Finite Group Databasebeta

← #14539668

Subgroups — #14539668

Order 2176.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763387C211yesmaximal, characteristic
2#11763380C211yesmaximal, characteristic, complemented
3#11763151C211yesmaximal, characteristic, complemented
4#11763145C211yesmaximal, characteristic, complemented
5#11763140C211yesmaximal, characteristic, complemented
6#11763134C211yesmaximal, characteristic, complemented
7#11763067C211yesmaximal, characteristic, complemented
8#11763055C211yesmaximal, characteristic, complemented
9#11763050C211yesmaximal, characteristic, complemented
10#11763038C211yesmaximal, characteristic, complemented
11#11762973C211yesmaximal, characteristic, complemented
12#11762973C211yesmaximal, characteristic, complemented
13#11762967C211yesmaximal, characteristic, complemented
14#11762950C211yesmaximal, characteristic, complemented
15C17×#408C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C22×Dic34C2211yescharacteristic
19C2×M16×C17C2211yescharacteristic
20C22×C8×C17C2211yescharacteristic
21#10587632—21no—
22#10587630—21no—
23#10587625—21no—
24#10587625—21no—
25#10587624—21no—
26#10587622—21no—
27#10587621—21no—
28#10587621—21no—
29#10587616C2211yescharacteristic
30#10587616C2211yescharacteristic
31#10587612C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33#10587610C2211yescharacteristic
34#10587610C2211yescharacteristic
35#10587610C2211yescharacteristic
36#10587610C2211yescharacteristic
37#10587610C2211yescharacteristic
38#10587610C2211yescharacteristic
39C2×Dic68C2211yescharacteristic
40C2×Dic68C2211yescharacteristic
41C2×#63219C2211yescharacteristic
42C2×#63219C2211yescharacteristic
43C17×C22⋊C8C2211yescharacteristic
44C17×C22⋊C8C2211yescharacteristic
45C17×C22⋊C8C2211yescharacteristic
46C17×C22⋊C8C2211yescharacteristic
47#10587492C2211yescharacteristic
48#10587492C2211yescharacteristic
49#10587489C2211yescharacteristic
50#10587489C2211yescharacteristic
51#10587488C2211yescharacteristic
52#10587488C2211yescharacteristic
53#10587487C2211yescharacteristic
54#10587487C2211yescharacteristic
55#10587487C2211yescharacteristic
56#10587487C2211yescharacteristic
57#10587487C2211yescharacteristic
58#10587487C2211yescharacteristic
59#41×C17—41no—
60#41×C17—41no—
61C2×Q8×C17D411yescharacteristic
62C2×D4×C17D411yescharacteristic
63C2×D4×C17D411yescharacteristic
64C2×D4×C17D411yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73#63252—41no—
74#63252—41no—
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
77C2×Dic34—21no—
78C2×Dic34—41no—
79C2×Dic34—41no—
80C2×Dic34C2311yescharacteristic
81C2×Dic34C2311yescharacteristic
82C2×Dic34C2311yescharacteristic
83M16×C17—41no—
84C2×C8×C17—41no—
85C2×C8×C17C2311yescharacteristic
86C2×C8×C17C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C2×C8×C17C2311yescharacteristic
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92C22⋊Dic17—21no—
93C22⋊Dic17—21no—
94C22⋊Dic17—21no—
95C22⋊Dic17—21no—
96#63227—21no—
97#63227—21no—
98C4⋊Dic17—21no—
99C4⋊Dic17C2311yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225—21no—
105#63225—21no—
106#63225—21no—
107#63225—21no—
108#63225—21no—
109C4×Dic17—21no—
110Dic68—41no—
111Dic68—41no—
112#63219—41no—
113#63219—41no—
114C23×C17C2×D411yescharacteristic
115C23×C17C2×D411yescharacteristic
116C23×C17C2×D411yescharacteristic
117C22×D17—21no—
118Q8×C17—41no—
119D4×C17—41no—
120D4×C17—41no—
121D4×C17—41no—
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×C17C2×D411yescharacteristic
129C2×C4×C17C2×D411yescharacteristic
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2411yescharacteristic
132C22⋊D17—42no—
133C22⋊D17—81no—
134C22⋊D17—81no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—42no—
143C2×Dic17—81no—
144D68—21no—
145D68—21no—
146C4×D17—42no—
147Dic34—21no—
148Dic34—21no—
149Dic34—21no—
150Dic34—21no—
151Dic34—21no—
152Dic34—21no—
153Dic34—42no—
154C8×C17—21no—
155C8×C17—21no—
156C8×C17—21no—
157C8×C17—21no—
158#3201—171nomaximal, Sylow
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17C22×D411yescharacteristic
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17H(5,𝔽2)11yescharacteristic
176C4×C17H(5,𝔽2)11yescharacteristic
177Dic17—41no—
