Finite Group Databasebeta

← #14539874

Subgroups — #14539874

Order 2176.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763426C211yesmaximal, characteristic, complemented
2#11763376C211yesmaximal, characteristic, complemented
3#11763197C211yesmaximal, characteristic, complemented
4#11763192C211yesmaximal, characteristic, complemented
5#11763167C211yesmaximal, characteristic, complemented
6#11763154C211yesmaximal, characteristic, complemented
7#11763030C211yesmaximal, characteristic, complemented
8#11763022C211yesmaximal, characteristic, complemented
9#11763008C211yesmaximal, characteristic, complemented
10#11763002C211yesmaximal, characteristic, complemented
11#11762973C211yesmaximal, characteristic, complemented
12#11762960C211yesmaximal, characteristic, complemented
13#11762952C211yesmaximal, characteristic, complemented
14#11762867C211yesmaximal, characteristic, complemented
15C17×#447C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C2×D8×C17C2211yescharacteristic
19C2×D8×C17C2211yescharacteristic
20C17×#121C2211yescharacteristic
21#10587639—21no—
22#10587639—21no—
23#10587626C2211yescharacteristic
24#10587626C2211yescharacteristic
25#10587624—21no—
26#10587624—21no—
27#10587622—21no—
28#10587622C2211yescharacteristic
29#10587622C2211yescharacteristic
30#10587622C2211yescharacteristic, complemented
31#10587621—21no—
32D4×Dic17C2211yescharacteristic, complemented
33C2×#63229C2211yescharacteristic
34C2×#63229C2211yescharacteristic
35C2×#63229C2211yescharacteristic, complemented
36C2×#63228C2211yescharacteristic, complemented
37#10587612C2211yescharacteristic
38#10587610C2211yescharacteristic
39C2×Dic68C2211yescharacteristic, complemented
40C2×#63219C2211yescharacteristic, complemented
41#10587574C2211yescharacteristic, complemented
42#10587573C2211yescharacteristic, complemented
43#10587565—21no—
44#10587564—21no—
45C17×#102C2211yescharacteristic
46C17×#102C2211yescharacteristic
47C17×C22⋊C8C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50C22⋊#3622C2211yescharacteristic
51#10587492C2211yescharacteristic, complemented
52#10587487C2211yescharacteristic, complemented
53#10587486C2211yescharacteristic, complemented
54#10587485C2211yescharacteristic, complemented
55#10587485C2211yescharacteristic, complemented
56C8×Dic17C2211yescharacteristic, complemented
57#10587481C2211yescharacteristic, complemented
58#10587480C2211yescharacteristic, complemented
59C24×C17—21no—
60C2×D4×C17—21no—
61C2×D4×C17—21no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C2×D4×C17C2311yescharacteristic
65C2×D4×C17C2311yescharacteristic
66C2×D4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C2×C22⋊D17—21no—
72C2×C22⋊D17—21no—
73C22×Dic17—21no—
74#63252—41no—
75#63252—41no—
76C2×D68C2311yescharacteristic
77C2×C4×D17—21no—
78C2×Dic34—21no—
79C2×Dic34C2311yescharacteristic
80C2×Dic34C2311yescharacteristic
81D8×C17—41no—
82D8×C17—41no—
83C2×C8×C17C2311yescharacteristic
84C2×C8×C17C2311yescharacteristic
85C4⋊C4×C17C2311yescharacteristic
86C22⋊C4×C17—21no—
87C22⋊Dic17—21no—
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92C22⋊Dic17—21no—
93C22⋊Dic17—21no—
94C22⋊Dic17—21no—
95C22⋊Dic17D411yescharacteristic
96C22⋊Dic17D411yescharacteristic
97#63229—41no—
98#63229—41no—
99#63229—41no—
100#63228—41no—
101C4⋊Dic17D411yescharacteristic
102C4⋊Dic17D411yescharacteristic
103C4⋊Dic17C2311yescharacteristic
104#63225—21no—
105#63225—21no—
106#63225—21no—
107C4×Dic17—21no—
108C4×Dic17C2311yescharacteristic
109C4×Dic17C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112Dic68—41no—
113#63219—41no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—42no—
118C23×C17—81no—
119C23×C17—81no—
120C23×C17C2×D411yescharacteristic
121C22×D17—21no—
122D4×C17—21no—
123D4×C17—21no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—42no—
129D4×C17—81no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2411yescharacteristic
135C22⋊D17—42no—
136C22⋊D17—81no—
137C22⋊D17—81no—
138C22⋊D17—81no—
139C22⋊D17—81no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—81no—
144C2×Dic17C2×D411yescharacteristic
145C2×Dic17C2×D411yescharacteristic
146C2×Dic17C2×D411yescharacteristic
147C2×Dic17C2×D411yescharacteristic
148D68—21no—
149D68—21no—
150C4×D17—42no—
151Dic34—21no—
152Dic34—21no—
153Dic34—21no—
154Dic34—21no—
155Dic34—81no—
156C8×C17—21no—
157C8×C17—21no—
158#3622—21no—
159#3622—21no—
160#3224—171nomaximal, Sylow
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—84no—
174C22×C17C22×D411yescharacteristic
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C4×C17—21no—
179C4×C17—41no—
180C4×C17H(5,𝔽2)11yescharacteristic
181C4×C17H(5,𝔽2)11yescharacteristic
182Dic17—21no—
