Finite Group Databasebeta

← #14535378

Subgroups — #14535378

Order 2176.

506 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587571C211yesmaximal, characteristic
2#11762737C211yesmaximal, characteristic, complemented
3#11762735C211yesmaximal, characteristic, complemented
4#11762731C211yesmaximal, characteristic, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762724C211yesmaximal, characteristic
7#11762720C211yesmaximal, characteristic, complemented
8#11762716C211yesmaximal, characteristic, complemented
9#11762714C211yesmaximal, characteristic, complemented
10#11762702C211yesmaximal, characteristic, complemented
11#11762699C211yesmaximal, characteristic, complemented
12#11762690C211yesmaximal, characteristic, complemented
13#11762688C211yesmaximal, characteristic
14#11762672C211yesmaximal, characteristic, complemented
15C2×C17×C22⋊Q8C211yesmaximal, characteristic, complemented
16C22×Q8×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C22×Dic34C2211yescharacteristic
20C17×C22⋊Q8—24no—
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C4⋊C4×C17C2211yescharacteristic
23C2×C4⋊C4×C17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×C4⋊Dic17C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×C4×Dic17C2211yescharacteristic
39C2×C4×Dic17C2211yescharacteristic
40#10587571—24no—
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
54C23×D17C2311yescharacteristic
55C2×Q8×C17—22no—
56C2×Q8×C17—22no—
57C2×Q8×C17—41no—
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×C4×C17C2311yescharacteristic
67C2×C22⋊D17—22no—
68C2×C22⋊D17—22no—
69C2×C22⋊D17—41no—
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C2×Dic34—22no—
78C2×Dic34—22no—
79C2×Dic34—41no—
80C4⋊C4×C17—22no—
81C4⋊C4×C17—22no—
82C4⋊C4×C17—22no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17D414yes—
85C22⋊Dic17—22no—
86C22⋊Dic17—41no—
87C22⋊Dic17—41no—
88#63227—22no—
89#63227—22no—
90#63227—22no—
91#63227—41no—
92#63227—41no—
93#63227—41no—
94C4⋊Dic17—41no—
95#63225—22no—
96#63225—41no—
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×C17C2×D412yes—
106C23×C17C2411yescharacteristic
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17—21no—
112C22×D17—21no—
113C22×D17—21no—
114Q8×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—21no—
130C2×C4×C17—44no—
131C2×C4×C17C2×D412yes—
132C2×C4×C17C2×D412yes—
133C2×C4×C17#4112yes—
134C22⋊D17—44no—
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—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17#4112yes—
156Dic34—44no—
157#2700—171nomaximal, Sylow
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C22×D411yescharacteristic
166C22×C17C2×#4111yescharacteristic
167C22×C17C2×#4111yescharacteristic
168C22×C17H(5,𝔽2)11yescharacteristic
169C22×C17H(5,𝔽2)11yescharacteristic
170C22×C172−1+411yescharacteristic
171C22×C172−1+411yescharacteristic
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—41no—
179C4×C17—22no—
180C4×C17—22no—
181C4×C17—22no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185Dic17—22no—
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189Dic17—41no—
190Dic17—41no—
191Dic17—41no—
192C2×#124—171no—
193C2×C22⋊Q8D1711yescharacteristic, complemented
194#397—171no—
195#397—171no—
196#396—171no—
197#395—171no—
198#394—171no—
199#393—171no—
200#391—171no—
201#391—171no—
202#388—171no—
203#388—171no—
204#387—171no—
205#386—171no—
206#381—171no—
207C2×C17—42no—
208C2×C17#53611yescharacteristic
209C2×C17#54211yescharacteristic
210C2×C17#54211yescharacteristic
211C2×C17#54611yescharacteristic
