Finite Group Databasebeta

← #14542374

Subgroups — #14542374

Order 2176.

1112 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#11763541C213yesmaximal, complemented
2#11763535C211yesmaximal, characteristic, complemented
3#11763530C213yesmaximal, complemented
4#11763511C213yesmaximal, complemented
5#11763505C213yesmaximal, complemented
6#11763478C211yesmaximal, characteristic, complemented
7#11763470C211yesmaximal, characteristic, complemented
8#11763354C213yesmaximal, complemented
9#11763345C213yesmaximal, complemented
10#11763338C213yesmaximal, complemented
11#41×Dic17C211yesmaximal, characteristic, complemented
12C4×#63256C211yesmaximal, characteristic, complemented
13C4×#63254C213yesmaximal, complemented
14C2×Q8×Dic17C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C2×#63256C2211yescharacteristic
19C2×#63254C2213yes—
20C17×#124C2213yes—
21C17×C22⋊Q8C2213yes—
22C17×#121C2213yes—
23C4×Q8×C17C2211yescharacteristic
24C4×D4×C17C2213yes—
25#10587633C2213yes—
26#10587632C2213yes—
27Q8×Dic17C2211yescharacteristic
28Q8×Dic17C2211yescharacteristic
29Q8×Dic17C2213yes—
30Q8×Dic17C2211yescharacteristic
31Q8×Dic17C2213yes—
32#10587626C2213yes—
33#10587625C2216yes—
34#10587624C2213yes—
35D4×Dic17C2213yes—
36D4×Dic17C2213yes—
37D4×Dic17C2213yes—
38#10587616C2213yes—
39#10587616C2213yes—
40C4×C22⋊D17C2213yes—
41C4×C22⋊D17C2213yes—
42#10587612C2213yes—
43C2×C4⋊Dic17C2213yes—
44C2×C4×Dic17C2213yes—
45C2×C4×Dic17C2213yes—
46#10587584C2216yes—
47#10587579C2213yes—
48#10587578C2213yes—
49#10587578C2213yes—
50#10587576C2213yes—
51#10587574C2213yes—
52#10587573C2213yes—
53#10587572C2216yes—
54#10587571C2216yes—
55#10587569C2216yes—
56#10587567C2213yes—
57#10587567C2213yes—
58#10587567C2216yes—
59#10587565C2216yes—
60#10587563C2211yescharacteristic
61#10587563C2213yes—
62C4×D68C2213yes—
63C42×D17C2213yes—
64#10587556C2213yes—
65C4×Dic34C2213yes—
66#41×C17—24no—
67C2×Q8×C17—24no—
68C2×Q8×C17C2313yes—
69C2×Q8×C17C2311yescharacteristic
70C2×D4×C17C2313yes—
71C2×D4×C17C2313yes—
72C22×C4×C17C2313yes—
73C22×C4×C17C2313yes—
74C2×C22⋊D17C2313yes—
75C2×C22⋊D17C2313yes—
76C2×C22⋊D17C2316yes—
77C22×Dic17C2313yes—
78C22×Dic17C2313yes—
79C22×Dic17C2313yes—
80C22×Dic17C2311yescharacteristic
81#63256—24no—
82#63254—212no—
83C2×D68C2313yes—
84C2×C4×D17C2313yes—
85C2×C4×D17C2313yes—
86C2×C4×D17C2313yes—
87C2×Dic34C2313yes—
88C4⋊C4×C17C2313yes—
89C4⋊C4×C17C2313yes—
90C22⋊C4×C17C2316yes—
91C22⋊C4×C17C2313yes—
92C22⋊C4×C17C2311yescharacteristic
93C42×C17C2313yes—
94C22⋊Dic17C2313yes—
95C22⋊Dic17C2311yescharacteristic
96C22⋊Dic17C2316yes—
97#63227C2316yes—
98#63227C2316yes—
99#63227C2313yes—
100#63227C2313yes—
101C4⋊Dic17C2313yes—
102C4⋊Dic17C2313yes—
103C4⋊Dic17C2313yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2316yes—
106#63225C2311yescharacteristic
107#63225C2313yes—
108#63225C2316yes—
