Finite Group Databasebeta

← #14537856

Subgroups — #14537856

Order 2176.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763356C211yesmaximal, characteristic
2#11762885C211yesmaximal, characteristic, complemented
3#11762884C211yesmaximal, characteristic, complemented
4#11762882C211yesmaximal, characteristic, complemented
5#11762881C211yesmaximal, characteristic, complemented
6#11762879C211yesmaximal, characteristic, complemented
7C8×C22⋊D17C211yesmaximal, characteristic, complemented
8#11762872C211yesmaximal, characteristic, complemented
9#11762864C211yesmaximal, characteristic, complemented
10#11762861C211yesmaximal, characteristic, complemented
11#11762777C211yesmaximal, characteristic, complemented
12#11762775C211yesmaximal, characteristic, complemented
13#11762774C211yesmaximal, characteristic, complemented
14#11762774C211yesmaximal, characteristic, complemented
15C17×#435C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C2×M16×C17C2211yescharacteristic
19C22×C8×C17C2211yescharacteristic
20C4×D4×C17C2211yescharacteristic
21#10587626C411yescharacteristic
22#10587624C411yescharacteristic
23C4×C22⋊D17C2211yescharacteristic
24C4×C22⋊D17C2211yescharacteristic
25C2×#63223C2211yescharacteristic
26C22×#3622C2211yescharacteristic
27C2×#63218C2211yescharacteristic
28C2×#63218C2211yescharacteristic
29C2×C8×D17C2211yescharacteristic
30C2×C8×D17C2211yescharacteristic
31#10587581C411yescharacteristic
32#10587574C411yescharacteristic
33#10587571C411yescharacteristic
34#10587571C411yescharacteristic
35#10587570C411yescharacteristic
36#10587570C411yescharacteristic
37C4×D68C2211yescharacteristic
38C42×D17C2211yescharacteristic
39C17×C4⋊C8C2211yescharacteristic
40C17×C22⋊C8C2211yescharacteristic
41C17×C22⋊C8C2211yescharacteristic
42C17×#97C2211yescharacteristic
43C22⋊#3622C2211yescharacteristic
44C22⋊#3622C2211yescharacteristic
45#10587491C2211yescharacteristic
46#10587491C2211yescharacteristic
47#10587491C2211yescharacteristic
48#10587491C2211yescharacteristic
49#10587491C2211yescharacteristic
50#10587491C2211yescharacteristic
51#10587491C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587485C2211yescharacteristic
54#10587485C2211yescharacteristic
55C8×Dic17C2211yescharacteristic
56C8×Dic17C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58#10587474C2211yescharacteristic
59C2×D4×C17C2×C411yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C2×C22⋊D17C2×C411yescharacteristic
63C2×C22⋊D17C2×C411yescharacteristic
64C2×C22⋊D17C2×C411yescharacteristic
65C2×C22⋊D17C2×C411yescharacteristic
66C2×C22⋊D17C2×C411yescharacteristic
67C2×C22⋊D17C2×C411yescharacteristic
68#63252—41no—
69#63252—41no—
70#63252—41no—
71#63252—41no—
72C2×D68C2×C411yescharacteristic
73C2×D68C2×C411yescharacteristic
74C2×D68C2×C411yescharacteristic
75C2×C4×D17C2×C411yescharacteristic
76C2×C4×D17C2×C411yescharacteristic
77C2×C4×D17C2311yescharacteristic
78C2×C4×D17C2311yescharacteristic
79C2×C4×D17C2311yescharacteristic
80C2×Dic34C2×C411yescharacteristic
81C2×Dic34C2×C411yescharacteristic
82M16×C17—41no—
83C2×C8×C17—41no—
84C2×C8×C17C2311yescharacteristic
85C2×C8×C17C2311yescharacteristic
86C2×C8×C17C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C4⋊C4×C17C2×C411yescharacteristic
89C22⋊C4×C17C2×C411yescharacteristic
90C22⋊C4×C17C2×C411yescharacteristic
