Finite Group Databasebeta

← #14532528

Subgroups — #14532528

Order 2176.

610 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587625C211yesmaximal, characteristic
2#11762747C211yesmaximal, characteristic, complemented
3#11762737C211yesmaximal, characteristic, complemented
4#11762735C211yesmaximal, characteristic, complemented
5#11762724C211yesmaximal, characteristic, complemented
6#11762700C211yesmaximal, characteristic, complemented
7#11762693C211yesmaximal, characteristic, complemented
8#11762689C211yesmaximal, characteristic, complemented
9#11762688C211yesmaximal, characteristic, complemented
10#11762688C211yesmaximal, characteristic, complemented
11#11762686C211yesmaximal, characteristic, complemented
12#11762681C211yesmaximal, characteristic, complemented
13#11762672C211yesmaximal, characteristic, complemented
14C2×#10587502C211yesmaximal, characteristic, complemented
15C2×C4×D4×C17C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C23×C4×C17C2211yescharacteristic
19C22×C22⋊D17C2211yescharacteristic
20C22×C22⋊D17C2211yescharacteristic
21C23×Dic17C2211yescharacteristic
22C23×Dic17C2211yescharacteristic
23C4×D4×C17—24no—
24C2×C4⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C42×C17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32#10587625—24no—
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C24×C17C2311yescharacteristic
55C23×D17C2311yescharacteristic
56C2×D4×C17—22no—
57C2×D4×C17—22no—
58C2×D4×C17D414yes—
59C22×C4×C17—22no—
60C22×C4×C17—22no—
61C22×C4×C17—22no—
62C22×C4×C17—22no—
63C22×C4×C17—41no—
64C22×C4×C17—41no—
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C22×C4×C17C2311yescharacteristic
69C22×C4×C17C2311yescharacteristic
70C2×C22⋊D17—22no—
71C2×C22⋊D17—22no—
72C2×C22⋊D17—22no—
73C2×C22⋊D17—22no—
74C2×C22⋊D17—22no—
75C2×C22⋊D17—41no—
76C22×Dic17—22no—
77C22×Dic17—22no—
78C22×Dic17—22no—
79C22×Dic17—41no—
80C22×Dic17—41no—
81C22×Dic17—41no—
82C22×Dic17C2311yescharacteristic
83C22×Dic17C2311yescharacteristic
84C22×Dic17C2311yescharacteristic
85C22×Dic17C2311yescharacteristic
86C22×Dic17C2311yescharacteristic
87C22×Dic17C2311yescharacteristic
88C22×Dic17C2311yescharacteristic
89C4⋊C4×C17—22no—
90C22⋊C4×C17—22no—
91C22⋊C4×C17—22no—
92C42×C17—22no—
93C22⋊Dic17—22no—
94C22⋊Dic17—22no—
95C22⋊Dic17—41no—
96C22⋊Dic17—41no—
97#63227—22no—
98#63227—22no—
99#63227—41no—
100#63227—41no—
101#63227—41no—
102#63225—22no—
103#63225—22no—
104#63225—41no—
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×C17C2×D412yes—
116C23×C17C2×D412yes—
117C23×C17#4112yes—
118C23×C17#4112yes—
119C23×C17C2411yescharacteristic
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17—21no—
127D4×C17—28no—
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—44no—
142C2×C4×C17—44no—
143C2×C4×C17C2×D412yes—
144C2×C4×C17#4112yes—
145C22⋊D17—44no—
146C22⋊D17—44no—
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—
164C2×Dic17—21no—
165C2×Dic17—21no—
166C2×Dic17—21no—
167C2×Dic17—44no—
168C2×Dic17—44no—
169C2×Dic17#4112yes—
170#2444—171nomaximal, Sylow
171C22×C17—22no—
172C22×C17—22no—
173C22×C17—22no—
174C22×C17—22no—
175C22×C17—22no—
176C22×C17—22no—
177C22×C17—22no—
178C22×C17—22no—
179C22×C17—22no—
180C22×C17—41no—
181C22×C17—41no—
182C22×C17—41no—
183C22×C17—41no—
184C22×C17#12314yes—
185C22×C17C22×D411yescharacteristic
186C22×C17C2×#4111yescharacteristic
187C22×C17C2×#4111yescharacteristic
188C22×C17C2×#4111yescharacteristic
189C22×C17C2×#4111yescharacteristic
190C22×C17H(5,𝔽2)11yescharacteristic
191C22×C17H(5,𝔽2)11yescharacteristic
192C2×D17—41no—
193C2×D17—41no—
194C2×D17—41no—
195C2×D17—41no—
196C2×D17—41no—
197C2×D17—41no—
198C2×D17—41no—
199C4×C17—22no—
200C4×C17—22no—
201C4×C17—41no—
202C4×C17—41no—
203C4×C17—41no—
204Dic17—22no—
205Dic17—41no—
206Dic17—41no—
207Dic17—41no—
208Dic17—41no—
209Dic17—41no—
210Dic17—41no—
211C2×#121—171no—
212C2×C4×D4D1711yescharacteristic, complemented
213#400—171no—
214#397—171no—
215#397—171no—
216#394—171no—
217#388—171no—
218#387—171no—
219#386—171no—
220#386—171no—
221#386—171no—
222#386—171no—
223#385—171no—
224#381—171no—
225C2×#95—171no—
226C2×C17—24no—
227C2×C17—42no—
228C2×C17C2×#12311yescharacteristic
229C2×C17#52911yescharacteristic
230C2×C17#53811yescharacteristic
