Finite Group Databasebeta

← #14539875

Subgroups — #14539875

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#11763424C211yesmaximal, characteristic, complemented
2#11763372C211yesmaximal, characteristic, complemented
3#11763197C211yesmaximal, characteristic, complemented
4#11763192C211yesmaximal, characteristic, complemented
5#11763167C211yesmaximal, characteristic, complemented
6#11763154C211yesmaximal, characteristic, complemented
7#11763032C211yesmaximal, characteristic, complemented
8#11763024C211yesmaximal, characteristic, complemented
9#11763006C211yesmaximal, characteristic, complemented
10#11762972C211yesmaximal, characteristic, complemented
11#11762961C211yesmaximal, characteristic, complemented
12#11762951C211yesmaximal, characteristic, complemented
13#11762941C211yesmaximal, characteristic, complemented
14#11762869C211yesmaximal, characteristic, complemented
15C17×#459C211yesmaximal, characteristic, complemented
16C2×#63254C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×D8×C17C2211yescharacteristic
19C2×D8×C17C2211yescharacteristic, complemented
20C17×#127C2211yescharacteristic
21C4×D4×C17C2211yescharacteristic, complemented
22#10587626C2211yescharacteristic
23#10587625—21no—
24#10587625—21no—
25#10587624C2211yescharacteristic
26#10587623—21no—
27#10587623—21no—
28#10587622C2211yescharacteristic
29#10587622C2211yescharacteristic, complemented
30C2×#63229C2211yescharacteristic
31C2×#63229C2211yescharacteristic, complemented
32C2×#63229C2211yescharacteristic, complemented
33C2×#63228C2211yescharacteristic
34#10587615—21no—
35#10587610—21no—
36C2×Dic68C2211yescharacteristic, complemented
37C2×#63219C2211yescharacteristic
38#10587583C2211yescharacteristic
39#10587574C2211yescharacteristic, complemented
40#10587571—21no—
41#10587565—21no—
42#10587561C2211yescharacteristic
43#10587558C2211yescharacteristic
44C4×Dic34C2211yescharacteristic, complemented
45C17×C4⋊C8C2211yescharacteristic, complemented
46C17×#102C2211yescharacteristic
47C17×#102C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic, complemented
50#10587492C2211yescharacteristic
51#10587491C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587487C2211yescharacteristic, complemented
54#10587486C2211yescharacteristic, complemented
55C8×Dic17C2211yescharacteristic, complemented
56#10587481C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic
58C4⋊#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×D4×C17—21no—
61C2×D4×C17—21no—
62C2×D4×C17C2311yescharacteristic
63C2×D4×C17C2311yescharacteristic
64C2×D4×C17C2311yescharacteristic
65C22×C4×C17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C22×Dic17—21no—
72#63254—41no—
73#63254—41no—
74D4×D17—41no—
75D4×D17—41no—
76C2×D68C2311yescharacteristic
77C2×C4×D17C2311yescharacteristic
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—
87C42×C17C2311yescharacteristic
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92C22⋊Dic17—21no—
93#63229—41no—
94#63229—41no—
95#63229—41no—
96#63228—41no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227D411yescharacteristic
101#63227D411yescharacteristic
102C4⋊Dic17—21no—
103C4⋊Dic17C2311yescharacteristic
104#63225—21no—
105#63225—21no—
106#63225D411yescharacteristic
107#63225D411yescharacteristic
108C4×Dic17—21no—
109C4×Dic17C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112Dic68—41no—
113#63219—41no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C22×D17—21no—
118C22×D17—42no—
119C22×D17—81no—
120C22×D17—81no—
121C22×D17C2×D411yescharacteristic
122D4×C17—21no—
123D4×C17—21no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—81no—
129D4×C17—81no—
130C2×C4×C17—21no—
131C2×C4×C17—81no—
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2411yescharacteristic
135C22⋊D17—42no—
136C22⋊D17—42no—
137C22⋊D17—81no—
138C22⋊D17—81no—
139C22⋊D17—81no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—42no—
144C2×Dic17C2×D411yescharacteristic
145C2×Dic17C2×D411yescharacteristic
146C2×Dic17C2×D411yescharacteristic
147D68—21no—
148D68—21no—
149C4×D17—21no—
150C4×D17—21no—
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—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17C22×D411yescharacteristic
171C2×D17—21no—
172C2×D17—21no—
173C2×D17—21no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—84no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—41no—
