Finite Group Databasebeta

← #14535821

Subgroups — #14535821

Order 2176.

552 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762745C211yesmaximal, characteristic, complemented
2#11762737C211yesmaximal, characteristic, complemented
3#11762737C211yesmaximal, characteristic, complemented
4#11762734C211yesmaximal, characteristic, complemented
5#11762732C211yesmaximal, characteristic, complemented
6#11762732C211yesmaximal, characteristic, complemented
7#11762703C211yesmaximal, characteristic, complemented
8#11762703C211yesmaximal, characteristic, complemented
9#11762702C211yesmaximal, characteristic, complemented
10#11762702C211yesmaximal, characteristic, complemented
11#11762697C211yesmaximal, characteristic, complemented
12#11762697C211yesmaximal, characteristic, complemented
13#11762665C211yesmaximal, characteristic
14#11762665C211yesmaximal, characteristic
15C17×#379C211yesmaximal, characteristic
16C25×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
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×C4⋊Dic17C2211yescharacteristic, complemented
36C2×C4⋊Dic17C2211yescharacteristic, complemented
37C2×#63225C2211yescharacteristic, complemented
38C2×#63225C2211yescharacteristic, complemented
39C2×#63225C2211yescharacteristic, complemented
40C2×#63225C2211yescharacteristic, complemented
41C2×C4×Dic17C2211yescharacteristic, complemented
42C2×C4×Dic17C2211yescharacteristic, 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×C17—22no—
52C24×C17—22no—
53C24×C17—22no—
54C24×C17—41no—
55C24×C17—41no—
56C24×C17—41no—
57C24×C17—41no—
58C24×C17C2311yescharacteristic
59C24×C17C2311yescharacteristic
60C24×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22⋊C4×C17—22no—
74C22⋊C4×C17—22no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78C22⋊C4×C17—41no—
79C22⋊Dic17—22no—
80C22⋊Dic17—22no—
81C22⋊Dic17—22no—
82C22⋊Dic17—22no—
83C22⋊Dic17—41no—
84C22⋊Dic17—41no—
85C22⋊Dic17—41no—
86C22⋊Dic17—41no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89C22⋊Dic17—41no—
90C22⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93#63225—41no—
94#63225—41no—
95#63225—41no—
96#63225—41no—
97C4×Dic17—41no—
98C4×Dic17—41no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—44no—
118C23×C17—44no—
119C23×C17—44no—
120C23×C17—44no—
121C23×C17—44no—
122C23×C17—44no—
123C23×C17—44no—
124C23×C17#4112yes—
125C23×C17#4112yes—
126C23×C17#4112yes—
127C23×C17C2411yescharacteristic
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×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—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—
164#2735—171nomaximal, Sylow
165C22×C17—22no—
166C22×C17—22no—
167C22×C17—22no—
168C22×C17—22no—
169C22×C17—22no—
170C22×C17—22no—
171C22×C17—22no—
172C22×C17—22no—
173C22×C17—22no—
174C22×C17—416no—
175C22×C17—41no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17—41no—
180C22×C17—41no—
181C22×C17—41no—
182C22×C17—41no—
183C22×C17—41no—
184C22×C17—41no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17C2×#4111yescharacteristic
188C22×C17C2×#4111yescharacteristic
189C22×C17C2×#4111yescharacteristic
190C22×C17H(5,𝔽2)11yescharacteristic
191C22×C17H(5,𝔽2)11yescharacteristic
192C22×C17H(5,𝔽2)11yescharacteristic
193C22×C17H(5,𝔽2)11yescharacteristic
194C4×C17—41no—
195C4×C17—41no—
196C4×C17—41no—
197C4×C17—41no—
198Dic17—41no—
199Dic17—41no—
200Dic17—41no—
201Dic17—41no—
202Dic17—41no—
203Dic17—41no—
204Dic17—41no—
205Dic17—41no—
206#399—171no—
207#397—171no—
208#397—171no—
209#397—171no—
210#396—171no—
211#396—171no—
212#388—171no—
213#388—171no—
214#388—171no—
215#388—171no—
216#388—171no—
217#388—171no—
218#379—171no—
219#379—171no—
220#379D1711yescharacteristic
221C2×C17—42no—
222C2×C17—42no—
223C2×C17—42no—
224C2×C17#53811yescharacteristic
225C2×C17#53811yescharacteristic
226C2×C17#53811yescharacteristic
