Finite Group Databasebeta

← #14541267

Subgroups — #14541267

Order 2176.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763514C211yesmaximal, characteristic, complemented
2#11763511C211yesmaximal, characteristic
3#11763216C211yesmaximal, characteristic, complemented
4#11763153C211yesmaximal, characteristic, complemented
5#11763145C211yesmaximal, characteristic, complemented
6#11763121C211yesmaximal, characteristic, complemented
7#11763119C211yesmaximal, characteristic, complemented
8#11763087C211yesmaximal, characteristic, complemented
9#11763054C211yesmaximal, characteristic, complemented
10#11763036C211yesmaximal, characteristic, complemented
11#11762906C211yesmaximal, characteristic, complemented
12#11762877C211yesmaximal, characteristic, complemented
13#11762848C211yesmaximal, characteristic, complemented
14C2×#10587478C211yesmaximal, characteristic, complemented
15C17×#481C211yesmaximal, characteristic, complemented
16C17×#121C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18#10587625—21no—
19#10587624C2211yescharacteristic
20D4×Dic17—21no—
21D4×Dic17C2211yescharacteristic
22C2×#63229C2211yescharacteristic
23C4×C22⋊D17—21no—
24C4×C22⋊D17C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic
26#10587610—21no—
27#10587610C2211yescharacteristic
28C22×#3622C2211yescharacteristic
29C2×#63218C2211yescharacteristic
30#10587583C2211yescharacteristic
31#10587582—21no—
32#10587578C2211yescharacteristic
33C4⋊C4×D17C2211yescharacteristic
34#10587576C2211yescharacteristic
35#10587575—21no—
36#10587574C2211yescharacteristic
37#10587573C2211yescharacteristic
38#10587572—21no—
39#10587565—21no—
40C17×#106C2211yescharacteristic
41C17×#106C2211yescharacteristic
42C17×#102C2211yescharacteristic
43C17×#102C2211yescharacteristic
44C17×C22⋊C8C2211yescharacteristic
45#10587503C2211yescharacteristic
46#10587491C2211yescharacteristic
47#10587489C2211yescharacteristic
48#10587489C2211yescharacteristic
49#10587487C2211yescharacteristic
50#10587487C2211yescharacteristic
51#10587486C2211yescharacteristic
52#10587485C2211yescharacteristic
53#10587481C2211yescharacteristic
54#10587479C2211yescharacteristic
55#10587478C2211yescharacteristic
56#10587478C2211yescharacteristic
57#10587478C2211yescharacteristic
58#10587478C2211yescharacteristic
59C2×D4×C17—21no—
60C2×D4×C17C2311yescharacteristic
61C22×C4×C17—21no—
62C22×C4×C17C2311yescharacteristic
63C2×C22⋊D17—21no—
64C2×C22⋊D17D411yescharacteristic
65C22×Dic17—21no—
66C22×Dic17—21no—
67C2×C4×D17—21no—
68C2×C4×D17C2311yescharacteristic
69C2×Dic34—21no—
70C2×Dic34C2311yescharacteristic
71C2×C8×C17C2311yescharacteristic
72C2×C8×C17C2311yescharacteristic
73C4⋊C4×C17—21no—
74C4⋊C4×C17C2311yescharacteristic
75C4⋊C4×C17C2311yescharacteristic
76C4⋊C4×C17C2311yescharacteristic
77C22⋊C4×C17—21no—
78C22⋊Dic17—21no—
79C22⋊Dic17—21no—
80C22⋊Dic17—21no—
81C22⋊Dic17D411yescharacteristic
82#63229—22no—
83#63227—21no—
84#63227—21no—
85#63227D411yescharacteristic
86C4⋊Dic17—21no—
87C4⋊Dic17—21no—
88C4⋊Dic17—21no—
89C4⋊Dic17C2311yescharacteristic
90C4⋊Dic17C2311yescharacteristic
91C4⋊Dic17C2311yescharacteristic
92#63225—21no—
93#63225—21no—
94#63225—21no—
95#63225—21no—
96#63225—21no—
97#63225D411yescharacteristic
98C4×Dic17—21no—
99C4×Dic17—21no—
100C4×Dic17C2311yescharacteristic
101C2×#3622—22no—
102C2×#3622C2311yescharacteristic
103C2×#3622C2311yescharacteristic
104#63218—22no—
105C23×C17—21no—
106C23×C17C2×D411yescharacteristic
107C22×D17C2×D411yescharacteristic
108D4×C17—21no—
109D4×C17—21no—
110D4×C17—42no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—81no—
117C2×C4×C17C2×D411yescharacteristic
118C2×C4×C17C2411yescharacteristic
119C22⋊D17—22no—
120C22⋊D17—81no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—81no—
126C2×Dic17—81no—
127C2×Dic17C2×D411yescharacteristic
128C2×Dic17C2×D411yescharacteristic
129C2×Dic17C2×D411yescharacteristic
130C4×D17—21no—
131C4×D17—21no—
132C4×D17—81no—
133Dic34—21no—
134Dic34—21no—
135Dic34—81no—
136C8×C17—21no—
137C8×C17—21no—
138#3622—21no—
139#3622#4112yes—
140#3346—171nomaximal, Sylow
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17#13712yes—
147C22×C17C22×D411yescharacteristic
148C2×D17—21no—
149C2×D17—21no—
150C2×D17—21no—
