Finite Group Databasebeta

← #14538968

Subgroups — #14538968

Order 2176.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763430C211yesmaximal, characteristic, complemented
2#11763150C211yesmaximal, characteristic, complemented
3#11763133C211yesmaximal, characteristic, complemented
4#11763099C211yesmaximal, characteristic, complemented
5#11763089C211yesmaximal, characteristic, complemented
6#11763024C211yesmaximal, characteristic, complemented
7#11763024C211yesmaximal, characteristic, complemented
8#11762960C211yesmaximal, characteristic, complemented
9#11762943C211yesmaximal, characteristic
10#11762919C211yesmaximal, characteristic, complemented
11#11762909C211yesmaximal, characteristic, complemented
12C4×#63229C211yesmaximal, characteristic, complemented
13C4×#63228C211yesmaximal, characteristic, complemented
14#11762870C211yesmaximal, characteristic, complemented
15C17×#518C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C4×D4×C17—21no—
18C4×D4×C17—21no—
19C4×D4×C17C2211yescharacteristic
20C4×D4×C17C2211yescharacteristic
21C4×D4×C17C2211yescharacteristic
22C4×D4×C17C2211yescharacteristic
23C17×#117C2211yescharacteristic
24C17×#117C2211yescharacteristic
25C2×C22⋊C4×C17—21no—
26C2×C22⋊C4×C17—21no—
27C2×#63229C2211yescharacteristic
28C2×#63229C2211yescharacteristic
29C2×#63228C2211yescharacteristic
30C2×#63228C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587610C2211yescharacteristic
33#10587584—21no—
34#10587582—21no—
35#10587571—21no—
36#10587569—21no—
37#10587561C2211yescharacteristic, complemented
38C4×D68C2211yescharacteristic, complemented
39#10587555C2211yescharacteristic, complemented
40C4×Dic34C2211yescharacteristic, complemented
41#10587503C2211yescharacteristic
42#10587503C2211yescharacteristic
43#10587503C2211yescharacteristic
44#10587503C2211yescharacteristic
45C22⋊#3622C2211yescharacteristic
46C22⋊#3622C2211yescharacteristic
47#10587481C2211yescharacteristic, complemented
48#10587481C2211yescharacteristic, complemented
49#10587480C2211yescharacteristic, complemented
50#10587480C2211yescharacteristic, complemented
51#10587479C2211yescharacteristic, complemented
52#10587479C2211yescharacteristic, complemented
53#10587478C2211yescharacteristic, complemented
54#10587478C2211yescharacteristic, complemented
55C4⋊#3622C2211yescharacteristic, complemented
56C4⋊#3622C2211yescharacteristic, complemented
57#10587474C2211yescharacteristic, complemented
58C4×#3622C2211yescharacteristic, complemented
59C24×C17—21no—
60C2×D4×C17—22no—
61C2×D4×C17—22no—
62C2×D4×C17—21no—
63C2×D4×C17C2311yescharacteristic
64C2×D4×C17C2311yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17C2311yescharacteristic
70C2×C22⋊D17—21no—
71C2×D68C2311yescharacteristic
72C2×C4×D17—21no—
73C2×Dic34—21no—
74C2×Dic34C2311yescharacteristic
75C4⋊C4×C17D411yescharacteristic
76C4⋊C4×C17D411yescharacteristic
77C4⋊C4×C17C2311yescharacteristic
78C4⋊C4×C17C2311yescharacteristic
79C22⋊C4×C17—21no—
80C22⋊C4×C17—21no—
81C22⋊C4×C17—21no—
82C22⋊C4×C17—21no—
83C22⋊C4×C17—21no—
84C22⋊C4×C17D411yescharacteristic
85C22⋊C4×C17D411yescharacteristic
86C42×C17—21no—
87C42×C17C2311yescharacteristic
88C42×C17C2311yescharacteristic
89C22⋊Dic17—21no—
90#63229—22no—
91#63229—22no—
92#63228—22no—
93#63228—22no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98C4⋊Dic17—21no—
99C4⋊Dic17C2311yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101#63225—21no—
102#63225—21no—
103C4×Dic17—21no—
104C2×#3622C2311yescharacteristic
105C2×#3622C2311yescharacteristic
106C2×#3622C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108C23×C17—24no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—81no—
112C23×C17—81no—
113C23×C17C2×D411yescharacteristic
114C22×D17—21no—
115D4×C17—24no—
116D4×C17—21no—
117D4×C17—21no—
118D4×C17#4112yes—
119D4×C17#4112yes—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—81no—
125C2×C4×C17—81no—
126C2×C4×C17—81no—
127C2×C4×C17—81no—
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2×D411yescharacteristic
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—81no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138D68—21no—
139D68—21no—
140C4×D17—81no—
141Dic34—21no—
142Dic34—21no—
143Dic34—81no—
144#3622—21no—
145#3622—21no—
146#3622—21no—
147#3622—21no—
148#3118—171nomaximal, Sylow
149C22×C17—21no—
150C22×C17—22no—
151C22×C17—22no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—84no—
159C22×C17C22×D411yescharacteristic
160C2×D17—41no—
161C2×D17—41no—
162C2×D17—41no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17C2×#4111yescharacteristic