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184#537—171no—
185#536—171no—
186#485—171no—
187#484—171no—
188#483—171no—
189#482—171no—
190#470—171no—
191#468—171no—
192#467—171no—
193#465—171no—
194#452—171no—
195#452—171no—
196#451—171no—
197#448—171no—
198#408D1711yescharacteristic, complemented
199C2×C17—41no—
200C2×C17—41no—
201C2×C17—41no—
202C2×C17#53411yescharacteristic
203C2×C17#57711yescharacteristic
204C2×C17#57811yescharacteristic
205D17—81no—
206C2×#41—171no—
207C2×#41C2×D1711yescharacteristic
208C22×Q8—171no—
209C2×Q16—171no—
210C2×Q16—171no—
211C2×SD16—171no—
212C2×SD16—171no—
213C2×M16C2×D1711yescharacteristic
214C22×C8C2×D1711yescharacteristic
215C4⋊Q8—341no—
216#124—341no—
217#123—341no—
218#123—341no—
219C22⋊Q8—171no—
220C22⋊Q8—171no—
221C22⋊Q8—171no—
222C22⋊Q8—171no—
223C22⋊Q8—171no—
224C22⋊Q8—171no—
225C22⋊Q8—341no—
226#121—171no—
227#121—171no—
228#121—341no—
229#121—341no—
230#121—341no—
231#117—171no—
232C2×C4⋊C4—171no—
233#107—171no—
234#107—171no—
235#106—171no—
236#106—171no—
237#103—171no—
238#103—171no—
239#103—171no—
240#103—171no—
241#103—171no—
242#103—171no—
243#102—171no—
244#102—171no—
245C22⋊C8C2×D1711yescharacteristic
246C22⋊C8C2×D1711yescharacteristic
247C22⋊C8C2×D1711yescharacteristic
248C22⋊C8C2×D1711yescharacteristic
249C17#320111yescharacteristic, complemented, Sylow
250#41—41no—
251#41—41no—
252#41—681no—
253#41—681no—
254C2×Q8—171no—
255C2×Q8—171no—
256C2×Q8—171no—
257C2×Q8—341no—
258C2×Q8—681no—
259C2×Q8—681no—
260C2×Q8D6811yescharacteristic
261C2×D4—171no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4—341no—
265C2×D4D6811yescharacteristic
266C2×D4D6811yescharacteristic
267C2×D4D6811yescharacteristic
268C22×C4—341no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4C22×D1711yescharacteristic
272C22×C4C22×D1711yescharacteristic
273C22×C4C22×D1711yescharacteristic
274Q16—681no—
275Q16—681no—
276SD16—681no—
277SD16—681no—
278M16—41no—
279C2×C8—41no—
280C2×C8C22×D1711yescharacteristic
281C2×C8C22×D1711yescharacteristic
282C2×C8C22×D1711yescharacteristic
283C2×C8C22×D1711yescharacteristic
284C4⋊C4—171no—
285C4⋊C4—171no—
286C4⋊C4—171no—
287C4⋊C4—171no—
288C4⋊C4—341no—
289C4⋊C4—341no—
290C4⋊C4—341no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C22⋊C4—341no—
298C22⋊C4—341no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C42—341no—
305C23—341no—
306C23C2×D6811yescharacteristic
307C23C2×D6811yescharacteristic
308C23C2×D6811yescharacteristic
309Q8—41no—
310Q8—341no—
311Q8—341no—
312Q8—341no—
313Q8—341no—
314Q8—341no—
315Q8—341no—
316Q8—682no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—341no—
321D4—341no—
322D4—682no—
323D4—1361no—
324D4—1361no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—682no—
339C2×C4—682no—
340C2×C4—1361no—
341C2×C4C2×D6811yescharacteristic
342C2×C4C2×D6811yescharacteristic
343C2×C4C2×D6811yescharacteristic
344C2×C4C23×D1711yescharacteristic
345C8—21no—
346C8—21no—
347C8—21no—
348C8—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—681no—
359C22—681no—
360C22—681no—
361C22C22×D6811yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—681no—
366C4—681no—
367C4—681no—
368C4—681no—
369C4—681no—
370C4—681no—
371C4—681no—
372C4#1058769111yescharacteristic
373C4#1058769111yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—1361no—
378C2#1176337111yescharacteristic
379C2#1176362111yescharacteristic
380C2#1176362611yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#408C2
socleC22×C17C22×D4
prime radical#11763387C2
hypercenter#408D17
hypocenterC17#3201
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #11763380C2semidirect
3. #11763151C2semidirect
4. #11763145C2semidirect
5. #11763140C2semidirect
6. #11763134C2semidirect
7. #11763067C2semidirect
8. #11763055C2semidirect
9. #11763050C2semidirect
10. #11763038C2semidirect
11. #11762973C2semidirect
12. #11762973C2semidirect
13. #11762967C2semidirect
14. #11762950C2semidirect
15. C17×#408C2semidirect
198. #408D17semidirect
249. C17#3201semidirect

Maximal subgroups

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

subgroupindex
#117633872
#117633802
#117631512
#117631452
#117631402
#117631342
#117630672
#117630552
#117630502
#117630382
#117629732
#117629672
#117629502
C17×#4082
#320117