151C4×C17—21no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17C2×#4111yescharacteristic
156C4×C172−1+411yescharacteristic
157Dic17—21no—
158Dic17—21no—
159Dic17—21no—
160Dic17—41no—
161Dic17—41no—
162Dic17—41no—
163Dic17—41no—
164#554—171no—
165#553—171no—
166#501—171no—
167#485—171no—
168#484—171no—
169#481D1711yescharacteristic, complemented
170#480—171no—
171#479—171no—
172#474—171no—
173#468—171no—
174#465—171no—
175#440—171no—
176#435—171no—
177#428—171no—
178C2×#107—171no—
179C2×C17—22no—
180C2×C17—81no—
181C2×C17#54711yescharacteristic
182C2×C17C2×#13711yescharacteristic
183C2×C17#57611yescharacteristic
184D17—41no—
185C2×SD16—171no—
186C2×M16—171no—
187C22×C8—171no—
188C4⋊Q8—171no—
189#126—341no—
190#125—171no—
191#125—341no—
192#123—341no—
193C22⋊Q8—171no—
194C22⋊Q8—171no—
195C22⋊Q8—341no—
196C22⋊Q8—341no—
197#121—171no—
198#121—341no—
199#121C2×D1711yescharacteristic
200C4×Q8—171no—
201C4×D4—171no—
202C4×D4—171no—
203C4×D4—341no—
204C4×D4—341no—
205#117—171no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—171no—
208C2×C4⋊C4C2×D1711yescharacteristic
209#107—171no—
210#107—171no—
211#107—171no—
212#107—171no—
213#107—171no—
214#107—171no—
215#106—171no—
216#106C2×D1711yescharacteristic
217#106C2×D1711yescharacteristic
218C4⋊C8—171no—
219#103—171no—
220#103—171no—
221#103—171no—
222#102—171no—
223#102C2×D1711yescharacteristic
224#102C2×D1711yescharacteristic
225C22⋊C8—171no—
226C22⋊C8C2×D1711yescharacteristic
227#97—171no—
228C17#334611yescharacteristic, complemented, Sylow
229C2×Q8—171no—
230C2×Q8—341no—
231C2×D4—21no—
232C2×D4—171no—
233C2×D4—341no—
234C2×D4C22×D1711yescharacteristic
235C22×C4—21no—
236C22×C4—171no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4C22×D1711yescharacteristic
241SD16—342no—
242M16—342no—
243C2×C8—171no—
244C2×C8—171no—
245C2×C8—342no—
246C2×C8C22×D1711yescharacteristic
247C2×C8C22×D1711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—171no—
250C4⋊C4—171no—
251C4⋊C4—171no—
252C4⋊C4—171no—
253C4⋊C4—341no—
254C4⋊C4—341no—
255C4⋊C4—341no—
256C4⋊C4—341no—
257C4⋊C4—341no—
258C4⋊C4—341no—
259C4⋊C4—341no—
260C4⋊C4—341no—
261C4⋊C4C22×D1711yescharacteristic
262C4⋊C4C22×D1711yescharacteristic
263C4⋊C4C22×D1711yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—171no—
266C22⋊C4—171no—
267C22⋊C4—341no—
268C22⋊C4—341no—
269C22⋊C4—341no—
270C22⋊C4—341no—
271C22⋊C4—341no—
272C42—171no—
273C42—341no—
274C42—341no—
275C23—21no—
276C23—171no—
277C23D4×D1711yescharacteristic
278Q8—341no—
279Q8—341no—
280Q8—1361no—
281D4—21no—
282D4—21no—
283D4—42no—
284D4—342no—
285D4—1361no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—81no—
292C2×C4—171no—
293C2×C4—171no—
294C2×C4—171no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—1361no—
302C2×C4—1361no—
303C2×C4—1361no—
304C2×C4D4×D1711yescharacteristic
305C2×C4C23×D1711yescharacteristic
306C8—21no—
307C8—21no—
308C8—172no—
309C8—341no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—341no—
316C22—341no—
317C22—341no—
318C22#1058760112yes—
319C22C2×D4×D1711yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—341no—
325C4—341no—
326C4—341no—
327C4—681no—
328C4—681no—
329C4—681no—
330C4—681no—
331C4#41×D1711yescharacteristic
332C4#1058769711yescharacteristic
333C2—22no—
334C2—81no—
335C2—681no—
336C2#1176345711yescharacteristic
337C2C2×#1058760111yescharacteristic
338C2#1176361311yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#481C2
socleC22×C17C22×D4
prime radical#11763511C2
hypercenter#481D17
hypocenterC17#3346
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #11763514C2semidirect
3. #11763216C2semidirect
4. #11763153C2semidirect
5. #11763145C2semidirect
6. #11763121C2semidirect
7. #11763119C2semidirect
8. #11763087C2semidirect
9. #11763054C2semidirect
10. #11763036C2semidirect
11. #11762906C2semidirect
12. #11762877C2semidirect
13. #11762848C2semidirect
14. C2×#10587478C2semidirect
15. C17×#481C2semidirect
169. #481D17semidirect
228. C17#3346semidirect

Maximal subgroups

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

subgroupindex
#117635142
#117635112
#117632162
#117631532
#117631452
#117631212
#117631192
#117630872
#117630542
#117630362
#117629062
#117628772
#117628482
C2×#105874782
C17×#4812
#334617