Finite Group Databasebeta

← #14540993

Subgroups — #14540993

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#11763501C211yesmaximal, characteristic, complemented
2#11763447C211yesmaximal, characteristic
3#11763165C211yesmaximal, characteristic, complemented
4#11763148C211yesmaximal, characteristic, complemented
5#11763128C211yesmaximal, characteristic, complemented
6#11763111C211yesmaximal, characteristic, complemented
7#11763064C211yesmaximal, characteristic, complemented
8#11763057C211yesmaximal, characteristic, complemented
9#11763033C211yesmaximal, characteristic, complemented
10#11762958C211yesmaximal, characteristic, complemented
11SD16×Dic17C211yesmaximal, characteristic, complemented
12#11762884C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14#11762810C211yesmaximal, characteristic, complemented
15C17×#452C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×Q16×C17C2211yescharacteristic, complemented
19C2×SD16×C17C2211yescharacteristic
20C17×C22⋊Q8C2211yescharacteristic, complemented
21#10587633C2211yescharacteristic
22#10587632C2211yescharacteristic
23#10587632C2211yescharacteristic
24Q8×Dic17C2211yescharacteristic, complemented
25C2×#63230C2211yescharacteristic
26#10587625—21no—
27#10587623—21no—
28D4×Dic17C2211yescharacteristic
29#10587616C2211yescharacteristic
30#10587615—21no—
31C4×C22⋊D17C2211yescharacteristic
32C2×#63227—21no—
33C2×C4⋊Dic17C2211yescharacteristic, complemented
34C2×#63223C2211yescharacteristic, complemented
35C2×#63218C2211yescharacteristic
36#10587584—21no—
37#10587578C2211yescharacteristic
38#10587576—21no—
39#10587576C2211yescharacteristic, complemented
40#10587572—21no—
41#10587563—21no—
42C17×#103C2211yescharacteristic, complemented
43C17×#102C2211yescharacteristic
44C17×C22⋊C8C2211yescharacteristic, complemented
45#10587506C2211yescharacteristic, complemented
46#10587506C2211yescharacteristic, complemented
47#10587506C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587492C2211yescharacteristic
51#10587492C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587488C2211yescharacteristic, complemented
54#10587488C2211yescharacteristic, complemented
55#10587485C2211yescharacteristic, complemented
56C8×Dic17C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60#41×C17—22no—
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—
70C2×C22⋊D17D411yescharacteristic
71C22×Dic17—21no—
72C22×Dic17—21no—
73D4×D17—22no—
74D4×D17—41no—
75C2×D68C2311yescharacteristic
76C2×C4×D17C2311yescharacteristic
77Q16×C17—41no—
78SD16×C17—22no—
79C2×C8×C17C2311yescharacteristic
80C2×C8×C17C2311yescharacteristic
81C4⋊C4×C17—21no—
82C4⋊C4×C17C2311yescharacteristic
83C22⋊C4×C17—21no—
84C22⋊Dic17—21no—
85C22⋊Dic17D411yescharacteristic
86#63230—41no—
87#63227—21no—
88#63227—21no—
89#63227—21no—
90#63227—21no—
91#63227—21no—
92#63227—21no—
93#63227D411yescharacteristic
94C4⋊Dic17—21no—
95C4⋊Dic17—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99C4⋊Dic17C2311yescharacteristic
100#63225—21no—
101#63225—21no—
102#63225—21no—
103#63225D411yescharacteristic
104C4×Dic17—21no—
105C4×Dic17—21no—
106C4×Dic17C2311yescharacteristic
107#63223—41no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110#63218—41no—
111C23×C17—21no—
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114C22×D17—42no—
115C22×D17—42no—
116C22×D17—81no—
117C22×D17C2×D411yescharacteristic
118Q8×C17—21no—
119Q8×C17—21no—
120Q8×C17—21no—
121Q8×C17—21no—
122D4×C17—21no—
123D4×C17—42no—
124D4×C17#4112yes—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—42no—
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2411yescharacteristic
133C22⋊D17—41no—
134C22⋊D17—42no—
135C22⋊D17—81no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—42no—
140C2×Dic17—81no—
141C2×Dic17C2×D411yescharacteristic
142C2×Dic17C2×D411yescharacteristic
143C2×Dic17C2×D411yescharacteristic
144D68—21no—
145D68—21no—
146C4×D17—21no—
147C4×D17—21no—
148C8×C17—21no—
149C8×C17—21no—
150#3622—21no—
151#3622—21no—
152#3323—171nomaximal, Sylow
153C22×C17—22no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17C22×D411yescharacteristic
