Finite Group Databasebeta

← #14535171

Subgroups — #14535171

Order 2176.

666 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587565C211yesmaximal, characteristic, complemented
2#11762745C211yesmaximal, characteristic, complemented
3#11762732C211yesmaximal, characteristic, complemented
4#11762723C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762723C211yesmaximal, characteristic, complemented
7#11762723C211yesmaximal, characteristic, complemented
8#11762712C211yesmaximal, characteristic, complemented
9#11762712C211yesmaximal, characteristic, complemented
10#11762703C211yesmaximal, characteristic, complemented
11#11762702C211yesmaximal, characteristic, complemented
12#11762697C211yesmaximal, characteristic, complemented
13#11762665C211yesmaximal, characteristic, complemented
14#11762665C211yesmaximal, characteristic, complemented
15C2×C17×#120C211yesmaximal, characteristic
16C25×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×D4×C17C2211yescharacteristic
19C22×D4×C17C2211yescharacteristic
20C17×#120—24no—
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32C2×C22⋊Dic17C2211yescharacteristic
33C2×C22⋊Dic17C2211yescharacteristic
34C2×C22⋊Dic17C2211yescharacteristic
35C2×C22⋊Dic17C2211yescharacteristic
36C2×C22⋊Dic17C2211yescharacteristic, complemented
37C2×C22⋊Dic17C2211yescharacteristic, complemented
38C2×C22⋊Dic17C2211yescharacteristic, complemented
39C2×C22⋊Dic17C2211yescharacteristic, complemented
40C2×C4⋊Dic17C2211yescharacteristic, complemented
41C2×C4⋊Dic17C2211yescharacteristic, complemented
42C2×#63225C2211yescharacteristic, complemented
43C2×#63225C2211yescharacteristic, complemented
44C2×#63225C2211yescharacteristic, complemented
45C2×#63225C2211yescharacteristic, complemented
46C2×C4×Dic17C2211yescharacteristic, complemented
47C2×C4×Dic17C2211yescharacteristic, complemented
48#10587565—24no—
49#10587502C2211yescharacteristic, complemented
50#10587502C2211yescharacteristic, complemented
51#10587502C2211yescharacteristic, complemented
52#10587502C2211yescharacteristic, complemented
53C24×C17—22no—
54C24×C17—22no—
55C24×C17—22no—
56C24×C17—41no—
57C24×C17—41no—
58C24×C17—41no—
59C24×C17—41no—
60C24×C17C2311yescharacteristic
61C24×C17C2311yescharacteristic
62C24×C17C2311yescharacteristic
63C24×C17C2311yescharacteristic
64C2×D4×C17—22no—
65C2×D4×C17—22no—
66C2×D4×C17—22no—
67C2×D4×C17—22no—
68C2×D4×C17—22no—
69C2×D4×C17—41no—
70C2×D4×C17—41no—
71C2×D4×C17—41no—
72C2×D4×C17—41no—
73C22×C4×C17C2311yescharacteristic
74C22×C4×C17C2311yescharacteristic
75C22×C4×C17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2311yescharacteristic
80C22×Dic17C2311yescharacteristic
81C22×Dic17C2311yescharacteristic
82C22×Dic17C2311yescharacteristic
83C22×Dic17C2311yescharacteristic
84C22⋊C4×C17—22no—
85C22⋊C4×C17—22no—
86C22⋊C4×C17D414yes—
87C22⋊Dic17—22no—
88C22⋊Dic17—22no—
89C22⋊Dic17—22no—
90C22⋊Dic17—22no—
91C22⋊Dic17—41no—
92C22⋊Dic17—41no—
93C22⋊Dic17—41no—
94C22⋊Dic17—41no—
95C22⋊Dic17—41no—
96C22⋊Dic17—41no—
97C22⋊Dic17—41no—
98C22⋊Dic17—41no—
99C22⋊Dic17—41no—
100C22⋊Dic17—41no—
101C22⋊Dic17—41no—
102C22⋊Dic17—41no—
103C4⋊Dic17—22no—
104C4⋊Dic17—41no—
105#63225—22no—
