Finite Group Databasebeta

← #14542781

Subgroups — #14542781

Order 2176.

1122 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763535C211yesmaximal, characteristic, complemented
2#11763506C213yesmaximal, complemented
3#11763505C213yesmaximal, complemented
4#11763479C213yesmaximal, complemented
5#11763478C211yesmaximal, characteristic, complemented
6#11763478C211yesmaximal, characteristic, complemented
7#11763459C213yesmaximal, complemented
8#11763413C213yesmaximal, complemented
9#11763412C213yesmaximal, complemented
10#11763354C213yesmaximal, complemented
11#11763339C213yesmaximal, complemented
12#11763292C211yesmaximal, characteristic, complemented
13#11763291C211yesmaximal, characteristic, complemented
14#41×Dic17C211yesmaximal, characteristic, complemented
15C17×#550C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#41×C17C2211yescharacteristic
18C2×#63256C2211yescharacteristic
19C2×#63254C2216yes—
20C17×#127C2213yes—
21C17×#121C2213yes—
22C17×#121C2213yes—
23C4×Q8×C17C2211yescharacteristic
24C4×D4×C17C2213yes—
25Q8×Dic17C2211yescharacteristic
26Q8×Dic17C2211yescharacteristic
27Q8×Dic17C2211yescharacteristic
28#10587626C2213yes—
29#10587626C2213yes—
30#10587625C2213yes—
31#10587625C2216yes—
32#10587625C2216yes—
33#10587624C2213yes—
34#10587624C2213yes—
35#10587624C2216yes—
36#10587621C2216yes—
37#10587621C2213yes—
38D4×Dic17C2213yes—
39D4×Dic17C2213yes—
40D4×Dic17C2213yes—
41D4×Dic17C2213yes—
42#10587616C2213yes—
43#10587616C2213yes—
44C4×C22⋊D17C2213yes—
45C4×C22⋊D17C2213yes—
46#10587612C2213yes—
47#10587612C2213yes—
48C2×C4⋊Dic17C2213yes—
49C2×C4×Dic17C2213yes—
50#10587584C2213yes—
51#10587579C2213yes—
52C4⋊C4×D17C2213yes—
53#10587576C2213yes—
54#10587575C2211yescharacteristic
55#10587574C2213yes—
56#10587572C2213yes—
57#10587571C2213yes—
58#10587569C2216yes—
59#10587567C2216yes—
60#10587565C2213yes—
61#10587564C2213yes—
62C4×D68C2213yes—
63#10587558C2213yes—
64#10587556C2213yes—
65C4×Dic34C2213yes—
66#41×C17—24no—
67#41×C17—24no—
68C2×Q8×C17C2311yescharacteristic
69C2×D4×C17C2313yes—
70C2×D4×C17C2313yes—
71C2×D4×C17C2313yes—
72C2×D4×C17C2316yes—
73C22×C4×C17C2313yes—
74C22×C4×C17C2313yes—
75C2×C22⋊D17C2313yes—
76C2×C22⋊D17C2316yes—
77C2×C22⋊D17C2316yes—
78C2×C22⋊D17C2313yes—
79C22×Dic17C2313yes—
80C22×Dic17C2316yes—
81C22×Dic17C2313yes—
82#63256—24no—
83#63254—412no—
84C2×D68C2313yes—
85C2×C4×D17C2313yes—
86C2×C4×D17C2316yes—
87C2×Dic34C2313yes—
88C4⋊C4×C17C2313yes—
89C4⋊C4×C17C2311yescharacteristic
90C22⋊C4×C17C2313yes—
91C22⋊C4×C17C2313yes—
92C42×C17C2313yes—
93C22⋊Dic17C2313yes—
94C22⋊Dic17C2313yes—
95C22⋊Dic17C2313yes—
96C22⋊Dic17C2316yes—
97C22⋊Dic17C2313yes—
98#63227C2313yes—
99#63227C2313yes—
100#63227C2313yes—
101#63227C2313yes—
102C4⋊Dic17C2313yes—
103C4⋊Dic17C2313yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2313yes—
106C4⋊Dic17C2313yes—