109#63225C2313yes—
110#63225C2311yescharacteristic
111C4×Dic17C2313yes—
112C4×Dic17C2313yes—
113C4×Dic17C2311yescharacteristic
114C4×Dic17C2313yes—
115C4×Dic17C2313yes—
116C4×Dic17C2313yes—
117C4×Dic17C2311yescharacteristic
118C4×Dic17C2313yes—
119C4×Dic17C2313yes—
120C23×C17C2413yes—
121C23×C17C2411yescharacteristic
122C22×D17C2413yes—
123Q8×C17—26no—
124Q8×C17#4114yescomplemented
125D4×C17—26no—
126D4×C17—26no—
127C2×C4×C17—26no—
128C2×C4×C17—26no—
129C2×C4×C17C2411yescharacteristic
130C2×C4×C17C2413yes—
131C2×C4×C17C2413yes—
132C2×C4×C17C2411yescharacteristic
133C2×C4×C17C2413yes—
134C22⋊D17—26no—
135C22⋊D17—26no—
136C22⋊D17—46no—
137C2×Dic17—26no—
138C2×Dic17—43no—
139C2×Dic17—43no—
140C2×Dic17#4114yescomplemented
141C2×Dic17C2411yescharacteristic
142C2×Dic17C2413yes—
143C2×Dic17C2413yes—
144C2×Dic17C2411yescharacteristic
145C2×Dic17C2413yes—
146C2×Dic17C2411yescharacteristic
147C2×Dic17C2411yescharacteristic
148D68—26no—
149C4×D17—26no—
150C4×D17—26no—
151C4×D17—43no—
152Dic34—26no—
153#3480—171nomaximal, Sylow
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—23no—
157C22×C17—23no—
158C22×C17—23no—
159C22×C17C2×#4112yes—
160C22×C17C2511yescharacteristic
161C2×D17—23no—
162C2×D17—23no—
163C2×D17—23no—
164C4×C17—23no—
165C4×C17—21no—
166C4×C17—23no—
167C4×C17—21no—
168C4×C17C2×#4116yes—
169Dic17—23no—
170Dic17—21no—
171Dic17—23no—
172Dic17—21no—
173Dic17—23no—
174Dic17C2×#4112yes—
175Dic17C2×#4112yes—
176#559—173no—
177#557—171no—
178#556—173no—
179#553—173no—
180#553—173no—
181#550—171no—
182#548—171no—
183#548D1711yescharacteristic, complemented
184#532—173no—
185#529—173no—
186#529—173no—
187C4×#41—171no—
188C4×#41—171no—
189C4×#41—173no—
190C2×C4×Q8—171no—
191C2×C17—22no—
192C2×C17—43no—
193C2×C17C22×#4111yescharacteristic
194C2×C17C22×#4111yescharacteristic
195C2×C17#58511yescharacteristic
196D17—43no—
197C2×#41—171no—
198C2×#41—173no—
199C2×#41C2×D1711yescharacteristic
200C22×Q8C2×D1711yescharacteristic
201C4⋊Q8—173no—
202#127—173no—
203#126—176no—
204#126—176no—
205#125—173no—
206#125—173no—
207#124—173no—
208#124—176no—
209#124C2×D1713yes—
210#123—176no—
211#123—176no—
212C22⋊Q8—173no—
213C22⋊Q8C2×D1713yes—
214#121—173no—
215#121—176no—
216#121—173no—
217#121—173no—
218#121C2×D1713yes—
219C4×Q8—171no—
220C4×Q8—173no—
221C4×Q8—171no—
222C4×Q8—173no—
223C4×Q8—171no—
224C4×Q8—173no—
225C4×Q8—173no—
226C4×Q8C2×D1711yescharacteristic
227C4×D4—173no—
228C4×D4—173no—
229C4×D4—173no—
230C4×D4—173no—
231C4×D4—173no—
232C4×D4—173no—
233C4×D4—173no—
234C4×D4—173no—
235C4×D4—176no—
236C4×D4—173no—
237C4×D4C2×D1713yes—
238#117—173no—
239#117—173no—
240#117—173no—
241#117—171no—
242#117—173no—
243C2×C4⋊C4—173no—
244C2×C42—173no—
245C2×C42—173no—
246C2×C42—173no—
247C17#348011yescharacteristic, complemented, Sylow
248#41—24no—
249#41—3412no—
250#41—344no—
251C2×Q8—24no—
