Finite Group Databasebeta

← #14542382

Subgroups — #14542382

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#11763542C213yesmaximal, complemented
2#11763535C211yesmaximal, characteristic, complemented
3#11763527C213yesmaximal, complemented
4#11763509C213yesmaximal, complemented
5#11763503C213yesmaximal, complemented
6#11763478C211yesmaximal, characteristic, complemented
7#11763471C211yesmaximal, characteristic, complemented
8#11763466C211yesmaximal, characteristic, complemented
9#11763356C213yesmaximal, complemented
10#11763343C213yesmaximal, complemented
11#11763341C213yesmaximal, complemented
12C4×#63256C211yesmaximal, characteristic, complemented
13C4×#63252C213yesmaximal, complemented
14C4×Q8×D17C211yesmaximal, characteristic, complemented
15C4×#41×C17C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#63256C2211yescharacteristic
18C2×Q8×D17C2211yescharacteristic
19C2×#63252C2213yes—
20C4×Q8×C17C2211yescharacteristic
21C4×Q8×C17C2211yescharacteristic
22C4×D4×C17C2213yes—
23C4×D4×C17C2213yes—
24C17×#117C2213yes—
25C2×C42×C17C2213yes—
26Q8×Dic17C2211yescharacteristic
27Q8×Dic17C2211yescharacteristic
28#10587626C2213yes—
29#10587624C2213yes—
30#10587624C2213yes—
31#10587622C2213yes—
32#10587616C2213yes—
33#10587615C2216yes—
34C4×C22⋊D17C2213yes—
35C4×C22⋊D17C2213yes—
36C4×C22⋊D17C2213yes—
37C4×C22⋊D17C2213yes—
38C4×C22⋊D17C2216yes—
39#10587612C2213yes—
40#10587610C2213yes—
41#10587583C2213yes—
42#10587581C2213yes—
43#10587579C2213yes—
44#10587578C2213yes—
45C4⋊C4×D17C2213yes—
46#10587576C2213yes—
47#10587574C2213yes—
48#10587573C2213yes—
49#10587571C2216yes—
50#10587570C2216yes—
51#10587569C2216yes—
52#10587565C2216yes—
53#10587562C2216yes—
54#10587561C2213yes—
55C4×D68C2213yes—
56C4×D68C2213yes—
57C4×D68C2213yes—
58#10587558C2211yescharacteristic
59#10587558C2213yes—
60C42×D17C2213yes—
61C42×D17C2213yes—
62#10587556C2213yes—
63C4×Dic34C2213yes—
64C4×Dic34C2213yes—
65C4×Dic34C2213yes—
66#41×C17—24no—
67C2×Q8×C17C2311yescharacteristic
68C2×D4×C17C2313yes—
69C22×C4×C17C2313yes—
70C22×C4×C17C2313yes—
71C22×C4×C17C2313yes—
72C2×C22⋊D17C2313yes—
73C2×C22⋊D17C2313yes—
74C2×C22⋊D17C2316yes—
75#63256—24no—
76Q8×D17—24no—
77#63252—212no—
78C2×D68C2313yes—
79C2×D68C2313yes—
80C2×C4×D17C2313yes—
81C2×C4×D17C2313yes—
82C2×C4×D17C2313yes—
83C2×C4×D17C2313yes—
84C2×C4×D17C2313yes—
85C2×C4×D17C2311yescharacteristic
86C2×Dic34C2313yes—
87C2×Dic34C2313yes—
88C4⋊C4×C17C2313yes—
89C4⋊C4×C17C2313yes—
90C22⋊C4×C17C2316yes—
91C42×C17C2313yes—
92C42×C17C2313yes—
93C42×C17C2313yes—
94C42×C17C2311yescharacteristic
95C22⋊Dic17C2313yes—
96C22⋊Dic17C2316yes—
97C22⋊Dic17C2313yes—
98#63227C2311yescharacteristic
99#63227C2311yescharacteristic
100#63227C2313yes—
101#63227C2313yes—
102#63227C2316yes—