181C4×C17H(5,𝔽2)11yescharacteristic
182C4×C17H(5,𝔽2)11yescharacteristic
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#459D1711yescharacteristic, complemented
199#452—171no—
200#450—171no—
201#449—171no—
202#447—171no—
203#433—171no—
204C2×C17—81no—
205C2×C17—81no—
206C2×C17—81no—
207C2×C17#53511yescharacteristic
208C2×C17#57611yescharacteristic
209C2×C17#57711yescharacteristic
210D17—41no—
211D17—81no—
212C2×#41—171no—
213C22×D4—171no—
214C2×Q16—171no—
215C2×SD16—171no—
216C2×SD16—171no—
217C2×SD16—171no—
218C2×SD16—171no—
219C2×D8—171no—
220C2×D8C2×D1711yescharacteristic
221C2×D8C2×D1711yescharacteristic, complemented
222C4⋊Q8—171no—
223#127—171no—
224#127C2×D1711yescharacteristic
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—171no—
234#121—341no—
235#121—341no—
236#120—341no—
237#120—341no—
238C4×Q8—171no—
239C4×D4C2×D1711yescharacteristic, complemented
240#117—171no—
241C4⋊C8—171no—
242C4⋊C8C2×D1711yescharacteristic, complemented
243#103—171no—
244#103—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102C2×D1711yescharacteristic
250#102C2×D1711yescharacteristic, complemented
251C22⋊C8—171no—
252C22⋊C8—171no—
253#97—171no—
254C4×C8—171no—
255C17#322411yescharacteristic, complemented, Sylow
256C24—341no—
257#41—681no—
258#41—681no—
259C2×Q8—171no—
260C2×Q8—171no—
261C2×Q8—341no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—171no—
265C2×D4—341no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4—681no—
271C2×D4—681no—
272C2×D4C22×D1711yescharacteristic
273C2×D4C22×D1711yescharacteristic
274C2×D4C22×D1711yescharacteristic
275C22×C4—21no—
276C22×C4—171no—
277C22×C4—341no—
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—341no—
310C42C22×D1711yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—171no—
315C23—341no—
316C23—682no—
317C23—1361no—
318C23—1361no—
319Q8—341no—
320Q8—341no—
321Q8—341no—
322Q8—341no—
323Q8—1361no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—81no—
331D4—81no—
332D4—341no—
333D4—341no—
334D4—682no—
335D4—682no—
336D4—1361no—
337D4—1361no—
338D4—1361no—
339C2×C4—21no—
340C2×C4—81no—
341C2×C4—171no—
342C2×C4—171no—
343C2×C4—171no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—682no—
350C2×C4D4×D1711yescharacteristic
351C2×C4D4×D1711yescharacteristic
352C2×C4C23×D1711yescharacteristic
353C8—21no—
354C8—21no—
355C8—341no—
356C8—341no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—341no—
367C22—341no—
368C22—341no—
369C22—681no—
370C22—681no—
371C22—681no—
372C22—1364no—
373C22C2×D4×D1711yescharacteristic
374C4—21no—
375C4—21no—
376C4—41no—
377C4—341no—
378C4—341no—
379C4—341no—
380C4—681no—
381C4—681no—
382C4—681no—
383C4#1058769111yescharacteristic
384C4#1058769611yescharacteristic
385C2—81no—
386C2—81no—
387C2—81no—
388C2—681no—
389C2—1361no—
390C2#1176337511yescharacteristic
391C2#1176361011yescharacteristic
392C2#1176361911yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#459C2
socleC22×C17C22×D4
hypercenter#459D17
hypocenterC17#3224
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #11763424C2semidirect
2. #11763372C2semidirect
3. #11763197C2semidirect
4. #11763192C2semidirect
5. #11763167C2semidirect
6. #11763154C2semidirect
7. #11763032C2semidirect
8. #11763024C2semidirect
9. #11763006C2semidirect
10. #11762972C2semidirect
11. #11762961C2semidirect
12. #11762951C2semidirect
13. #11762941C2semidirect
14. #11762869C2semidirect
15. C17×#459C2semidirect
19. C2×D8×C17C22semidirect
21. C4×D4×C17C22semidirect
29. #10587622C22semidirect
31. C2×#63229C22semidirect
32. C2×#63229C22semidirect
36. C2×Dic68C22semidirect
39. #10587574C22semidirect
44. C4×Dic34C22semidirect
45. C17×C4⋊C8C22semidirect
47. C17×#102C22semidirect
49. #10587503C22semidirect
53. #10587487C22semidirect
54. #10587486C22semidirect
55. C8×Dic17C22semidirect
56. #10587481C22semidirect
58. C4⋊#3622C22semidirect
198. #459D17semidirect
221. C2×D8C2×D17semidirect
239. C4×D4C2×D17semidirect
242. C4⋊C8C2×D17semidirect
250. #102C2×D17semidirect
255. C17#3224semidirect

Maximal subgroups

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

subgroupindex
#117634242
#117633722
#117631972
#117631922
#117631672
#117631542
#117630322
#117630242
#117630062
#117629722
#117629612
#117629512
#117629412
#117628692
C17×#4592
#322417