106#63225—22no—
107#63225—41no—
108#63225—41no—
109C4×Dic17—22no—
110C4×Dic17—41no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
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×C17—21no—
130C23×C17—21no—
131C23×C17—21no—
132C23×C17—21no—
133C23×C17—21no—
134C23×C17—21no—
135C23×C17—21no—
136C23×C17—44no—
137C23×C17—44no—
138C23×C17—44no—
139C23×C17—44no—
140C23×C17—44no—
141C23×C17—44no—
142C23×C17—44no—
143C23×C17C2×D412yes—
144C23×C17#4112yes—
145C23×C17#4112yes—
146C23×C17C2411yescharacteristic
147D4×C17—44no—
148D4×C17—44no—
149D4×C17—44no—
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×C17C2×D412yes—
158C2×C4×C17C2×D412yes—
159C2×Dic17—21no—
160C2×Dic17—21no—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17—21no—
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—
181C2×Dic17—21no—
182C2×Dic17—21no—
183#2685—171nomaximal, Sylow
184C22×C17—22no—
185C22×C17—22no—
186C22×C17—22no—
187C22×C17—22no—
188C22×C17—22no—
189C22×C17—22no—
190C22×C17—22no—
191C22×C17—22no—
192C22×C17—22no—
193C22×C17—416no—
194C22×C17—41no—
195C22×C17—41no—
196C22×C17—41no—
197C22×C17—41no—
198C22×C17—41no—
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×C17—41no—
208C22×C17—41no—
209C22×C17—41no—
210C22×C17—41no—
211C22×C17—41no—
212C22×C17—41no—
213C22×C17C22×D411yescharacteristic
214C22×C17C2×#4111yescharacteristic
215C22×C17C2×#4111yescharacteristic
216C22×C17H(5,𝔽2)11yescharacteristic
217C22×C17H(5,𝔽2)11yescharacteristic
218C22×C17H(5,𝔽2)11yescharacteristic
219C22×C17H(5,𝔽2)11yescharacteristic
220C4×C17—22no—
221C4×C17—22no—
222C4×C17—41no—
223Dic17—41no—
224Dic17—41no—
225Dic17—41no—
226Dic17—41no—
227Dic17—41no—
228Dic17—41no—
229Dic17—41no—
230Dic17—41no—
231C2×#126—171no—
232C2×#120D1711yescharacteristic
233#399—171no—
234#396—171no—
235#394—171no—
236#394—171no—
237#394—171no—
238#394—171no—
239#390—171no—
240#390—171no—
241#388—171no—
242#388—171no—
243#388—171no—
244#379—171no—
245#379—171no—
246C2×C17—42no—
247C2×C17—42no—
248C2×C17—42no—
249C2×C17—81no—
250C2×C17#53511yescharacteristic
251C2×C17#53811yescharacteristic
252C2×C17#53811yescharacteristic
253C2×C17#54611yescharacteristic
254C2×C17#54611yescharacteristic
255C2×C17#55111yescharacteristic
256C2×C17#56011yescharacteristic
257C25C2×D1711yescharacteristic
258C22×D4C2×D1711yescharacteristic
259C22×D4C2×D1711yescharacteristic
260C22×D4C2×D1711yescharacteristic
261#126—344no—
262#120—24no—
263C2×C4⋊C4—171no—
264C2×C4⋊C4—171no—
265C2×C4⋊C4—171no—
266C2×C4⋊C4—171no—
267C2×C4⋊C4—171no—
268C2×C4⋊C4—171no—
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⋊C4—171no—
285C2×C22⋊C4C2×D1711yescharacteristic
286C2×C22⋊C4C2×D1711yescharacteristic
287C2×C22⋊C4C2×D1711yescharacteristic
288C2×C42—171no—
289C2×C42—171no—
290#95—171no—
291#95—171no—
292#95—171no—
293#95—171no—
294C17#268511yescharacteristic, complemented, Sylow
295C24—22no—
296C24—22no—
297C24—22no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24C22×D1711yescharacteristic
303C24C22×D1711yescharacteristic
304C24C22×D1711yescharacteristic
305C24C22×D1711yescharacteristic
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C22×C4—171no—
316C22×C4—171no—
