Finite Group Databasebeta

← #14535829

Subgroups — #14535829

Order 2176.

560 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762738C211yesmaximal, characteristic, complemented
2#11762737C211yesmaximal, characteristic, complemented
3#11762735C211yesmaximal, characteristic, complemented
4#11762722C211yesmaximal, characteristic, complemented
5#11762722C211yesmaximal, characteristic, complemented
6#11762721C211yesmaximal, characteristic, complemented
7#11762700C211yesmaximal, characteristic, complemented
8#11762700C211yesmaximal, characteristic, complemented
9#11762699C211yesmaximal, characteristic, complemented
10#11762698C211yesmaximal, characteristic, complemented
11#11762697C211yesmaximal, characteristic, complemented
12#11762664C211yesmaximal, characteristic
13#11762664C211yesmaximal, characteristic
14#11762664C211yesmaximal, characteristic
15C17×#388C211yesmaximal, characteristic, complemented
16C24×D17C2211yescharacteristic
17C22×Dic34C2211yescharacteristic, complemented
18C2×C4⋊C4×C17C2211yescharacteristic, complemented
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C42×C17C2211yescharacteristic, complemented
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63225C2211yescharacteristic, complemented
39C2×#63225C2211yescharacteristic, complemented
40C2×C4×Dic17C2211yescharacteristic, complemented
41C17×#95C2211yescharacteristic, complemented
42C17×#95C2211yescharacteristic, complemented
43#10587502C2211yescharacteristic, complemented
44#10587502C2211yescharacteristic, complemented
45#10587502C2211yescharacteristic, complemented
46#10587502C2211yescharacteristic, complemented
47#10587502C2211yescharacteristic, complemented
48#10587502C2211yescharacteristic, complemented
49#10587502C2211yescharacteristic, complemented
50#10587502C2211yescharacteristic, complemented
51C24×C17C2311yescharacteristic
52C23×D17—22no—
53C23×D17—22no—
54C23×D17—22no—
55C23×D17—41no—
56C23×D17—41no—
57C23×D17—41no—
58C23×D17—41no—
59C23×D17C2311yescharacteristic
60C23×D17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C2×Dic34—41no—
74C2×Dic34—41no—
75C2×Dic34—41no—
76C4⋊C4×C17—41no—
77C22⋊C4×C17—22no—
78C22⋊C4×C17—41no—
79C22⋊C4×C17—41no—
80C42×C17—41no—
81C22⋊Dic17—22no—
82C22⋊Dic17—41no—
83C22⋊Dic17—41no—
84#63227—22no—
85#63227—22no—
86#63227—22no—
87#63227—22no—
88#63227—41no—
89#63227—41no—
90#63227—41no—
91#63227—41no—
92#63227—41no—
93#63227—41no—
94#63227—41no—
95#63227—41no—
96#63225—41no—
97#63225—41no—
98C4×Dic17—41no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17#4112yes—
106C23×C17C2411yescharacteristic
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17—21no—
112C22×D17—21no—
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—44no—
120C22×D17—44no—
121C22×D17—44no—
122C22×D17—44no—
123C22×D17—44no—
124C22×D17—44no—
125C22×D17—44no—
126C22×D17#4112yes—
127C22×D17#4112yes—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—21no—
157C2×Dic17—21no—
158C2×Dic17—21no—
159C2×Dic17—21no—
160C2×Dic17—21no—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164Dic34—44no—
