Finite Group Databasebeta

← #14535165

Subgroups — #14535165

Order 2176.

688 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587561C211yesmaximal, characteristic, complemented
2C2×#10587568C211yesmaximal, characteristic
3#11762738C211yesmaximal, characteristic, complemented
4#11762735C211yesmaximal, characteristic, complemented
5#11762726C211yesmaximal, characteristic, complemented
6#11762724C211yesmaximal, characteristic, complemented
7#11762722C211yesmaximal, characteristic, complemented
8#11762721C211yesmaximal, characteristic, complemented
9#11762700C211yesmaximal, characteristic, complemented
10#11762687C211yesmaximal, characteristic, complemented
11#11762687C211yesmaximal, characteristic, complemented
12#11762685C211yesmaximal, characteristic, complemented
13#11762664C211yesmaximal, characteristic, complemented
14#11762664C211yesmaximal, characteristic, complemented
15C17×#388C211yesmaximal, characteristic, complemented
16C24×D17C2211yescharacteristic
17C22×C22⋊D17C2211yescharacteristic
18C22×D68C2211yescharacteristic
19C22×D68C2211yescharacteristic
20C22×C4×D17C2211yescharacteristic, complemented
21C22×Dic34C2211yescharacteristic, complemented
22C2×C4⋊C4×C17C2211yescharacteristic, complemented
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C42×C17C2211yescharacteristic, complemented
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63227C2211yescharacteristic
39C2×#63227C2211yescharacteristic, complemented
40C2×#63227C2211yescharacteristic, complemented
41C2×#63227C2211yescharacteristic, complemented
42C2×#63225C2211yescharacteristic, complemented
43#10587568—24no—
44#10587561—24no—
45C17×#95C2211yescharacteristic, complemented
46C17×#95C2211yescharacteristic, complemented
47#10587502C2211yescharacteristic, complemented
48#10587502C2211yescharacteristic, complemented
49#10587502C2211yescharacteristic, complemented
50#10587502C2211yescharacteristic, complemented
51#10587502C2211yescharacteristic, complemented
52#10587502C2211yescharacteristic, complemented
53C24×C17C2311yescharacteristic
54C23×D17—22no—
55C23×D17—22no—
56C23×D17—22no—
57C23×D17—41no—
58C23×D17—41no—
59C23×D17—41no—
60C23×D17—41no—
61C23×D17C2311yescharacteristic
62C23×D17C2311yescharacteristic
63C23×D17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C22×C4×C17C2311yescharacteristic
69C22×C4×C17C2311yescharacteristic
70C2×C22⋊D17—22no—
71C2×C22⋊D17—22no—
72C2×C22⋊D17—41no—
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C2×D68—22no—
79C2×D68—22no—
80C2×D68—41no—
81C2×D68—41no—
82C2×D68—41no—
83C2×D68D414yes—
84C2×C4×D17—41no—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×Dic34—22no—
88C2×Dic34—41no—
89C2×Dic34—41no—
90C4⋊C4×C17—41no—
91C22⋊C4×C17—22no—
92C22⋊C4×C17—22no—
93C22⋊C4×C17—41no—
94C42×C17—22no—
95C22⋊Dic17—41no—
96C22⋊Dic17—41no—
97#63227—22no—
98#63227—22no—
99#63227—22no—
100#63227—22no—
101#63227—22no—
102#63227—22no—
103#63227—22no—
104#63227—41no—
105#63227—41no—
106#63227—41no—
107#63227—41no—
108#63227—41no—
109#63227—41no—
110#63225—41no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17#4112yes—
118C23×C17C2411yescharacteristic
119C22×D17—21no—
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17—21no—
127C22×D17—21no—
128C22×D17—21no—
129C22×D17—21no—
130C22×D17—21no—
131C22×D17—21no—
132C22×D17—21no—
133C22×D17—21no—
134C22×D17—21no—
135C22×D17—21no—
136C22×D17—21no—
137C22×D17—44no—
138C22×D17—44no—
139C22×D17—44no—
140C22×D17—44no—
141C22×D17—44no—
142C22×D17—44no—
143C22×D17—44no—
144C22×D17C2×D412yes—
145C22×D17C2×D412yes—
