Finite Group Databasebeta

← #14539681

Subgroups — #14539681

Order 2176.

402 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#11763387C211yesmaximal, characteristic, complemented
2#11763144C211yesmaximal, characteristic, complemented
3#11763142C211yesmaximal, characteristic, complemented
4#11763122C211yesmaximal, characteristic, complemented
5#11763120C211yesmaximal, characteristic, complemented
6#11763057C211yesmaximal, characteristic, complemented
7#11763053C211yesmaximal, characteristic, complemented
8#11763048C211yesmaximal, characteristic, complemented
9#11763043C211yesmaximal, characteristic, complemented
10#11762971C211yesmaximal, characteristic, complemented
11#11762969C211yesmaximal, characteristic, complemented
12#11762954C211yesmaximal, characteristic, complemented
13#11762952C211yesmaximal, characteristic, complemented
14#11762776C211yesmaximal, characteristic, complemented
15C17×#537C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#41×C17C2211yescharacteristic
18C2×#63252C2211yescharacteristic
19C17×C22⋊Q8—21no—
20C17×C22⋊Q8C2211yescharacteristic
21C17×C22⋊Q8C2211yescharacteristic
22C17×#121—21no—
23C17×#121—21no—
24C17×#121—21no—
25C17×#121C2211yescharacteristic
26C17×#121C2211yescharacteristic
27C2×C4⋊C4×C17C2211yescharacteristic
28#10587632—21no—
29C2×#63231C2211yescharacteristic
30C2×#63230C2211yescharacteristic
31#10587625—21no—
32#10587625—21no—
33#10587624—21no—
34C2×#63229C2211yescharacteristic
35C2×#63228C2211yescharacteristic
36#10587616C2211yescharacteristic
37#10587616C2211yescharacteristic
38C2×C4⋊Dic17C2211yescharacteristic
39#10587610C2211yescharacteristic
40#10587610C2211yescharacteristic
41C2×#63223C2211yescharacteristic
42C22×#3622C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47C22⋊#3622C2211yescharacteristic
48C22⋊#3622C2211yescharacteristic
49C22⋊#3622C2211yescharacteristic
50C22⋊#3622C2211yescharacteristic
51#10587481C2211yescharacteristic
52#10587481C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587480C2211yescharacteristic
55#10587478C2211yescharacteristic
56#10587478C2211yescharacteristic
57#10587478C2211yescharacteristic
58#10587478C2211yescharacteristic
59#41×C17—41no—
60#41×C17—41no—
61#41×C17—41no—
62#41×C17—41no—
63C2×Q8×C17D411yescharacteristic
64C2×Q8×C17C2311yescharacteristic
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17D411yescharacteristic
69C2×D4×C17D411yescharacteristic
70C2×D4×C17D411yescharacteristic
71C2×D4×C17C2311yescharacteristic
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17C2311yescharacteristic
75C22×C4×C17C2311yescharacteristic
76C22×C4×C17C2311yescharacteristic
77C2×C22⋊D17—21no—
78C2×C22⋊D17—21no—
79C2×C22⋊D17—21no—
80C22×Dic17—21no—
81#63252—41no—
82#63252—41no—
83C2×D68C2311yescharacteristic
84C2×C4×D17—21no—
85C2×Dic34C2311yescharacteristic
86C4⋊C4×C17—21no—
87C4⋊C4×C17—21no—
88C4⋊C4×C17—21no—
89C4⋊C4×C17C2311yescharacteristic
90C4⋊C4×C17C2311yescharacteristic
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C22⋊C4×C17—21no—
94C22⋊C4×C17—21no—
95C22⋊Dic17—21no—
96C22⋊Dic17—21no—
97#63231—41no—
98#63230—41no—
99#63229—41no—
100#63228—41no—
101#63227—21no—
102#63227—21no—
103C4⋊Dic17—21no—
104C4⋊Dic17C2311yescharacteristic
105C4⋊Dic17C2311yescharacteristic
106#63225—21no—
107#63225—21no—
108#63223—41no—
109C2×#3622—41no—
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113C2×#3622C2311yescharacteristic
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C23×C17C2×D411yescharacteristic
117C23×C17C2×D411yescharacteristic
118C22×D17—21no—
119Q8×C17—21no—
120Q8×C17—21no—
121Q8×C17—41no—
122D4×C17—21no—
123D4×C17—21no—
124D4×C17—41no—
125D4×C17—41no—
126D4×C17—41no—
127D4×C17—42no—
128D4×C17—81no—
129D4×C17—81no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—42no—
140C2×C4×C17—81no—
141C2×C4×C17C2×D411yescharacteristic
142C2×C4×C17C2×D411yescharacteristic
143C2×C4×C17C2×D411yescharacteristic
144C2×C4×C17C2411yescharacteristic
145C22⋊D17—42no—
146C22⋊D17—81no—
147C22⋊D17—81no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—81no—
152D68—21no—
153D68—21no—
154C4×D17—42no—
155Dic34—21no—
156Dic34—21no—
157#3622—21no—
158#3622—21no—
159#3622—21no—
160#3622—21no—
161#3203—171nomaximal, Sylow
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17C22×D411yescharacteristic
