Finite Group Databasebeta

← #14537613

Subgroups — #14537613

Order 2176.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587597C211yesmaximal, characteristic
2#11763288C211yesmaximal, characteristic, complemented
3#11763281C211yesmaximal, characteristic, complemented
4#11762918C211yesmaximal, characteristic, complemented
5#11762911C211yesmaximal, characteristic, complemented
6#11762906C211yesmaximal, characteristic, complemented
7#11762896C211yesmaximal, characteristic, complemented
8C4×#63229C211yesmaximal, characteristic, complemented
9D8×Dic17C211yesmaximal, characteristic, complemented
10C4×#63228C211yesmaximal, characteristic, complemented
11#11762848C211yesmaximal, characteristic, complemented
12#11762802C211yesmaximal, characteristic, complemented
13#102×D17C211yesmaximal, characteristic, complemented
14#11762766C211yesmaximal, characteristic, complemented
15C17×#442C211yesmaximal, characteristic, complemented
16C2×#63254C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×D8×C17C2211yescharacteristic
19C4×D4×C17C2211yescharacteristic
20C4×D4×C17C2211yescharacteristic, complemented
21D4×Dic17C2211yescharacteristic
22D4×Dic17C2211yescharacteristic, complemented
23C2×#63229C2211yescharacteristic
24C2×#63228C2211yescharacteristic
25C4×C22⋊D17—21no—
26C4×C22⋊D17—21no—
27C2×#63227—21no—
28C2×#63225—21no—
29#10587597C418yescomplemented
30C2×#63219C2211yescharacteristic
31C2×#63218C2211yescharacteristic
32C2×C8×D17C2211yescharacteristic
33#10587579C2211yescharacteristic
34#10587578C2211yescharacteristic
35C4⋊C4×D17C2211yescharacteristic
36#10587573C2211yescharacteristic, complemented
37#10587567—21no—
38#10587567—21no—
39C22⋊C4×D17—21no—
40#10587563—21no—
41C4×D68C2211yescharacteristic
42#10587558C2211yescharacteristic
43C4×Dic34C2211yescharacteristic, complemented
44C17×#106C2211yescharacteristic, complemented
45C17×#102C2211yescharacteristic
46C17×#102C2211yescharacteristic, complemented
47C17×#97C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic, complemented
50#10587492C2211yescharacteristic
51#10587489C2211yescharacteristic, complemented
52#10587487C2211yescharacteristic, complemented
53#10587486C2211yescharacteristic, complemented
54C8×Dic17C2211yescharacteristic, complemented
55#10587481C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic
57#10587479C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C4×#3622C2211yescharacteristic, complemented
60C23×D17—21no—
61C2×D4×C17C2311yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70#63254—22no—
71#63254C2×C414yescomplemented
72D4×D17—22no—
73D4×D17C2×C414yescomplemented
74C2×D68C2311yescharacteristic
75C2×C4×D17—21no—
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17C2311yescharacteristic
79C2×Dic34C2311yescharacteristic
80D8×C17C2×C414yescomplemented
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C4⋊C4×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C42×C17C2311yescharacteristic
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90#63229C2×C414yescomplemented
91#63228C2×C414yescomplemented
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227D411yescharacteristic
96#63227D411yescharacteristic
97C4⋊Dic17C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99#63225—21no—
100#63225—21no—
101#63225—21no—
102#63225D411yescharacteristic