160C2×D17—21no—
161C2×D17—21no—
162C2×D17—21no—
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×C17C2×#4111yescharacteristic
172C4×C17H(5,𝔽2)11yescharacteristic
173Dic17—21no—
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#464—171no—
188#452D1711yescharacteristic, complemented
189#450—171no—
190C4×SD16—171no—
191#436—171no—
192#418—171no—
193#417—171no—
194C2×C17—42no—
195C2×C17—41no—
196C2×C17#54611yescharacteristic
197C2×C17#57511yescharacteristic
198C2×C17#57711yescharacteristic
199D17—41no—
200D17—81no—
201C2×#41C2×D1711yescharacteristic
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#124—171no—
212#123—341no—
213#123—341no—
214C22⋊Q8—171no—
215C22⋊Q8—171no—
216C22⋊Q8C2×D1711yescharacteristic, complemented
217#121—171no—
218#121—341no—
219#120—341no—
220C4×Q8—171no—
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⋊C8—171no—
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⋊C8C2×D1711yescharacteristic, complemented
243C4×C8—171no—
244C17#332311yescharacteristic, complemented, Sylow
245C24—341no—
246#41—22no—
247#41—41no—
248C2×Q8C22×D1711yescharacteristic
249C2×Q8C22×D1711yescharacteristic
250C2×D4—21no—
251C2×D4—171no—
252C2×D4—171no—
253C2×D4—341no—
254C2×D4—342no—
255C2×D4—341no—
256C2×D4—681no—
257C2×D4C22×D1711yescharacteristic
258C22×C4—21no—
259C22×C4—171no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4C22×D1711yescharacteristic
263Q16—41no—
264SD16—22no—
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—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—171no—
286C22⋊C4—171no—
287C22⋊C4—341no—
288C22⋊C4—341no—
289C22⋊C4—341no—
290C22⋊C4—341no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C42—171no—
295C42—341no—
296C42—341no—
297C23—21no—
298C23—171no—
299C23—341no—
300C23—682no—
301C23—682no—
302C23—1361no—
303C23D4×D1711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—42no—
310D4—341no—
311D4—341no—
312D4—681no—
313D4—682no—
314D4—1361no—
315D4#6325412yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—171no—
323C2×C4—171no—
324C2×C4—171no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
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—22no—
342C22—41no—
343C22—41no—
344C22—341no—
345C22—341no—
346C22—341no—
347C22—681no—
348C22—681no—
349C22—681no—
350C22—1364no—
351C22C2×D4×D1711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—341no—
357C4—341no—
358C4—341no—
359C4—681no—
360C4—681no—
361C4—681no—
362C4C2×#6325411yescharacteristic
363C4#1058769111yescharacteristic
364C2—41no—
365C2—42no—
366C2—681no—
367C2—1361no—
368C2#1176344611yescharacteristic
369C2#1176360811yescharacteristic
370C2#1176361611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#452C2
socleC22×C17C22×D4
prime radical#11763447C2
hypercenter#452D17
hypocenterC17#3323
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
1. #11763501C2semidirect
3. #11763165C2semidirect
4. #11763148C2semidirect
5. #11763128C2semidirect
6. #11763111C2semidirect
7. #11763064C2semidirect
8. #11763057C2semidirect
9. #11763033C2semidirect
10. #11762958C2semidirect
11. SD16×Dic17C2semidirect
12. #11762884C2semidirect
13. #11762812C2semidirect
14. #11762810C2semidirect
15. C17×#452C2semidirect
18. C2×Q16×C17C22semidirect
20. C17×C22⋊Q8C22semidirect
24. Q8×Dic17C22semidirect
33. C2×C4⋊Dic17C22semidirect
34. C2×#63223C22semidirect
39. #10587576C22semidirect
42. C17×#103C22semidirect
44. C17×C22⋊C8C22semidirect
45. #10587506C22semidirect
46. #10587506C22semidirect
47. #10587506C22semidirect
53. #10587488C22semidirect
54. #10587488C22semidirect
55. #10587485C22semidirect
56. C8×Dic17C22semidirect
58. #10587478C22semidirect
188. #452D17semidirect
203. C2×Q16C2×D17semidirect
216. C22⋊Q8C2×D17semidirect
234. #103C2×D17semidirect
242. C22⋊C8C2×D17semidirect
244. C17#3323semidirect

Maximal subgroups

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

subgroupindex
#117635012
#117634472
#117631652
#117631482
#117631282
#117631112
#117630642
#117630572
#117630332
#117629582
SD16×Dic172
#117628842
#117628122
#117628102
C17×#4522
#332317