165#2736—171nomaximal, Sylow
166C22×C17—22no—
167C22×C17—22no—
168C22×C17—22no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17C2×#4111yescharacteristic
174C22×C17C2×#4111yescharacteristic
175C22×C17C2×#4111yescharacteristic
176C22×C17H(5,𝔽2)11yescharacteristic
177C22×C17H(5,𝔽2)11yescharacteristic
178C22×C17H(5,𝔽2)11yescharacteristic
179C22×C17H(5,𝔽2)11yescharacteristic
180C2×D17—22no—
181C2×D17—22no—
182C2×D17—22no—
183C2×D17—22no—
184C2×D17—22no—
185C2×D17—22no—
186C2×D17—416no—
187C2×D17—41no—
188C2×D17—41no—
189C2×D17—41no—
190C2×D17—41no—
191C2×D17—41no—
192C2×D17—41no—
193C2×D17—41no—
194C2×D17—41no—
195C4×C17—41no—
196C4×C17—41no—
197C4×C17—41no—
198C4×C17—41no—
199C4×C17—41no—
200C4×C17—41no—
201Dic17—41no—
202Dic17—41no—
203Dic17—41no—
204Dic17—41no—
205Dic17—41no—
206Dic17—41no—
207#397—171no—
208#397—171no—
209#397—171no—
210#393—171no—
211#393—171no—
212#393—171no—
213#388—171no—
214#388—171no—
215#388—171no—
216#388—171no—
217#388—171no—
218#388D1711yescharacteristic, complemented
219#379—171no—
220#379—171no—
221#379—171no—
222C2×C17—42no—
223C2×C17#53811yescharacteristic
224C2×C17#53811yescharacteristic
225C2×C17#53811yescharacteristic
226C2×C17#55111yescharacteristic
227C2×C17#55111yescharacteristic
228C2×C17#55111yescharacteristic
229C2×C17H(3,𝔽4)11yescharacteristic
230D17—42no—
231D17—42no—
232C25—171no—
233C22×Q8—171no—
234C2×C4⋊C4—171no—
235C2×C4⋊C4—171no—
236C2×C4⋊C4C2×D1711yescharacteristic, complemented
237C2×C22⋊C4—171no—
238C2×C22⋊C4—171no—
239C2×C22⋊C4—171no—
240C2×C22⋊C4—171no—
241C2×C22⋊C4—171no—
242C2×C22⋊C4—171no—
243C2×C22⋊C4—171no—
244C2×C22⋊C4—171no—
245C2×C22⋊C4—171no—
246C2×C22⋊C4—171no—
247C2×C22⋊C4—171no—
248C2×C22⋊C4—171no—
249C2×C22⋊C4—171no—
250C2×C22⋊C4—171no—
251C2×C22⋊C4—171no—
252C2×C22⋊C4C2×D1711yescharacteristic
253C2×C22⋊C4C2×D1711yescharacteristic
254C2×C22⋊C4C2×D1711yescharacteristic
255C2×C42—171no—
256C2×C42C2×D1711yescharacteristic, complemented
257#95—171no—
258#95—171no—
259#95—171no—
260#95—171no—
261#95—171no—
262#95—171no—
263#95—171no—
264#95—171no—
265#95C2×D1711yescharacteristic, complemented
266#95C2×D1711yescharacteristic, complemented
267C17#273611yescharacteristic, complemented, Sylow
268C24—171no—
269C24—171no—
270C24—342no—
271C24—342no—
272C24—342no—
273C24—681no—
274C24—681no—
275C24—681no—
276C24—681no—
277C24C22×D1711yescharacteristic
278C2×Q8—681no—
279C2×Q8—681no—
280C2×Q8—681no—
281C22×C4—171no—
282C22×C4—171no—
283C22×C4—171no—
284C22×C4—171no—
285C22×C4—171no—
286C22×C4—171no—
287C22×C4C22×D1711yescharacteristic
288C22×C4C22×D1711yescharacteristic
289C22×C4C22×D1711yescharacteristic
290C22×C4C22×D1711yescharacteristic
291C22×C4C22×D1711yescharacteristic
292C22×C4C22×D1711yescharacteristic
293C4⋊C4—41no—
294C4⋊C4—681no—
295C4⋊C4—681no—
296C22⋊C4—22no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—342no—
300C22⋊C4—342no—
301C22⋊C4—342no—
302C22⋊C4—342no—
303C22⋊C4—342no—
304C22⋊C4—681no—
305C22⋊C4—681no—
306C22⋊C4—681no—
307C22⋊C4—681no—
308C22⋊C4—681no—
