Finite Group Databasebeta

← #14535120

Subgroups — #14535120

Order 2176.

568 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587572C211yesmaximal, characteristic
2C2×#10587569C211yesmaximal, characteristic
3C2×#10587582C211yesmaximal, characteristic, complemented
4C2×#10587564C211yesmaximal, characteristic, complemented
5#11762743C211yesmaximal, characteristic, complemented
6#11762741C211yesmaximal, characteristic, complemented
7#11762738C211yesmaximal, characteristic, complemented
8#11762730C211yesmaximal, characteristic, complemented
9#11762724C211yesmaximal, characteristic
10#11762703C211yesmaximal, characteristic, complemented
11#11762698C211yesmaximal, characteristic, complemented
12#11762683C211yesmaximal, characteristic, complemented
13#11762681C211yesmaximal, characteristic, complemented
14#11762678C211yesmaximal, characteristic, complemented
15C17×#397C211yesmaximal, characteristic, complemented
16C22×C22⋊D17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C22×Dic34C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×C4⋊Dic17C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33C2×C4⋊Dic17C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C2×#63225C2211yescharacteristic
42C2×C4×Dic17C2211yescharacteristic
43#10587582—24no—
44#10587572—24no—
45#10587569—24no—
46#10587564—24no—
47C17×#95C2211yescharacteristic
48C17×#95C2211yescharacteristic
49C17×#95C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53#10587502C2211yescharacteristic
54#10587502C2211yescharacteristic
55C24×C17C2311yescharacteristic
56C23×D17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C2×C22⋊D17—22no—
64C2×C22⋊D17—22no—
65C2×C22⋊D17—41no—
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—41no—
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—41no—
79C2×Dic34—22no—
80C2×Dic34—22no—
81C2×Dic34—41no—
82C4⋊C4×C17—22no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—22no—
85C22⋊C4×C17—22no—
86C22⋊Dic17—22no—
87C22⋊Dic17D414yes—
88#63227—22no—
89#63227—22no—
90#63227—22no—
91#63227—22no—
92#63227D414yes—
93C4⋊Dic17—22no—
94C4⋊Dic17—22no—
95C4⋊Dic17—22no—
96C4⋊Dic17—22no—
97#63225—22no—
98#63225—22no—
99#63225—22no—
100#63225—22no—
101#63225—22no—
102#63225—41no—
103#63225—41no—
104C4×Dic17—22no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17C2×D412yes—
112C23×C17C2411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17C2×D412yes—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
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×C17C2×D412yes—
138C22⋊D17—44no—
139C2×Dic17—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—44no—
157C2×Dic17C2×D412yes—
158C2×Dic17C2×D412yes—
159C2×Dic17C2×D412yes—
160C2×Dic17#4112yes—
161C4×D17—44no—
162Dic34—44no—
163#2680—171nomaximal, Sylow
164C22×C17—22no—
165C22×C17—22no—
166C22×C17—22no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17C22×D411yescharacteristic
172C22×C17C22×D411yescharacteristic
173C22×C17C2×#4111yescharacteristic
174C22×C17H(5,𝔽2)11yescharacteristic
175C22×C172−1+411yescharacteristic
176C22×C172−1+411yescharacteristic
177C22×C172−1+411yescharacteristic
178C2×D17—22no—
179C2×D17—22no—
180C2×D17—22no—
181C2×D17—41no—
182C2×D17—41no—
183C2×D17—41no—
184C2×D17—41no—
185C4×C17—22no—
186C4×C17—41no—
187C4×C17—41no—
188C4×C17—41no—
189C4×C17—41no—
190C4×C17—41no—
191Dic17—22no—
192Dic17—22no—
193Dic17—22no—
194Dic17—22no—
195Dic17—41no—
196Dic17—41no—
197Dic17—41no—
198C2×#123—171no—
199C2×#123—171no—
200C2×C22⋊Q8—171no—
201C2×C22⋊Q8—171no—
202#398—171no—
203#398—171no—
204#397—171no—
205#397D1711yescharacteristic, complemented
206#395—171no—
207#394—171no—
208#388—171no—
209#388—171no—
210#385—171no—
211#385—171no—
212#384—171no—
213C2×C17—42no—
214C2×C17#53611yescharacteristic
215C2×C17#53611yescharacteristic
216C2×C17#53911yescharacteristic
