Finite Group Databasebeta

← #14540320

Subgroups — #14540320

Order 2176.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587618C211yesmaximal, characteristic, complemented
2C2×#10587636C211yesmaximal, characteristic, complemented
3#11763480C211yesmaximal, characteristic, complemented
4#11763199C211yesmaximal, characteristic, complemented
5#11763057C211yesmaximal, characteristic, complemented
6#11763043C211yesmaximal, characteristic, complemented
7#11763028C211yesmaximal, characteristic, complemented
8#11763004C211yesmaximal, characteristic, complemented
9#11762969C211yesmaximal, characteristic, complemented
10#11762954C211yesmaximal, characteristic, complemented
11#11762952C211yesmaximal, characteristic, complemented
12#11762943C211yesmaximal, characteristic, complemented
13#11762909C211yesmaximal, characteristic, complemented
14#11762881C211yesmaximal, characteristic, complemented
15C17×#546C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C2×#63252C2211yescharacteristic
19C2×#63252C2211yescharacteristic, complemented
20C17×#124C2211yescharacteristic
21C17×#123—21no—
22C17×C22⋊Q8C2211yescharacteristic
23C17×#121—21no—
24C17×#121C2211yescharacteristic
25C17×#120—21no—
26C4×D4×C17C2211yescharacteristic
27C4×D4×C17C2211yescharacteristic
28C2×C22⋊C4×C17—21no—
29#10587636—24no—
30C2×#63231C2211yescharacteristic, complemented
31C2×#63230C2211yescharacteristic, complemented
32C2×#63230C2211yescharacteristic, complemented
33#10587626—21no—
34C2×#63229C2211yescharacteristic
35C2×#63229C2211yescharacteristic
36#10587618—24no—
37C2×#63228C2211yescharacteristic
38C2×#63228C2211yescharacteristic
39C2×#63228C2211yescharacteristic
40C2×#63228C2211yescharacteristic
41#10587616C2211yescharacteristic
42#10587616C2211yescharacteristic, complemented
43C2×#63223C2211yescharacteristic
44C22×#3622C2211yescharacteristic, complemented
45#10587579—21no—
46#10587570—21no—
47#10587570—21no—
48#10587560C2211yescharacteristic, complemented
49C4×Dic34C2211yescharacteristic, complemented
50#10587506C2211yescharacteristic, complemented
51#10587503C2211yescharacteristic
52#10587503C2211yescharacteristic
53C22⋊#3622C2211yescharacteristic
54C22⋊#3622C2211yescharacteristic, complemented
55#10587481C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C4⋊#3622C2211yescharacteristic, complemented
60#10587474C2211yescharacteristic, complemented
61C24×C17—21no—
62#41×C17—22no—
63#41×C17—41no—
64C2×Q8×C17C2311yescharacteristic
65C2×D4×C17—22no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17—21no—
69C2×D4×C17—41no—
70C2×D4×C17D411yescharacteristic
71C2×D4×C17C2311yescharacteristic
72C2×D4×C17C2311yescharacteristic
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17C2311yescharacteristic
76C22×C4×C17C2311yescharacteristic
77C2×C22⋊D17—21no—
78C2×C22⋊D17—21no—
79C2×C22⋊D17—21no—
80C2×C22⋊D17—21no—
81#63252—22no—
82#63252—22no—
83#63252—41no—
84#63252—41no—
85C2×D68—21no—
86C2×D68—21no—
87C2×D68C2311yescharacteristic
88C2×D68C2311yescharacteristic
89C2×C4×D17—21no—
90C2×C4×D17—21no—
91C2×Dic34C2311yescharacteristic
92C4⋊C4×C17—21no—
93C4⋊C4×C17D411yescharacteristic
94C4⋊C4×C17C2311yescharacteristic
95C22⋊C4×C17—21no—
96C22⋊C4×C17—21no—
97C22⋊C4×C17—21no—
98C22⋊C4×C17—21no—
99C22⋊C4×C17—21no—
100C22⋊C4×C17D411yescharacteristic
101C22⋊C4×C17D411yescharacteristic
102C42×C17C2311yescharacteristic
103#63231—22no—
104#63230—22no—
105#63230—41no—
