Finite Group Databasebeta

← #14532243

Subgroups — #14532243

Order 2176.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587623C211yesmaximal, characteristic
2C2×C4×C22⋊D17C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D17C211yesmaximal, characteristic, complemented
4#11762744C211yesmaximal, characteristic, complemented
5#11762737C211yesmaximal, characteristic, complemented
6#11762725C211yesmaximal, characteristic, complemented
7#11762723C211yesmaximal, characteristic, complemented
8#11762702C211yesmaximal, characteristic, complemented
9#11762699C211yesmaximal, characteristic, complemented
10#11762688C211yesmaximal, characteristic, complemented
11#11762685C211yesmaximal, characteristic, complemented
12#11762680C211yesmaximal, characteristic, complemented
13#11762666C211yesmaximal, characteristic, complemented
14#11762664C211yesmaximal, characteristic, complemented
15C2×C17×#123C211yesmaximal, characteristic, complemented
16C24×D17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C23×C4×C17C2211yescharacteristic, complemented
19C22×C22⋊D17C2211yescharacteristic
20C22×C22⋊D17C2211yescharacteristic
21C22×C4×D17C2211yescharacteristic
22C22×C4×D17C2211yescharacteristic
23C17×#123—24no—
24C2×C4⋊C4×C17C2211yescharacteristic, complemented
25C2×C4⋊C4×C17C2211yescharacteristic, complemented
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C22⋊C4×C17C2211yescharacteristic, complemented
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32C2×C22⋊Dic17C2211yescharacteristic, complemented
33C2×C22⋊Dic17C2211yescharacteristic, complemented
34#10587623—24no—
35C4×C22⋊D17—24no—
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63227C2211yescharacteristic
39C2×#63227C2211yescharacteristic
40C2×#63227C2211yescharacteristic
41C2×#63227C2211yescharacteristic
42C2×#63227C2211yescharacteristic
43C2×#63227C2211yescharacteristic
44C2×C4⋊Dic17C2211yescharacteristic, complemented
45C2×#63225C2211yescharacteristic, complemented
46C2×#63225C2211yescharacteristic, complemented
47C2×C4×Dic17C2211yescharacteristic, complemented
48C22⋊C4×D17—24no—
49#10587502C2211yescharacteristic, complemented
50#10587502C2211yescharacteristic, complemented
51#10587502C2211yescharacteristic, complemented
52#10587502C2211yescharacteristic, complemented
53#10587502C2211yescharacteristic, complemented
54#10587502C2211yescharacteristic, complemented
55C24×C17C2311yescharacteristic
56C24×C17C2311yescharacteristic
57C23×D17—22no—
58C23×D17—22no—
59C23×D17—22no—
60C23×D17—22no—
61C23×D17—22no—
62C23×D17—41no—
63C23×D17—41no—
64C23×D17C2311yescharacteristic
65C23×D17C2311yescharacteristic
66C2×D4×C17—22no—
67C2×D4×C17—22no—
68C2×D4×C17—41no—
69C22×C4×C17—22no—
70C22×C4×C17—22no—
71C22×C4×C17—41no—
72C22×C4×C17C2311yescharacteristic
73C22×C4×C17C2311yescharacteristic
74C22×C4×C17C2311yescharacteristic
75C22×C4×C17C2311yescharacteristic
76C22×C4×C17C2311yescharacteristic
77C2×C22⋊D17—22no—
78C2×C22⋊D17—22no—
79C2×C22⋊D17—22no—
80C2×C22⋊D17—22no—