107C4⋊Dic17C2311yescharacteristic
108#63225C2311yescharacteristic
109#63225C2313yes—
110#63225C2316yes—
111#63225C2311yescharacteristic
112#63225C2313yes—
113#63225C2311yescharacteristic
114#63225C2311yescharacteristic
115C4×Dic17C2313yes—
116C4×Dic17C2313yes—
117C4×Dic17C2313yes—
118C4×Dic17C2313yes—
119C4×Dic17C2311yescharacteristic
120C23×C17C2413yes—
121C23×C17C2413yes—
122C22×D17C2413yes—
123Q8×C17#4114yescomplemented
124D4×C17—26no—
125D4×C17—26no—
126D4×C17—46no—
127D4×C17—43no—
128C2×C4×C17—26no—
129C2×C4×C17—26no—
130C2×C4×C17C2411yescharacteristic
131C2×C4×C17C2413yes—
132C2×C4×C17C2411yescharacteristic
133C2×C4×C17C2411yescharacteristic
134C2×C4×C17C2413yes—
135C22⋊D17—26no—
136C22⋊D17—46no—
137C22⋊D17—46no—
138C22⋊D17—43no—
139C2×Dic17—26no—
140C2×Dic17—46no—
141C2×Dic17—43no—
142C2×Dic17C2411yescharacteristic
143C2×Dic17C2413yes—
144C2×Dic17C2411yescharacteristic
145C2×Dic17C2413yes—
146C2×Dic17C2411yescharacteristic
147C2×Dic17C2413yes—
148C2×Dic17C2411yescharacteristic
149D68—26no—
150C4×D17—26no—
151C4×D17—46no—
152Dic34—26no—
153#3517—171nomaximal, Sylow
154C22×C17—23no—
155C22×C17—23no—
156C22×C17—23no—
157C22×C17—23no—
158C22×C17—23no—
159C22×C17—23no—
160C22×C17C2511yescharacteristic
161C2×D17—23no—
162C2×D17—23no—
163C2×D17—23no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—23no—
167C4×C17—21no—
168C4×C17C2×#4116yes—
169Dic17—21no—
170Dic17—23no—
171Dic17—23no—
172Dic17—21no—
173Dic17—23no—
174Dic17—21no—
175Dic17H(5,𝔽2)12yes—
176#557—171no—
177#553—173no—
178#553—173no—
179#550—171no—
180#550—171no—
181#550—173no—
182#550D1711yescharacteristic, complemented
183#547—173no—
184#540—173no—
185#540—173no—
186#532—173no—
187#529—173no—
188#520—171no—
189#520—171no—
190C4×#41—171no—
191C2×C17—43no—
192C2×C17—43no—
193C2×C17C22×#4111yescharacteristic
194C2×C17C2×H(5,𝔽2)11yescharacteristic
195C2×C17#58511yescharacteristic
196D17—43no—
197C2×#41—171no—
198C2×#41—176no—
199C2×#41C2×D1711yescharacteristic
200C2×#41C2×D1711yescharacteristic
201C4⋊Q8—173no—
202#127—173no—
203#127—173no—
204#127C2×D1713yes—
205#126—173no—
206#126—173no—
207#125—171no—
208#125—173no—
209#125—173no—
210#124—173no—
211#123—173no—
212#123—176no—
213#123—173no—
214#123—176no—
215C22⋊Q8—173no—
216#121—173no—
217#121—176no—
218#121—173no—
219#121—176no—
220#121—173no—
221#121—176no—
222#121—173no—
223#121—173no—
224#121C2×D1713yes—
225#121C2×D1713yes—
226C4×Q8—171no—
227C4×Q8—171no—
228C4×Q8—171no—
229C4×Q8—173no—
230C4×Q8C2×D1711yescharacteristic
231C4×D4—173no—
232C4×D4—173no—
233C4×D4—173no—
234C4×D4—173no—
235C4×D4—173no—
236C4×D4—173no—
237C4×D4—173no—
238C4×D4—176no—
239C4×D4—173no—
240C4×D4C2×D1713yes—
241#117—173no—
242#117—173no—
243#117—173no—
244C2×C4⋊C4—173no—
245C2×C4⋊C4—173no—
246C2×C42—173no—
247C17#351711yescharacteristic, complemented, Sylow
248#41—24no—
249#41—24no—
