Finite Group Databasebeta

← #14540739

Subgroups — #14540739

Order 2176.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763455C211yesmaximal, characteristic, complemented
2#11763171C211yesmaximal, characteristic, complemented
3#11763139C211yesmaximal, characteristic, complemented
4#11763127C211yesmaximal, characteristic, complemented
5#11763096C211yesmaximal, characteristic, complemented
6#11763060C211yesmaximal, characteristic, complemented
7#11763036C211yesmaximal, characteristic, complemented
8#11763024C211yesmaximal, characteristic, complemented
9#11762952C211yesmaximal, characteristic, complemented
10#11762942C211yesmaximal, characteristic, complemented
11#11762919C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762808C211yesmaximal, characteristic, complemented
14C2×#10587480C211yesmaximal, characteristic, complemented
15C17×#547C211yesmaximal, characteristic, complemented
16C2×#41×C17—21no—
17C2×#63252C2211yescharacteristic
18C22×D68C2211yescharacteristic
19C17×C4⋊Q8C2211yescharacteristic, complemented
20C17×#123—21no—
21C17×#123—21no—
22C17×C22⋊Q8—21no—
23C17×#121C2211yescharacteristic
24C17×#121C2211yescharacteristic, complemented
25C4×D4×C17C2211yescharacteristic
26C4×D4×C17C2211yescharacteristic, complemented
27C2×C4⋊C4×C17C2211yescharacteristic
28C2×C4⋊C4×C17C2211yescharacteristic, complemented
29#10587623—21no—
30C2×#63229C2211yescharacteristic, complemented
31C2×#63228C2211yescharacteristic
32C2×#63228C2211yescharacteristic
33#10587616C2211yescharacteristic
34#10587610C2211yescharacteristic
35C2×#63223C2211yescharacteristic, complemented
36C22×#3622C2211yescharacteristic
37#10587580—21no—
38#10587570—21no—
39C22⋊C4×D17—21no—
40#10587561C2211yescharacteristic
41C4×D68C2211yescharacteristic
42#10587503C2211yescharacteristic, complemented
43C22⋊#3622C2211yescharacteristic
44C22⋊#3622C2211yescharacteristic, complemented
45#10587481C2211yescharacteristic, complemented
46#10587481C2211yescharacteristic, complemented
47#10587481C2211yescharacteristic, complemented
48#10587480C2211yescharacteristic
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587480C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587479C2211yescharacteristic, complemented
55#10587479C2211yescharacteristic, complemented
56#10587479C2211yescharacteristic, complemented
57C4⋊#3622C2211yescharacteristic, complemented
58#10587474C2211yescharacteristic, complemented
59C23×D17—21no—
60#41×C17—44no—
61C2×Q8×C17—21no—
62C2×D4×C17—21no—
63C2×D4×C17—21no—
64C2×D4×C17D411yescharacteristic
65C2×D4×C17C2311yescharacteristic
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17C2311yescharacteristic
70C22×C4×C17C2311yescharacteristic
71C2×C22⋊D17—21no—
72C2×C22⋊D17—21no—
73#63252—22no—
74#63252—41no—
75C2×D68—22no—
76C2×D68—21no—
77C2×D68—41no—
78C2×D68C2311yescharacteristic
79C2×D68C2311yescharacteristic
80C2×C4×D17—21no—
81C2×C4×D17—21no—
82C2×Dic34C2311yescharacteristic
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17D411yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C4⋊C4×C17C2311yescharacteristic
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C22⋊C4×C17D411yescharacteristic
94C22⋊C4×C17D411yescharacteristic
95C42×C17C2311yescharacteristic
96C22⋊Dic17—21no—
97#63229—41no—
98#63228—22no—
99#63228—41no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103#63227—21no—
104C4⋊Dic17C2311yescharacteristic
105#63225—21no—
106#63223—41no—
107C2×#3622—41no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C23×C17—21no—
113C23×C17C2×D411yescharacteristic
114C23×C17C2×D411yescharacteristic
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17—81no—
120Q8×C17—42no—
121D4×C17—22no—
122D4×C17—21no—
123D4×C17—21no—
124D4×C17—42no—
125D4×C17—42no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—42no—
134C2×C4×C17—42no—
135C2×C4×C17—42no—
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2×D411yescharacteristic
140C2×C4×C17C2411yescharacteristic
141C22⋊D17—42no—
142C22⋊D17—81no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145D68—21no—
146D68—21no—
147D68—21no—
148D68—42no—
149D68#4112yescomplemented
150C4×D17—42no—
151C4×D17—81no—
152Dic34—21no—
153Dic34—21no—
154#3622—21no—
155#3622—21no—
156#3622—21no—
157#3622—21no—
158#3306—171nomaximal, Sylow
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17AGL(1,ℤ/8)12yes—
168C22×C17C22×D411yescharacteristic
169C2×D17—22no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—84no—
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17C2×#4111yescharacteristic
184C4×C17H(5,𝔽2)11yescharacteristic
185Dic17—41no—
186Dic17—41no—
187#547D1711yescharacteristic, complemented
188#546—171no—
189#489—171no—
190#483—171no—
191#481—171no—
