Finite Group Databasebeta

← #14536007

Subgroups — #14536007

Order 2176.

498 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762749C211yesmaximal, characteristic, complemented
2#11762738C211yesmaximal, characteristic, complemented
3#11762738C211yesmaximal, characteristic, complemented
4#11762732C211yesmaximal, characteristic, complemented
5#11762725C211yesmaximal, characteristic, complemented
6#11762717C211yesmaximal, characteristic
7#11762709C211yesmaximal, characteristic, complemented
8#11762703C211yesmaximal, characteristic, complemented
9#11762689C211yesmaximal, characteristic, complemented
10#11762688C211yesmaximal, characteristic, complemented
11#11762686C211yesmaximal, characteristic, complemented
12#11762681C211yesmaximal, characteristic, complemented
13#11762676C211yesmaximal, characteristic, complemented
14#11762672C211yesmaximal, characteristic, complemented
15C17×#386C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C22×C22⋊D17C2211yescharacteristic
20C23×Dic17C2211yescharacteristic
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×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×C4×Dic17C2211yescharacteristic
39C17×#95C2211yescharacteristic
40C17×#95C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C24×C17C2311yescharacteristic
53C23×D17C2311yescharacteristic
54C2×D4×C17—22no—
55C2×D4×C17—22no—
56C2×D4×C17—41no—
57C22×C4×C17—22no—
58C22×C4×C17—41no—
59C22×C4×C17—41no—
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C2×C22⋊D17—22no—
66C2×C22⋊D17—22no—
67C2×C22⋊D17—22no—
68C2×C22⋊D17—41no—
69C2×C22⋊D17—41no—
70C2×C22⋊D17—41no—
71C22×Dic17—41no—
72C22×Dic17—41no—
73C22×Dic17—41no—
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2311yescharacteristic
80C22×Dic17C2311yescharacteristic
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C22⋊Dic17—22no—
85C22⋊Dic17—22no—
86C22⋊Dic17—41no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89#63227—41no—
90#63227—41no—
91#63227—41no—
92#63227—41no—
93#63227—41no—
94C4⋊Dic17—41no—
95#63225—41no—
96#63225—41no—
97#63225—41no—
98C4×Dic17—22no—
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#4112yes—
112C23×C17#4112yes—
113C23×C17C2411yescharacteristic
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17—21no—
121D4×C17—44no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
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—44no—
138C22⋊D17—44no—
139C22⋊D17—44no—
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—44no—
161C2×Dic17#4112yes—
162#2749—171nomaximal, Sylow
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—22no—
166C22×C17—22no—
167C22×C17—22no—
168C22×C17—22no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17—41no—
177C22×C17C2×#4111yescharacteristic
178C22×C17C2×#4111yescharacteristic
179C22×C17C2×#4111yescharacteristic
180C22×C17H(5,𝔽2)11yescharacteristic
181C22×C17H(5,𝔽2)11yescharacteristic
182C22×C17H(5,𝔽2)11yescharacteristic
183C22×C17H(5,𝔽2)11yescharacteristic
184C2×D17—41no—
185C2×D17—41no—
186C2×D17—41no—
187C2×D17—41no—
188C2×D17—41no—
189C2×D17—41no—
190C2×D17—41no—
191C4×C17—41no—
192C4×C17—41no—
193C4×C17—41no—
194C4×C17—41no—
195C4×C17—41no—
196Dic17—22no—
197Dic17—41no—
198Dic17—41no—
199Dic17—41no—
200Dic17—41no—
201Dic17—41no—
202Dic17—41no—
203#400—171no—
204#397—171no—
205#397—171no—
206#396—171no—
207#394—171no—
208#392—171no—
209#390—171no—
210#388—171no—
211#386—171no—
212#386—171no—
213#386—171no—
214#386D1711yescharacteristic, complemented
215#385—171no—
216#383—171no—
217#381—171no—
218C2×C17—42no—
219C2×C17—42no—
220C2×C17#53811yescharacteristic
221C2×C17#53811yescharacteristic
222C2×C17#54011yescharacteristic
223C2×C17#55111yescharacteristic
224C2×C17#55111yescharacteristic