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17H(5,𝔽2)11yescharacteristic
185C4×C17H(5,𝔽2)11yescharacteristic
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189#537—171no—
190#537D1711yescharacteristic, complemented
191#484—171no—
192#484—171no—
193#480—171no—
194#480—171no—
195#468—171no—
196#468—171no—
197#467—171no—
198#466—171no—
199#452—171no—
200#452—171no—
201#449—171no—
202#449—171no—
203#408—171no—
204C2×C17—41no—
205C2×C17—41no—
206C2×C17—41no—
207C2×C17—81no—
208C2×C17#53411yescharacteristic
209C2×C17#57611yescharacteristic
210C2×C17#57811yescharacteristic
211D17—81no—
212C2×#41—171no—
213C2×#41C2×D1711yescharacteristic
214C2×#41C2×D1711yescharacteristic
215C2×Q16—171no—
216C2×SD16—171no—
217C2×SD16—171no—
218C2×D8—171no—
219C2×M16—171no—
220C22×C8—171no—
221C22⋊Q8—21no—
222C22⋊Q8—171no—
223C22⋊Q8—171no—
224C22⋊Q8—341no—
225C22⋊Q8C2×D1711yescharacteristic
226C22⋊Q8C2×D1711yescharacteristic
227#121—21no—
228#121—21no—
229#121—21no—
230#121—171no—
231#121—171no—
232#121—341no—
233#121—341no—
234#121—341no—
235#121C2×D1711yescharacteristic
236#121C2×D1711yescharacteristic
237C2×C4⋊C4—171no—
238C2×C4⋊C4C2×D1711yescharacteristic
239#107—171no—
240#107—171no—
241#107—171no—
242#107—171no—
243#103—171no—
244#103—171no—
245#103—171no—
246#103—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250#102—171no—
251C22⋊C8—171no—
252C22⋊C8—171no—
253C22⋊C8—171no—
254C22⋊C8—171no—
255C17#320311yescharacteristic, complemented, Sylow
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—681no—
261#41—681no—
262C2×Q8—171no—
263C2×Q8C22⋊D1711yescharacteristic
264C2×Q8C22×D1711yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—171no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4—341no—
272C2×D4C22⋊D1711yescharacteristic
273C2×D4C22⋊D1711yescharacteristic
274C2×D4C22⋊D1711yescharacteristic
275C2×D4C22×D1711yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—341no—
279C22×C4—341no—
280C22×C4C22×D1711yescharacteristic
281C22×C4C22×D1711yescharacteristic
282C22×C4C22×D1711yescharacteristic
283Q16—681no—
284SD16—681no—
285SD16—681no—
286D8—681no—
287M16—681no—
288C2×C8—171no—
289C2×C8—171no—
290C2×C8—171no—
291C2×C8—171no—
292C2×C8—681no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—171no—
297C4⋊C4—171no—
298C4⋊C4—341no—
299C4⋊C4—341no—
300C4⋊C4—341no—
301C4⋊C4C22×D1711yescharacteristic
302C4⋊C4C22×D1711yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—341no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C22⋊C4—341no—
311C23—21no—
312C23—341no—
313C23C2×C22⋊D1711yescharacteristic
314C23C2×C22⋊D1711yescharacteristic
315C23C2×C22⋊D1711yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—41no—
319Q8—341no—
320Q8—341no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—341no—
330D4—341no—
331D4—682no—
332D4—1361no—
333D4—1361no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—682no—
349C2×C4—1361no—
350C2×C4C2×C22⋊D1711yescharacteristic
351C2×C4C2×C22⋊D1711yescharacteristic
352C2×C4C2×C22⋊D1711yescharacteristic
353C2×C4C23×D1711yescharacteristic
354C8—341no—
355C8—341no—
356C8—341no—
357C8—341no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—681no—
371C22—681no—
372C22—681no—
373C22C22×C22⋊D1711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—681no—
381C4—681no—
382C4—681no—
383C4#1058769111yescharacteristic
384C4#1058769111yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—81no—
389C2—1361no—
390C2#1176336811yescharacteristic
391C2#1176361211yescharacteristic
392C2#1176362411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#537C2
socleC22×C17C22×D4
hypercenter#537D17
hypocenterC17#3203
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #11763387C2semidirect
2. #11763144C2semidirect
3. #11763142C2semidirect
4. #11763122C2semidirect
5. #11763120C2semidirect
6. #11763057C2semidirect
7. #11763053C2semidirect
8. #11763048C2semidirect
9. #11763043C2semidirect
10. #11762971C2semidirect
11. #11762969C2semidirect
12. #11762954C2semidirect
13. #11762952C2semidirect
14. #11762776C2semidirect
15. C17×#537C2semidirect
190. #537D17semidirect
255. C17#3203semidirect

Maximal subgroups

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

subgroupindex
#117633872
#117631442
#117631422
#117631222
#117631202
#117630572
#117630532
#117630482
#117630432
#117629712
#117629692
#117629542
#117629522
#117627762
C17×#5372
#320317