Finite Group Databasebeta

← #14539612

Subgroups — #14539612

Order 2176.

414 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763316C211yesmaximal, characteristic
2C22⋊Q8×D17C211yesmaximal, characteristic, complemented
3#11763151C211yesmaximal, characteristic, complemented
4#11763147C211yesmaximal, characteristic, complemented
5#11763143C211yesmaximal, characteristic, complemented
6#11763126C211yesmaximal, characteristic, complemented
7#11763066C211yesmaximal, characteristic, complemented
8#11763064C211yesmaximal, characteristic, complemented
9#11763054C211yesmaximal, characteristic, complemented
10#11763052C211yesmaximal, characteristic, complemented
11#11762971C211yesmaximal, characteristic, complemented
12#11762963C211yesmaximal, characteristic, complemented
13#11762946C211yesmaximal, characteristic, complemented
14#11762771C211yesmaximal, characteristic, complemented
15C17×#452C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×Q8×D17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C2×Q16×C17C2211yescharacteristic, complemented
20C2×SD16×C17C2211yescharacteristic, complemented
21C17×C22⋊Q8C2211yescharacteristic
22#10587632C2211yescharacteristic
23#10587632C2211yescharacteristic
24#10587632C2211yescharacteristic
25C2×#63231C2211yescharacteristic, complemented
26#10587624C2211yescharacteristic
27#10587624C2211yescharacteristic
28#10587623—21no—
29#10587621—21no—
30C2×#63229C2211yescharacteristic, complemented
31C4×C22⋊D17—21no—
32C4×C22⋊D17—21no—
33#10587612C2211yescharacteristic
34#10587610C2211yescharacteristic
35C2×#63218C2211yescharacteristic
36C2×#63218C2211yescharacteristic
37#10587583C2211yescharacteristic
38#10587582—21no—
39C4⋊C4×D17—21no—
40C4⋊C4×D17C2211yescharacteristic
41C22⋊C4×D17—21no—
42#10587564—21no—
43C17×#103C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic, complemented
45C17×C22⋊C8C2211yescharacteristic
46#10587506C2211yescharacteristic, complemented
47#10587506C2211yescharacteristic, complemented
48#10587506C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic, complemented
50C22⋊#3622C2211yescharacteristic
51#10587491C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587489C2211yescharacteristic, complemented
54#10587488C2211yescharacteristic, complemented
55#10587487C2211yescharacteristic, complemented
56#10587487C2211yescharacteristic, complemented
57#10587479C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17D411yescharacteristic
60#41×C17—41no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×Q8×C17C2311yescharacteristic
64C2×D4×C17—21no—
65C2×D4×C17C2311yescharacteristic
66C22×C4×C17—21no—
67C22×C4×C17C2311yescharacteristic
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C22×Dic17D411yescharacteristic
71Q8×D17—41no—
72Q8×D17—41no—
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17—41no—
76C2×C4×D17—41no—
77C2×C4×D17D411yescharacteristic
78C2×C4×D17D411yescharacteristic
79C2×C4×D17C2311yescharacteristic
80C2×C4×D17C2311yescharacteristic
81C2×Dic34—21no—
82C2×Dic34C2311yescharacteristic
83Q16×C17—41no—
84SD16×C17—41no—
85C2×C8×C17C2311yescharacteristic
86C2×C8×C17C2311yescharacteristic
87C4⋊C4×C17—21no—
88C4⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92C22⋊Dic17—21no—
93C22⋊Dic17—21no—
94#63231—41no—
95#63229—41no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103C4⋊Dic17C2311yescharacteristic
104#63225—21no—
105#63225—21no—
106#63225—21no—
107#63225—21no—
108#63225—21no—
109C4×Dic17—21no—
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112#63218—41no—
113#63218—41no—
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C22×D17—22no—
117C22×D17—41no—
118C22×D17—41no—
119C22×D17C2×D411yescharacteristic
120C22×D17C2×D411yescharacteristic
121Q8×C17—21no—
122Q8×C17—21no—
123Q8×C17—21no—
124Q8×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—42no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—42no—
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2411yescharacteristic
136C22⋊D17—42no—
137C22⋊D17—42no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—41no—
143C2×Dic17C2×D411yescharacteristic
144C2×Dic17C2×D411yescharacteristic
145C4×D17—21no—
146C4×D17—21no—
147C4×D17—21no—
148C4×D17—21no—
149C4×D17—41no—
150C4×D17—41no—
151C4×D17—42no—
152C4×D17—42no—
153Dic34—21no—
154Dic34—21no—
155Dic34—42no—
156C8×C17—21no—
157C8×C17—21no—
158#3622—21no—
159#3622—21no—
160#3196—171nomaximal, Sylow
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17C22×D411yescharacteristic
168C2×D17—21no—
169C2×D17—21no—
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—21no—
173C2×D17—44no—
174C2×D17#13712yes—
175C4×C17—21no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17H(5,𝔽2)11yescharacteristic
180C4×C17H(5,𝔽2)11yescharacteristic
181Dic17—21no—
182Dic17—21no—