252C2×Q8—173no—
253C2×Q8C22×D1713yes—
254C2×Q8C22×D1711yescharacteristic
255C2×D4—173no—
256C2×D4—173no—
257C2×D4—176no—
258C2×D4—173no—
259C2×D4C22×D1713yes—
260C2×D4C22×D1713yes—
261C22×C4—171no—
262C22×C4—173no—
263C22×C4—173no—
264C22×C4—173no—
265C22×C4—173no—
266C22×C4—173no—
267C22×C4—173no—
268C22×C4C22×D1713yes—
269C22×C4C22×D1713yes—
270C4⋊C4—173no—
271C4⋊C4—173no—
272C4⋊C4—176no—
273C4⋊C4—171no—
274C4⋊C4—171no—
275C4⋊C4—173no—
276C4⋊C4—173no—
277C4⋊C4—173no—
278C4⋊C4—173no—
279C4⋊C4—176no—
280C4⋊C4C22×D1713yes—
281C4⋊C4C22×D1713yes—
282C22⋊C4—171no—
283C22⋊C4—173no—
284C22⋊C4—176no—
285C22⋊C4—173no—
286C22⋊C4—176no—
287C22⋊C4—173no—
288C22⋊C4—176no—
289C22⋊C4C22×D1711yescharacteristic
290C22⋊C4C22×D1713yes—
291C22⋊C4C22×D1716yes—
292C42—171no—
293C42—171no—
294C42—173no—
295C42—173no—
296C42—173no—
297C42—173no—
298C42—173no—
299C42—173no—
300C42—173no—
301C42C22×D1713yes—
302C23—173no—
303C23C23×D1713yes—
304C23C23×D1711yescharacteristic
305Q8—26no—
306Q8—346no—
307Q8#6325414yescomplemented
308D4—26no—
309D4—26no—
310D4—346no—
311D4—346no—
312D4—346no—
313D4—686no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—174no—
317C2×C4—171no—
318C2×C4—171no—
319C2×C4—171no—
320C2×C4—171no—
321C2×C4—173no—
322C2×C4—173no—
323C2×C4—173no—
324C2×C4—346no—
325C2×C4—346no—
326C2×C4—346no—
327C2×C4—683no—
328C2×C4—683no—
329C2×C4—683no—
330C2×C4C23×D1711yescharacteristic
331C2×C4C23×D1713yes—
332C2×C4C23×D1711yescharacteristic
333C2×C4C23×D1713yes—
334C2×C4C23×D1713yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—343no—
341C22—343no—
342C22—343no—
343C22#41×D1712yes—
344C22C24×D1711yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—172no—
350C4—172no—
351C4—343no—
352C4—341no—
353C4—343no—
354C4—341no—
355C4—343no—
356C4C2×#6325416yes—
357C2—22no—
358C2—43no—
359C2—683no—
360C2C2×#41×D1711yescharacteristic
361C2C22×#6325411yescharacteristic
362C2#1176364611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#548C2
hypercenter#548D17
hypocenterC17#3480
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #11763541C2semidirect
2. #11763535C2semidirect
3. #11763530C2semidirect
4. #11763511C2semidirect
5. #11763505C2semidirect
6. #11763478C2semidirect
7. #11763470C2semidirect
8. #11763354C2semidirect
9. #11763345C2semidirect
10. #11763338C2semidirect
11. #41×Dic17C2semidirect
12. C4×#63256C2semidirect
13. C4×#63254C2semidirect
14. C2×Q8×Dic17C2semidirect
15. C17×#548C2semidirect
124. Q8×C17#41semidirect
140. C2×Dic17#41semidirect
183. #548D17semidirect
247. C17#3480semidirect
307. Q8#63254semidirect

Maximal subgroups

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

subgroupindex
#117635412
#117635352
#117635302
#117635112
#117635052
#117634782
#117634702
#117633542
#117633452
#117633382
#41×Dic172
C4×#632562
C4×#632542
C2×Q8×Dic172
C17×#5482
#348017