Finite Group Databasebeta

← #14539872

Subgroups — #14539872

Order 2176.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763378C211yesmaximal, characteristic, complemented
2#11763199C211yesmaximal, characteristic, complemented
3#11763191C211yesmaximal, characteristic, complemented
4#11763171C211yesmaximal, characteristic, complemented
5#11763157C211yesmaximal, characteristic, complemented
6#11763031C211yesmaximal, characteristic, complemented
7#11763024C211yesmaximal, characteristic, complemented
8#11763009C211yesmaximal, characteristic, complemented
9#11763004C211yesmaximal, characteristic, complemented
10#11762969C211yesmaximal, characteristic, complemented
11#11762959C211yesmaximal, characteristic, complemented
12#11762952C211yesmaximal, characteristic, complemented
13#11762942C211yesmaximal, characteristic, complemented
14#11762870C211yesmaximal, characteristic, complemented
15C17×#542C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C17×C4⋊Q8C2211yescharacteristic, complemented
19C17×#126—21no—
20C17×#124—21no—
21C17×#124C2211yescharacteristic, complemented
22C17×#123—21no—
23C17×C22⋊Q8—21no—
24C17×C22⋊Q8C2211yescharacteristic
25C17×#121C2211yescharacteristic
26C4×Q8×C17C2211yescharacteristic, complemented
27C4×D4×C17C2211yescharacteristic, complemented
28C17×#117C2211yescharacteristic
29C2×#63231C2211yescharacteristic, complemented
30C2×#63230C2211yescharacteristic
31C2×#63230C2211yescharacteristic
32C2×#63230C2211yescharacteristic, complemented
33C2×#63229C2211yescharacteristic, complemented
34C2×#63228C2211yescharacteristic
35C2×#63228C2211yescharacteristic
36C2×#63228C2211yescharacteristic, complemented
37#10587581—21no—
38#10587581—21no—
39#10587568—21no—
40#10587568—21no—
41#10587561C2211yescharacteristic
42#10587561C2211yescharacteristic
43#10587560C2211yescharacteristic
44#10587560C2211yescharacteristic
45C22⋊#3622C2211yescharacteristic
46C22⋊#3622C2211yescharacteristic
47#10587481C2211yescharacteristic, complemented
48#10587481C2211yescharacteristic, complemented
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587480C2211yescharacteristic
53#10587480C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic, complemented
55C4⋊#3622C2211yescharacteristic, complemented
56C4⋊#3622C2211yescharacteristic, complemented
57#10587474C2211yescharacteristic, complemented
58C4×#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17—21no—
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17—21no—
63C2×D4×C17C2311yescharacteristic
64C22×C4×C17—21no—
65C22×C4×C17C2311yescharacteristic
66C2×C22⋊D17—21no—
67#63252—41no—
68#63252—41no—
69C2×D68—21no—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68—41no—
75C2×D68—41no—
76C2×D68C2311yescharacteristic
77C2×D68C2311yescharacteristic
78C2×D68C2311yescharacteristic
79C2×C4×D17—21no—
80C2×Dic34C2311yescharacteristic
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17D411yescharacteristic
85C4⋊C4×C17D411yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C22⋊C4×C17D411yescharacteristic
94C22⋊C4×C17D411yescharacteristic
95C42×C17—21no—
96C42×C17C2311yescharacteristic
97C42×C17C2311yescharacteristic
98#63231—41no—
99#63230—41no—
100#63230—41no—
101#63230—41no—
102#63229—41no—
103#63228—41no—
104#63228—41no—
105#63228—41no—
106#63227—21no—
107#63227—21no—
108#63227—21no—
109#63227—21no—
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113C2×#3622C2311yescharacteristic
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—42no—
120C22×D17—81no—
121C22×D17—81no—
122Q8×C17—21no—
123Q8×C17—21no—
124Q8×C17—81no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—81no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—81no—
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2411yescharacteristic
140C22⋊D17—42no—
141C2×Dic17—21no—
142D68—21no—
143D68—21no—
144D68—21no—
145D68—21no—
146D68—21no—
147D68—21no—
148D68—42no—
149D68—81no—
150D68—81no—
151D68—81no—
152D68—81no—
153C4×D17—42no—
154Dic34—21no—
155Dic34—21no—
156#3622—21no—
157#3622—21no—
158#3622—21no—
159#3622—21no—
160#3224—171nomaximal, Sylow
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17C22×D411yescharacteristic
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—84no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17—41no—
186C4×C17H(5,𝔽2)11yescharacteristic
187C4×C17H(5,𝔽2)11yescharacteristic
188Dic17—41no—
189#542D1711yescharacteristic, complemented
190#535—171no—
191#496—171no—
192#495—171no—
193#489—171no—
194#486—171no—
195#464—171no—
196#463—171no—
197#460—171no—
198#459—171no—
199#452—171no—
200#450—171no—