231C2×C17#54611yescharacteristic
232C2×C17#54611yescharacteristic
233C2×C17#55311yescharacteristic
234C2×C17#55311yescharacteristic
235D17—81no—
236C22×D4—171no—
237C22×D4—171no—
238C22×D4C2×D1711yescharacteristic
239C23×C4—171no—
240C23×C4—171no—
241C23×C4C2×D1711yescharacteristic
242C23×C4C2×D1711yescharacteristic
243#121—344no—
244C4×D4—24no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4—171no—
247C2×C4⋊C4—171no—
248C2×C4⋊C4C2×D1711yescharacteristic
249C2×C22⋊C4—171no—
250C2×C22⋊C4—171no—
251C2×C22⋊C4—171no—
252C2×C22⋊C4—171no—
253C2×C22⋊C4—171no—
254C2×C22⋊C4—171no—
255C2×C22⋊C4—171no—
256C2×C22⋊C4—171no—
257C2×C22⋊C4—171no—
258C2×C22⋊C4C2×D1711yescharacteristic
259C2×C22⋊C4C2×D1711yescharacteristic
260C2×C42C2×D1711yescharacteristic
261#95—171no—
262#95—171no—
263#95—171no—
264#95—171no—
265#95—171no—
266#95—171no—
267#95—171no—
268#95—171no—
269#95—171no—
270#95—171no—
271#95—171no—
272#95—171no—
273C17#244411yescharacteristic, complemented, Sylow
274C24—171no—
275C24C22×D1711yescharacteristic
276C24C22×D1711yescharacteristic
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—342no—
280C2×D4—342no—
281C2×D4—342no—
282C2×D4—342no—
283C2×D4—342no—
284C2×D4—681no—
285C2×D4C22⋊D1714yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—171no—
293C22×C4—171no—
294C22×C4—171no—
295C22×C4—171no—
296C22×C4—171no—
297C22×C4—171no—
298C22×C4—171no—
299C22×C4—342no—
300C22×C4—342no—
301C22×C4—342no—
302C22×C4—681no—
303C22×C4—681no—
304C22×C4—681no—
305C22×C4C22×D1711yescharacteristic
306C22×C4C22×D1711yescharacteristic
307C22×C4C22×D1711yescharacteristic
308C22×C4C22×D1711yescharacteristic
309C22×C4C22×D1711yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—342no—
312C4⋊C4—342no—
313C4⋊C4—681no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—342no—
317C22⋊C4—342no—
318C22⋊C4—342no—
319C22⋊C4—342no—
320C22⋊C4—681no—
321C22⋊C4—681no—
322C22⋊C4—681no—
323C22⋊C4—681no—
324C22⋊C4—681no—
325C42—22no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—341no—
337C23—341no—
338C23—341no—
339C23—341no—
340C23—341no—
341C23—341no—
342C23—341no—
343C23#6325212yes—
344C23#6325212yes—
345C23C2×C22⋊D1712yes—
346C23C2×C22⋊D1712yes—
347C23C23×D1711yescharacteristic
348D4—28no—
349D4—684no—
350D4—684no—
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—44no—
365C2×C4—44no—
366C2×C4—172no—
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—
382C2×C4—341no—
383C2×C4—341no—
384C2×C4—341no—
385C2×C4—341no—
386C2×C4—341no—
387C2×C4—684no—
388C2×C4—684no—
389C2×C4#6325412yes—
390C2×C4C2×C22⋊D1712yes—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—681no—
405C22—681no—
406C22—681no—
407C22—681no—
408C22—681no—
409C22—681no—
410C22—681no—
411C22#1058761514yes—
412C22C2×#6325211yescharacteristic
413C22C2×#6325211yescharacteristic
414C22C2×#6325411yescharacteristic
415C22#1058769111yescharacteristic
416C22#41×D1711yescharacteristic
417C22#1058769611yescharacteristic
418C22C22×C22⋊D1711yescharacteristic
419C4—22no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—342no—
425C4—681no—
426C4—681no—
427C4—681no—
428C4—681no—
429C4—681no—
430C4—681no—
431C2—24no—
432C2—42no—
433C2—1361no—
434C2C2×#1058761511yescharacteristic
435C2#1176333911yescharacteristic
436C2#1176339211yescharacteristic
437C2#1176344711yescharacteristic
438C2#1176345311yescharacteristic
439C2#1176350511yescharacteristic
440C2#1176350811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C4×D4×C17C2
prime radicalC2×#10587625C2
hypercenterC2×C4×D4D17
hypocenterC17#2444
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #11762747C2semidirect
3. #11762737C2semidirect
4. #11762735C2semidirect
5. #11762724C2semidirect
6. #11762700C2semidirect
7. #11762693C2semidirect
8. #11762689C2semidirect
9. #11762688C2semidirect
10. #11762688C2semidirect
11. #11762686C2semidirect
12. #11762681C2semidirect
13. #11762672C2semidirect
14. C2×#10587502C2semidirect
15. C2×C4×D4×C17C2semidirect
212. C2×C4×D4D17semidirect
273. C17#2444semidirect

Maximal subgroups

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

subgroupindex
C2×#105876252
#117627472
#117627372
#117627352
#117627242
#117627002
#117626932
#117626892
#117626882
#117626862
#117626812
#117626722
C2×#105875022
C2×C4×D4×C172
#244417