Finite Group Databasebeta

← #14540290

Subgroups — #14540290

Order 2176.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587637C211yesmaximal, characteristic, complemented
2C22×#63229C211yesmaximal, characteristic, complemented
3D4×Dic34C211yesmaximal, characteristic, complemented
4#11763204C211yesmaximal, characteristic, complemented
5#11763053C211yesmaximal, characteristic, complemented
6#11763048C211yesmaximal, characteristic, complemented
7#11763028C211yesmaximal, characteristic, complemented
8#11763014C211yesmaximal, characteristic, complemented
9#11762971C211yesmaximal, characteristic, complemented
10#11762964C211yesmaximal, characteristic, complemented
11#11762960C211yesmaximal, characteristic, complemented
12#11762947C211yesmaximal, characteristic, complemented
13#11762909C211yesmaximal, characteristic, complemented
14#11762881C211yesmaximal, characteristic, complemented
15C17×#546C211yesmaximal, characteristic
16C2×#41×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×Dic34C2211yescharacteristic
19C22×Dic34C2211yescharacteristic, complemented
20C17×#124C2211yescharacteristic
21C17×#123—21no—
22C17×C22⋊Q8C2211yescharacteristic
23C17×#121—21no—
24C17×#121C2211yescharacteristic
25C17×#120—21no—
26C4×D4×C17C2211yescharacteristic
27C4×D4×C17C2211yescharacteristic
28C2×C22⋊C4×C17—21no—
29#10587637—24no—
30C2×#63231C2211yescharacteristic, complemented
31C2×#63231C2211yescharacteristic, complemented
32C2×#63231C2211yescharacteristic, complemented
33D4×Dic17—21no—
34C2×#63229—24no—
35C2×#63229C2211yescharacteristic
36C2×#63229C2211yescharacteristic
37C2×#63229C2211yescharacteristic
38C2×#63229C2211yescharacteristic
39C2×#63229C2211yescharacteristic
40C2×#63229C2211yescharacteristic
41#10587610C2211yescharacteristic
42#10587610C2211yescharacteristic, complemented
43C2×#63223C2211yescharacteristic, complemented
44C22×#3622C2211yescharacteristic
45#10587574—21no—
46#10587564—21no—
47#10587564—21no—
48#10587555C2211yescharacteristic, complemented
49C4×Dic34C2211yescharacteristic, complemented
50#10587506C2211yescharacteristic, complemented
51#10587503C2211yescharacteristic
52#10587503C2211yescharacteristic
53C22⋊#3622C2211yescharacteristic
54C22⋊#3622C2211yescharacteristic, complemented
55#10587481C2211yescharacteristic, complemented
56#10587481C2211yescharacteristic, complemented
57#10587481C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C4⋊#3622C2211yescharacteristic, complemented
60#10587474C2211yescharacteristic, complemented
61C24×C17—21no—
62#41×C17—22no—
63#41×C17—41no—
64C2×Q8×C17C2311yescharacteristic
65C2×D4×C17—22no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17—21no—
69C2×D4×C17—41no—
70C2×D4×C17D411yescharacteristic
71C2×D4×C17C2311yescharacteristic
72C2×D4×C17C2311yescharacteristic
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17C2311yescharacteristic
76C22×C4×C17C2311yescharacteristic
77C22×Dic17—21no—
78C22×Dic17—21no—
79C2×Dic34—22no—
80C2×Dic34—22no—
81C2×Dic34—21no—
82C2×Dic34—21no—
83C2×Dic34—41no—
84C2×Dic34—41no—
85C2×Dic34C2311yescharacteristic
86C2×Dic34C2311yescharacteristic
87C2×Dic34C2311yescharacteristic
88C4⋊C4×C17—21no—
89C4⋊C4×C17D411yescharacteristic
90C4⋊C4×C17C2311yescharacteristic
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C22⋊C4×C17—21no—
94C22⋊C4×C17—21no—
95C22⋊C4×C17—21no—
96C22⋊C4×C17D411yescharacteristic