103#63227C2316yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2313yes—
106C4⋊Dic17C2313yes—
107C4⋊Dic17C2313yes—
108#63225C2311yescharacteristic
109#63225C2311yescharacteristic
110#63225C2313yes—
111#63225C2313yes—
112#63225C2316yes—
113#63225C2316yes—
114C4×Dic17C2313yes—
115C4×Dic17C2313yes—
116C4×Dic17C2313yes—
117C4×Dic17C2313yes—
118C4×Dic17C2313yes—
119C4×Dic17C2311yescharacteristic
120C23×C17C2413yes—
121C22×D17C2413yes—
122C22×D17C2411yescharacteristic
123Q8×C17#4114yescomplemented
124D4×C17—26no—
125C2×C4×C17—26no—
126C2×C4×C17—26no—
127C2×C4×C17—43no—
128C2×C4×C17C2411yescharacteristic
129C2×C4×C17C2411yescharacteristic
130C2×C4×C17C2413yes—
131C2×C4×C17C2413yes—
132C2×C4×C17C2413yes—
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—26no—
135C22⋊D17—26no—
136C22⋊D17—46no—
137C2×Dic17C2413yes—
138C2×Dic17C2411yescharacteristic
139C2×Dic17C2411yescharacteristic
140C2×Dic17C2413yes—
141C2×Dic17C2413yes—
142C2×Dic17C2411yescharacteristic
143D68—26no—
144D68—26no—
145C4×D17—26no—
146C4×D17—26no—
147C4×D17—26no—
148C4×D17—43no—
149C4×D17—43no—
150C4×D17#4114yescomplemented
151Dic34—26no—
152Dic34—26no—
153#3480—171nomaximal, Sylow
154C22×C17—23no—
155C22×C17—23no—
156C22×C17—23no—
157C22×C17C2511yescharacteristic
158C2×D17—21no—
159C2×D17—21no—
160C2×D17—23no—
161C2×D17—23no—
162C2×D17—23no—
163C2×D17C2×#4112yes—
164C4×C17—23no—
165C4×C17—21no—
166C4×C17—23no—
167C4×C17—21no—
168C4×C17C2×#4116yes—
169C4×C17C2×#4112yes—
170Dic17—23no—
171Dic17—23no—
172Dic17—21no—
173Dic17—21no—
174Dic17—23no—
175Dic17C2×#4112yes—
176#559—173no—
177#557—171no—
178#556—173no—
179#553—173no—
180#553—173no—
181#550—171no—
182#548—171no—
183#548—171no—
184#532—173no—
185#529—173no—
186#529—173no—
187C4×#41—171no—
188C4×#41—173no—
189C4×#41D1711yescharacteristic, complemented
190C2×C4×Q8—171no—
191C2×C17—43no—
192C2×C17C22×#4111yescharacteristic
193C2×C17C22×#4111yescharacteristic
194C2×C17#58511yescharacteristic
195D17—22no—
196D17—43no—
197C2×#41—171no—
198C2×#41—173no—
199C2×#41C2×D1711yescharacteristic
200C22×Q8—171no—
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#124—173no—
210#123—176no—
211#123—176no—
212C22⋊Q8—173no—
213C22⋊Q8—173no—
214#121—173no—
215#121—173no—
216#121—173no—
217#121—173no—
218#121—176no—
219C4×Q8—171no—
220C4×Q8—171no—
221C4×Q8—173no—
222C4×Q8—173no—
223C4×Q8—173no—
224C4×Q8—173no—
225C4×Q8C2×D1711yescharacteristic
226C4×Q8C2×D1711yescharacteristic
227C4×D4—173no—
228C4×D4—173no—
229C4×D4—173no—
230C4×D4—173no—
231C4×D4—176no—
232C4×D4—173no—
233C4×D4—173no—
234C4×D4—173no—
235C4×D4—173no—
236C4×D4C2×D1713yes—
237C4×D4C2×D1713yes—
238#117—173no—
239#117—173no—
240#117—171no—
241#117—173no—
242#117C2×D1713yes—
243C2×C4⋊C4—173no—
244C2×C42—173no—
245C2×C42—173no—
246C2×C42C2×D1713yes—
247C17#348011yescharacteristic, complemented, Sylow