81C2×C22⋊D17—41no—
82C2×C22⋊D17D414yes—
83C22×Dic17C2311yescharacteristic
84C22×Dic17C2311yescharacteristic
85C22×Dic17C2311yescharacteristic
86C22×Dic17C2311yescharacteristic
87C22×Dic17C2311yescharacteristic
88C22×Dic17C2311yescharacteristic
89C2×C4×D17—22no—
90C2×C4×D17—22no—
91C2×C4×D17—22no—
92C2×C4×D17—22no—
93C2×C4×D17—41no—
94C2×C4×D17—41no—
95C4⋊C4×C17—22no—
96C4⋊C4×C17—22no—
97C22⋊C4×C17—22no—
98C22⋊C4×C17—22no—
99C22⋊C4×C17D414yescomplemented
100C22⋊Dic17—22no—
101C22⋊Dic17—22no—
102C22⋊Dic17—22no—
103C22⋊Dic17—41no—
104C22⋊Dic17—41no—
105#63227—22no—
106#63227—22no—
107#63227—22no—
108#63227—22no—
109#63227—22no—
110#63227—22no—
111#63227—22no—
112#63227—41no—
113C4⋊Dic17—41no—
114#63225—22no—
115#63225—41no—
116C4×Dic17—22no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—21no—
120C23×C17—21no—
121C23×C17—21no—
122C23×C17—21no—
123C23×C17—21no—
124C23×C17—21no—
125C23×C17—21no—
126C23×C17—21no—
127C23×C17—21no—
128C23×C17—21no—
129C23×C17C2×D412yes—
130C23×C17C2×D412yes—
131C23×C17C2411yescharacteristic
132C22×D17—21no—
133C22×D17—21no—
134C22×D17—21no—
135C22×D17—21no—
136C22×D17—28no—
137C22×D17—21no—
138C22×D17—21no—
139C22×D17—21no—
140C22×D17—21no—
141C22×D17—21no—
142C22×D17—21no—
143C22×D17—44no—
144C22×D17—44no—
145C22×D17—44no—
146C22×D17—44no—
147C22×D17—44no—
148C22×D17—44no—
149C22×D17C2×D412yes—
150C22×D17#4112yes—
151C22×D17#4112yes—
152C22×D17#4112yes—
153D4×C17—44no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17—21no—
159C2×C4×C17—21no—
160C2×C4×C17—21no—
161C2×C4×C17—21no—
162C2×C4×C17—21no—
163C2×C4×C17—21no—
164C2×C4×C17—21no—
165C2×C4×C17—21no—
166C2×C4×C17—21no—
167C2×C4×C17—44no—
168C2×C4×C17C2×D412yes—
169C2×C4×C17C2×D412yes—
170C22⋊D17—28no—
171C22⋊D17—44no—
172C2×Dic17—21no—
173C2×Dic17—21no—
174C2×Dic17—21no—
175C2×Dic17—21no—
176C2×Dic17—21no—
177C2×Dic17—21no—
178C2×Dic17—21no—
179C2×Dic17—21no—
180C2×Dic17—21no—
181C2×Dic17—21no—
182C2×Dic17—21no—
183C2×Dic17—21no—
184C2×Dic17—21no—
185C2×Dic17—21no—
186C2×Dic17—21no—
187C2×Dic17—21no—
188C2×Dic17—21no—
189C2×Dic17C2×D412yes—
190C4×D17—44no—
191C4×D17—44no—
192#2418—171nomaximal, Sylow
193C22×C17—22no—
194C22×C17—22no—
195C22×C17—22no—
196C22×C17—22no—
197C22×C17—22no—
198C22×C17—22no—
199C22×C17—41no—
200C22×C17—41no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—41no—
204C22×C17—41no—
205C22×C17—41no—
206C22×C17—41no—
207C22×C17C22×D411yescharacteristic
208C22×C17C22×D411yescharacteristic
209C22×C17C2×#4111yescharacteristic
210C22×C17C2×#4111yescharacteristic
211C22×C17C2×#4111yescharacteristic
212C22×C17H(5,𝔽2)11yescharacteristic
213C22×C17H(5,𝔽2)11yescharacteristic
214C2×D17—22no—
215C2×D17—22no—
216C2×D17—22no—
217C2×D17—22no—
218C2×D17—22no—
219C2×D17—22no—
220C2×D17—22no—
221C2×D17—22no—
222C2×D17—22no—
223C2×D17—416no—
224C2×D17—41no—
225C2×D17—41no—
226C2×D17—41no—
227C2×D17—41no—
228C2×D17#12314yescomplemented
229C4×C17—22no—
230C4×C17—22no—
231C4×C17—41no—
232C4×C17—41no—
233C4×C17—41no—