97C22⋊C4×C17D411yescharacteristic
98C42×C17C2311yescharacteristic
99C22⋊Dic17—21no—
100C22⋊Dic17—21no—
101#63231—22no—
102#63231—22no—
103#63231—41no—
104#63229—22no—
105#63229—22no—
106#63229—22no—
107#63229—22no—
108#63229—41no—
109#63229D414yescomplemented
110C4⋊Dic17—21no—
111C4⋊Dic17—21no—
112C4⋊Dic17C2311yescharacteristic
113#63225—21no—
114#63225—21no—
115#63225—21no—
116C4×Dic17—21no—
117#63223—22no—
118C2×#3622—22no—
119C2×#3622C2311yescharacteristic
120C2×#3622C2311yescharacteristic
121C2×#3622C2311yescharacteristic
122C2×#3622C2311yescharacteristic
123C23×C17—21no—
124C23×C17—21no—
125C23×C17—42no—
126C23×C17—42no—
127C23×C17—81no—
128C23×C17C2×D411yescharacteristic
129C23×C17C2×D411yescharacteristic
130Q8×C17—21no—
131Q8×C17—21no—
132D4×C17—22no—
133D4×C17—21no—
134D4×C17—21no—
135D4×C17—21no—
136D4×C17—42no—
137D4×C17—42no—
138D4×C17—81no—
139D4×C17C2×D412yes—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—42no—
147C2×C4×C17—81no—
148C2×C4×C17C2×D411yescharacteristic
149C2×C4×C17C2×D411yescharacteristic
150C2×C4×C17C2×D411yescharacteristic
151C2×C4×C17C2×D411yescharacteristic
152C2×C4×C17C2411yescharacteristic
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—21no—
157C2×Dic17—42no—
158C2×Dic17—42no—
159Dic34—21no—
160Dic34—21no—
161Dic34—21no—
162Dic34—21no—
163Dic34—21no—
164Dic34—42no—
165Dic34—42no—
166Dic34C2×D412yes—
167#3622—21no—
168#3622—21no—
169#3622—21no—
170#3622C2×D412yes—
171#3266—171nomaximal, Sylow
172C22×C17—21no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—22no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—41no—
179C22×C17—41no—
180C22×C17—41no—
181C22×C17—41no—
182C22×C17—41no—
183C22×C17—84no—
184C22×C17#13712yescomplemented
185C22×C17C22×D411yescharacteristic
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—21no—
190C4×C17—41no—
191C4×C17—41no—
192C4×C17C22×D411yescharacteristic
193C4×C17H(5,𝔽2)11yescharacteristic
194Dic17—22no—
195Dic17—41no—
196Dic17—41no—
197Dic17—41no—
198C2×#137—171no—
199C22×SD16—171no—
200D4×Q8—171no—
201#546D1711yescharacteristic
202#497—171no—
203#468—171no—
204#467—171no—
205#464—171no—
206#461—171no—
207#452—171no—
208#451—171no—
209#450—171no—
210#448—171no—
211#440—171no—
212#435—171no—
213C2×C17—22no—
214C2×C17—41no—
215C2×C17—42no—
216C2×C17—81no—
217C2×C17D4211yescharacteristic
218C2×C17C2×#13711yescharacteristic
219C2×C17#57711yescharacteristic
220C2×#41C2×D1711yescharacteristic
221C22×Q8—171no—
222C22×Q8—171no—
223C22×D4C2×D1711yescharacteristic
224#137—344no—
225C2×Q16—171no—
226C2×Q16—171no—
227C2×Q16—171no—
228C2×SD16—171no—
229C2×SD16—171no—
230C2×SD16—171no—
231C2×SD16—171no—
232C2×SD16—171no—
233C2×SD16—171no—
234C2×SD16—344no—
235C2×M16—171no—
236C22×C8—171no—
237C4⋊Q8—171no—
238C4⋊Q8—341no—
239#124C2×D1711yescharacteristic
240#123—21no—
241C22⋊Q8—171no—
242C22⋊Q8—171no—
243C22⋊Q8—341no—
244C22⋊Q8—341no—
245C22⋊Q8C2×D1711yescharacteristic
246#121—21no—
247#121C2×D1711yescharacteristic
248#120—21no—
249C4×Q8—171no—
250C4×D4—341no—
251C4×D4C2×D1711yescharacteristic
252C4×D4C2×D1711yescharacteristic
253C2×C22⋊C4—21no—
254#107—171no—
255C4⋊C8—171no—
256#103—171no—
257#103—171no—
258#103—171no—
259#103—171no—
260#102—171no—
261#102—171no—
262C22⋊C8—171no—
263C22⋊C8—171no—
264#97—171no—
265C17#326611yescharacteristic, complemented, Sylow
266C24—21no—
267#41—22no—
268#41—41no—
269C2×Q8—171no—
270C2×Q8—171no—