192#476—171no—
193#469—171no—
194#465—171no—
195#463—171no—
196#449—171no—
197#447—171no—
198#442—171no—
199#435—171no—
200#417—171no—
201C2×#102—171no—
202C2×C17—22no—
203C2×C17—41no—
204C2×C17—81no—
205C2×C17#54611yescharacteristic
206C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
207C2×C17#57711yescharacteristic
208D17—42no—
209D17—81no—
210C2×#41—21no—
211C2×#41—171no—
212C22×D4—171no—
213C2×SD16—171no—
214C2×D8—171no—
215C2×D8—171no—
216C2×M16—171no—
217C22×C8—171no—
218C4⋊Q8C2×D1711yescharacteristic, complemented
219#124—171no—
220#123—21no—
221#123—21no—
222#123—341no—
223C22⋊Q8—21no—
224C22⋊Q8—171no—
225#121—171no—
226#121—341no—
227#121C2×D1711yescharacteristic
228#121C2×D1711yescharacteristic, complemented
229#120—341no—
230C4×D4—171no—
231C4×D4C2×D1711yescharacteristic
232C4×D4C2×D1711yescharacteristic, complemented
233C2×C4⋊C4C2×D1711yescharacteristic
234C2×C4⋊C4C2×D1711yescharacteristic, complemented
235C2×C22⋊C4—341no—
236#106—171no—
237#106—171no—
238#106—171no—
239C4⋊C8—171no—
240#103—171no—
241#103—171no—
242#103—171no—
243#102—171no—
244#102—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250C22⋊C8—171no—
251C22⋊C8—171no—
252#97—171no—
253C17#330611yescharacteristic, complemented, Sylow
254C24—341no—
255#41—44no—
256#41—342no—
257#41—681no—
258C2×Q8—21no—
259C2×Q8—171no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—171no—
263C2×D4—171no—
264C2×D4—342no—
265C2×D4—341no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—681no—
269C2×D4C22⋊D1711yescharacteristic
270C2×D4C22×D1711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4C22×D1711yescharacteristic
277C22×C4C22×D1711yescharacteristic
278SD16—681no—
279D8—342no—
280D8—681no—
281M16—681no—
282C2×C8—171no—
283C2×C8—171no—
284C2×C8—171no—
285C2×C8—171no—
286C2×C8—681no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—171no—
291C4⋊C4—341no—
292C4⋊C4C22⋊D1711yescharacteristic
293C4⋊C4C22×D1711yescharacteristic
294C4⋊C4C22×D1711yescharacteristic
295C4⋊C4C22×D1711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4C22⋊D1711yescharacteristic
305C22⋊C4C22⋊D1711yescharacteristic
306C42C22×D1711yescharacteristic
307C23—21no—
308C23—341no—
309C23—341no—
310C23—682no—
311C23—682no—
312C23—1361no—
313C23C2×C22⋊D1711yescharacteristic
314C23C2×C22⋊D1711yescharacteristic
315Q8—42no—
316Q8—341no—
317Q8—341no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—172no—
324D4—341no—
325D4—341no—
326D4—341no—
327D4—682no—
328D4—682no—
329D4—1361no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—682no—
343C2×C4—1361no—
344C2×C4C2×C22⋊D1711yescharacteristic
345C2×C4C2×C22⋊D1711yescharacteristic
346C2×C4C2×C22⋊D1711yescharacteristic
347C2×C4C2×C22⋊D1711yescharacteristic
348C2×C4C23×D1711yescharacteristic
349C8—341no—
350C8—341no—
351C8—341no—
352C8—341no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—342no—
362C22—681no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—1364no—
368C22#1058761812yes—
369C22C22×C22⋊D1711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—681no—
378C4—681no—
379C4#41×D1711yescharacteristic
380C4#1058769611yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—682no—
385C2—1361no—
386C2#1176344711yescharacteristic
387C2C2×#1058761811yescharacteristic
388C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#547C2
socleC22×C17C22×D4
hypercenter#547D17
hypocenterC17#3306
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #11763455C2semidirect
2. #11763171C2semidirect
3. #11763139C2semidirect
4. #11763127C2semidirect
5. #11763096C2semidirect
6. #11763060C2semidirect
7. #11763036C2semidirect
8. #11763024C2semidirect
9. #11762952C2semidirect
10. #11762942C2semidirect
11. #11762919C2semidirect
12. #11762881C2semidirect
13. #11762808C2semidirect
14. C2×#10587480C2semidirect
15. C17×#547C2semidirect
19. C17×C4⋊Q8C22semidirect
24. C17×#121C22semidirect
26. C4×D4×C17C22semidirect
28. C2×C4⋊C4×C17C22semidirect
30. C2×#63229C22semidirect
35. C2×#63223C22semidirect
42. #10587503C22semidirect
44. C22⋊#3622C22semidirect
45. #10587481C22semidirect
46. #10587481C22semidirect
47. #10587481C22semidirect
54. #10587479C22semidirect
55. #10587479C22semidirect
56. #10587479C22semidirect
57. C4⋊#3622C22semidirect
58. #10587474C22semidirect
149. D68#41semidirect
187. #547D17semidirect
218. C4⋊Q8C2×D17semidirect
228. #121C2×D17semidirect
232. C4×D4C2×D17semidirect
234. C2×C4⋊C4C2×D17semidirect
253. C17#3306semidirect

Maximal subgroups

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

subgroupindex
#117634552
#117631712
#117631392
#117631272
#117630962
#117630602
#117630362
#117630242
#117629522
#117629422
#117629192
#117628812
#117628082
C2×#105874802
C17×#5472
#330617