106#63229—22no—
107#63229D414yescomplemented
108#63228—22no—
109#63228—22no—
110#63228—22no—
111#63228—41no—
112#63227—21no—
113#63227—21no—
114C4⋊Dic17C2311yescharacteristic
115#63225—21no—
116C4×Dic17—21no—
117#63223—22no—
118C2×#3622—22no—
119C2×#3622C2311yescharacteristic
120C2×#3622C2311yescharacteristic
121C2×#3622C2311yescharacteristic
122C2×#3622C2311yescharacteristic
123C23×C17—21no—
124C23×C17—21no—
125C23×C17—42no—
126C23×C17—42no—
127C23×C17—81no—
128C23×C17C2×D411yescharacteristic
129C23×C17C2×D411yescharacteristic
130C22×D17—21no—
131C22×D17—21no—
132Q8×C17—21no—
133Q8×C17—21no—
134D4×C17—21no—
135D4×C17—21no—
136D4×C17—21no—
137D4×C17—41no—
138D4×C17—42no—
139D4×C17—42no—
140D4×C17—81no—
141D4×C17C2×D412yes—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—42no—
149C2×C4×C17—81no—
150C2×C4×C17C2×D411yescharacteristic
151C2×C4×C17C2×D411yescharacteristic
152C2×C4×C17C2×D411yescharacteristic
153C2×C4×C17C2×D411yescharacteristic
154C2×C4×C17C2411yescharacteristic
155C22⋊D17—42no—
156C22⋊D17—42no—
157C22⋊D17—81no—
158C22⋊D17—81no—
159C2×Dic17—21no—
160C2×Dic17—21no—
161D68—21no—
162D68—21no—
163D68—21no—
164D68—21no—
165D68—42no—
166D68—81no—
167C4×D17—42no—
168C4×D17—42no—
169Dic34—21no—
170Dic34C2×D412yes—
171#3622—21no—
172#3622—21no—
173#3622—21no—
174#3622C2×D412yes—
175#3268—171nomaximal, Sylow
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—22no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—41no—
184C22×C17—41no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—84no—
189C22×C17C22×D411yescharacteristic
190C2×D17—41no—
191C2×D17—41no—
192C2×D17—41no—
193C2×D17—41no—
194C2×D17—41no—
195C2×D17—41no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—21no—
200C4×C17—41no—
201C4×C17—41no—
202C4×C17C22×D411yescharacteristic
203C4×C17H(5,𝔽2)11yescharacteristic
204Dic17—22no—
205Dic17—41no—
206C2×AGL(1,ℤ/8)—171no—
207C2×#135—171no—
208#550—171no—
209#546D1711yescharacteristic, complemented
210#496—171no—
211#468—171no—
212#466—171no—
213#464—171no—
214#459—171no—
215#452—171no—
216#449—171no—
217#449—171no—
218#447—171no—
219#440—171no—
220#435—171no—
221C2×C17—41no—
222C2×C17—41no—
223C2×C17—42no—
224C2×C17—81no—
225C2×C17D4211yescharacteristic
226C2×C17#57511yescharacteristic
227C2×C17#57611yescharacteristic
228D17—81no—
229D17—81no—
230C2×#41—171no—
231C2×#41—171no—
232C2×#41C2×D1711yescharacteristic
233C22×D4C2×D1711yescharacteristic
234AGL(1,ℤ/8)—344no—
235#135—344no—
236C2×Q16—171no—
237C2×SD16—171no—
238C2×SD16—171no—
239C2×SD16—171no—
240C2×SD16—171no—
241C2×D8—171no—
242C2×D8—171no—
243C2×D8—171no—
244C2×D8—171no—
245C2×M16—171no—
246C22×C8—171no—
247#127—171no—
248#127—341no—
249#124C2×D1711yescharacteristic
250#123—21no—
251C22⋊Q8C2×D1711yescharacteristic
252#121—21no—
253#121—171no—
254#121—171no—
255#121—341no—
256#121—341no—
257#121C2×D1711yescharacteristic
258#120—21no—
259C4×Q8—171no—
260C4×D4—341no—
261C4×D4C2×D1711yescharacteristic
262C4×D4C2×D1711yescharacteristic
263C2×C22⋊C4—21no—
264#107—171no—
265C4⋊C8—171no—
266#103—171no—
267#103—171no—
268#102—171no—
269#102—171no—
270#102—171no—
271#102—171no—
272C22⋊C8—171no—
273C22⋊C8—171no—
274#97—171no—
275C17#326811yescharacteristic, complemented, Sylow
276C24—21no—
277#41—22no—
278#41—41no—
279#41—342no—
280#41—342no—
281#41—681no—
282#41—681no—
283C2×Q8—171no—
284C2×Q8C22×D1711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—171no—
