Finite Group Databasebeta

← #14541859

Subgroups — #14541859

Order 2176.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763532C211yesmaximal, characteristic, complemented
2#11763204C211yesmaximal, characteristic, complemented
3#11763201C211yesmaximal, characteristic, complemented
4#11763191C211yesmaximal, characteristic
5#11763031C211yesmaximal, characteristic, complemented
6#11763028C211yesmaximal, characteristic, complemented
7#11763024C211yesmaximal, characteristic
8#11763021C211yesmaximal, characteristic
9#11763019C211yesmaximal, characteristic, complemented
10#11763016C211yesmaximal, characteristic, complemented
11#11762940C211yesmaximal, characteristic, complemented
12#11762909C211yesmaximal, characteristic, complemented
13#11762907C211yesmaximal, characteristic, complemented
14C4×#63231C211yesmaximal, characteristic, complemented
15C17×#556C211yesmaximal, characteristic, complemented
16C17×C4⋊Q8C2211yescharacteristic
17C17×#126—21no—
18C17×#126—21no—
19C17×#124C2211yescharacteristic
20C17×#124C2211yescharacteristic
21C17×C22⋊Q8—21no—
22C4×Q8×C17C2211yescharacteristic
23C4×Q8×C17C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17C2211yescharacteristic
26C17×#117—21no—
27#10587632—21no—
28C2×#63231C2211yescharacteristic
29C2×#63231C2211yescharacteristic
30C2×#63231C2211yescharacteristic
31C2×#63231C2211yescharacteristic
32C2×#63230C2211yescharacteristic
33C2×#63230C2211yescharacteristic
34C2×#63229C2211yescharacteristic
35C2×#63229C2211yescharacteristic
36C2×#63228C2211yescharacteristic
37#10587584—21no—
38#10587584—21no—
39#10587578—21no—
40#10587561C2211yescharacteristic
41#10587561C2211yescharacteristic
42C4×D68C2211yescharacteristic
43#10587555C2211yescharacteristic
44C4×Dic34C2211yescharacteristic
45C4×Dic34C2211yescharacteristic
46#10587506C2211yescharacteristic
47#10587506C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587481C2211yescharacteristic
50#10587481C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587478C2211yescharacteristic
53C4⋊#3622C2211yescharacteristic
54C4⋊#3622C2211yescharacteristic
55C4⋊#3622C2211yescharacteristic
56#10587474C2211yescharacteristic
57#10587474C2211yescharacteristic
58C4×#3622C2211yescharacteristic
59C2×Q8×C17D411yescharacteristic
60C2×Q8×C17C2311yescharacteristic
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C22×C4×C17—21no—
64C2×D68C2311yescharacteristic
65C2×C4×D17—21no—
66C2×Dic34C2311yescharacteristic
67C2×Dic34C2311yescharacteristic
68C4⋊C4×C17—21no—
69C4⋊C4×C17—21no—
70C4⋊C4×C17—21no—
71C4⋊C4×C17—21no—
72C4⋊C4×C17D411yescharacteristic
73C4⋊C4×C17D411yescharacteristic
74C4⋊C4×C17D411yescharacteristic
75C4⋊C4×C17C2311yescharacteristic
76C22⋊C4×C17—21no—
77C22⋊C4×C17—21no—
78C22⋊C4×C17—21no—
79C22⋊C4×C17—21no—
80C22⋊C4×C17—21no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17C2311yescharacteristic
84C42×C17C2311yescharacteristic
85C42×C17C2311yescharacteristic
86#63231—22no—
87#63231—41no—
88#63231—41no—
89#63231—41no—
90#63230—22no—
91#63230—41no—
92#63229—22no—
93#63229—41no—
94#63228—41no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17—21no—
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225—21no—
105C4×Dic17—21no—
106C4×Dic17—21no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17—21no—
112C22×D17—21no—
113Q8×C17—21no—
114Q8×C17—21no—
115Q8×C17—21no—
116Q8×C17—21no—
117Q8×C17—41no—
118D4×C17—21no—
119D4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—42no—
124C2×C4×C17C2×D411yescharacteristic
125C2×C4×C17C2×D411yescharacteristic
126C2×C4×C17C2×D411yescharacteristic
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2×D411yescharacteristic
130C2×C4×C17C2411yescharacteristic
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134D68—21no—
135D68—21no—
136C4×D17—42no—
137Dic34—21no—
138Dic34—21no—
139Dic34—21no—
140Dic34—21no—
141#3622—21no—
142#3622—21no—
143#3622—21no—
144#3622#4112yes—
145#3402—171nomaximal, Sylow
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17C22×D411yescharacteristic
150C2×D17—41no—
151C2×D17—41no—
152C2×D17—41no—