309C22⋊C4—681no—
310C22⋊C4—681no—
311C22⋊C4—681no—
312C22⋊C4—681no—
313C22⋊C4—681no—
314C42—41no—
315C42—681no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—172no—
323C23—172no—
324C23—341no—
325C23—341no—
326C23—341no—
327C23—341no—
328C23—341no—
329C23—341no—
330C23—341no—
331C23—341no—
332C23—341no—
333C23—341no—
334C23—341no—
335C23—341no—
336C23—684no—
337C23—684no—
338C23—684no—
339C23—684no—
340C23—684no—
341C23—684no—
342C23—684no—
343C23#6325212yes—
344C23C23×D1711yescharacteristic
345Q8—684no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—341no—
365C2×C4—341no—
366C2×C4—341no—
367C2×C4—341no—
368C2×C4—341no—
369C2×C4—341no—
370C2×C4—341no—
371C2×C4—341no—
372C2×C4—341no—
373C2×C4—341no—
374C2×C4—341no—
375C2×C4—341no—
376C2×C4—341no—
377C2×C4—341no—
378C2×C4—341no—
379C2×C4—341no—
380C2×C4—341no—
381C2×C4—341no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—342no—
390C22—342no—
391C22—342no—
392C22—342no—
393C22—342no—
394C22—342no—
395C22—6816no—
396C22—681no—
397C22—681no—
398C22—681no—
399C22—681no—
400C22—681no—
401C22—681no—
402C22—681no—
403C22—681no—
404C22C2×#6325211yescharacteristic
405C22#1058769111yescharacteristic
406C22#1058769111yescharacteristic
407C22#41×D1711yescharacteristic
408C22#41×D1711yescharacteristic
409C22#1058769611yescharacteristic
410C22#1058769611yescharacteristic
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—681no—
418C4—681no—
419C4—681no—
420C4—681no—
421C4—681no—
422C4—681no—
423C2—42no—
424C2—682no—
425C2—682no—
426C2#1176339111yescharacteristic
427C2#1176339211yescharacteristic
428C2#1176339511yescharacteristic
429C2#1176348311yescharacteristic
430C2#1176348411yescharacteristic
431C2#1176348811yescharacteristic
432C2#1176355111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#388C2
prime radicalC24×D17C22
hypercenter#388D17
hypocenterC17#2736
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. #11762738C2semidirect
2. #11762737C2semidirect
3. #11762735C2semidirect
4. #11762722C2semidirect
5. #11762722C2semidirect
6. #11762721C2semidirect
7. #11762700C2semidirect
8. #11762700C2semidirect
9. #11762699C2semidirect
10. #11762698C2semidirect
11. #11762697C2semidirect
15. C17×#388C2semidirect
17. C22×Dic34C22semidirect
18. C2×C4⋊C4×C17C22semidirect
22. C2×C42×C17C22semidirect
38. C2×#63225C22semidirect
39. C2×#63225C22semidirect
40. C2×C4×Dic17C22semidirect
41. C17×#95C22semidirect
42. C17×#95C22semidirect
43. #10587502C22semidirect
44. #10587502C22semidirect
45. #10587502C22semidirect
46. #10587502C22semidirect
47. #10587502C22semidirect
48. #10587502C22semidirect
49. #10587502C22semidirect
50. #10587502C22semidirect
218. #388D17semidirect
236. C2×C4⋊C4C2×D17semidirect
256. C2×C42C2×D17semidirect
265. #95C2×D17semidirect
266. #95C2×D17semidirect
267. C17#2736semidirect

Maximal subgroups

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

subgroupindex
#117627382
#117627372
#117627352
#117627222
#117627212
#117627002
#117626992
#117626982
#117626972
#117626642
C17×#3882
#273617