91C42×C17C2311yescharacteristic
92C22⋊Dic17C2×C411yescharacteristic
93C22⋊Dic17C2×C411yescharacteristic
94#63227C2×C411yescharacteristic
95#63227C2×C411yescharacteristic
96#63227C2×C411yescharacteristic
97#63227C2×C411yescharacteristic
98C4⋊Dic17C2×C411yescharacteristic
99#63225C2×C411yescharacteristic
100#63225C2×C411yescharacteristic
101C4×Dic17C2×C411yescharacteristic
102C4×Dic17C2×C411yescharacteristic
103C4×Dic17C2311yescharacteristic
104#63223—41no—
105C2×#3622—41no—
106C2×#3622C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110#63218—41no—
111#63218—41no—
112C8×D17—41no—
113C8×D17—41no—
114C23×C17C22×C411yescharacteristic
115C23×C17C22×C411yescharacteristic
116C22×D17C22×C411yescharacteristic
117C22×D17C22×C411yescharacteristic
118C22×D17C22×C411yescharacteristic
119D4×C17—41no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17C22×C411yescharacteristic
125C2×C4×C17C22×C411yescharacteristic
126C2×C4×C17C22×C411yescharacteristic
127C2×C4×C17C22×C411yescharacteristic
128C2×C4×C17C2411yescharacteristic
129C22⋊D17—22no—
130C22⋊D17—22no—
131C22⋊D17—41no—
132C22⋊D17—41no—
133C22⋊D17—41no—
134C22⋊D17—41no—
135C2×Dic17C22×C411yescharacteristic
136C2×Dic17C22×C411yescharacteristic
137C2×Dic17C22×C411yescharacteristic
138C2×Dic17C22×C411yescharacteristic
139C2×Dic17C22×C411yescharacteristic
140D68—41no—
141D68—41no—
142D68—41no—
143C4×D17—21no—
144C4×D17—21no—
145C4×D17—21no—
146C4×D17—21no—
147C4×D17—21no—
148C4×D17—21no—
149C4×D17—22no—
150C4×D17—41no—
151Dic34—41no—
152Dic34—41no—
153C8×C17—21no—
154C8×C17—21no—
155C8×C17—21no—
156C8×C17—21no—
157#3622—21no—
158#3622—21no—
159#3622—21no—
160#3622—21no—
161#2981—171nomaximal, Sylow
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17C23×C411yescharacteristic
169C2×D17—21no—
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—21no—
173C2×D17—21no—
174C2×D17—21no—
175C2×D17—21no—
176C2×D17—21no—
177C2×D17—21no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17H(5,𝔽2)11yescharacteristic
183C4×C17H(5,𝔽2)11yescharacteristic
184Dic17—21no—
185Dic17—21no—
186Dic17—21no—
187Dic17—21no—
188Dic17#13112yes—
189#532—171no—
190#436—171no—
191#436—171no—
192#435—171no—
193#435—171no—
194#435—171no—
195#435D1711yescharacteristic, complemented
196C8×D4—171no—
197C8×D4—171no—
198#432—171no—
199#431—171no—
200#408—171no—
201#408—171no—
202#408—171no—
203#408—171no—
204C2×C17—41no—
205C2×C17—41no—
206C2×C17#51811yescharacteristic
207C2×C17C2×#13111yescharacteristic
208C2×C17#56811yescharacteristic
209D17—41no—
210D17—41no—
211D17—41no—
212C2×#41—171no—
213C2×#41—171no—
214C2×M16—171no—
215C2×M16—171no—
216C2×M16—171no—
217C2×M16C2×D1711yescharacteristic
218C22×C8—171no—
219C22×C8—171no—
220C22×C8—171no—
221C22×C8C2×D1711yescharacteristic
222C4⋊Q8—171no—
223#127—171no—
224#124—171no—
225#124—171no—
226#121—171no—
227#121—171no—
228#121—171no—
229#121—171no—
230C4×D4—171no—
231C4×D4—171no—
232C4×D4—171no—
233C4×D4C2×D1711yescharacteristic
234C2×C42—171no—
235C4⋊C8—171no—
236C4⋊C8—171no—
237C4⋊C8—171no—
238C4⋊C8C2×D1711yescharacteristic
239C22⋊C8—171no—
240C22⋊C8—171no—
241C22⋊C8—171no—
242C22⋊C8—171no—
243C22⋊C8—171no—
244C22⋊C8—171no—
245C22⋊C8—171no—
246C22⋊C8—171no—
247C22⋊C8—171no—
248C22⋊C8—171no—
249C22⋊C8C2×D1711yescharacteristic
250C22⋊C8C2×D1711yescharacteristic
251#97—171no—
