Finite Group Databasebeta

← #14541007

Subgroups — #14541007

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#11763500C211yesmaximal, characteristic, complemented
2#11763452C211yesmaximal, characteristic
3#11763163C211yesmaximal, characteristic, complemented
4#11763149C211yesmaximal, characteristic, complemented
5#11763126C211yesmaximal, characteristic, complemented
6#11763110C211yesmaximal, characteristic, complemented
7#11763064C211yesmaximal, characteristic, complemented
8#11763052C211yesmaximal, characteristic, complemented
9#11762970C211yesmaximal, characteristic, complemented
10#11762958C211yesmaximal, characteristic, complemented
11C4×#63230C211yesmaximal, characteristic, complemented
12#11762886C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14#11762809C211yesmaximal, characteristic, complemented
15C17×#464C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×Q16×C17C2211yescharacteristic, complemented
19C2×SD16×C17C2211yescharacteristic
20C17×#124C2211yescharacteristic
21C4×Q8×C17C2211yescharacteristic, complemented
22#10587632C2211yescharacteristic
23#10587632C2211yescharacteristic
24#10587632C2211yescharacteristic, complemented
25C2×#63230C2211yescharacteristic
26#10587624C2211yescharacteristic
27C2×#63218C2211yescharacteristic
28C2×#63218C2211yescharacteristic, complemented
29#10587580—21no—
30#10587578C2211yescharacteristic
31C4⋊C4×D17C2211yescharacteristic, complemented
32#10587575—21no—
33#10587570—21no—
34#10587569—21no—
35#10587568—21no—
36C22⋊C4×D17—21no—
37#10587562—21no—
38C4×D68C2211yescharacteristic
39C4×D68C2211yescharacteristic
40#10587558—21no—
41#10587556C2211yescharacteristic, complemented
42C17×C4⋊C8C2211yescharacteristic, complemented
43C17×#103C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic
45#10587506C2211yescharacteristic, complemented
46#10587506C2211yescharacteristic, complemented
47#10587506C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587492C2211yescharacteristic
51#10587492C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587491C2211yescharacteristic, complemented
54#10587489C2211yescharacteristic, complemented
55#10587488C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic
57#10587479C2211yescharacteristic, complemented
58C4×#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C2×C22⋊D17—21no—
64C2×C22⋊D17—21no—
65#63256—22no—
66#63256—41no—
67D4×D17—22no—
68D4×D17—41no—
69C2×D68—21no—
70C2×D68D411yescharacteristic
71C2×D68C2311yescharacteristic
72C2×C4×D17—21no—
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17C2311yescharacteristic
76C2×C4×D17C2311yescharacteristic
77Q16×C17—41no—
78SD16×C17—41no—
79C2×C8×C17C2311yescharacteristic
80C2×C8×C17C2311yescharacteristic
81C4⋊C4×C17—21no—
82C4⋊C4×C17C2311yescharacteristic
83C22⋊C4×C17—21no—
84C22⋊C4×C17—21no—
85C42×C17—21no—
86C42×C17C2311yescharacteristic
87C22⋊Dic17—21no—
88#63230—22no—
89#63227—21no—
90#63227—21no—
91#63227—21no—
92#63227—21no—
93#63227—21no—
94#63227D411yescharacteristic
95#63227D411yescharacteristic
96C4⋊Dic17D411yescharacteristic
97C4⋊Dic17C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99C4⋊Dic17C2311yescharacteristic
100#63225—21no—
101#63225—21no—
102#63225—21no—
103#63225—21no—
104#63225—21no—
105#63225—21no—
106C4×Dic17—21no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109#63218—41no—
110#63218—41no—
111C23×C17—21no—
112C22×D17—21no—
113C22×D17—42no—
114C22×D17—42no—
115C22×D17—81no—
116C22×D17C2×D411yescharacteristic
117C22×D17C2×D411yescharacteristic
118Q8×C17—21no—
119Q8×C17—21no—
120Q8×C17—21no—
121Q8×C17—21no—
122D4×C17—21no—
123D4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2411yescharacteristic
130C22⋊D17—42no—
131C22⋊D17—81no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17C2×D411yescharacteristic
136C2×Dic17C2×D411yescharacteristic
137D68—21no—
138D68—41no—
139D68—42no—
140D68#4112yes—
141C4×D17—21no—
142C4×D17—21no—
143C4×D17—21no—
144C4×D17—21no—
145C4×D17—42no—
146C4×D17—42no—
147C4×D17—81no—
148C8×C17—21no—
149C8×C17—21no—
150#3622—21no—
151#3622—21no—
152#3323—171nomaximal, Sylow
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17C22×D411yescharacteristic
157C2×D17—21no—
158C2×D17—21no—
159C2×D17—22no—
160C2×D17—21no—
161C2×D17—21no—
162C2×D17—21no—
163C2×D17—21no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—84no—
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—41no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17C2×#4111yescharacteristic