183Dic17—41no—
184Dic17—41no—
185Dic17—41no—
186Dic17—41no—
187#525—171no—
188C2×C22⋊Q8—171no—
189#485—171no—
190#485—171no—
191#484—171no—
192#481—171no—
193#470—171no—
194#470—171no—
195#468—171no—
196#468—171no—
197#452—171no—
198#452D1711yescharacteristic, complemented
199#451—171no—
200#448—171no—
201#407—171no—
202C2×C17—41no—
203C2×C17—81no—
204C2×C17#53411yescharacteristic
205C2×C17C2×#13711yescharacteristic
206C2×C17#57511yescharacteristic
207D17—22no—
208D17—41no—
209C2×#41C2×D1711yescharacteristic
210C22×Q8—171no—
211C23×C4—171no—
212C2×Q16—171no—
213C2×Q16C2×D1711yescharacteristic, complemented
214C2×SD16—171no—
215C2×SD16C2×D1711yescharacteristic, complemented
216C2×M16—171no—
217C2×M16—171no—
218#123—341no—
219C22⋊Q8—171no—
220C22⋊Q8—171no—
221C22⋊Q8—171no—
222C22⋊Q8—171no—
223C22⋊Q8—171no—
224C22⋊Q8—341no—
225C22⋊Q8—341no—
226C22⋊Q8C2×D1711yescharacteristic
227#121—171no—
228#121—171no—
229#120—341no—
230C4×D4—341no—
231C4×D4—341no—
232#117—171no—
233C2×C4⋊C4—171no—
234C2×C4⋊C4—341no—
235C2×C22⋊C4—341no—
236#107—171no—
237#107—171no—
238#106—171no—
239#106—171no—
240#103—171no—
241#103—171no—
242#103—171no—
243#103—171no—
244#103—171no—
245#103C2×D1711yescharacteristic, complemented
246#102—171no—
247#102C2×D1711yescharacteristic, complemented
248C22⋊C8—171no—
249C22⋊C8—171no—
250C22⋊C8—171no—
251C22⋊C8C2×D1711yescharacteristic
252C17#319611yescharacteristic, complemented, Sylow
253C24—171no—
254#41—41no—
255#41—41no—
256C2×Q8—171no—
257C2×Q8—341no—
258C2×Q8—681no—
259C2×Q8—681no—
260C2×Q8C22×D1711yescharacteristic
261C2×Q8C22×D1711yescharacteristic
262C2×D4—21no—
263C2×D4—341no—
264C2×D4—341no—
265C2×D4C22×D1711yescharacteristic
266C22×C4—21no—
267C22×C4—171no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—171no—
271C22×C4—171no—
272C22×C4—341no—
273C22×C4—341no—
274C22×C4—681no—
275C22×C4—681no—
276C22×C4C22×D1711yescharacteristic
277Q16—41no—
278Q16—681no—
279SD16—41no—
280SD16—681no—
281M16—681no—
282M16—681no—
283C2×C8—171no—
284C2×C8—171no—
285C2×C8C22×D1711yescharacteristic
286C2×C8C22×D1711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—171no—
289C4⋊C4—171no—
290C4⋊C4—171no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4—341no—
295C4⋊C4—341no—
296C4⋊C4—341no—
297C4⋊C4C22×D1711yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4—341no—
307C42—341no—
308C23—21no—
309C23—171no—
310C23—171no—
311C23—342no—
312C23—681no—
313C23—681no—
314C23D4×D1711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—341no—
320Q8—341no—
321Q8—682no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—682no—
326D4—682no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—171no—
334C2×C4—171no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—682no—
346C2×C4—682no—
347C2×C4—681no—
348C2×C4D4×D1711yescharacteristic
349C2×C4C23×D1711yescharacteristic
350C8—21no—
351C8—21no—
352C8—341no—
353C8—341no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—172no—
361C22—341no—
362C22—341no—
363C22—341no—
364C22—341no—
365C22—341no—
366C22—684no—
367C22C2×D4×D1711yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—341no—
373C4—341no—
374C4—681no—
375C4—681no—
376C4—681no—
377C4—681no—
378C4#1058769111yescharacteristic
379C4#1058769111yescharacteristic
380C2—41no—
381C2—81no—
382C2—342no—
383C2—681no—
384C2#1176336911yescharacteristic
385C2#137×D1711yescharacteristic
386C2#1176360611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#452C2
socleC22×C17C22×D4
prime radical#11763316C2
hypercenter#452D17
hypocenterC17#3196
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D17C2semidirect
3. #11763151C2semidirect
4. #11763147C2semidirect
5. #11763143C2semidirect
6. #11763126C2semidirect
7. #11763066C2semidirect
8. #11763064C2semidirect
9. #11763054C2semidirect
10. #11763052C2semidirect
11. #11762971C2semidirect
12. #11762963C2semidirect
13. #11762946C2semidirect
14. #11762771C2semidirect
15. C17×#452C2semidirect
19. C2×Q16×C17C22semidirect
20. C2×SD16×C17C22semidirect
25. C2×#63231C22semidirect
30. C2×#63229C22semidirect
43. C17×#103C22semidirect
44. C17×#102C22semidirect
46. #10587506C22semidirect
47. #10587506C22semidirect
48. #10587506C22semidirect
49. #10587503C22semidirect
53. #10587489C22semidirect
54. #10587488C22semidirect
55. #10587487C22semidirect
56. #10587487C22semidirect
57. #10587479C22semidirect
58. #10587478C22semidirect
198. #452D17semidirect
213. C2×Q16C2×D17semidirect
215. C2×SD16C2×D17semidirect
245. #103C2×D17semidirect
247. #102C2×D17semidirect
252. C17#3196semidirect

Maximal subgroups

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

subgroupindex
#117633162
C22⋊Q8×D172
#117631512
#117631472
#117631432
#117631262
#117630662
#117630642
#117630542
#117630522
#117629712
#117629632
#117629462
#117627712
C17×#4522
#319617