227C2×C17#55111yescharacteristic
228C2×C17#55111yescharacteristic
229C2×C17#55111yescharacteristic
230C2×C17#56011yescharacteristic
231C25C2×D1711yescharacteristic
232C2×C4⋊C4—171no—
233C2×C4⋊C4—171no—
234C2×C4⋊C4—171no—
235C2×C4⋊C4—171no—
236C2×C4⋊C4—171no—
237C2×C4⋊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⋊C4C2×D1711yescharacteristic
251C2×C22⋊C4C2×D1711yescharacteristic
252C2×C22⋊C4C2×D1711yescharacteristic
253C2×C22⋊C4C2×D1711yescharacteristic
254C2×C22⋊C4C2×D1711yescharacteristic
255C2×C22⋊C4C2×D1711yescharacteristic
256C2×C42—171no—
257C2×C42—171no—
258#95—171no—
259#95—171no—
260#95—171no—
261#95—171no—
262#95—171no—
263#95—171no—
264#95—171no—
265#95—171no—
266C17#273511yescharacteristic, complemented, Sylow
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24C22×D1711yescharacteristic
275C24C22×D1711yescharacteristic
276C24C22×D1711yescharacteristic
277C22×C4—171no—
278C22×C4—171no—
279C22×C4—171no—
280C22×C4—171no—
281C22×C4—171no—
282C22×C4—171no—
283C22×C4—171no—
284C22×C4—171no—
285C22×C4C22×D1711yescharacteristic
286C22×C4C22×D1711yescharacteristic
287C22×C4C22×D1711yescharacteristic
288C22×C4C22×D1711yescharacteristic
289C4⋊C4—681no—
290C4⋊C4—681no—
291C4⋊C4—681no—
292C4⋊C4—681no—
293C4⋊C4—681no—
294C4⋊C4—681no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—342no—
302C22⋊C4—342no—
303C22⋊C4—342no—
304C22⋊C4—342no—
305C22⋊C4—681no—
306C22⋊C4—681no—
307C22⋊C4—681no—
308C22⋊C4—681no—
309C22⋊C4—681no—
310C22⋊C4—681no—
311C22⋊C4—681no—
312C22⋊C4—681no—
313C42—681no—
314C42—681no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23—44no—
340C23#6325212yes—
341C23#6325412yes—
342C23#6325412yes—
343C23C23×D1711yescharacteristic
344C2×C4—21no—
345C2×C4—21no—
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—341no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—341no—
362C2×C4—341no—
363C2×C4—341no—
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—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—416no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22C2×#6325211yescharacteristic
403C22C2×#6325411yescharacteristic
404C22C2×#6325411yescharacteristic
405C22#1058769111yescharacteristic
406C22#1058769111yescharacteristic
407C22#1058769111yescharacteristic
408C22#1058769111yescharacteristic
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—681no—
414C4—681no—
415C4—681no—
416C4—681no—
417C4—681no—
418C4—681no—
419C4—681no—
420C4—681no—
421C2—42no—
422C2—42no—
423C2—42no—
424C2#1176339411yescharacteristic
425C2#1176339411yescharacteristic
426C2#1176339711yescharacteristic
427C2#1176348511yescharacteristic
428C2#1176348511yescharacteristic
429C2#1176349011yescharacteristic
430C2#1176354811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#379C2
prime radicalC25×C17C22
hypercenter#379D17
hypocenterC17#2735
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #11762745C2semidirect
2. #11762737C2semidirect
3. #11762737C2semidirect
4. #11762734C2semidirect
5. #11762732C2semidirect
6. #11762732C2semidirect
7. #11762703C2semidirect
8. #11762703C2semidirect
9. #11762702C2semidirect
10. #11762702C2semidirect
11. #11762697C2semidirect
12. #11762697C2semidirect
35. C2×C4⋊Dic17C22semidirect
36. C2×C4⋊Dic17C22semidirect
37. C2×#63225C22semidirect
38. C2×#63225C22semidirect
39. C2×#63225C22semidirect
40. C2×#63225C22semidirect
41. C2×C4×Dic17C22semidirect
42. C2×C4×Dic17C22semidirect
43. #10587502C22semidirect
44. #10587502C22semidirect
45. #10587502C22semidirect
46. #10587502C22semidirect
47. #10587502C22semidirect
48. #10587502C22semidirect
49. #10587502C22semidirect
50. #10587502C22semidirect
266. C17#2735semidirect

Maximal subgroups

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

subgroupindex
#117627452
#117627372
#117627342
#117627322
#117627032
#117627022
#117626972
#117626652
C17×#3792
#273517