252#97C2×D1711yescharacteristic
253C4×C8—171no—
254C4×C8—171no—
255C17#298111yescharacteristic, complemented, Sylow
256#41—681no—
257#41—681no—
258#41—681no—
259#41—681no—
260C2×Q8—171no—
261C2×Q8—171no—
262C2×D4—171no—
263C2×D4—171no—
264C2×D4—171no—
265C2×D4—171no—
266C2×D4—171no—
267C2×D4—171no—
268C2×D4—171no—
269C2×D4—171no—
270C2×D4—171no—
271C2×D4C4×D1711yescharacteristic
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—171no—
277C22×C4C22×D1711yescharacteristic
278C22×C4C22×D1711yescharacteristic
279M16—41no—
280M16—681no—
281M16—681no—
282M16—681no—
283C2×C8—41no—
284C2×C8—171no—
285C2×C8—171no—
286C2×C8—171no—
287C2×C8—171no—
288C2×C8—681no—
289C2×C8—681no—
290C2×C8—681no—
291C2×C8C22×D1711yescharacteristic
292C2×C8C22×D1711yescharacteristic
293C2×C8C22×D1711yescharacteristic
294C2×C8C22×D1711yescharacteristic
295C4⋊C4—171no—
296C4⋊C4—171no—
297C4⋊C4—171no—
298C4⋊C4C4×D1711yescharacteristic
299C22⋊C4—171no—
300C22⋊C4—171no—
301C22⋊C4—171no—
302C22⋊C4—171no—
303C22⋊C4—171no—
304C22⋊C4—171no—
305C22⋊C4C4×D1711yescharacteristic
306C22⋊C4C4×D1711yescharacteristic
307C42—171no—
308C42—171no—
309C42—171no—
310C42C22×D1711yescharacteristic
311C23—171no—
312C23—171no—
313C23—171no—
314C23C2×C4×D1711yescharacteristic
315C23C2×C4×D1711yescharacteristic
316Q8—681no—
317Q8—681no—
318D4—41no—
319D4—342no—
320D4—342no—
321D4—681no—
322D4—681no—
323D4—681no—
324D4—681no—
325D4—681no—
326D4—681no—
327D4—681no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—171no—
333C2×C4—171no—
334C2×C4—171no—
335C2×C4—171no—
336C2×C4—171no—
337C2×C4—342no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—681no—
345C2×C4C2×C4×D1711yescharacteristic
346C2×C4C2×C4×D1711yescharacteristic
347C2×C4C2×C4×D1711yescharacteristic
348C2×C4C2×C4×D1711yescharacteristic
349C2×C4C23×D1711yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—341no—
355C8—341no—
356C8—341no—
357C8—341no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—341no—
365C22—341no—
366C22—341no—
367C22—341no—
368C22—341no—
369C22—341no—
370C22—341no—
371C22—341no—
372C22—341no—
373C22C22×C4×D1711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—172no—
379C4—341no—
380C4—341no—
381C4—341no—
382C4—341no—
383C4#1058769111yescharacteristic
384C4#1058769611yescharacteristic
385C2—41no—
386C2—41no—
387C2—681no—
388C2—681no—
389C2—681no—
390C2#1176328111yescharacteristic
391C2#131×D1711yescharacteristic
392C2#1176357511yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D17
commutator, socleC22×C17C23×C4
FittingC17×#435C2
prime radical#11763356C2
cosocleC2×C4×C17C24
hypercenter#435D17
hypocenterC17#2981
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #11762885C2semidirect
3. #11762884C2semidirect
4. #11762882C2semidirect
5. #11762881C2semidirect
6. #11762879C2semidirect
7. C8×C22⋊D17C2semidirect
8. #11762872C2semidirect
9. #11762864C2semidirect
10. #11762861C2semidirect
11. #11762777C2semidirect
12. #11762775C2semidirect
13. #11762774C2semidirect
14. #11762774C2semidirect
15. C17×#435C2semidirect
195. #435D17semidirect
255. C17#2981semidirect

Maximal subgroups

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

subgroupindex
#117633562
#117628852
#117628842
#117628822
#117628812
#117628792
C8×C22⋊D172
#117628722
#117628642
#117628612
#117627772
#117627752
#117627742
C17×#4352
#298117