146C22×D17#4112yes—
147C2×C4×C17—21no—
148C2×C4×C17—21no—
149C2×C4×C17—21no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
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×C17C2×D412yes—
165C22⋊D17—44no—
166C2×Dic17—21no—
167C2×Dic17—21no—
168C2×Dic17—21no—
169C2×Dic17—21no—
170C2×Dic17—21no—
171C2×Dic17—21no—
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—
181D68—44no—
182D68—44no—
183C4×D17—44no—
184Dic34—44no—
185#2684—171nomaximal, Sylow
186C22×C17—22no—
187C22×C17—22no—
188C22×C17—22no—
189C22×C17—41no—
190C22×C17—41no—
191C22×C17—41no—
192C22×C17—41no—
193C22×C17C22×D411yescharacteristic
194C22×C17C2×#4111yescharacteristic
195C22×C17C2×#4111yescharacteristic
196C22×C17H(5,𝔽2)11yescharacteristic
197C22×C17H(5,𝔽2)11yescharacteristic
198C22×C17H(5,𝔽2)11yescharacteristic
199C22×C17H(5,𝔽2)11yescharacteristic
200C2×D17—22no—
201C2×D17—22no—
202C2×D17—22no—
203C2×D17—22no—
204C2×D17—22no—
205C2×D17—22no—
206C2×D17—22no—
207C2×D17—22no—
208C2×D17—22no—
209C2×D17—416no—
210C2×D17—41no—
211C2×D17—41no—
212C2×D17—41no—
213C2×D17—41no—
214C2×D17—41no—
215C2×D17—41no—
216C2×D17—41no—
217C2×D17—41no—
218C2×D17—41no—
219C2×D17—41no—
220C2×D17—41no—
221C2×D17—41no—
222C4×C17—22no—
223C4×C17—41no—
224C4×C17—41no—
225C4×C17—41no—
226C4×C17—41no—
227C4×C17—41no—
228Dic17—41no—
229Dic17—41no—
230Dic17—41no—
231Dic17—41no—
232Dic17—41no—
233C2×#124—171no—
234C2×#120—171no—
235#397—171no—
236#397—171no—
237#394—171no—
238#394—171no—
239#393—171no—
240#393—171no—
241#388—171no—
242#388D1711yescharacteristic, complemented
243#386—171no—
244#386—171no—
245#386—171no—
246#379—171no—
247#379—171no—
248C2×C17—42no—
249C2×C17#53411yescharacteristic
250C2×C17#53811yescharacteristic
251C2×C17#53811yescharacteristic
252C2×C17#54611yescharacteristic
253C2×C17#54611yescharacteristic
254C2×C17#55111yescharacteristic
255C2×C17H(3,𝔽4)11yescharacteristic
256D17—42no—
257D17—42no—
258D17—42no—
259C25—171no—
260C22×Q8—171no—
261C22×D4—171no—
262C22×D4—171no—
263C22×D4—171no—
264C23×C4—171no—
265#124—344no—
266#120—344no—
267C2×C4⋊C4—171no—
268C2×C4⋊C4C2×D1711yescharacteristic, complemented
269C2×C22⋊C4—171no—
270C2×C22⋊C4—171no—
271C2×C22⋊C4—171no—
272C2×C22⋊C4—171no—
273C2×C22⋊C4—171no—
274C2×C22⋊C4—171no—
275C2×C22⋊C4—171no—
276C2×C22⋊C4—171no—
277C2×C22⋊C4—171no—
278C2×C22⋊C4—171no—
279C2×C22⋊C4—171no—
280C2×C22⋊C4—171no—
281C2×C22⋊C4—171no—
282C2×C22⋊C4—171no—
283C2×C22⋊C4—171no—
284C2×C22⋊C4C2×D1711yescharacteristic
285C2×C22⋊C4C2×D1711yescharacteristic
286C2×C22⋊C4C2×D1711yescharacteristic
287C2×C42C2×D1711yescharacteristic, complemented
288#95—171no—
289#95—171no—
290#95—171no—
291#95—171no—
292#95—171no—
293#95—171no—
294#95C2×D1711yescharacteristic, complemented
295#95C2×D1711yescharacteristic, complemented
296C17#268411yescharacteristic, complemented, Sylow
297C24—171no—
298C24—171no—
299C24—171no—
300C24—342no—
301C24—342no—
302C24—342no—
303C24—681no—
304C24—681no—
305C24—681no—
306C24—681no—
307C24C22×D1711yescharacteristic
308C2×Q8—342no—
309C2×Q8—681no—
310C2×Q8—681no—
311C2×D4—174no—
312C2×D4—342no—
313C2×D4—342no—
314C2×D4—342no—
315C2×D4—342no—
316C2×D4—681no—
317C2×D4—681no—
318C2×D4—681no—
319C2×D4—681no—
320C22×C4—171no—
321C22×C4—171no—
322C22×C4—171no—
323C22×C4—171no—
324C22×C4—171no—
325C22×C4—681no—
326C22×C4—681no—
327C22×C4—681no—
328C22×C4C22×D1711yescharacteristic
329C22×C4C22×D1711yescharacteristic
330C22×C4C22×D1711yescharacteristic
331C22×C4C22×D1711yescharacteristic
332C22×C4C22×D1711yescharacteristic
333C22×C4C22×D1711yescharacteristic