291C2×D4—171no—
292C2×D4—341no—
293C2×D4—341no—
294C2×D4—341no—
295C2×D4—341no—
296C2×D4—341no—
297C2×D4—341no—
298C2×D4C22⋊D1711yescharacteristic
299C2×D4C22×D1711yescharacteristic
300C2×D4C22×D1711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—341no—
304C22×C4—341no—
305C22×C4C22×D1711yescharacteristic
306C22×C4C22×D1711yescharacteristic
307Q16—342no—
308SD16—174no—
309SD16—342no—
310SD16—342no—
311SD16—681no—
312D8—342no—
313D8—342no—
314D8—342no—
315D8—681no—
316M16—342no—
317C2×C8—171no—
318C2×C8—171no—
319C2×C8—171no—
320C2×C8—171no—
321C2×C8—342no—
322C4⋊C4—21no—
323C4⋊C4—171no—
324C4⋊C4—341no—
325C4⋊C4C22⋊D1711yescharacteristic
326C4⋊C4C22×D1711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—341no—
333C22⋊C4—341no—
334C22⋊C4C22⋊D1711yescharacteristic
335C22⋊C4C22⋊D1711yescharacteristic
336C42—341no—
337C42C22×D1711yescharacteristic
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—81no—
343C23—341no—
344C23—341no—
345C23C2×C22⋊D1711yescharacteristic
346C23C2×C22⋊D1711yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—172no—
350Q8—341no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—42no—
356D4—42no—
357D4—81no—
358D4—341no—
359D4—341no—
360D4—341no—
361D4—341no—
362D4—682no—
363D4—682no—
364D4—682no—
365D4—1361no—
366D4—1361no—
367D4—1361no—
368D4D4×D1712yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—341no—
378C2×C4—341no—
379C2×C4—682no—
380C2×C4—682no—
381C2×C4C2×C22⋊D1711yescharacteristic
382C2×C4C2×C22⋊D1711yescharacteristic
383C2×C4C2×C22⋊D1711yescharacteristic
384C2×C4C2×C22⋊D1711yescharacteristic
385C2×C4C23×D1711yescharacteristic
386C8—172no—
387C8—341no—
388C8—341no—
389C8—341no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—681no—
404C22—681no—
405C22—681no—
406C22—681no—
407C22—681no—
408C22—681no—
409C22C22×C22⋊D1711yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—342no—
417C4—681no—
418C4C2×D4×D1711yescharacteristic
419C4#1058769111yescharacteristic
420C2—41no—
421C2—41no—
422C2—42no—
423C2—81no—
424C2—1361no—
425C2—1361no—
426C2D4×C22⋊D1711yescharacteristic
427C2#1176360711yescharacteristic
428C2#1176361211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#546C2
socleC22×C17C22×D4
hypercenter#546D17
hypocenterC17#3268
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#10587618C2semidirect
2. C2×#10587636C2semidirect
3. #11763480C2semidirect
4. #11763199C2semidirect
5. #11763057C2semidirect
6. #11763043C2semidirect
7. #11763028C2semidirect
8. #11763004C2semidirect
9. #11762969C2semidirect
10. #11762954C2semidirect
11. #11762952C2semidirect
12. #11762943C2semidirect
13. #11762909C2semidirect
14. #11762881C2semidirect
15. C17×#546C2semidirect
19. C2×#63252C22semidirect
30. C2×#63231C22semidirect
31. C2×#63230C22semidirect
32. C2×#63230C22semidirect
42. #10587616C22semidirect
44. C22×#3622C22semidirect
48. #10587560C22semidirect
49. C4×Dic34C22semidirect
50. #10587506C22semidirect
54. C22⋊#3622C22semidirect
55. #10587481C22semidirect
56. #10587480C22semidirect
57. #10587480C22semidirect
58. #10587478C22semidirect
59. C4⋊#3622C22semidirect
60. #10587474C22semidirect
107. #63229D4semidirect
209. #546D17semidirect
275. C17#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#105876182
C2×#105876362
#117634802
#117631992
#117630572
#117630432
#117630282
#117630042
#117629692
#117629542
#117629522
#117629432
#117629092
#117628812
C17×#5462
#326817