171C4×C17C2×#4111yescharacteristic
172Dic17—41no—
173Dic17—41no—
174Dic17—41no—
175#542—171no—
176#518D1711yescharacteristic, complemented
177#485—171no—
178#482—171no—
179#476—171no—
180#474—171no—
181#463—171no—
182#463—171no—
183#450—171no—
184#447—171no—
185#442—171no—
186#440—171no—
187C4×SD16—171no—
188C4×D8—171no—
189#433—171no—
190C2×C17—41no—
191C2×C17—42no—
192C2×C17—42no—
193C2×C17#52511yescharacteristic
194C2×C17#57611yescharacteristic
195C2×C17#57711yescharacteristic
196D17—81no—
197C22×D4C2×D1711yescharacteristic
198C2×SD16—171no—
199C2×SD16—171no—
200C2×D8—171no—
201C2×D8—171no—
202C4⋊Q8—171no—
203#126—341no—
204#124—171no—
205#124—341no—
206#123—341no—
207C22⋊Q8—171no—
208C22⋊Q8—341no—
209#121—171no—
210C4×Q8—171no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—171no—
214C4×D4C2×D1711yescharacteristic
215C4×D4C2×D1711yescharacteristic
216C4×D4C2×D1711yescharacteristic
217C4×D4C2×D1711yescharacteristic
218#117C2×D1711yescharacteristic
219#117C2×D1711yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222#107—171no—
223#107—171no—
224#106—171no—
225#106—171no—
226C4⋊C8—171no—
227C4⋊C8—171no—
228#103—171no—
229#103—171no—
230#102—171no—
231#102—171no—
232#102—171no—
233#102—171no—
234#102—171no—
235#102—171no—
236C22⋊C8—171no—
237C22⋊C8—171no—
238#97—171no—
239C4×C8—171no—
240C17#311811yescharacteristic, complemented, Sylow
241C24—21no—
242C2×Q8—171no—
243C2×Q8—341no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—171no—
248C2×D4—341no—
249C2×D4C22×D1711yescharacteristic
250C2×D4C22×D1711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—341no—
256C22×C4C22×D1711yescharacteristic
257SD16—342no—
258SD16—342no—
259D8—342no—
260D8—342no—
261C2×C8—171no—
262C2×C8—171no—
263C2×C8—171no—
264C2×C8—171no—
265C4⋊C4—171no—
266C4⋊C4—171no—
267C4⋊C4—341no—
268C4⋊C4—341no—
269C4⋊C4—341no—
270C4⋊C4C22⋊D1711yescharacteristic
271C4⋊C4C22⋊D1711yescharacteristic
272C4⋊C4C22×D1711yescharacteristic
273C4⋊C4C22×D1711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—341no—
280C22⋊C4—341no—
281C22⋊C4—341no—
282C22⋊C4—341no—
283C22⋊C4—341no—
284C22⋊C4C22⋊D1711yescharacteristic
285C22⋊C4C22⋊D1711yescharacteristic
286C42—21no—
287C42—341no—
288C42C22×D1711yescharacteristic
289C42C22×D1711yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—341no—
296C23C2×C22⋊D1711yescharacteristic
297Q8—341no—
298Q8—341no—
299Q8—1361no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—341no—
304D4—341no—
305D4—1361no—
306D4#6325212yes—
307D4#6325212yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—341no—
317C2×C4—341no—
318C2×C4—341no—
319C2×C4—1361no—
320C2×C4C2×C22⋊D1711yescharacteristic
321C2×C4C2×C22⋊D1711yescharacteristic
322C2×C4C2×C22⋊D1711yescharacteristic
323C2×C4C2×C22⋊D1711yescharacteristic
324C2×C4C2×C22⋊D1711yescharacteristic
325C2×C4C23×D1711yescharacteristic
326C8—341no—
327C8—341no—
328C8—341no—
329C8—341no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—681no—
341C22—681no—
342C22—681no—
343C22C22×C22⋊D1711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—681no—
352C4—681no—
353C4—681no—
354C4C2×#6325211yescharacteristic
355C4C2×#6325211yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—1361no—
360C2#1176332011yescharacteristic
361C2#1176361211yescharacteristic
362C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#518C2
socleC22×C17C22×D4
prime radical#11762943C2
hypercenter#518D17
hypocenterC17#3118
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #11763430C2semidirect
2. #11763150C2semidirect
3. #11763133C2semidirect
4. #11763099C2semidirect
5. #11763089C2semidirect
6. #11763024C2semidirect
7. #11763024C2semidirect
8. #11762960C2semidirect
10. #11762919C2semidirect
11. #11762909C2semidirect
12. C4×#63229C2semidirect
13. C4×#63228C2semidirect
14. #11762870C2semidirect
15. C17×#518C2semidirect
37. #10587561C22semidirect
38. C4×D68C22semidirect
39. #10587555C22semidirect
40. C4×Dic34C22semidirect
47. #10587481C22semidirect
48. #10587481C22semidirect
49. #10587480C22semidirect
50. #10587480C22semidirect
51. #10587479C22semidirect
52. #10587479C22semidirect
53. #10587478C22semidirect
54. #10587478C22semidirect
55. C4⋊#3622C22semidirect
56. C4⋊#3622C22semidirect
57. #10587474C22semidirect
58. C4×#3622C22semidirect
176. #518D17semidirect
240. C17#3118semidirect

Maximal subgroups

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

subgroupindex
#117634302
#117631502
#117631332
#117630992
#117630892
#117630242
#117629602
#117629432
#117629192
#117629092
C4×#632292
C4×#632282
#117628702
C17×#5182
#311817