14538406

Order 2176.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#10587635C211yesmaximal, characteristic, complemented
2C2×M16×D17C211yesmaximal, characteristic, complemented
3#11763316C211yesmaximal, characteristic, complemented
4C2×#10587581C211yesmaximal, characteristic, complemented
5#11763061C212yesmaximal, complemented
6#11763058C212yesmaximal, complemented
7#11763044C212yesmaximal, complemented
8#11763041C212yesmaximal, complemented
9#11762849C211yesmaximal, characteristic, complemented
10#11762815C211yesmaximal, characteristic, complemented
11C17×#417C211yesmaximal, characteristic, complemented
12C2×#41×C17C2211yescharacteristic
13C22×D68—21no—
14C22×D68C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C2×M16×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18#10587635—24no—
19#10587632C2211yescharacteristic, complemented
20#10587632C2211yescharacteristic, complemented
21C2×#63230C2212yescomplemented
22#10587624C2211yescharacteristic, complemented
23#10587624C2211yescharacteristic, complemented
24#10587623—21no—
25#10587621—21no—
26C2×#63228C2212yescomplemented
27C4×C22⋊D17—21no—
28C4×C22⋊D17—21no—
29C2×#63227—21no—
30#10587612C2211yescharacteristic
31C2×#63223C2211yescharacteristic
32M16×D17—24no—
33C2×#63218C2212yescomplemented
34C2×C8×D17C2212yescomplemented
35#10587581—21no—
36#10587581—21no—
37#10587581C2212yes—
38#10587581C2212yes—
39C17×#103C2212yescomplemented
40C17×#102C2212yescomplemented
41#10587492C2212yescomplemented
42#10587492C2212yescomplemented
43#10587479C2212yescomplemented
44#10587478C2212yescomplemented
45C23×D17—21no—
46C23×D17D411yescharacteristic
47#41×C17—22no—
48#41×C17—22no—
49C2×Q8×C17C2311yescharacteristic
50C2×D4×C17—21no—
51C2×D4×C17C2311yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17C2311yescharacteristic
55C2×C22⋊D17—21no—
56C2×C22⋊D17—21no—
57C22×Dic17D411yescharacteristic
58C2×D68—22no—
59C2×D68—22no—
60C2×D68—22no—
61C2×D68—22no—
62C2×D68—21no—
63C2×D68—44no—
64C2×D68C2312yes—
65C2×C4×D17—22no—
66C2×C4×D17—22no—
67C2×C4×D17D411yescharacteristic
68C2×C4×D17D411yescharacteristic
69C2×C4×D17C2311yescharacteristic
70C2×C4×D17C2311yescharacteristic
71M16×C17—22no—
72C2×C8×C17C2312yes—
73C4⋊C4×C17—21no—
74C4⋊C4×C17C2312yes—
75C22⋊Dic17—21no—
76C22⋊Dic17—21no—
77C22⋊Dic17—21no—
78#63230—24no—
79#63228—24no—
80#63227—21no—
81#63227—21no—
82#63227—22no—
83#63227—22no—
84C4⋊Dic17C2311yescharacteristic
85C4⋊Dic17C2311yescharacteristic
86#63225—21no—
87#63225—21no—
88C4×Dic17—21no—
89#63223D414yescomplemented
90C2×#3622C2312yes—
91#63218—24no—
92C8×D17—24no—
93C23×C17—21no—
94C23×C17C2×D411yescharacteristic
95C22×D17—22no—
96C22×D17—22no—
97C22×D17—22no—
98C22×D17—41no—
99C22×D17—42no—
100C22×D17—42no—
101C22×D17—42no—
102C22×D17C2×D411yescharacteristic
103C22×D17C2×D411yescharacteristic
104Q8×C17—21no—
105Q8×C17—21no—
106D4×C17—21no—
107D4×C17—21no—
108D4×C17—42no—
109C2×C4×C17—22no—
110C2×C4×C17—21no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17C2×D411yescharacteristic
114C2×C4×C17C2×D412yes—
115C2×C4×C17#4112yes—
116C2×C4×C17C2411yescharacteristic
117C22⋊D17—42no—
118C22⋊D17—42no—
119C2×Dic17—22no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17C2×D411yescharacteristic
123C2×Dic17C2×D411yescharacteristic
124D68—22no—
125D68—22no—
126D68—44no—
127D68—44no—
128D68—42no—
129C4×D17—22no—
130C4×D17—22no—
131C4×D17—21no—
132C4×D17—21no—
133C4×D17—21no—
134C4×D17—21no—
135C8×C17—22no—
136#3622C2×D414yescomplemented
137#3055—171nomaximal, Sylow
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—41no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17#12112yes—
144C22×C17C22×D411yescharacteristic
145C2×D17—21no—
146C2×D17—21no—
147C2×D17—21no—
148C2×D17—21no—
149C2×D17—21no—
150C2×D17—44no—
151C2×D17—42no—
152C2×D17—42no—
153C2×D17—42no—
154C2×D17—84no—
155C2×D17AGL(1,ℤ/8)12yescomplemented
156C4×C17—41no—
157C4×C17—42no—
158C4×C17#12112yes—
159C4×C17C22×D411yescharacteristic
160C4×C17C2×#4111yescharacteristic
161Dic17—21no—
162Dic17—21no—
163Dic17—41no—
164Dic17—41no—
165C2×AGL(1,ℤ/8)—171no—
166C22×M16—171no—
167#525—171no—
168C2×#121—171no—
169#469—172no—
170#468—172no—
171#466—172no—
