Finite Group Databasebeta

← #14540889

Subgroups — #14540889

Order 2176.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763468C211yesmaximal, characteristic
2#11763171C211yesmaximal, characteristic, complemented
3#11763139C211yesmaximal, characteristic, complemented
4#11763127C211yesmaximal, characteristic, complemented
5#11763091C211yesmaximal, characteristic, complemented
6#11763065C211yesmaximal, characteristic, complemented
7#11763040C211yesmaximal, characteristic, complemented
8#11763031C211yesmaximal, characteristic, complemented
9#11762969C211yesmaximal, characteristic, complemented
10#11762964C211yesmaximal, characteristic, complemented
11#11762914C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762808C211yesmaximal, characteristic, complemented
14C2×#10587481C211yesmaximal, characteristic, complemented
15C17×#554C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C22×Dic34C2211yescharacteristic
18C17×C4⋊Q8C2211yescharacteristic
19C17×#125—21no—
20C17×C22⋊Q8—21no—
21C17×C22⋊Q8—21no—
22C17×C22⋊Q8C2211yescharacteristic
23C17×C22⋊Q8C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17—21no—
26C4×D4×C17C2211yescharacteristic
27C2×C4⋊C4×C17C2211yescharacteristic
28C2×C4⋊C4×C17C2211yescharacteristic
29C2×#63231C2211yescharacteristic
30C2×#63231C2211yescharacteristic
31C2×#63230C2211yescharacteristic
32#10587625—21no—
33#10587616C2211yescharacteristic
34#10587610C2211yescharacteristic
35C2×#63223C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587582—21no—
38#10587571—21no—
39#10587567—21no—
40#10587561C2211yescharacteristic
41C4×Dic34C2211yescharacteristic
42#10587506C2211yescharacteristic
43C22⋊#3622C2211yescharacteristic
44C22⋊#3622C2211yescharacteristic
45#10587481C2211yescharacteristic
46#10587481C2211yescharacteristic
47#10587481C2211yescharacteristic
48#10587481C2211yescharacteristic
49#10587481C2211yescharacteristic
50#10587481C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587480C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587479C2211yescharacteristic
55#10587479C2211yescharacteristic
56#10587479C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58#10587474C2211yescharacteristic
59C2×Q8×C17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×D4×C17D411yescharacteristic
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C22×Dic17—21no—
69#63252—22no—
70#63252—41no—
71C2×D68C2311yescharacteristic
72C2×C4×D17—21no—
73C2×Dic34—22no—
74C2×Dic34—21no—
75C2×Dic34—41no—
76C2×Dic34C2311yescharacteristic
77C2×Dic34C2311yescharacteristic
78C4⋊C4×C17—21no—
79C4⋊C4×C17—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17D411yescharacteristic
85C4⋊C4×C17C2311yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C22⋊C4×C17D411yescharacteristic
91C22⋊C4×C17D411yescharacteristic
92C42×C17—21no—
93C42×C17C2311yescharacteristic
94C22⋊Dic17—21no—
95#63231—22no—
96#63231—41no—
97#63230—41no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101C4⋊Dic17C2311yescharacteristic
102#63225—21no—
103#63225—21no—
104C4×Dic17—21no—
105#63223—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17C2×D411yescharacteristic
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114Q8×C17—21no—
115Q8×C17—21no—
116Q8×C17—81no—
117D4×C17—22no—
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—81no—
127C2×C4×C17—81no—
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2×D411yescharacteristic
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2×D411yescharacteristic
132C2×C4×C17C2411yescharacteristic
133C22⋊D17—42no—
134C22⋊D17—42no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—42no—
139D68—21no—
140D68—21no—
141C4×D17—42no—
142Dic34—21no—
143Dic34—21no—
144Dic34—21no—
145Dic34—42no—
146Dic34#4112yes—
147#3622—21no—
148#3622—21no—
149#3622—21no—
150#3622—21no—
151#3316—171nomaximal, Sylow
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17#13712yes—
158C22×C17C22×D411yescharacteristic
159C2×D17—41no—
160C2×D17—41no—
161C2×D17—41no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17C2×#4111yescharacteristic
171C4×C17H(5,𝔽2)11yescharacteristic
172Dic17—22no—
173Dic17—41no—
174Dic17—41no—