225C2×C17#55311yescharacteristic
226C2×C17#56011yescharacteristic
227D17—81no—
228C22×D4—171no—
229C22×D4—171no—
230C22×D4C2×D1711yescharacteristic
231C23×C4—171no—
232C23×C4C2×D1711yescharacteristic
233C2×C4⋊C4—171no—
234C2×C4⋊C4—171no—
235C2×C4⋊C4—171no—
236C2×C4⋊C4—171no—
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⋊C4C2×D1711yescharacteristic
248C2×C22⋊C4C2×D1711yescharacteristic
249C2×C22⋊C4C2×D1711yescharacteristic
250C2×C42—171no—
251#95—171no—
252#95—171no—
253#95—171no—
254#95—171no—
255#95—171no—
256#95—171no—
257#95—171no—
258#95—171no—
259#95—171no—
260#95—171no—
261#95C2×D1711yescharacteristic
262#95C2×D1711yescharacteristic
263C17#274911yescharacteristic, complemented, Sylow
264C24—171no—
265C24C22×D1711yescharacteristic
266C24C22×D1711yescharacteristic
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—41no—
270C2×D4—342no—
271C2×D4—342no—
272C2×D4—342no—
273C2×D4—681no—
274C2×D4—681no—
275C2×D4—681no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—171no—
280C22×C4—171no—
281C22×C4—171no—
282C22×C4—171no—
283C22×C4—171no—
284C22×C4—171no—
285C22×C4—171no—
286C22×C4—681no—
287C22×C4—681no—
288C22×C4—681no—
289C22×C4C22×D1711yescharacteristic
290C22×C4C22×D1711yescharacteristic
291C22×C4C22×D1711yescharacteristic
292C22×C4C22×D1711yescharacteristic
293C22×C4C22×D1711yescharacteristic
294C4⋊C4—681no—
295C4⋊C4—681no—
296C4⋊C4—681no—
297C4⋊C4—681no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—342no—
302C22⋊C4—342no—
303C22⋊C4—681no—
304C22⋊C4—681no—
305C22⋊C4—681no—
306C22⋊C4—681no—
307C22⋊C4—681no—
308C22⋊C4—681no—
309C22⋊C4—681no—
310C22⋊C4—681no—
311C42—342no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—341no—
325C23—341no—
326C23—341no—
327C23—341no—
328C23—341no—
329C23—341no—
330C23—341no—
331C23#6325212yes—
332C23#6325412yes—
333C23C23×D1711yescharacteristic
334D4—44no—
335D4—684no—
336D4—684no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
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—44no—
353C2×C4—172no—
354C2×C4—341no—
355C2×C4—341no—
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—684no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—681no—
390C22—681no—
391C22—681no—
392C22—681no—
393C22—681no—
394C22—681no—
395C22—681no—
396C22C2×#6325211yescharacteristic
397C22C2×#6325411yescharacteristic
398C22#1058769111yescharacteristic
399C22#1058769111yescharacteristic
400C22#1058769111yescharacteristic
401C22#41×D1711yescharacteristic
402C22#1058769611yescharacteristic
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—342no—
409C4—681no—
410C4—681no—
411C4—681no—
412C4—681no—
413C4—681no—
414C4—681no—
415C2—42no—
416C2—42no—
417C2—1361no—
418C2#1176339211yescharacteristic
419C2#1176339411yescharacteristic
420C2#1176340911yescharacteristic
421C2#1176348211yescharacteristic
422C2#1176348511yescharacteristic
423C2#1176350811yescharacteristic
424C2#1176354711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radical#11762717C2
hypercenter#386D17
hypocenterC17#2749
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #11762749C2semidirect
2. #11762738C2semidirect
3. #11762738C2semidirect
4. #11762732C2semidirect
5. #11762725C2semidirect
7. #11762709C2semidirect
8. #11762703C2semidirect
9. #11762689C2semidirect
10. #11762688C2semidirect
11. #11762686C2semidirect
12. #11762681C2semidirect
13. #11762676C2semidirect
14. #11762672C2semidirect
15. C17×#386C2semidirect
214. #386D17semidirect
263. C17#2749semidirect

Maximal subgroups

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

subgroupindex
#117627492
#117627382
#117627322
#117627252
#117627172
#117627092
#117627032
#117626892
#117626882
#117626862
#117626812
#117626762
#117626722
C17×#3862
#274917