172#465—172no—
173#428—171no—
174#418—171no—
175#417D1711yescharacteristic, complemented
176C2×C17—22no—
177C2×C17—81no—
178C2×C17C2×#12111yescharacteristic
179C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C17#57511yescharacteristic
181D17—22no—
182D17—41no—
183D17—82no—
184C2×#41C2×D1711yescharacteristic
185C22×D4—171no—
186C22×D4—341no—
187C23×C4—171no—
188AGL(1,ℤ/8)—344no—
189C2×SD16—172no—
190C2×D8—172no—
191C2×M16—171no—
192C2×M16—172no—
193C2×M16—344no—
194C2×M16C2×D1711yescharacteristic
195C22×C8—172no—
196#123—341no—
197C22⋊Q8—171no—
198C22⋊Q8—171no—
199#121—171no—
200#121—171no—
201#121—172no—
202#121—172no—
203#121—341no—
204#121—341no—
205#120—341no—
206C4×D4—341no—
207C4×D4—341no—
208#117—171no—
209C2×C4⋊C4C2×D1711yescharacteristic
210C2×C22⋊C4—341no—
211#107—172no—
212#106—172no—
213#103C2×D1712yescomplemented
214#102—172no—
215#102—172no—
216#102C2×D1712yescomplemented
217C17#305511yescharacteristic, complemented, Sylow
218C24—171no—
219C24—341no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D1711yescharacteristic
223C2×D4—21no—
224C2×D4—172no—
225C2×D4—342no—
226C2×D4—342no—
227C2×D4—341no—
228C2×D4—341no—
229C2×D4—342no—
230C2×D4—342no—
231C2×D4—341no—
232C2×D4—684no—
233C2×D4C22×D1711yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—342no—
242C22×C4—342no—
243C22×C4C22×D1711yescharacteristic
244SD16—344no—
245D8—344no—
246M16—22no—
247M16—174no—
248M16—344no—
249C2×C8—172no—
250C2×C8—344no—
251C2×C8C22×D1712yes—
252C4⋊C4—21no—
253C4⋊C4—171no—
254C4⋊C4—171no—
255C4⋊C4—341no—
256C4⋊C4—341no—
257C4⋊C4C22×D1712yes—
258C22⋊C4—342no—
259C22⋊C4—342no—
260C22⋊C4—341no—
261C22⋊C4—341no—
262C22⋊C4—341no—
263C22⋊C4—341no—
264C22⋊C4—341no—
265C42—341no—
266C23—21no—
267C23—171no—
268C23—171no—
269C23—342no—
270C23—342no—
271C23—342no—
272C23—681no—
273C23—682no—
274C23—682no—
275C23—682no—
276C23D4×D1711yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—342no—
283D4—342no—
284D4—684no—
285D4—684no—
286D4—682no—
287D4—682no—
288D4—682no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—171no—
294C2×C4—171no—
295C2×C4—342no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—342no—
301C2×C4—342no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4#6325212yes—
305C2×C4D4×D1712yes—
306C2×C4D4×D1711yescharacteristic
307C2×C4C23×D1711yescharacteristic
308C8—22no—
309C8—174no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—172no—
316C22—341no—
317C22—341no—
318C22—341no—
319C22—341no—
320C22—341no—
321C22—684no—
322C22—682no—
323C22—682no—
324C22—682no—
325C22—1364no—
326C22#1058757012yes—
327C22C2×D4×D1711yescharacteristic
328C4—41no—
329C4—42no—
330C4—341no—
331C4—341no—
332C4—681no—
333C4—681no—
334C4#1058757012yes—
335C4C2×#6325211yescharacteristic
336C4C2×D4×D1711yescharacteristic
337C2—22no—
338C2—81no—
339C2—342no—
340C2—681no—
341C2—1362no—
342C2C2×#1058757011yescharacteristic
343C2AGL(1,ℤ/8)×D1711yescharacteristic
344C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#417C2
socleC22×C17C22×D4
hypercenter#417D17
hypocenterC17#3055
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#10587635C2semidirect
2. C2×M16×D17C2semidirect
3. #11763316C2semidirect
4. C2×#10587581C2semidirect
5. #11763061C2semidirect
6. #11763058C2semidirect
7. #11763044C2semidirect
8. #11763041C2semidirect
9. #11762849C2semidirect
10. #11762815C2semidirect
11. C17×#417C2semidirect
19. #10587632C22semidirect
20. #10587632C22semidirect
21. C2×#63230C22semidirect
22. #10587624C22semidirect
23. #10587624C22semidirect
26. C2×#63228C22semidirect
33. C2×#63218C22semidirect
34. C2×C8×D17C22semidirect
39. C17×#103C22semidirect
40. C17×#102C22semidirect
41. #10587492C22semidirect
42. #10587492C22semidirect
43. #10587479C22semidirect
44. #10587478C22semidirect
89. #63223D4semidirect
136. #3622C2×D4semidirect
155. C2×D17AGL(1,ℤ/8)semidirect
175. #417D17semidirect
213. #103C2×D17semidirect
216. #102C2×D17semidirect
217. C17#3055semidirect

Maximal subgroups

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

subgroupindex
C2×#105876352
C2×M16×D172
#117633162
C2×#105875812
#117630612
#117630582
#117630442
#117630412
#117628492
#117628152
C17×#4172
#305517