234Dic17—22no—
235Dic17—41no—
236Dic17—41no—
237Dic17—41no—
238Dic17—41no—
239Dic17—41no—
240C2×#123D1711yescharacteristic, complemented
241C2×#120—171no—
242C2×C4×D4—171no—
243C22×C22⋊C4—171no—
244#399—171no—
245#397—171no—
246#394—171no—
247#394—171no—
248#388—171no—
249#388—171no—
250#386—171no—
251#386—171no—
252#385—171no—
253#380—171no—
254#379—171no—
255C2×C17—42no—
256C2×C17—42no—
257C2×C17C2×#12311yescharacteristic
258C2×C17#52511yescharacteristic
259C2×C17#53811yescharacteristic
260C2×C17D4211yescharacteristic
261C2×C17#54611yescharacteristic
262C2×C17#54611yescharacteristic
263C2×C17#55111yescharacteristic
264D17—24no—
265D17—42no—
266C25—171no—
267C22×D4—171no—
268C22×D4—171no—
269C22×D4C2×D1711yescharacteristic
270C23×C4—171no—
271C23×C4—171no—
272C23×C4C2×D1711yescharacteristic, complemented
273#123—24no—
274#120—344no—
275C4×D4—344no—
276C2×C4⋊C4—171no—
277C2×C4⋊C4—171no—
278C2×C4⋊C4—171no—
279C2×C4⋊C4C2×D1711yescharacteristic, complemented
280C2×C4⋊C4C2×D1711yescharacteristic, complemented
281C2×C22⋊C4—171no—
282C2×C22⋊C4—171no—
283C2×C22⋊C4—171no—
284C2×C22⋊C4—171no—
285C2×C22⋊C4—171no—
286C2×C22⋊C4—171no—
287C2×C22⋊C4—171no—
288C2×C22⋊C4—171no—
289C2×C22⋊C4—171no—
290C2×C22⋊C4—171no—
291C2×C22⋊C4—171no—
292C2×C22⋊C4—171no—
293C2×C22⋊C4—171no—
294C2×C22⋊C4—344no—
295C2×C22⋊C4C2×D1711yescharacteristic
296C2×C22⋊C4C2×D1711yescharacteristic
297C2×C22⋊C4C2×D1711yescharacteristic, complemented
298C2×C42—171no—
299#95—171no—
300#95—171no—
301#95—171no—
302#95—171no—
303#95—171no—
304#95—171no—
305C17#241811yescharacteristic, complemented, Sylow
306C24—171no—
307C24—171no—
308C24—342no—
309C24—342no—
310C24—342no—
311C24—342no—
312C24—342no—
313C24—681no—
314C24—681no—
315C24C22×D1711yescharacteristic
316C24C22×D1711yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—174no—
321C2×D4—342no—
322C2×D4—342no—
323C2×D4—342no—
324C2×D4—342no—
325C2×D4—681no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—171no—
330C22×C4—171no—
331C22×C4—171no—
332C22×C4—171no—
333C22×C4—171no—
334C22×C4—171no—
335C22×C4—342no—
336C22×C4—342no—
337C22×C4—342no—
338C22×C4—342no—
339C22×C4—681no—
340C22×C4—681no—
341C22×C4C22×D1711yescharacteristic
342C22×C4C22×D1711yescharacteristic
343C22×C4C22×D1711yescharacteristic
344C22×C4C22×D1711yescharacteristic
345C22×C4C22×D1711yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—342no—
349C4⋊C4—681no—
350C4⋊C4—681no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—342no—
354C22⋊C4—342no—
355C22⋊C4—342no—
356C22⋊C4—342no—
357C22⋊C4—342no—
358C22⋊C4—342no—
359C22⋊C4—342no—
360C22⋊C4—342no—
361C22⋊C4—342no—
362C22⋊C4—342no—
363C22⋊C4—681no—
364C22⋊C4—681no—
365C22⋊C4—681no—
366C22⋊C4C22⋊D1714yescomplemented
367C42—342no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—172no—
381C23—172no—
382C23—172no—
383C23—172no—
384C23—348no—
385C23—341no—
386C23—341no—
387C23—341no—
388C23—341no—
389C23—341no—
390C23—341no—
391C23—341no—
392C23—341no—
393C23—341no—