250#41—344no—
251#41—6812no—
252C2×Q8—173no—
253C2×Q8C22×D1711yescharacteristic
254C2×D4—173no—
255C2×D4—176no—
256C2×D4—173no—
257C2×D4—176no—
258C2×D4—173no—
259C2×D4C22×D1713yes—
260C2×D4C22×D1713yes—
261C2×D4C22×D1713yes—
262C2×D4C22×D1716yes—
263C22×C4—173no—
264C22×C4—173no—
265C22×C4—176no—
266C22×C4—173no—
267C22×C4—176no—
268C22×C4C22×D1713yes—
269C22×C4C22×D1713yes—
270C4⋊C4—173no—
271C4⋊C4—173no—
272C4⋊C4—173no—
273C4⋊C4—173no—
274C4⋊C4—171no—
275C4⋊C4—171no—
276C4⋊C4—173no—
277C4⋊C4—171no—
278C4⋊C4—171no—
279C4⋊C4—176no—
280C4⋊C4—171no—
281C4⋊C4—173no—
282C4⋊C4—173no—
283C4⋊C4C22×D1713yes—
284C4⋊C4C22×D1711yescharacteristic
285C22⋊C4—173no—
286C22⋊C4—176no—
287C22⋊C4—173no—
288C22⋊C4—173no—
289C22⋊C4—173no—
290C22⋊C4—173no—
291C22⋊C4—173no—
292C22⋊C4—173no—
293C22⋊C4—173no—
294C22⋊C4C22×D1713yes—
295C22⋊C4C22×D1713yes—
296C42—171no—
297C42—173no—
298C42—173no—
299C42—173no—
300C42—173no—
301C42C22×D1713yes—
302C23—173no—
303C23C23×D1713yes—
304C23C23×D1713yes—
305Q8—346no—
306Q8#6325414yescomplemented
307D4—26no—
308D4—26no—
309D4—46no—
310D4—43no—
311D4—346no—
312D4—346no—
313D4—686no—
314D4—686no—
315D4—683no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—171no—
319C2×C4—171no—
320C2×C4—171no—
321C2×C4—173no—
322C2×C4—173no—
323C2×C4—173no—
324C2×C4—171no—
325C2×C4—346no—
326C2×C4—346no—
327C2×C4—686no—
328C2×C4—686no—
329C2×C4—683no—
330C2×C4C23×D1711yescharacteristic
331C2×C4C23×D1711yescharacteristic
332C2×C4C23×D1713yes—
333C2×C4C23×D1711yescharacteristic
334C2×C4C23×D1713yes—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—23no—
341C22—343no—
342C22—343no—
343C22—343no—
344C22C24×D1711yescharacteristic
345C4—21no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—172no—
350C4—341no—
351C4—343no—
352C4—343no—
353C4—341no—
354C4—343no—
355C4—341no—
356C4C2×#6325416yes—
357C2—43no—
358C2—43no—
359C2—683no—
360C2C22×#6325411yescharacteristic
361C2H(5,𝔽2)×D1711yescharacteristic
362C2#1176364511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#550C2
hypercenter#550D17
hypocenterC17#3517
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #11763535C2semidirect
2. #11763506C2semidirect
3. #11763505C2semidirect
4. #11763479C2semidirect
5. #11763478C2semidirect
6. #11763478C2semidirect
7. #11763459C2semidirect
8. #11763413C2semidirect
9. #11763412C2semidirect
10. #11763354C2semidirect
11. #11763339C2semidirect
12. #11763292C2semidirect
13. #11763291C2semidirect
14. #41×Dic17C2semidirect
15. C17×#550C2semidirect
123. Q8×C17#41semidirect
182. #550D17semidirect
247. C17#3517semidirect
306. Q8#63254semidirect

Maximal subgroups

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

subgroupindex
#117635352
#117635062
#117635052
#117634792
#117634782
#117634592
#117634132
#117634122
#117633542
#117633392
#117632922
#117632912
#41×Dic172
C17×#5502
#351717