Finite Group Databasebeta

← #14533112

Subgroups — #14533112

Order 2176.

546 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587621C211yesmaximal, characteristic
2C2×#10587564C211yesmaximal, characteristic, complemented
3#11762751C211yesmaximal, characteristic, complemented
4#11762750C211yesmaximal, characteristic, complemented
5#11762737C211yesmaximal, characteristic, complemented
6#11762721C211yesmaximal, characteristic, complemented
7#11762713C211yesmaximal, characteristic, complemented
8#11762702C211yesmaximal, characteristic, complemented
9#11762697C211yesmaximal, characteristic, complemented
10#11762689C211yesmaximal, characteristic
11#11762673C211yesmaximal, characteristic, complemented
12#11762670C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic17C211yesmaximal, characteristic, complemented
14C2×#10587502C211yesmaximal, characteristic, complemented
15C17×#386C211yesmaximal, characteristic
16C22×D4×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C23×Dic17C2211yescharacteristic
19C23×Dic17C2211yescharacteristic
20C22×Dic34C2211yescharacteristic
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
30#10587621—24no—
31C2×C4⋊Dic17C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×#63225C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×C4×Dic17C2211yescharacteristic
37C2×C4×Dic17C2211yescharacteristic
38#10587564—24no—
39C17×#95C2211yescharacteristic
40C17×#95C2211yescharacteristic
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
55C2×D4×C17—22no—
56C2×D4×C17—22no—
57C2×D4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—41no—
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—41no—
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2311yescharacteristic
80C2×Dic34—22no—
81C2×Dic34—22no—
82C2×Dic34—41no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—22no—
85C22⋊C4×C17—41no—
86C22⋊Dic17—22no—
87C22⋊Dic17—22no—
88C22⋊Dic17—22no—
89C22⋊Dic17—22no—
90C22⋊Dic17—41no—
91C22⋊Dic17D414yes—
92C4⋊Dic17—22no—
93#63225—22no—
94#63225—22no—
95#63225—22no—
96#63225—22no—
97C4×Dic17—22no—
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×C17C2×D412yes—
111C23×C17#4112yes—
112C23×C17#4112yes—
113C23×C17C2411yescharacteristic
114D4×C17—44no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
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—44no—
130C2×C4×C17#4112yes—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
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—44no—
153C2×Dic17—44no—
154C2×Dic17C2×D412yes—
155C2×Dic17C2×D412yes—
156C2×Dic17#4112yes—
157Dic34—44no—
158#2488—171nomaximal, Sylow
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—22no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17#12414yes—
173C22×C17C22×D411yescharacteristic
174C22×C17C2×#4111yescharacteristic
175C22×C17C2×#4111yescharacteristic
176C22×C17C2×#4111yescharacteristic
177C22×C17C2×#4111yescharacteristic
178C22×C17H(5,𝔽2)11yescharacteristic
179C22×C172−1+411yescharacteristic
180C4×C17—22no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185Dic17—22no—
186Dic17—22no—
187Dic17—22no—
188Dic17—41no—
189Dic17—41no—
190Dic17—41no—
191Dic17—41no—
192Dic17—41no—
193C2×#123—171no—
194C2×C22⋊Q8—171no—
195#400—171no—
196#400—171no—
197#397—171no—
198#393—171no—
199#391—171no—
200#388—171no—
201#388—171no—
202#386—171no—
203#386D1711yescharacteristic
204#382—171no—
205#381—171no—
206C4×C22⋊C4—171no—
207C2×#95—171no—
208C2×C17—24no—
209C2×C17—81no—
210C2×C17C2×#12411yescharacteristic
211C2×C17#52911yescharacteristic
212C2×C17#53911yescharacteristic