103#63225D411yescharacteristic
104C4×Dic17—21no—
105C4×Dic17—21no—
106C4×Dic17—21no—
107C4×Dic17C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110#63219C2×C414yescomplemented
111#63218C2×C414yescomplemented
112C8×D17—22no—
113C23×C17—21no—
114C23×C17—21no—
115C22×D17—21no—
116C22×D17—24no—
117C22×D17—81no—
118C22×D17—81no—
119C22×D17C2×D411yescharacteristic
120D4×C17—21no—
121D4×C17—21no—
122D4×C17C22×C412yes—
123D4×C17C22×C412yes—
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×C17C2411yescharacteristic
131C22⋊D17—24no—
132C22⋊D17—24no—
133C2×Dic17—21no—
134C2×Dic17—24no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—81no—
138C2×Dic17—81no—
139C2×Dic17C2×D411yescharacteristic
140C2×Dic17C2×D411yescharacteristic
141C2×Dic17C2×D411yescharacteristic
142D68—21no—
143D68C22×C412yes—
144C4×D17—81no—
145C4×D17—81no—
146C4×D17—81no—
147C4×D17C22×C412yes—
148C4×D17#4112yes—
149Dic34—21no—
150Dic34C22×C412yes—
151C8×C17—21no—
152C8×C17C22×C412yes—
153#3622—21no—
154#3622C22×C412yes—
155#2946—171nomaximal, Sylow
156C22×C17—22no—
157C22×C17—22no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17C22×D411yescharacteristic
163C2×D17—21no—
164C2×D17—21no—
165C2×D17—22no—
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—84no—
169C2×D17C4×D412yes—
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17C23×C411yescharacteristic
175C4×C17C2×#4111yescharacteristic
176Dic17—22no—
177Dic17—21no—
178Dic17—21no—
179Dic17—41no—
180Dic17—41no—
181Dic17C4×D412yes—
182C2×AGL(1,ℤ/8)—171no—
183#520—171no—
184#518—171no—
185#442—171no—
186#442D1711yescharacteristic, complemented
187#440—171no—
188#440—171no—
189C4×SD16—171no—
190C4×SD16—171no—
191C4×D8—171no—
192C4×D8—171no—
193#428—171no—
194#416—171no—
195C2×#102—171no—
196#405—171no—
197C2×C17—42no—
198C2×C17—42no—
199C2×C17C2×C4×D411yescharacteristic
200C2×C17#57611yescharacteristic
201C2×C17#57711yescharacteristic
202D17—22no—
203D17—42no—
204C2×#41—171no—
205C22×D4—171no—
206AGL(1,ℤ/8)—178no—
207C2×SD16—171no—
208C2×SD16—171no—
209C2×D8—171no—
210C2×D8C2×D1711yescharacteristic
211C2×M16—171no—
212C22×C8—171no—
213C4×Q8—171no—
214C4×Q8—171no—
215C4×D4—171no—
216C4×D4—171no—
217C4×D4—171no—
218C4×D4—171no—
219C4×D4—341no—
220C4×D4—341no—
221C4×D4—341no—
222C4×D4—341no—
223C4×D4C2×D1711yescharacteristic
224C4×D4C2×D1711yescharacteristic, complemented
225#117—171no—
226#117—171no—
227#117—341no—
228C2×C4⋊C4—171no—
229C2×C4⋊C4—341no—
230C2×C22⋊C4—341no—
231C2×C22⋊C4—341no—
232#107—171no—
233#107—171no—
234#106—171no—
235#106C2×D1711yescharacteristic, complemented
236#103—171no—
237#103—171no—
238#102—171no—
239#102—171no—
240#102—171no—
241#102—171no—
242#102C2×D1711yescharacteristic
243#102C2×D1711yescharacteristic, complemented
244#97—171no—
245#97C2×D1711yescharacteristic, complemented
246C4×C8—171no—
247C4×C8—171no—
248C17#294611yescharacteristic, complemented, Sylow
249C24—341no—
250#41—174no—
251#41—342no—
252C2×Q8—171no—
253C2×D4—174no—
254C2×D4—171no—
255C2×D4—342no—
256C2×D4—341no—
257C2×D4—341no—
258C2×D4C22×D1711yescharacteristic
259C2×D4C22×D1711yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—171no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—341no—
267C22×C4—341no—
268C22×C4—341no—
269SD16—174no—
270SD16—174no—
271D8—174no—
272D8C4×D1714yescomplemented
273M16—174no—