183Dic17—21no—
184Dic17—21no—
185Dic17—21no—
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189#542—171no—
190#535—171no—
191#496—171no—
192#495—171no—
193#489—171no—
194#486—171no—
195#464—171no—
196#463—171no—
197#460—171no—
198#459—171no—
199#452—171no—
200#450—171no—
201#449—171no—
202#447D1711yescharacteristic, complemented
203#433—171no—
204C2×C17—41no—
205C2×C17—81no—
206C2×C17—81no—
207C2×C17—81no—
208C2×C17#53511yescharacteristic
209C2×C17#57611yescharacteristic
210C2×C17#57711yescharacteristic
211D17—81no—
212C2×#41—171no—
213C22×D4C2×D1711yescharacteristic
214C2×Q16—171no—
215C2×SD16—171no—
216C2×SD16—171no—
217C2×SD16—171no—
218C2×SD16—171no—
219C2×D8—171no—
220C2×D8C2×D1711yescharacteristic
221C2×D8C2×D1711yescharacteristic
222C4⋊Q8—171no—
223#127—171no—
224#127—171no—
225#126—341no—
226#124—171no—
227#124—171no—
228#124—171no—
229#124—341no—
230#123—341no—
231C22⋊Q8—171no—
232C22⋊Q8—341no—
233#121—341no—
234#121—341no—
235#121C2×D1711yescharacteristic
236#120—341no—
237#120—341no—
238C4×Q8—171no—
239C4×D4—171no—
240#117—171no—
241C4⋊C8—171no—
242C4⋊C8—171no—
243#103—171no—
244#103—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102C2×D1711yescharacteristic
250#102C2×D1711yescharacteristic
251C22⋊C8—171no—
252C22⋊C8C2×D1711yescharacteristic
253#97—171no—
254C4×C8—171no—
255C17#322411yescharacteristic, complemented, Sylow
256C24—21no—
257#41—681no—
258#41—681no—
259C2×Q8—171no—
260C2×Q8—171no—
261C2×Q8—341no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—171no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4—341no—
272C2×D4C22×D1711yescharacteristic
273C2×D4C22×D1711yescharacteristic
274C2×D4C22×D1711yescharacteristic
275C22×C4—341no—
276C22×C4—341no—
277C22×C4C22×D1711yescharacteristic
278Q16—681no—
279SD16—681no—
280SD16—681no—
281SD16—681no—
282SD16—681no—
283D8—41no—
284D8—41no—
285D8—681no—
286C2×C8—171no—
287C2×C8—171no—
288C2×C8C22×D1711yescharacteristic
289C2×C8C22×D1711yescharacteristic
290C4⋊C4—171no—
291C4⋊C4—171no—
292C4⋊C4—171no—
293C4⋊C4—341no—
294C4⋊C4—341no—
295C4⋊C4—341no—
296C4⋊C4C22×D1711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—171no—
299C22⋊C4—171no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4—341no—
307C22⋊C4—341no—
308C42—171no—
309C42—171no—
310C42—341no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—341no—
318C23D4×D1711yescharacteristic
319Q8—341no—
320Q8—341no—
321Q8—341no—
322Q8—341no—
323Q8—1361no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—81no—
332D4—341no—
333D4—341no—
334D4—682no—
335D4—1361no—
336D4—1361no—
337D4—1361no—
338D4—1361no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—171no—
343C2×C4—171no—
344C2×C4—171no—
345C2×C4—171no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—682no—
350C2×C4—1361no—
351C2×C4D4×D1711yescharacteristic
352C2×C4C23×D1711yescharacteristic
353C8—21no—
354C8—21no—
355C8—341no—
356C8—341no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—84no—
370C22—681no—
371C22—681no—
372C22—681no—
373C22C2×D4×D1711yescharacteristic
374C4—21no—
375C4—41no—
376C4—341no—
377C4—341no—
378C4—341no—
379C4—341no—
380C4—681no—
381C4—681no—
382C4—681no—
383C4#1058769111yescharacteristic
384C4#1058769111yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—81no—
389C2—1361no—
390C2#1176337411yescharacteristic
391C2#1176361011yescharacteristic
392C2#1176361911yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#447C2
socleC22×C17C22×D4
hypercenter#447D17
hypocenterC17#3224
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #11763426C2semidirect
2. #11763376C2semidirect
3. #11763197C2semidirect
4. #11763192C2semidirect
5. #11763167C2semidirect
6. #11763154C2semidirect
7. #11763030C2semidirect
8. #11763022C2semidirect
9. #11763008C2semidirect
10. #11763002C2semidirect
11. #11762973C2semidirect
12. #11762960C2semidirect
13. #11762952C2semidirect
14. #11762867C2semidirect
15. C17×#447C2semidirect
30. #10587622C22semidirect
32. D4×Dic17C22semidirect
35. C2×#63229C22semidirect
36. C2×#63228C22semidirect
39. C2×Dic68C22semidirect
40. C2×#63219C22semidirect
41. #10587574C22semidirect
42. #10587573C22semidirect
51. #10587492C22semidirect
52. #10587487C22semidirect
53. #10587486C22semidirect
54. #10587485C22semidirect
55. #10587485C22semidirect
56. C8×Dic17C22semidirect
57. #10587481C22semidirect
58. #10587480C22semidirect
202. #447D17semidirect
255. C17#3224semidirect

Maximal subgroups

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

subgroupindex
#117634262
#117633762
#117631972
#117631922
#117631672
#117631542
#117630302
#117630222
#117630082
#117630022
#117629732
#117629602
#117629522
#117628672
C17×#4472
#322417