271C2×Q8—171no—
272C2×Q8—342no—
273C2×Q8—342no—
274C2×Q8—341no—
275C2×Q8—341no—
276C2×Q8—681no—
277C2×Q8—681no—
278C2×Q8C22×D1711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4C22⋊D1711yescharacteristic
285C2×D4C22×D1711yescharacteristic
286C2×D4C22×D1711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—341no—
290C22×C4—341no—
291C22×C4C22×D1711yescharacteristic
292C22×C4C22×D1711yescharacteristic
293Q16—342no—
294Q16—342no—
295Q16—681no—
296SD16—174no—
297SD16—342no—
298SD16—342no—
299SD16—342no—
300SD16—342no—
301SD16—681no—
302M16—342no—
303C2×C8—171no—
304C2×C8—171no—
305C2×C8—171no—
306C2×C8—171no—
307C2×C8—342no—
308C4⋊C4—21no—
309C4⋊C4—171no—
310C4⋊C4—341no—
311C4⋊C4—341no—
312C4⋊C4—341no—
313C4⋊C4—341no—
314C4⋊C4—341no—
315C4⋊C4C22⋊D1711yescharacteristic
316C4⋊C4C22×D1711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—341no—
323C22⋊C4—341no—
324C22⋊C4C22⋊D1711yescharacteristic
325C22⋊C4C22⋊D1711yescharacteristic
326C42—341no—
327C42C22×D1711yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊D1711yescharacteristic
334C23C2×C22⋊D1711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—172no—
338Q8—341no—
339Q8—341no—
340Q8—341no—
341Q8—341no—
342Q8—341no—
343Q8—682no—
344Q8—682no—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4D4×D1712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—341no—
362C2×C4—341no—
363C2×C4—341no—
364C2×C4—341no—
365C2×C4—682no—
366C2×C4—682no—
367C2×C4C2×C22⋊D1711yescharacteristic
368C2×C4C2×C22⋊D1711yescharacteristic
369C2×C4C2×C22⋊D1711yescharacteristic
370C2×C4C2×C22⋊D1711yescharacteristic
371C2×C4C23×D1711yescharacteristic
372C8—172no—
373C8—341no—
374C8—341no—
375C8—341no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#1058763712yescomplemented
389C22C22×C22⋊D1711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—342no—
397C4—681no—
398C4—681no—
399C4—681no—
400C4C2×D4×D1711yescharacteristic
401C4#1058769111yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×C22⋊D1711yescharacteristic
407C2C2×#1058763711yescharacteristic
408C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#546C2
socleC22×C17C22×D4
hypercenter#546D17
hypocenterC17#3266
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#10587637C2semidirect
2. C22×#63229C2semidirect
3. D4×Dic34C2semidirect
4. #11763204C2semidirect
5. #11763053C2semidirect
6. #11763048C2semidirect
7. #11763028C2semidirect
8. #11763014C2semidirect
9. #11762971C2semidirect
10. #11762964C2semidirect
11. #11762960C2semidirect
12. #11762947C2semidirect
13. #11762909C2semidirect
14. #11762881C2semidirect
19. C22×Dic34C22semidirect
30. C2×#63231C22semidirect
31. C2×#63231C22semidirect
32. C2×#63231C22semidirect
42. #10587610C22semidirect
43. C2×#63223C22semidirect
48. #10587555C22semidirect
49. C4×Dic34C22semidirect
50. #10587506C22semidirect
54. C22⋊#3622C22semidirect
55. #10587481C22semidirect
56. #10587481C22semidirect
57. #10587481C22semidirect
58. #10587478C22semidirect
59. C4⋊#3622C22semidirect
60. #10587474C22semidirect
109. #63229D4semidirect
184. C22×C17#137semidirect
265. C17#3266semidirect
388. C22#10587637semidirect

Maximal subgroups

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

subgroupindex
C2×#105876372
C22×#632292
D4×Dic342
#117632042
#117630532
#117630482
#117630282
#117630142
#117629712
#117629642
#117629602
#117629472
#117629092
#117628812
C17×#5462
#326617