317C22×C4—171no—
318C22×C4—171no—
319C22×C4—171no—
320C22×C4—171no—
321C22×C4—171no—
322C22×C4—171no—
323C22×C4C22×D1711yescharacteristic
324C22×C4C22×D1711yescharacteristic
325C22×C4C22×D1711yescharacteristic
326C4⋊C4—342no—
327C4⋊C4—342no—
328C4⋊C4—342no—
329C4⋊C4—681no—
330C4⋊C4—681no—
331C4⋊C4—681no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—342no—
335C22⋊C4—342no—
336C22⋊C4—342no—
337C22⋊C4—342no—
338C22⋊C4—681no—
339C22⋊C4—681no—
340C22⋊C4—681no—
341C22⋊C4—681no—
342C22⋊C4—681no—
343C22⋊C4—681no—
344C22⋊C4—681no—
345C22⋊C4—681no—
346C22⋊C4—681no—
347C22⋊C4—681no—
348C22⋊C4—681no—
349C22⋊C4—681no—
350C22⋊C4C22⋊D1714yes—
351C42—342no—
352C42—681no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23#6325412yes—
386C23#6325412yes—
387C23C2×C22⋊D1712yes—
388C23C23×D1711yescharacteristic
389D4—44no—
390D4—44no—
391D4—44no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—341no—
400C2×C4—341no—
401C2×C4—341no—
402C2×C4—341no—
403C2×C4—341no—
404C2×C4—341no—
405C2×C4—341no—
406C2×C4—341no—
407C2×C4—341no—
408C2×C4—341no—
409C2×C4—341no—
410C2×C4—341no—
411C2×C4—341no—
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×C4C2×C22⋊D1712yes—
424C2×C4C2×C22⋊D1712yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—416no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22C2×#6325411yescharacteristic
455C22C2×#6325411yescharacteristic
456C22#1058769111yescharacteristic
457C22#1058769111yescharacteristic
458C22#1058769111yescharacteristic
459C22#1058769111yescharacteristic
460C22C22×C22⋊D1711yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—681no—
465C4—681no—
466C4—681no—
467C4—681no—
468C4—681no—
469C4—681no—
470C4—681no—
471C4—681no—
472C2—42no—
473C2—42no—
474C2—42no—
475C2—81no—
476C2#1176337611yescharacteristic
477C2#1176339411yescharacteristic
478C2#1176339411yescharacteristic
479C2#1176345311yescharacteristic
480C2#1176345311yescharacteristic
481C2#1176349011yescharacteristic
482C2#1176354811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC2×C17×#120C2
hypercenterC2×#120D17
hypocenterC17#2685
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#10587565C2semidirect
2. #11762745C2semidirect
3. #11762732C2semidirect
4. #11762723C2semidirect
5. #11762723C2semidirect
6. #11762723C2semidirect
7. #11762723C2semidirect
8. #11762712C2semidirect
9. #11762712C2semidirect
10. #11762703C2semidirect
11. #11762702C2semidirect
12. #11762697C2semidirect
13. #11762665C2semidirect
14. #11762665C2semidirect
36. C2×C22⋊Dic17C22semidirect
37. C2×C22⋊Dic17C22semidirect
38. C2×C22⋊Dic17C22semidirect
39. C2×C22⋊Dic17C22semidirect
40. C2×C4⋊Dic17C22semidirect
41. C2×C4⋊Dic17C22semidirect
42. C2×#63225C22semidirect
43. C2×#63225C22semidirect
44. C2×#63225C22semidirect
45. C2×#63225C22semidirect
46. C2×C4×Dic17C22semidirect
47. C2×C4×Dic17C22semidirect
49. #10587502C22semidirect
50. #10587502C22semidirect
51. #10587502C22semidirect
52. #10587502C22semidirect
294. C17#2685semidirect

Maximal subgroups

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

subgroupindex
C2×#105875652
#117627452
#117627322
#117627232
#117627122
#117627032
#117627022
#117626972
#117626652
C2×C17×#1202
#268517