175#554D1711yescharacteristic, complemented
176#548—171no—
177#489—171no—
178#483—171no—
179#481—171no—
180#475—171no—
181#470—171no—
182#465—171no—
183#464—171no—
184#452—171no—
185#451—171no—
186#441—171no—
187#435—171no—
188#417—171no—
189C2×#103—171no—
190C2×C17—22no—
191C2×C17—41no—
192C2×C17#54611yescharacteristic
193C2×C17C2×#13711yescharacteristic
194C2×C17#57711yescharacteristic
195D17—81no—
196C2×#41—171no—
197C22×Q8—171no—
198C2×Q16—171no—
199C2×Q16—171no—
200C2×SD16—171no—
201C2×M16—171no—
202C22×C8—171no—
203C4⋊Q8C2×D1711yescharacteristic
204#125—21no—
205#124—171no—
206#124—341no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—171no—
210C22⋊Q8—341no—
211C22⋊Q8C2×D1711yescharacteristic
212C22⋊Q8C2×D1711yescharacteristic
213#121—171no—
214#121—341no—
215C4×Q8—171no—
216C4×Q8C2×D1711yescharacteristic
217C4×D4—21no—
218C4×D4—341no—
219C4×D4C2×D1711yescharacteristic
220C2×C4⋊C4C2×D1711yescharacteristic
221C2×C4⋊C4C2×D1711yescharacteristic
222#106—171no—
223#106—171no—
224#106—171no—
225C4⋊C8—171no—
226#103—171no—
227#103—171no—
228#103—171no—
229#103—171no—
230#103—171no—
231#103—171no—
232#103—171no—
233#102—171no—
234#102—171no—
235#102—171no—
236C22⋊C8—171no—
237C22⋊C8—171no—
238#97—171no—
239C17#331611yescharacteristic, complemented, Sylow
240#41—342no—
241#41—681no—
242C2×Q8—21no—
243C2×Q8—171no—
244C2×Q8—171no—
245C2×Q8—342no—
246C2×Q8—341no—
247C2×Q8—681no—
248C2×Q8C22×D1711yescharacteristic
249C2×D4—171no—
250C2×D4—341no—
251C2×D4—341no—
252C2×D4C22⋊D1711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—341no—
256C22×C4—341no—
257C22×C4C22×D1711yescharacteristic
258C22×C4C22×D1711yescharacteristic
259Q16—342no—
260Q16—681no—
261SD16—681no—
262M16—681no—
263C2×C8—171no—
264C2×C8—171no—
265C2×C8—171no—
266C2×C8—171no—
267C2×C8—681no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—171no—
275C4⋊C4—341no—
276C4⋊C4—341no—
277C4⋊C4C22⋊D1711yescharacteristic
278C4⋊C4C22×D1711yescharacteristic
279C4⋊C4C22×D1711yescharacteristic
280C4⋊C4C22×D1711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—341no—
284C22⋊C4—341no—
285C22⋊C4—341no—
286C22⋊C4—341no—
287C22⋊C4C22⋊D1711yescharacteristic
288C22⋊C4C22⋊D1711yescharacteristic
289C42—21no—
290C42—341no—
291C42C22×D1711yescharacteristic
292C23—341no—
293C23C2×C22⋊D1711yescharacteristic
294C23C2×C22⋊D1711yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—81no—
298Q8—172no—
299Q8—341no—
300Q8—341no—
301Q8—341no—
302Q8—682no—
303D4—22no—
304D4—341no—
305D4—341no—
306D4—682no—
307D4—682no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—341no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—682no—
322C2×C4—682no—
323C2×C4C2×C22⋊D1711yescharacteristic
324C2×C4C2×C22⋊D1711yescharacteristic
325C2×C4C2×C22⋊D1711yescharacteristic
326C2×C4C2×C22⋊D1711yescharacteristic
327C2×C4C23×D1711yescharacteristic
328C8—341no—
329C8—341no—
330C8—341no—
331C8—341no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—681no—
338C22—681no—
339C22—681no—
340C22#1058762812yes—
341C22C22×C22⋊D1711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—342no—
351C4—681no—
352C4—681no—
353C4#41×D1711yescharacteristic
354C4#1058769611yescharacteristic
355C2—22no—
356C2—41no—
357C2—1361no—
358C2#1176344711yescharacteristic
359C2C2×#1058762811yescharacteristic
360C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#554C2
socleC22×C17C22×D4
prime radical#11763468C2
hypercenter#554D17
hypocenterC17#3316
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #11763171C2semidirect
3. #11763139C2semidirect
4. #11763127C2semidirect
5. #11763091C2semidirect
6. #11763065C2semidirect
7. #11763040C2semidirect
8. #11763031C2semidirect
9. #11762969C2semidirect
10. #11762964C2semidirect
11. #11762914C2semidirect
12. #11762881C2semidirect
13. #11762808C2semidirect
14. C2×#10587481C2semidirect
15. C17×#554C2semidirect
175. #554D17semidirect
239. C17#3316semidirect

Maximal subgroups

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

subgroupindex
#117634682
#117631712
#117631392
#117631272
#117630912
#117630652
#117630402
#117630312
#117629692
#117629642
#117629142
#117628812
#117628082
C2×#105874812
C17×#5542
#331617