Finite Group Databasebeta

← #14541001

Subgroups — #14541001

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#11763494C211yesmaximal, characteristic, complemented
2#11763447C211yesmaximal, characteristic
3#11763165C211yesmaximal, characteristic, complemented
4#11763149C211yesmaximal, characteristic, complemented
5#11763127C211yesmaximal, characteristic, complemented
6#11763107C211yesmaximal, characteristic, complemented
7#11763066C211yesmaximal, characteristic, complemented
8#11763055C211yesmaximal, characteristic, complemented
9#11763034C211yesmaximal, characteristic, complemented
10#11762971C211yesmaximal, characteristic, complemented
11#11762961C211yesmaximal, characteristic, complemented
12#11762900C211yesmaximal, characteristic, complemented
13#11762885C211yesmaximal, characteristic, complemented
14#11762812C211yesmaximal, characteristic, complemented
15C17×#417C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×M16×C17C2211yescharacteristic, complemented
19C2×C4⋊C4×C17C2211yescharacteristic, complemented
20#10587633C2211yescharacteristic
21#10587632C2211yescharacteristic
22C2×#63231C2211yescharacteristic, complemented
23#10587625—21no—
24#10587623—21no—
25D4×Dic17C2211yescharacteristic
26C2×#63229C2211yescharacteristic
27#10587616C2211yescharacteristic
28#10587615—21no—
29#10587615—21no—
30C4×C22⋊D17C2211yescharacteristic
31C2×#63227—21no—
32#10587612—21no—
33#10587610C2211yescharacteristic, complemented
34C2×#63219C2211yescharacteristic
35C2×#63218C2211yescharacteristic
36#10587584—21no—
37#10587583C2211yescharacteristic
38#10587578C2211yescharacteristic
39#10587576C2211yescharacteristic, complemented
40#10587575—21no—
41#10587573C2211yescharacteristic, complemented
42C17×#103C2211yescharacteristic, complemented
43C17×#103C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic
45C17×#102C2211yescharacteristic
46#10587506C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic
48C22⋊#3622C2211yescharacteristic, complemented
49#10587492C2211yescharacteristic
50#10587491C2211yescharacteristic
51#10587489C2211yescharacteristic, complemented
52#10587487C2211yescharacteristic, complemented
53#10587485C2211yescharacteristic, complemented
54C8×Dic17C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic
56#10587480C2211yescharacteristic
57#10587479C2211yescharacteristic, complemented
58#10587479C2211yescharacteristic, complemented
59C23×D17—21no—
60#41×C17—22no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×D4×C17—21no—
64C2×D4×C17C2311yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17C2311yescharacteristic
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17D411yescharacteristic
71C22×Dic17—21no—
72D4×D17—22no—
73D4×D17—41no—
74C2×D68C2311yescharacteristic
75C2×C4×D17C2311yescharacteristic
76C2×Dic34C2311yescharacteristic
77M16×C17—41no—
78C2×C8×C17C2311yescharacteristic
79C2×C8×C17C2311yescharacteristic
80C4⋊C4×C17—21no—
81C4⋊C4×C17C2311yescharacteristic
82C4⋊C4×C17C2311yescharacteristic
83C22⋊Dic17—21no—
84C22⋊Dic17—21no—
85C22⋊Dic17D411yescharacteristic
86#63231—41no—
87#63229—22no—
88#63227—21no—
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
98C4⋊Dic17C2311yescharacteristic
99#63225—21no—
100#63225—21no—
101#63225—21no—
102#63225—21no—
103#63225D411yescharacteristic
104C4×Dic17—21no—
105C4×Dic17—21no—
106C4×Dic17C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109#63219—41no—
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—
120D4×C17—21no—
121D4×C17—42no—
122D4×C17#4112yes—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2411yescharacteristic
132C22⋊D17—41no—
133C22⋊D17—42no—
134C22⋊D17—81no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—81no—
139C2×Dic17C2×D411yescharacteristic
140C2×Dic17C2×D411yescharacteristic
141C2×Dic17C2×D411yescharacteristic
142D68—21no—
143D68—21no—
144C4×D17—21no—
145C4×D17—21no—
146Dic34—21no—
147Dic34—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#452—171no—
189#450—171no—
190C4×SD16—171no—
191#436—171no—
192#418—171no—
193#417D1711yescharacteristic, complemented
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×Q16—171no—
204C2×SD16—171no—
205C2×SD16—171no—
206C2×M16—171no—
207C2×M16C2×D1711yescharacteristic, complemented
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⋊Q8—171no—
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⋊C4C2×D1711yescharacteristic, complemented
226C2×C22⋊C4—341no—
227#107—171no—
228#106—171no—
229#106—171no—
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#102C2×D1711yescharacteristic
240#102C2×D1711yescharacteristic
241C22⋊C8—171no—
242C22⋊C8—171no—
243C4×C8—171no—
244C17#332311yescharacteristic, complemented, Sylow
245C24—341no—
246#41—22no—
247#41—41no—
248C2×Q8—171no—
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—21no—
260C22×C4—171no—
261C22×C4—341no—
262C22×C4C22×D1711yescharacteristic
263Q16—681no—
264SD16—342no—
265SD16—681no—
266M16—41no—
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—341no—
277C4⋊C4—341no—
278C4⋊C4—341no—
279C4⋊C4—341no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4C22×D1711yescharacteristic
283C4⋊C4C22×D1711yescharacteristic
284C22⋊C4—171no—
285C22⋊C4—171no—
286C22⋊C4—341no—
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—341no—
307Q8—341no—
308D4—21no—
309D4—42no—
310D4—341no—
311D4—341no—
312D4—681no—
313D4—682no—
314D4—1361no—
315D4#6325212yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—171no—
324C2×C4—171no—
325C2×C4—171no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
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×#6325211yescharacteristic
363C4#1058769111yescharacteristic
364C2—41no—
365C2—42no—
366C2—681no—
367C2—1361no—
368C2#1176344911yescharacteristic
369C2#1176360511yescharacteristic
370C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#417C2
socleC22×C17C22×D4
prime radical#11763447C2
hypercenter#417D17
hypocenterC17#3323
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #11763494C2semidirect
3. #11763165C2semidirect
4. #11763149C2semidirect
5. #11763127C2semidirect
6. #11763107C2semidirect
7. #11763066C2semidirect
8. #11763055C2semidirect
9. #11763034C2semidirect
10. #11762971C2semidirect
11. #11762961C2semidirect
12. #11762900C2semidirect
13. #11762885C2semidirect
14. #11762812C2semidirect
15. C17×#417C2semidirect
18. C2×M16×C17C22semidirect
19. C2×C4⋊C4×C17C22semidirect
22. C2×#63231C22semidirect
33. #10587610C22semidirect
39. #10587576C22semidirect
41. #10587573C22semidirect
42. C17×#103C22semidirect
43. C17×#103C22semidirect
46. #10587506C22semidirect
48. C22⋊#3622C22semidirect
51. #10587489C22semidirect
52. #10587487C22semidirect
53. #10587485C22semidirect
54. C8×Dic17C22semidirect
57. #10587479C22semidirect
58. #10587479C22semidirect
193. #417D17semidirect
207. C2×M16C2×D17semidirect
225. C2×C4⋊C4C2×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
#117634942
#117634472
#117631652
#117631492
#117631272
#117631072
#117630662
#117630552
#117630342
#117629712
#117629612
#117629002
#117628852
#117628122
C17×#4172
#332317