274C2×C8—171no—
275C2×C8—171no—
276C2×C8—342no—
277C2×C8C22×D1711yescharacteristic
278C2×C8C22×D1711yescharacteristic
279C4⋊C4—171no—
280C4⋊C4—171no—
281C4⋊C4—171no—
282C4⋊C4—171no—
283C4⋊C4—341no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4C22×D1711yescharacteristic
287C4⋊C4C22×D1711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—171no—
291C22⋊C4—171no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C42—171no—
298C42—341no—
299C42—341no—
300C42—341no—
301C42C22×D1711yescharacteristic
302C23—21no—
303C23—21no—
304C23—171no—
305C23—344no—
306C23—341no—
307C23—1361no—
308C23—1361no—
309Q8—172no—
310Q8—341no—
311D4—21no—
312D4—21no—
313D4—172no—
314D4—344no—
315D4—344no—
316D4—341no—
317D4C2×C4×D1712yes—
318D4C2×C4×D1712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—171no—
324C2×C4—172no—
325C2×C4—172no—
326C2×C4—171no—
327C2×C4—171no—
328C2×C4—344no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—1361no—
333C2×C4—1361no—
334C2×C4—1361no—
335C2×C4—1361no—
336C2×C4—1361no—
337C2×C4D4×D1711yescharacteristic
338C2×C4D4×D1711yescharacteristic
339C2×C4C23×D1711yescharacteristic
340C8—21no—
341C8—172no—
342C8—341no—
343C8C2×C4×D1712yes—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—172no—
351C22—341no—
352C22—341no—
353C22—342no—
354C22—681no—
355C22—681no—
356C22—1364no—
357C22C2×D4×D1711yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—172no—
363C4—342no—
364C4—341no—
365C4—341no—
366C4—681no—
367C4—681no—
368C4C22×C4×D1711yescharacteristic
369C4#41×D1711yescharacteristic
370C2—42no—
371C2—42no—
372C2—342no—
373C2—682no—
374C2C4×D4×D1711yescharacteristic
375C2#1176361311yescharacteristic
376C2#1176361911yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutatorC4×C17C23×C4
FrattiniC2×C4C23×D17
FittingC17×#442C2
socleC22×C17C22×D4
prime radicalC2×#10587597C2
cosocleC2×C4×C17C24
hypercenter#442D17
hypocenterC17#2946
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #11763288C2semidirect
3. #11763281C2semidirect
4. #11762918C2semidirect
5. #11762911C2semidirect
6. #11762906C2semidirect
7. #11762896C2semidirect
8. C4×#63229C2semidirect
9. D8×Dic17C2semidirect
10. C4×#63228C2semidirect
11. #11762848C2semidirect
12. #11762802C2semidirect
13. #102×D17C2semidirect
14. #11762766C2semidirect
15. C17×#442C2semidirect
20. C4×D4×C17C22semidirect
22. D4×Dic17C22semidirect
29. #10587597C4semidirect
36. #10587573C22semidirect
43. C4×Dic34C22semidirect
44. C17×#106C22semidirect
46. C17×#102C22semidirect
47. C17×#97C22semidirect
49. #10587503C22semidirect
51. #10587489C22semidirect
52. #10587487C22semidirect
53. #10587486C22semidirect
54. C8×Dic17C22semidirect
55. #10587481C22semidirect
57. #10587479C22semidirect
58. #10587478C22semidirect
59. C4×#3622C22semidirect
71. #63254C2×C4semidirect
73. D4×D17C2×C4semidirect
80. D8×C17C2×C4semidirect
90. #63229C2×C4semidirect
91. #63228C2×C4semidirect
110. #63219C2×C4semidirect
111. #63218C2×C4semidirect
186. #442D17semidirect
224. C4×D4C2×D17semidirect
235. #106C2×D17semidirect
243. #102C2×D17semidirect
245. #97C2×D17semidirect
248. C17#2946semidirect
272. D8C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875972
#117632882
#117632812
#117629182
#117629112
#117629062
#117628962
C4×#632292
D8×Dic172
C4×#632282
#117628482
#117628022
#102×D172
#117627662
C17×#4422
#294617