212C2×C17#54611yescharacteristic
213C2×C17#55611yescharacteristic
214C2×C17#56211yescharacteristic
215D17—81no—
216C22×Q8—171no—
217C22×Q8C2×D1711yescharacteristic
218C22×D4—171no—
219C23×C4C2×D1711yescharacteristic
220#124—344no—
221C22⋊Q8—24no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4C2×D1711yescharacteristic
226C2×C4⋊C4C2×D1711yescharacteristic
227C2×C4⋊C4C2×D1711yescharacteristic
228C2×C22⋊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⋊C4—171no—
236C2×C22⋊C4—171no—
237C2×C22⋊C4C2×D1711yescharacteristic
238C2×C22⋊C4C2×D1711yescharacteristic
239C2×C42—171no—
240C2×C42—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#95—171no—
253C17#270011yescharacteristic, complemented, Sylow
254C24—171no—
255C24C22×D1711yescharacteristic
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—342no—
260C2×Q8—342no—
261C2×Q8—681no—
262C2×D4—342no—
263C2×D4—342no—
264C2×D4—681no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—171no—
271C22×C4—171no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4C22×D1711yescharacteristic
276C22×C4C22×D1711yescharacteristic
277C22×C4C22×D1711yescharacteristic
278C22×C4C22×D1711yescharacteristic
279C22×C4C22×D1711yescharacteristic
280C22×C4C22×D1711yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—342no—
285C4⋊C4—681no—
286C4⋊C4—681no—
287C22⋊C4—22no—
288C22⋊C4—342no—
289C22⋊C4—342no—
290C22⋊C4—342no—
291C22⋊C4—342no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C22⋊C4—681no—
295C22⋊C4—681no—
296C22⋊C4—681no—
297C22⋊C4C22⋊D1714yes—
298C42—342no—
299C42—681no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—341no—
307C23—341no—
308C23—341no—
309C23—341no—
310C23—341no—
311C23—341no—
312C23—341no—
313C23C2×C22⋊D1712yes—
314C23C23×D1711yescharacteristic
315Q8—44no—
316Q8—684no—
317D4—684no—
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—21no—
333C2×C4—44no—
334C2×C4—172no—
335C2×C4—341no—
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#6325412yes—
356C2×C4C2×C22⋊D1712yes—
357C2×C4C2×C22⋊D1712yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—681no—
369C22—681no—
370C22—681no—
371C22—681no—
372C22C2×#6325411yescharacteristic
373C22#1058769111yescharacteristic
374C22#1058769411yescharacteristic
375C22#1058769411yescharacteristic
376C22#41×D1711yescharacteristic
377C22#1058769611yescharacteristic
378C22C22×C22⋊D1711yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—342no—
386C4—681no—
387C4—681no—
388C4—681no—
389C4—681no—
390C4—681no—
391C4—681no—
392C2—42no—
393C2—1361no—
394C2#1176338311yescharacteristic
395C2#1176342911yescharacteristic
396C2#1176343211yescharacteristic
397C2#1176344711yescharacteristic
398C2#1176345311yescharacteristic
399C2#1176353011yescharacteristic
400C2#1176355511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×C22⋊Q8C2
prime radicalC22×C22⋊D17C22
hypercenterC2×C22⋊Q8D17
hypocenterC17#2700
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11762737C2semidirect
3. #11762735C2semidirect
4. #11762731C2semidirect
5. #11762729C2semidirect
7. #11762720C2semidirect
8. #11762716C2semidirect
9. #11762714C2semidirect
10. #11762702C2semidirect
11. #11762699C2semidirect
12. #11762690C2semidirect
14. #11762672C2semidirect
15. C2×C17×C22⋊Q8C2semidirect
193. C2×C22⋊Q8D17semidirect
253. C17#2700semidirect

Maximal subgroups

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

subgroupindex
C2×#105875712
#117627372
#117627352
#117627312
#117627292
#117627242
#117627202
#117627162
#117627142
#117627022
#117626992
#117626902
#117626882
#117626722
C2×C17×C22⋊Q82
#270017