Finite Group Databasebeta

← #14540998

Subgroups — #14540998

Order 2176.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763498C211yesmaximal, characteristic, complemented
2#11763450C211yesmaximal, characteristic
3#11763165C211yesmaximal, characteristic, complemented
4#11763129C211yesmaximal, characteristic, complemented
5#11763112C211yesmaximal, characteristic, complemented
6#11763065C211yesmaximal, characteristic, complemented
7#11763056C211yesmaximal, characteristic, complemented
8#11763034C211yesmaximal, characteristic, complemented
9#11762974C211yesmaximal, characteristic, complemented
10#11762962C211yesmaximal, characteristic, complemented
11#11762896C211yesmaximal, characteristic, complemented
12#11762884C211yesmaximal, characteristic, complemented
13#11762813C211yesmaximal, characteristic, complemented
14#11762809C211yesmaximal, characteristic, complemented
15C17×#485C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C17×C22⋊Q8C2211yescharacteristic
19C17×#117C2211yescharacteristic
20#10587633C2211yescharacteristic
21C2×#63231C2211yescharacteristic, complemented
22C4×C22⋊D17C2211yescharacteristic
23#10587610C2211yescharacteristic
24C2×#63223C2211yescharacteristic
25C2×#63219C2211yescharacteristic
26C2×#63219C2211yescharacteristic
27C2×#63218C2211yescharacteristic, complemented
28#10587583C2211yescharacteristic, complemented
29#10587581—21no—
30#10587581C2211yescharacteristic
31#10587580—21no—
32#10587579C2211yescharacteristic
33C4⋊C4×D17C2211yescharacteristic, complemented
34#10587576C2211yescharacteristic, complemented
35#10587573C2211yescharacteristic, complemented
36#10587569—21no—
37#10587568—21no—
38C22⋊C4×D17—21no—
39#10587565—21no—
40#10587558—21no—
41#10587556—21no—
42C17×#107C2211yescharacteristic, complemented
43C17×#106C2211yescharacteristic, complemented
44C17×#103C2211yescharacteristic, complemented
45C17×#103C2211yescharacteristic, complemented
46C17×C22⋊C8C2211yescharacteristic
47#10587506C2211yescharacteristic, complemented
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic
50#10587491C2211yescharacteristic, complemented
51#10587485C2211yescharacteristic, complemented
52C8×Dic17C2211yescharacteristic, complemented
53#10587481C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic
55#10587480C2211yescharacteristic
56#10587480C2211yescharacteristic
57#10587480C2211yescharacteristic
58#10587479C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C2×C22⋊D17D411yescharacteristic
63#63256—22no—
64#63256—41no—
65C2×D68—22no—
66C2×D68—21no—
67C2×D68—21no—
68C2×D68—21no—
69C2×D68—41no—
70C2×D68C2311yescharacteristic
71C2×D68C2311yescharacteristic
72C2×C4×D17—21no—
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17C2311yescharacteristic
76C2×Dic34C2311yescharacteristic
77C2×C8×C17C2311yescharacteristic
78C2×C8×C17C2311yescharacteristic
79C4⋊C4×C17—21no—
80C4⋊C4×C17C2311yescharacteristic
81C4⋊C4×C17C2311yescharacteristic
82C4⋊C4×C17C2311yescharacteristic
83C22⋊C4×C17—21no—
84C22⋊C4×C17—21no—
85C42×C17—21no—
86C22⋊Dic17—21no—
87C22⋊Dic17D411yescharacteristic
88#63231—41no—
89#63227—21no—
90#63227—21no—
91#63227—21no—
92#63227—21no—
93#63227—21no—
94#63227D411yescharacteristic
95C4⋊Dic17—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17C2311yescharacteristic
98#63225—21no—
99#63225—21no—
100#63225—21no—
101#63225—21no—
102#63225D411yescharacteristic
103C4×Dic17—21no—
104C4×Dic17C2311yescharacteristic
105#63223—41no—
106C2×#3622C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108#63219—22no—
109#63219—41no—
110#63218—41no—
111C23×C17C2×D411yescharacteristic
112C22×D17—21no—
113C22×D17—21no—
114C22×D17—42no—
115C22×D17—42no—
116C22×D17—81no—
117C22×D17C2×D411yescharacteristic
118Q8×C17—21no—
119Q8×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17C2×D411yescharacteristic
127C2×C4×C17C2411yescharacteristic
128C22⋊D17—41no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17C2×D411yescharacteristic
132C2×Dic17C2×D411yescharacteristic
133C2×Dic17C2×D411yescharacteristic
134D68—21no—
135D68—21no—
136D68—21no—
137D68—42no—
138D68—42no—
139D68—81no—
140D68#4112yes—
141C4×D17—21no—
142C4×D17—21no—
143C4×D17—42no—
144C4×D17—42no—
145C4×D17—81no—
146Dic34—21no—
147Dic34—21no—
148C8×C17—21no—
149C8×C17—21no—
150#3622—21no—
151#3622—21no—
152#3323—171nomaximal, Sylow
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17C22×D411yescharacteristic
157C2×D17—21no—
158C2×D17—22no—
159C2×D17—21no—
160C2×D17—21no—
161C2×D17—41no—
162C2×D17—41no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—84no—
167C4×C17—21no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17C2×#4111yescharacteristic
173C4×C17H(5,𝔽2)11yescharacteristic
174Dic17—21no—
175Dic17—21no—
176Dic17—21no—
177Dic17—41no—
178Dic17—41no—
179#552—171no—