334C4⋊C4—41no—
335C4⋊C4—681no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—41no—
339C22⋊C4—342no—
340C22⋊C4—342no—
341C22⋊C4—342no—
342C22⋊C4—342no—
343C22⋊C4—342no—
344C22⋊C4—342no—
345C22⋊C4—342no—
346C22⋊C4—681no—
347C22⋊C4—681no—
348C22⋊C4—681no—
349C22⋊C4—681no—
350C22⋊C4—681no—
351C22⋊C4—681no—
352C22⋊C4—681no—
353C22⋊C4—681no—
354C42—22no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—172no—
362C23—172no—
363C23—172no—
364C23—341no—
365C23—341no—
366C23—341no—
367C23—341no—
368C23—341no—
369C23—341no—
370C23—341no—
371C23—341no—
372C23—341no—
373C23—341no—
374C23—341no—
375C23—341no—
376C23—341no—
377C23—341no—
378C23—341no—
379C23—341no—
380C23—341no—
381C23—341no—
382C23—684no—
383C23—684no—
384C23—684no—
385C23—684no—
386C23—684no—
387C23—684no—
388C23—684no—
389C23#6325212yes—
390C23C23×D1711yescharacteristic
391Q8—684no—
392D4—684no—
393D4—684no—
394D4—684no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—341no—
413C2×C4—341no—
414C2×C4—341no—
415C2×C4—341no—
416C2×C4—341no—
417C2×C4—341no—
418C2×C4—341no—
419C2×C4—341no—
420C2×C4—341no—
421C2×C4—341no—
422C2×C4—341no—
423C2×C4—341no—
424C2×C4—341no—
425C2×C4—341no—
426C2×C4—341no—
427C2×C4—684no—
428C2×C4D4×D1712yes—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—342no—
437C22—342no—
438C22—342no—
439C22—342no—
440C22—342no—
441C22—342no—
442C22—342no—
443C22—342no—
444C22—342no—
445C22—6816no—
446C22—681no—
447C22—681no—
448C22—681no—
449C22—681no—
450C22—681no—
451C22—681no—
452C22—681no—
453C22—681no—
454C22—681no—
455C22—681no—
456C22—681no—
457C22—681no—
458C22C2×#6325211yescharacteristic
459C22C2×D4×D1711yescharacteristic
460C22#1058769111yescharacteristic
461C22#41×D1711yescharacteristic
462C22#1058769611yescharacteristic
463C22#1058769611yescharacteristic
464C22#1058769611yescharacteristic
465C4—22no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—681no—
472C4—681no—
473C4—681no—
474C4—681no—
475C4—681no—
476C2—42no—
477C2—682no—
478C2—682no—
479C2—682no—
480C2#1176337011yescharacteristic
481C2#1176339211yescharacteristic
482C2#1176339811yescharacteristic
483C2#1176345511yescharacteristic
484C2#1176345611yescharacteristic
485C2#1176348311yescharacteristic
486C2#1176355111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#388C2
prime radicalC2×#10587568C2
hypercenter#388D17
hypocenterC17#2684
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#10587561C2semidirect
3. #11762738C2semidirect
4. #11762735C2semidirect
5. #11762726C2semidirect
6. #11762724C2semidirect
7. #11762722C2semidirect
8. #11762721C2semidirect
9. #11762700C2semidirect
10. #11762687C2semidirect
11. #11762687C2semidirect
12. #11762685C2semidirect
13. #11762664C2semidirect
14. #11762664C2semidirect
15. C17×#388C2semidirect
20. C22×C4×D17C22semidirect
21. C22×Dic34C22semidirect
22. C2×C4⋊C4×C17C22semidirect
26. C2×C42×C17C22semidirect
39. C2×#63227C22semidirect
40. C2×#63227C22semidirect
41. C2×#63227C22semidirect
42. C2×#63225C22semidirect
45. C17×#95C22semidirect
46. C17×#95C22semidirect
47. #10587502C22semidirect
48. #10587502C22semidirect
49. #10587502C22semidirect
50. #10587502C22semidirect
51. #10587502C22semidirect
52. #10587502C22semidirect
242. #388D17semidirect
268. C2×C4⋊C4C2×D17semidirect
287. C2×C42C2×D17semidirect
294. #95C2×D17semidirect
295. #95C2×D17semidirect
296. C17#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#105875612
C2×#105875682
#117627382
#117627352
#117627262
#117627242
#117627222
#117627212
#117627002
#117626872
#117626852
#117626642
C17×#3882
#268417