213C2×C17#54611yescharacteristic
214C2×C17#54711yescharacteristic
215C2×C17#55111yescharacteristic
216C2×C17#55211yescharacteristic
217C22×Q8—171no—
218C22×D4C2×D1711yescharacteristic
219C23×C4—171no—
220C23×C4—171no—
221C23×C4C2×D1711yescharacteristic
222#123—344no—
223C22⋊Q8—344no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4—171no—
228C2×C4⋊C4—171no—
229C2×C22⋊C4—171no—
230C2×C22⋊C4—171no—
231C2×C22⋊C4—171no—
232C2×C22⋊C4—171no—
233C2×C22⋊C4—171no—
234C2×C22⋊C4—171no—
235C2×C22⋊C4C2×D1711yescharacteristic
236C2×C22⋊C4C2×D1711yescharacteristic
237C2×C22⋊C4C2×D1711yescharacteristic
238C2×C42—171no—
239C2×C42—171no—
240#95—171no—
241#95—171no—
242#95—171no—
243#95—171no—
244#95—171no—
245#95—171no—
246#95—171no—
247#95—171no—
248#95—171no—
249#95—171no—
250#95—171no—
251#95—171no—
252#95C2×D1711yescharacteristic
253#95C2×D1711yescharacteristic
254C17#248811yescharacteristic, complemented, Sylow
255C24C22×D1711yescharacteristic
256C24C22×D1711yescharacteristic
257C2×Q8—342no—
258C2×Q8—342no—
259C2×Q8—681no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—171no—
267C22×C4—171no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—171no—
271C22×C4—171no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—342no—
275C22×C4—342no—
276C22×C4—342no—
277C22×C4—342no—
278C22×C4—342no—
279C22×C4—681no—
280C22×C4C22×D1711yescharacteristic
281C22×C4C22×D1711yescharacteristic
282C22×C4C22×D1711yescharacteristic
283C22×C4C22×D1711yescharacteristic
284C22×C4C22×D1711yescharacteristic
285C4⋊C4—342no—
286C4⋊C4—342no—
287C4⋊C4—342no—
288C4⋊C4—342no—
289C4⋊C4—342no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C22⋊C4—174no—
294C22⋊C4—342no—
295C22⋊C4—342no—
296C22⋊C4—342no—
297C22⋊C4—342no—
298C22⋊C4—681no—
299C42—342no—
300C42—681no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23#6325212yes—
313C23D4×D1712yes—
314C23#6325412yes—
315C23C23×D1711yescharacteristic
316Q8—684no—
317D4—44no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—44no—
333C2×C4—172no—
334C2×C4—172no—
335C2×C4—172no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—684no—
358C2×C4—684no—
359C2×C4#6325412yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#1058757114yes—
374C22C2×#6325211yescharacteristic
375C22C2×D4×D1711yescharacteristic
376C22C2×#6325411yescharacteristic
377C22C2×#6325411yescharacteristic
378C22#1058769111yescharacteristic
379C22#41×D1711yescharacteristic
380C22#1058769711yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—342no—
387C4—342no—
388C4—342no—
389C4—681no—
390C4—681no—
391C4—681no—
392C4—681no—
393C4—681no—
394C2—24no—
395C2—81no—
396C2C2×#1058757111yescharacteristic
397C2#1176333911yescharacteristic
398C2#1176340111yescharacteristic
399C2#1176344811yescharacteristic
400C2#1176345911yescharacteristic
401C2#1176349011yescharacteristic
402C2#1176349611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC22×D4×C17C22
hypercenter#386D17
hypocenterC17#2488
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#10587564C2semidirect
3. #11762751C2semidirect
4. #11762750C2semidirect
5. #11762737C2semidirect
6. #11762721C2semidirect
7. #11762713C2semidirect
8. #11762702C2semidirect
9. #11762697C2semidirect
11. #11762673C2semidirect
12. #11762670C2semidirect
13. C4×C22⋊Dic17C2semidirect
14. C2×#10587502C2semidirect
254. C17#2488semidirect

Maximal subgroups

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

subgroupindex
C2×#105876212
C2×#105875642
#117627512
#117627502
#117627372
#117627212
#117627132
#117627022
#117626972
#117626892
#117626732
#117626702
C4×C22⋊Dic172
C2×#105875022
C17×#3862
#248817