153C4×C17—21no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17C2×#4111yescharacteristic
163C4×C17H(5,𝔽2)11yescharacteristic
164Dic17—41no—
165Dic17—41no—
166Dic17—41no—
167#556—171no—
168#556D1711yescharacteristic, complemented
169#497—171no—
170#497—171no—
171#495—171no—
172#464—171no—
173#464—171no—
174#463—171no—
175#463—171no—
176#462—171no—
177#462—171no—
178#446—171no—
179#440—171no—
180#440—171no—
181C4×Q16—171no—
182C2×C17—81no—
183C2×C17#55011yescharacteristic
184C2×C17#57511yescharacteristic
185C2×C17#57811yescharacteristic
186D17—81no—
187C2×Q16—171no—
188C2×Q16—171no—
189C2×Q16—171no—
190C2×Q16—171no—
191C2×SD16—171no—
192C2×SD16—171no—
193C2×SD16—171no—
194C2×SD16—171no—
195C2×D8—171no—
196C4⋊Q8—171no—
197C4⋊Q8C2×D1711yescharacteristic
198#126—21no—
199#126—21no—
200#126—341no—
201#126—341no—
202#124—171no—
203#124—171no—
204#124C2×D1711yescharacteristic
205#124C2×D1711yescharacteristic
206C22⋊Q8—21no—
207C22⋊Q8—341no—
208C4×Q8—171no—
209C4×Q8—171no—
210C4×Q8C2×D1711yescharacteristic
211C4×Q8C2×D1711yescharacteristic
212C4×Q8C2×D1711yescharacteristic
213C4×D4—171no—
214C4×D4C2×D1711yescharacteristic
215#117—21no—
216#117—341no—
217#107—171no—
218C4⋊C8—171no—
219C4⋊C8—171no—
220C4⋊C8—171no—
221#103—171no—
222#103—171no—
223#103—171no—
224#103—171no—
225#102—171no—
226#102—171no—
227#97—171no—
228#97—171no—
229C4×C8—171no—
230C17#340211yescharacteristic, complemented, Sylow
231C2×Q8—171no—
232C2×Q8—171no—
233C2×Q8C22⋊D1711yescharacteristic
234C2×Q8C22×D1711yescharacteristic
235C2×Q8C22×D1711yescharacteristic
236C2×D4—171no—
237C2×D4C22×D1711yescharacteristic
238C22×C4—21no—
239C22×C4—341no—
240Q16—342no—
241Q16—681no—
242Q16—681no—
243Q16—681no—
244SD16—342no—
245SD16—342no—
246SD16—681no—
247SD16—681no—
248D8—681no—
249C2×C8—171no—
250C2×C8—171no—
251C2×C8—171no—
252C2×C8—171no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—171no—
258C4⋊C4—341no—
259C4⋊C4—341no—
260C4⋊C4—341no—
261C4⋊C4—341no—
262C4⋊C4C22⋊D1711yescharacteristic
263C4⋊C4C22⋊D1711yescharacteristic
264C4⋊C4C22⋊D1711yescharacteristic
265C4⋊C4C22×D1711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—341no—
272C22⋊C4—341no—
273C22⋊C4—341no—
274C22⋊C4—341no—
275C22⋊C4—341no—
276C42—21no—
277C42—21no—
278C42—341no—
279C42—341no—
280C42C22×D1711yescharacteristic
281C42C22×D1711yescharacteristic
282C42C22×D1711yescharacteristic
283C23—21no—
284C23—341no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—41no—
290Q8—341no—
291Q8—341no—
292Q8—341no—
293Q8—341no—
294D4—21no—
295D4—21no—
296D4—341no—
297D4—341no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—682no—
306C2×C4C2×C22⋊D1711yescharacteristic
307C2×C4C2×C22⋊D1711yescharacteristic
308C2×C4C2×C22⋊D1711yescharacteristic
309C2×C4C2×C22⋊D1711yescharacteristic
310C2×C4C2×C22⋊D1711yescharacteristic
311C2×C4C2×C22⋊D1711yescharacteristic
312C2×C4C23×D1711yescharacteristic
313C8—172no—
314C8—341no—
315C8—341no—
316C8—341no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—681no—
321C22—681no—
322C22—681no—
323C22C22×C22⋊D1711yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—681no—
334C4—681no—
335C4—681no—
336C4#41×D1711yescharacteristic
337C4#1058769611yescharacteristic
338C2—81no—
339C2—1361no—
340C2#1176347811yescharacteristic
341C2#1176360711yescharacteristic
342C2#1176362411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#556C2
socleC22×C17C22×D4
prime radicalC2×#63228C22
hypercenter#556D17
hypocenterC17#3402
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #11763532C2semidirect
2. #11763204C2semidirect
3. #11763201C2semidirect
5. #11763031C2semidirect
6. #11763028C2semidirect
9. #11763019C2semidirect
10. #11763016C2semidirect
11. #11762940C2semidirect
12. #11762909C2semidirect
13. #11762907C2semidirect
14. C4×#63231C2semidirect
15. C17×#556C2semidirect
168. #556D17semidirect
230. C17#3402semidirect

Maximal subgroups

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

subgroupindex
#117635322
#117632042
#117632012
#117631912
#117630312
#117630282
#117630242
#117630212
#117630192
#117630162
#117629402
#117629092
#117629072
C4×#632312
C17×#5562
#340217