394C23—341no—
395C23—684no—
396C23—684no—
397C23—684no—
398C23—684no—
399C23—684no—
400C23—684no—
401C23D4×D1712yes—
402C23C2×C22⋊D1712yes—
403C23C23×D1711yescharacteristic
404D4—44no—
405D4—348no—
406D4—684no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—44no—
421C2×C4—172no—
422C2×C4—341no—
423C2×C4—341no—
424C2×C4—341no—
425C2×C4—341no—
426C2×C4—341no—
427C2×C4—341no—
428C2×C4—341no—
429C2×C4—341no—
430C2×C4—341no—
431C2×C4—341no—
432C2×C4—341no—
433C2×C4—341no—
434C2×C4—341no—
435C2×C4—341no—
436C2×C4—341no—
437C2×C4—341no—
438C2×C4—341no—
439C2×C4—684no—
440C2×C4—684no—
441C2×C4C2×C22⋊D1712yes—
442C2×C4C2×C22⋊D1712yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—174no—
458C22—342no—
459C22—342no—
460C22—342no—
461C22—342no—
462C22—342no—
463C22—342no—
464C22—342no—
465C22—342no—
466C22—342no—
467C22—6816no—
468C22—681no—
469C22—681no—
470C22—681no—
471C22—681no—
472C22C2×D4×D1711yescharacteristic
473C22#1058769111yescharacteristic
474C22#41×D1711yescharacteristic
475C22#41×D1711yescharacteristic
476C22#41×D1711yescharacteristic
477C22#1058769611yescharacteristic
478C22C22×C22⋊D1711yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—342no—
485C4—681no—
486C4—681no—
487C4—681no—
488C4—681no—
489C4—681no—
490C2—42no—
491C2—42no—
492C2—344no—
493C2—682no—
494C2#123×D1711yescharacteristic
495C2#1176331611yescharacteristic
496C2#1176339511yescharacteristic
497C2D4×C22⋊D1711yescharacteristic
498C2#1176344711yescharacteristic
499C2#1176345111yescharacteristic
500C2#1176348711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#123C2
prime radicalC2×#10587623C2
hypercenterC2×#123D17
hypocenterC17#2418
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D17C2semidirect
3. C2×C22⋊C4×D17C2semidirect
4. #11762744C2semidirect
5. #11762737C2semidirect
6. #11762725C2semidirect
7. #11762723C2semidirect
8. #11762702C2semidirect
9. #11762699C2semidirect
10. #11762688C2semidirect
11. #11762685C2semidirect
12. #11762680C2semidirect
13. #11762666C2semidirect
14. #11762664C2semidirect
15. C2×C17×#123C2semidirect
18. C23×C4×C17C22semidirect
24. C2×C4⋊C4×C17C22semidirect
25. C2×C4⋊C4×C17C22semidirect
28. C2×C22⋊C4×C17C22semidirect
32. C2×C22⋊Dic17C22semidirect
33. C2×C22⋊Dic17C22semidirect
44. C2×C4⋊Dic17C22semidirect
45. C2×#63225C22semidirect
46. C2×#63225C22semidirect
47. C2×C4×Dic17C22semidirect
49. #10587502C22semidirect
50. #10587502C22semidirect
51. #10587502C22semidirect
52. #10587502C22semidirect
53. #10587502C22semidirect
54. #10587502C22semidirect
99. C22⋊C4×C17D4semidirect
228. C2×D17#123semidirect
240. C2×#123D17semidirect
272. C23×C4C2×D17semidirect
279. C2×C4⋊C4C2×D17semidirect
280. C2×C4⋊C4C2×D17semidirect
297. C2×C22⋊C4C2×D17semidirect
305. C17#2418semidirect
366. C22⋊C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876232
C2×C4×C22⋊D172
C2×C22⋊C4×D172
#117627442
#117627372
#117627252
#117627232
#117627022
#117626992
#117626882
#117626852
#117626802
#117626662
#117626642
C2×C17×#1232
#241817