180#546—171no—
181#488—171no—
182#485D1711yescharacteristic, complemented
183#481—171no—
184#478—171no—
185#470—171no—
186#468—171no—
187#464—171no—
188#452—171no—
189#450—171no—
190C4×SD16—171no—
191#436—171no—
192#418—171no—
193#417—171no—
194C2×C17—41no—
195C2×C17#54611yescharacteristic
196C2×C17#57511yescharacteristic
197C2×C17#57711yescharacteristic
198D17—41no—
199D17—42no—
200D17—81no—
201C2×#41—171no—
202C22×D4—171no—
203C2×Q16—171no—
204C2×SD16—171no—
205C2×SD16—171no—
206C2×M16—171no—
207C2×M16—171no—
208#126—341no—
209#125—171no—
210#125—341no—
211#124—171no—
212#123—341no—
213#123—341no—
214C22⋊Q8—171no—
215C22⋊Q8—171no—
216C22⋊Q8C2×D1711yescharacteristic
217#121—171no—
218#121—341no—
219#120—341no—
220C4×Q8—171no—
221C4×D4—171no—
222C4×D4—171no—
223#117—341no—
224#117C2×D1711yescharacteristic
225C2×C4⋊C4—171no—
226C2×C22⋊C4—341no—
227#107C2×D1711yescharacteristic, complemented
228#106—171no—
229#106C2×D1711yescharacteristic, complemented
230C4⋊C8—171no—
231#103—171no—
232#103—171no—
233#103C2×D1711yescharacteristic, complemented
234#103C2×D1711yescharacteristic, complemented
235#102—171no—
236#102—171no—
237#102—171no—
238#102—171no—
239#102—171no—
240#102—171no—
241C22⋊C8—171no—
242C22⋊C8C2×D1711yescharacteristic
243C4×C8—171no—
244C17#332311yescharacteristic, complemented, Sylow
245C24—341no—
246#41—342no—
247#41—681no—
248C2×Q8—171no—
249C2×Q8C22×D1711yescharacteristic
250C2×D4—171no—
251C2×D4—171no—
252C2×D4—171no—
253C2×D4—342no—
254C2×D4—341no—
255C2×D4—341no—
256C2×D4—341no—
257C2×D4—681no—
258C22×C4—171no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4C22×D1711yescharacteristic
263Q16—681no—
264SD16—342no—
265SD16—681no—
266M16—681no—
267M16—681no—
268C2×C8—171no—
269C2×C8—171no—
270C2×C8C22×D1711yescharacteristic
271C2×C8C22×D1711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—171no—
274C4⋊C4—171no—
275C4⋊C4—341no—
276C4⋊C4—341no—
277C4⋊C4—341no—
278C4⋊C4—341no—
279C4⋊C4—341no—
280C4⋊C4—341no—
281C4⋊C4C22×D1711yescharacteristic
282C4⋊C4C22×D1711yescharacteristic
283C4⋊C4C22×D1711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—171no—
287C22⋊C4—171no—
288C22⋊C4—341no—
289C22⋊C4—341no—
290C22⋊C4—341no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C42—21no—
295C42—171no—
296C42—341no—
297C23—171no—
298C23—341no—
299C23—341no—
300C23—682no—
301C23—682no—
302C23—1361no—
303C23D4×D1711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—341no—
307Q8—341no—
308D4—172no—
309D4—341no—
310D4—341no—
311D4—341no—
312D4—682no—
313D4—682no—
314D4—681no—
315D4—1361no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—171no—
323C2×C4—171no—
324C2×C4—171no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—682no—
330C2×C4—682no—
331C2×C4—1361no—
332C2×C4D4×D1711yescharacteristic
333C2×C4C23×D1711yescharacteristic
334C8—21no—
335C8—21no—
336C8—341no—
337C8—341no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—341no—
342C22—341no—
343C22—341no—
344C22—342no—
345C22—681no—
346C22—681no—
347C22—681no—
348C22—681no—
349C22—681no—
350C22—1364no—
351C22C2×D4×D1711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—341no—
358C4—341no—
359C4—341no—
360C4—681no—
361C4—681no—
362C4#41×D1711yescharacteristic
363C4#1058769611yescharacteristic
364C2—41no—
365C2—681no—
366C2—682no—
367C2—1361no—
368C2#1176345111yescharacteristic
369C2#1176360811yescharacteristic
370C2#1176361611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#485C2
socleC22×C17C22×D4
prime radical#11763450C2
hypercenter#485D17
hypocenterC17#3323
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #11763498C2semidirect
3. #11763165C2semidirect
4. #11763129C2semidirect
5. #11763112C2semidirect
6. #11763065C2semidirect
7. #11763056C2semidirect
8. #11763034C2semidirect
9. #11762974C2semidirect
10. #11762962C2semidirect
11. #11762896C2semidirect
12. #11762884C2semidirect
13. #11762813C2semidirect
14. #11762809C2semidirect
15. C17×#485C2semidirect
21. C2×#63231C22semidirect
27. C2×#63218C22semidirect
28. #10587583C22semidirect
33. C4⋊C4×D17C22semidirect
34. #10587576C22semidirect
35. #10587573C22semidirect
42. C17×#107C22semidirect
43. C17×#106C22semidirect
44. C17×#103C22semidirect
45. C17×#103C22semidirect
47. #10587506C22semidirect
50. #10587491C22semidirect
51. #10587485C22semidirect
52. C8×Dic17C22semidirect
53. #10587481C22semidirect
58. #10587479C22semidirect
182. #485D17semidirect
227. #107C2×D17semidirect
229. #106C2×D17semidirect
233. #103C2×D17semidirect
234. #103C2×D17semidirect
244. C17#3323semidirect

Maximal subgroups

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

subgroupindex
#117634982
#117634502
#117631652
#117631292
#117631122
#117630652
#117630562
#117630342
#117629742
#117629622
#117628962
#117628842
#117628132
#117628092
C17×#4852
#332317