217C2×C17D4211yescharacteristic
218C2×C17#54711yescharacteristic
219C2×C17#54711yescharacteristic
220C2×C17#56311yescharacteristic
221D17—42no—
222C22×Q8—171no—
223C22×D4—171no—
224C23×C4—171no—
225C23×C4—171no—
226#123—344no—
227#123—344no—
228C22⋊Q8—344no—
229C22⋊Q8—344no—
230C2×C4⋊C4—171no—
231C2×C4⋊C4—171no—
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×C4⋊C4—171no—
239C2×C4⋊C4—171no—
240C2×C4⋊C4—171no—
241C2×C4⋊C4C2×D1711yescharacteristic
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⋊C4C2×D1711yescharacteristic
250C2×C22⋊C4C2×D1711yescharacteristic
251C2×C22⋊C4C2×D1711yescharacteristic
252C2×C42—171no—
253#95—171no—
254#95—171no—
255#95—171no—
256#95—171no—
257#95—171no—
258#95C2×D1711yescharacteristic
259#95C2×D1711yescharacteristic
260#95C2×D1711yescharacteristic
261C17#268011yescharacteristic, complemented, Sylow
262C24—171no—
263C24C22×D1711yescharacteristic
264C2×Q8—342no—
265C2×Q8—342no—
266C2×Q8—681no—
267C2×D4—342no—
268C2×D4—342no—
269C2×D4—681no—
270C22×C4—171no—
271C22×C4—171no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—171no—
277C22×C4—342no—
278C22×C4—342no—
279C22×C4—342no—
280C22×C4—342no—
281C22×C4—681no—
282C22×C4—681no—
283C22×C4C22×D1711yescharacteristic
284C22×C4C22×D1711yescharacteristic
285C22×C4C22×D1711yescharacteristic
286C22×C4C22×D1711yescharacteristic
287C22×C4C22×D1711yescharacteristic
288C22×C4C22×D1711yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—342no—
291C4⋊C4—342no—
292C4⋊C4—342no—
293C4⋊C4—342no—
294C4⋊C4—342no—
295C4⋊C4—342no—
296C4⋊C4—342no—
297C4⋊C4—342no—
298C4⋊C4—342no—
299C4⋊C4—681no—
300C4⋊C4—681no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—174no—
305C22⋊C4—174no—
306C22⋊C4—342no—
307C22⋊C4—342no—
308C22⋊C4—342no—
309C22⋊C4—342no—
310C22⋊C4—342no—
311C42—342no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—172no—
319C23—341no—
320C23—341no—
321C23—341no—
322C23—341no—
323C23—341no—
324C23—341no—
325C23D4×D1712yes—
326C23C23×D1711yescharacteristic
327Q8—684no—
328D4—684no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
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—172no—
347C2×C4—172no—
348C2×C4—172no—
349C2×C4—172no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
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—684no—
368C2×C4—684no—
369C2×C4D4×D1712yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—342no—
378C22—342no—
379C22—342no—
380C22—681no—
381C22—681no—
382C22—681no—
383C22—681no—
384C22C2×D4×D1711yescharacteristic
385C22C2×D4×D1711yescharacteristic
386C22#41×D1711yescharacteristic
387C22#1058769611yescharacteristic
388C22#1058769711yescharacteristic
389C22#1058769711yescharacteristic
390C22#1058769711yescharacteristic
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—342no—
398C4—342no—
399C4—342no—
400C4—342no—
401C4—681no—
402C4—681no—
403C4—681no—
404C2—42no—
405C2—682no—
406C2#1176337911yescharacteristic
407C2#1176338111yescharacteristic
408C2#1176340011yescharacteristic
409C2#1176344311yescharacteristic
410C2#1176345711yescharacteristic
411C2#1176345711yescharacteristic
412C2#1176356111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC22×C22⋊D17C22
hypercenter#397D17
hypocenterC17#2680
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
3. C2×#10587582C2semidirect
4. C2×#10587564C2semidirect
5. #11762743C2semidirect
6. #11762741C2semidirect
7. #11762738C2semidirect
8. #11762730C2semidirect
10. #11762703C2semidirect
11. #11762698C2semidirect
12. #11762683C2semidirect
13. #11762681C2semidirect
14. #11762678C2semidirect
15. C17×#397C2semidirect
205. #397D17semidirect
261. C17#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#105875722
C2×#105875692
C2×#105875822
C2×#105875642
#117627432
#117627412
#117627382
#117627302
#117627242
#117627032
#117626982
#117626832
#117626812
#117626782
C17×#3972
#268017