Finite Group Databasebeta

← #14539744

Subgroups — #14539744

Order 2176.

410 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763338C211yesmaximal, characteristic, complemented
2#124×D17C211yesmaximal, characteristic, complemented
3#11763191C211yesmaximal, characteristic, complemented
4#11763164C211yesmaximal, characteristic, complemented
5#11763163C211yesmaximal, characteristic, complemented
6#11763033C211yesmaximal, characteristic, complemented
7#11763029C211yesmaximal, characteristic, complemented
8#11763026C211yesmaximal, characteristic, complemented
9#11763022C211yesmaximal, characteristic, complemented
10#11762963C211yesmaximal, characteristic, complemented
11#11762958C211yesmaximal, characteristic, complemented
12#11762946C211yesmaximal, characteristic, complemented
13#11762941C211yesmaximal, characteristic
14#11762860C211yesmaximal, characteristic, complemented
15C17×#495C211yesmaximal, characteristic, complemented
16C2×Q8×D17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×Q16×C17C2211yescharacteristic, complemented
19C2×SD16×C17C2211yescharacteristic
20C2×SD16×C17C2211yescharacteristic, complemented
21C2×D8×C17C2211yescharacteristic
22C17×#124C2211yescharacteristic
23C17×#124C2211yescharacteristic, complemented
24#10587633C2211yescharacteristic
25#10587632C2211yescharacteristic
26Q8×Dic17C2211yescharacteristic, complemented
27C2×#63231C2211yescharacteristic, complemented
28C2×#63230C2211yescharacteristic
29#10587624C2211yescharacteristic
30#10587622C2211yescharacteristic
31D4×Dic17C2211yescharacteristic, complemented
32C2×#63229C2211yescharacteristic, complemented
33C2×#63228C2211yescharacteristic
34#10587571—21no—
35#10587571—21no—
36#10587569—21no—
37#10587567—21no—
38C22⋊C4×D17—21no—
39C22⋊C4×D17—21no—
40#10587565—21no—
41#10587563—21no—
42#10587561C2211yescharacteristic
43C42×D17C2211yescharacteristic
44#10587556C2211yescharacteristic, complemented
45C4×C8×C17C2211yescharacteristic, complemented
46#10587506C2211yescharacteristic, complemented
47#10587506C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587492C2211yescharacteristic
51#10587492C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587491C2211yescharacteristic
54#10587487C2211yescharacteristic, complemented
55#10587487C2211yescharacteristic, complemented
56#10587485C2211yescharacteristic, complemented
57#10587485C2211yescharacteristic, complemented
58C4×#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C2×D4×C17C2311yescharacteristic
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C22×Dic17—21no—
67Q8×D17—41no—
68Q8×D17—41no—
69D4×D17—41no—
70D4×D17—41no—
71C2×D68C2311yescharacteristic
72C2×C4×D17—21no—
73C2×C4×D17D411yescharacteristic
74C2×C4×D17D411yescharacteristic
75C2×C4×D17C2311yescharacteristic
76C2×Dic34—21no—
77C2×Dic34C2311yescharacteristic
78Q16×C17—41no—
79SD16×C17—41no—
80SD16×C17—41no—
81D8×C17—41no—
82C2×C8×C17C2311yescharacteristic
83C2×C8×C17C2311yescharacteristic
84C22⋊C4×C17—21no—
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C22⋊C4×C17—21no—
88C42×C17C2311yescharacteristic
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92C22⋊Dic17—21no—
93#63231—41no—
94#63230—41no—
95#63229—41no—
96#63228—41no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17C2311yescharacteristic
104C4⋊Dic17C2311yescharacteristic
105#63225—21no—
106#63225—21no—
107#63225—21no—
108C4×Dic17—21no—
109C4×Dic17D411yescharacteristic
110C4×Dic17D411yescharacteristic
111C4×Dic17C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113C2×#3622C2311yescharacteristic
114C23×C17—21no—
115C23×C17—21no—
116C22×D17—21no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—42no—
120C22×D17C2×D411yescharacteristic
121Q8×C17—21no—
122Q8×C17—21no—
123Q8×C17—21no—
124Q8×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—42no—
135C22⋊D17—42no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—42no—
140C2×Dic17C2×D411yescharacteristic
141C2×Dic17C2×D411yescharacteristic
142C2×Dic17C2×D411yescharacteristic
143D68—21no—
144D68—21no—
145C4×D17—22no—
146C4×D17—21no—
147C4×D17—21no—
148C4×D17—41no—
149C4×D17—42no—
150Dic34—21no—
151Dic34—21no—
152Dic34—42no—
153C8×C17—21no—
154C8×C17—21no—
155#3622—21no—
156#3622—21no—
157#3211—171nomaximal, Sylow
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17C22×D411yescharacteristic
165C2×D17—21no—
166C2×D17—21no—
167C2×D17—41no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—84no—
171C2×D17#13512yes—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17H(5,𝔽2)11yescharacteristic
177C4×C17H(5,𝔽2)11yescharacteristic
178Dic17—21no—
179Dic17—21no—
180Dic17—21no—
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184#529—171no—
185C2×#124—171no—
186#495—171no—