201#449—171no—
202#447—171no—
203#433—171no—
204C2×C17—41no—
205C2×C17—81no—
206C2×C17#53511yescharacteristic
207C2×C17#57611yescharacteristic
208C2×C17#57711yescharacteristic
209D17—81no—
210D17—81no—
211D17—81no—
212C2×#41—171no—
213C22×D4—171no—
214C2×Q16—171no—
215C2×SD16—171no—
216C2×SD16—171no—
217C2×SD16—171no—
218C2×SD16—171no—
219C2×D8—171no—
220C2×D8—171no—
221C2×D8—171no—
222C4⋊Q8C2×D1711yescharacteristic, complemented
223#127—171no—
224#127—171no—
225#126—21no—
226#124—21no—
227#124—171no—
228#124—171no—
229#124C2×D1711yescharacteristic, complemented
230#123—21no—
231C22⋊Q8—21no—
232C22⋊Q8C2×D1711yescharacteristic
233#121—341no—
234#121—341no—
235#121C2×D1711yescharacteristic
236#120—341no—
237#120—341no—
238C4×Q8C2×D1711yescharacteristic, complemented
239C4×D4C2×D1711yescharacteristic, complemented
240#117C2×D1711yescharacteristic
241C4⋊C8—171no—
242C4⋊C8—171no—
243#103—171no—
244#103—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250#102—171no—
251C22⋊C8—171no—
252C22⋊C8—171no—
253#97—171no—
254C4×C8—171no—
255C17#322411yescharacteristic, complemented, Sylow
256C24—341no—
257#41—681no—
258#41—681no—
259C2×Q8—21no—
260C2×Q8—171no—
261C2×Q8C22×D1711yescharacteristic
262C2×D4—21no—
263C2×D4—171no—
264C2×D4—171no—
265C2×D4—171no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4—341no—
272C2×D4—681no—
273C2×D4—681no—
274C2×D4C22×D1711yescharacteristic
275C22×C4—21no—
276C22×C4—341no—
277C22×C4C22×D1711yescharacteristic
278Q16—681no—
279SD16—681no—
280SD16—681no—
281SD16—681no—
282SD16—681no—
283D8—681no—
284D8—681no—
285D8—681no—
286C2×C8—171no—
287C2×C8—171no—
288C2×C8—171no—
289C2×C8—171no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4C22⋊D1711yescharacteristic
294C4⋊C4C22⋊D1711yescharacteristic
295C4⋊C4C22×D1711yescharacteristic
296C4⋊C4C22×D1711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4C22⋊D1711yescharacteristic
307C22⋊C4C22⋊D1711yescharacteristic
308C42—21no—
309C42C22×D1711yescharacteristic
310C42C22×D1711yescharacteristic
311C23—21no—
312C23—341no—
313C23—341no—
314C23—341no—
315C23—682no—
316C23—1361no—
317C23—1361no—
318C23C2×C22⋊D1711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—81no—
322Q8—341no—
323Q8—341no—
324D4—21no—
325D4—21no—
326D4—81no—
327D4—341no—
328D4—341no—
329D4—341no—
330D4—341no—
331D4—341no—
332D4—341no—
333D4—682no—
334D4—682no—
335D4—1361no—
336D4—1361no—
337D4—1361no—
338D4—1361no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—341no—
346C2×C4—682no—
347C2×C4C2×C22⋊D1711yescharacteristic
348C2×C4C2×C22⋊D1711yescharacteristic
349C2×C4C2×C22⋊D1711yescharacteristic
350C2×C4C2×C22⋊D1711yescharacteristic
351C2×C4C2×C22⋊D1711yescharacteristic
352C2×C4C23×D1711yescharacteristic
353C8—341no—
354C8—341no—
355C8—341no—
356C8—341no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—681no—
369C22—681no—
370C22—681no—
371C22—681no—
372C22—1364no—
373C22C22×C22⋊D1711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—681no—
383C4#1058769611yescharacteristic
384C4#1058769611yescharacteristic
385C2—41no—
386C2—81no—
387C2—1361no—
388C2—1361no—
389C2—1361no—
390C2#1176337311yescharacteristic
391C2#1176361211yescharacteristic
392C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#542C2
socleC22×C17C22×D4
hypercenter#542D17
hypocenterC17#3224
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #11763378C2semidirect
2. #11763199C2semidirect
3. #11763191C2semidirect
4. #11763171C2semidirect
5. #11763157C2semidirect
6. #11763031C2semidirect
7. #11763024C2semidirect
8. #11763009C2semidirect
9. #11763004C2semidirect
10. #11762969C2semidirect
11. #11762959C2semidirect
12. #11762952C2semidirect
13. #11762942C2semidirect
14. #11762870C2semidirect
15. C17×#542C2semidirect
18. C17×C4⋊Q8C22semidirect
21. C17×#124C22semidirect
26. C4×Q8×C17C22semidirect
27. C4×D4×C17C22semidirect
29. C2×#63231C22semidirect
32. C2×#63230C22semidirect
33. C2×#63229C22semidirect
36. C2×#63228C22semidirect
47. #10587481C22semidirect
48. #10587481C22semidirect
53. #10587480C22semidirect
54. #10587480C22semidirect
55. C4⋊#3622C22semidirect
56. C4⋊#3622C22semidirect
57. #10587474C22semidirect
58. C4×#3622C22semidirect
189. #542D17semidirect
222. C4⋊Q8C2×D17semidirect
229. #124C2×D17semidirect
238. C4×Q8C2×D17semidirect
239. C4×D4C2×D17semidirect
255. C17#3224semidirect

Maximal subgroups

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

subgroupindex
#117633782
#117631992
#117631912
#117631712
#117631572
#117630312
#117630242
#117630092
#117630042
#117629692
#117629592
#117629522
#117629422
#117628702
C17×#5422
#322417