173C4×C17H(5,𝔽2)11yescharacteristic
174Dic17—21no—
175Dic17—21no—
176Dic17—41no—
177Dic17—41no—
178Dic17—41no—
179#552—171no—
180#546—171no—
181#488—171no—
182#485—171no—
183#481—171no—
184#478—171no—
185#470—171no—
186#468—171no—
187#464D1711yescharacteristic, complemented
188#452—171no—
189#450—171no—
190C4×SD16—171no—
191#436—171no—
192#418—171no—
193#417—171no—
194C2×C17—81no—
195C2×C17#54611yescharacteristic
196C2×C17#57511yescharacteristic
197C2×C17#57711yescharacteristic
198D17—41no—
199D17—41no—
200D17—42no—
201C2×#41—171no—
202C22×D4—171no—
203C2×Q16C2×D1711yescharacteristic, complemented
204C2×SD16—171no—
205C2×SD16C2×D1711yescharacteristic
206C2×M16—171no—
207C2×M16—171no—
208#126—341no—
209#125—171no—
210#125—341no—
211#124C2×D1711yescharacteristic
212#123—341no—
213#123—341no—
214C22⋊Q8—171no—
215C22⋊Q8—171no—
216C22⋊Q8—171no—
217#121—171no—
218#121—341no—
219#120—341no—
220C4×Q8C2×D1711yescharacteristic, complemented
221C4×D4—171no—
222C4×D4—171no—
223#117—171no—
224#117—341no—
225C2×C4⋊C4—171no—
226C2×C22⋊C4—341no—
227#107—171no—
228#106—171no—
229#106—171no—
230C4⋊C8C2×D1711yescharacteristic, complemented
231#103—171no—
232#103—171no—
233#103—171no—
234#103C2×D1711yescharacteristic, complemented
235#102—171no—
236#102—171no—
237#102—171no—
238#102—171no—
239#102—171no—
240#102C2×D1711yescharacteristic
241C22⋊C8—171no—
242C22⋊C8—171no—
243C4×C8—171no—
244C17#332311yescharacteristic, complemented, Sylow
245C24—341no—
246#41—342no—
247#41—681no—
248C2×Q8C22×D1711yescharacteristic
249C2×Q8C22×D1711yescharacteristic
250C2×D4—171no—
251C2×D4—171no—
252C2×D4—341no—
253C2×D4—341no—
254C2×D4—342no—
255C2×D4—341no—
256C2×D4—681no—
257C2×D4C22×D1711yescharacteristic
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263Q16—41no—
264SD16—41no—
265SD16—342no—
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—171no—
276C4⋊C4—171no—
277C4⋊C4—341no—
278C4⋊C4—341no—
279C4⋊C4—341no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4—341no—
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—341no—
296C42C22×D1711yescharacteristic
297C23—21no—
298C23—171no—
299C23—171no—
300C23—341no—
301C23—682no—
302C23—682no—
303C23—1361no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—21no—
310D4—172no—
311D4—341no—
312D4—681no—
313D4—682no—
314D4—682no—
315D4—1361no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—171no—
320C2×C4—171no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—682no—
329C2×C4—682no—
330C2×C4—1361no—
331C2×C4D4×D1711yescharacteristic
332C2×C4D4×D1711yescharacteristic
333C2×C4C23×D1711yescharacteristic
334C8—21no—
335C8—21no—
336C8—341no—
337C8—341no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—341no—
342C22—341no—
343C22—342no—
344C22—341no—
345C22—341no—
346C22—341no—
347C22—341no—
348C22—681no—
349C22—681no—
350C22—1364no—
351C22C2×D4×D1711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—341no—
358C4—341no—
359C4—681no—
360C4—681no—
361C4—681no—
362C4#1058769111yescharacteristic
363C4#41×D1711yescharacteristic
364C2—81no—
365C2—681no—
366C2—681no—
367C2—682no—
368C2#1176345611yescharacteristic
369C2#1176360611yescharacteristic
370C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#464C2
socleC22×C17C22×D4
prime radical#11763452C2
hypercenter#464D17
hypocenterC17#3323
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #11763500C2semidirect
3. #11763163C2semidirect
4. #11763149C2semidirect
5. #11763126C2semidirect
6. #11763110C2semidirect
7. #11763064C2semidirect
8. #11763052C2semidirect
9. #11762970C2semidirect
10. #11762958C2semidirect
11. C4×#63230C2semidirect
12. #11762886C2semidirect
13. #11762812C2semidirect
14. #11762809C2semidirect
15. C17×#464C2semidirect
18. C2×Q16×C17C22semidirect
21. C4×Q8×C17C22semidirect
24. #10587632C22semidirect
28. C2×#63218C22semidirect
31. C4⋊C4×D17C22semidirect
41. #10587556C22semidirect
42. C17×C4⋊C8C22semidirect
43. C17×#103C22semidirect
45. #10587506C22semidirect
46. #10587506C22semidirect
47. #10587506C22semidirect
53. #10587491C22semidirect
54. #10587489C22semidirect
55. #10587488C22semidirect
57. #10587479C22semidirect
58. C4×#3622C22semidirect
187. #464D17semidirect
203. C2×Q16C2×D17semidirect
220. C4×Q8C2×D17semidirect
230. C4⋊C8C2×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
#117635002
#117634522
#117631632
#117631492
#117631262
#117631102
#117630642
#117630522
#117629702
#117629582
C4×#632302
#117628862
#117628122
#117628092
C17×#4642
#332317