187#495D1711yescharacteristic, complemented
188#488—171no—
189#488—171no—
190#464—171no—
191#464—171no—
192#463—171no—
193#463—171no—
194#451—171no—
195#450—171no—
196#448—171no—
197#447—171no—
198#431—171no—
199C2×C17—81no—
200C2×C17—81no—
201C2×C17#53511yescharacteristic
202C2×C17C2×#13511yescharacteristic
203C2×C17#57511yescharacteristic
204D17—22no—
205D17—81no—
206C22×Q8—171no—
207C22×D4—171no—
208C2×Q16—171no—
209C2×Q16C2×D1711yescharacteristic, complemented
210C2×SD16—171no—
211C2×SD16—171no—
212C2×SD16C2×D1711yescharacteristic
213C2×SD16C2×D1711yescharacteristic, complemented
214C2×D8—171no—
215C2×D8C2×D1711yescharacteristic
216#126—341no—
217#125—171no—
218#124—171no—
219#124—171no—
220#124—171no—
221#124—341no—
222#124—341no—
223#124C2×D1711yescharacteristic
224#124C2×D1711yescharacteristic, complemented
225#123—341no—
226C22⋊Q8—171no—
227#121—171no—
228C4×Q8—171no—
229C4×D4—171no—
230C4×D4—341no—
231#117—341no—
232C2×C22⋊C4—341no—
233C2×C22⋊C4—341no—
234C2×C42—171no—
235C4⋊C8—171no—
236C4⋊C8—171no—
237#103—171no—
238#103—171no—
239#103—171no—
240#103—171no—
241#102—171no—
242#102—171no—
243#102—171no—
244#102—171no—
245C22⋊C8—171no—
246C22⋊C8—171no—
247C4×C8—171no—
248C4×C8C2×D1711yescharacteristic, complemented
249C17#321111yescharacteristic, complemented, Sylow
250C24—341no—
251C2×Q8—171no—
252C2×Q8—341no—
253C2×Q8—681no—
254C2×Q8—681no—
255C2×Q8C22×D1711yescharacteristic
256C2×Q8C22×D1711yescharacteristic
257C2×D4—171no—
258C2×D4—341no—
259C2×D4—341no—
260C2×D4—681no—
261C2×D4—681no—
262C2×D4C22×D1711yescharacteristic
263C2×D4C22×D1711yescharacteristic
264C22×C4—171no—
265C22×C4—171no—
266C22×C4—171no—
267C22×C4—341no—
268C22×C4—341no—
269Q16—41no—
270Q16—681no—
271SD16—41no—
272SD16—41no—
273SD16—681no—
274SD16—681no—
275D8—41no—
276D8—681no—
277C2×C8—171no—
278C2×C8—171no—
279C2×C8C22×D1711yescharacteristic
280C2×C8C22×D1711yescharacteristic
281C4⋊C4—171no—
282C4⋊C4—171no—
283C4⋊C4—341no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4—341no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C22⋊C4—341no—
298C22⋊C4—341no—
299C22⋊C4—341no—
300C42—171no—
301C42—171no—
302C42—171no—
303C42—341no—
304C42C22×D1711yescharacteristic
305C23—21no—
306C23—21no—
307C23—171no—
308C23—341no—
309C23—682no—
310C23—682no—
311C23—682no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—341no—
317Q8—341no—
318Q8—682no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—341no—
324D4—341no—
325D4—682no—
326D4—682no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—171no—
330C2×C4—171no—
331C2×C4—171no—
332C2×C4—342no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—681no—
339C2×C4—682no—
340C2×C4—682no—
341C2×C4D4×D1711yescharacteristic
342C2×C4D4×D1711yescharacteristic
343C2×C4C23×D1711yescharacteristic
344C8—21no—
345C8—21no—
346C8—341no—
347C8—341no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—172no—
355C22—341no—
356C22—341no—
357C22—681no—
358C22—681no—
359C22—681no—
360C22—1364no—
361C22C2×D4×D1711yescharacteristic
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—341no—
367C4—341no—
368C4—341no—
369C4—681no—
370C4—681no—
371C4—681no—
372C4#1058769111yescharacteristic
373C4#1058769111yescharacteristic
374C2—81no—
375C2—81no—
376C2—342no—
377C2—1361no—
378C2#1176337211yescharacteristic
379C2#135×D1711yescharacteristic
380C2#1176360611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#495C2
socleC22×C17C22×D4
prime radical#11762941C2
hypercenter#495D17
hypocenterC17#3211
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #11763338C2semidirect
2. #124×D17C2semidirect
3. #11763191C2semidirect
4. #11763164C2semidirect
5. #11763163C2semidirect
6. #11763033C2semidirect
7. #11763029C2semidirect
8. #11763026C2semidirect
9. #11763022C2semidirect
10. #11762963C2semidirect
11. #11762958C2semidirect
12. #11762946C2semidirect
14. #11762860C2semidirect
15. C17×#495C2semidirect
18. C2×Q16×C17C22semidirect
20. C2×SD16×C17C22semidirect
23. C17×#124C22semidirect
26. Q8×Dic17C22semidirect
27. C2×#63231C22semidirect
31. D4×Dic17C22semidirect
32. C2×#63229C22semidirect
44. #10587556C22semidirect
45. C4×C8×C17C22semidirect
46. #10587506C22semidirect
47. #10587506C22semidirect
54. #10587487C22semidirect
55. #10587487C22semidirect
56. #10587485C22semidirect
57. #10587485C22semidirect
58. C4×#3622C22semidirect
187. #495D17semidirect
209. C2×Q16C2×D17semidirect
213. C2×SD16C2×D17semidirect
224. #124C2×D17semidirect
248. C4×C8C2×D17semidirect
249. C17#3211semidirect

Maximal subgroups

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

subgroupindex
#117633382
#124×D172
#117631912
#117631642
#117631632
#117630332
#117630292
#117630262
#117630222
#117629632
#117629582
#117629462
#117629412
#117628602
C17×#4952
#321117