248#41—24no—
249#41—3412no—
250#41—344no—
251C2×Q8—173no—
252C2×Q8—173no—
253C2×Q8—344no—
254C2×Q8C22×D1711yescharacteristic
255C2×D4—173no—
256C2×D4—173no—
257C2×D4—176no—
258C2×D4—173no—
259C2×D4—173no—
260C2×D4C22×D1713yes—
261C22×C4—171no—
262C22×C4—173no—
263C22×C4—173no—
264C22×C4—173no—
265C22×C4—173no—
266C22×C4—173no—
267C22×C4C22×D1713yes—
268C22×C4C22×D1713yes—
269C22×C4C22×D1713yes—
270C4⋊C4—173no—
271C4⋊C4—176no—
272C4⋊C4—171no—
273C4⋊C4—171no—
274C4⋊C4—173no—
275C4⋊C4—173no—
276C4⋊C4—173no—
277C4⋊C4—173no—
278C4⋊C4—173no—
279C4⋊C4—176no—
280C4⋊C4C22×D1713yes—
281C4⋊C4C22×D1713yes—
282C22⋊C4—173no—
283C22⋊C4—173no—
284C22⋊C4—176no—
285C22⋊C4—171no—
286C22⋊C4—171no—
287C22⋊C4—173no—
288C22⋊C4—176no—
289C22⋊C4—173no—
290C22⋊C4—176no—
291C22⋊C4C22×D1716yes—
292C42—173no—
293C42—173no—
294C42—173no—
295C42—173no—
296C42—171no—
297C42—173no—
298C42C22×D1713yes—
299C42C22×D1713yes—
300C42C22×D1713yes—
301C42C22×D1711yescharacteristic
302C23—171no—
303C23—173no—
304C23C23×D1713yes—
305Q8—346no—
306Q8—346no—
307Q8#6325214yescomplemented
308D4—26no—
309D4—346no—
310D4—346no—
311D4—346no—
312D4—346no—
313D4—686no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—43no—
317C2×C4—171no—
318C2×C4—174no—
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×C4C23×D1713yes—
330C2×C4C23×D1711yescharacteristic
331C2×C4C23×D1711yescharacteristic
332C2×C4C23×D1713yes—
333C2×C4C23×D1713yes—
334C2×C4C23×D1711yescharacteristic
335C22—23no—
336C22—23no—
337C22—23no—
338C22—172no—
339C22—341no—
340C22—341no—
341C22—343no—
342C22—343no—
343C22—343no—
344C22C24×D1711yescharacteristic
345C4—23no—
346C4—21no—
347C4—21no—
348C4—23no—
349C4—172no—
350C4—343no—
351C4—341no—
352C4—343no—
353C4—341no—
354C4—343no—
355C4C2×#6325216yes—
356C4#41×D1712yes—
357C2—43no—
358C2—342no—
359C2—683no—
360C2C2×#41×D1711yescharacteristic
361C2C22×#6325211yescharacteristic
362C2#1176364511yescharacteristic
subgroupG/H
centerC2×C4C23×D17
commutator, socle, cosocleC22×C17C25
FrattiniC22C24×D17
FittingC4×#41×C17C2
hypercenterC4×#41D17
hypocenterC17#3480
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #11763542C2semidirect
2. #11763535C2semidirect
3. #11763527C2semidirect
4. #11763509C2semidirect
5. #11763503C2semidirect
6. #11763478C2semidirect
7. #11763471C2semidirect
8. #11763466C2semidirect
9. #11763356C2semidirect
10. #11763343C2semidirect
11. #11763341C2semidirect
12. C4×#63256C2semidirect
13. C4×#63252C2semidirect
14. C4×Q8×D17C2semidirect
15. C4×#41×C17C2semidirect
123. Q8×C17#41semidirect
150. C4×D17#41semidirect
189. C4×#41D17semidirect
247. C17#3480semidirect
307. Q8#63252semidirect

Maximal subgroups

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

subgroupindex
#117635422
#117635352
#117635272
#117635092
#117635032
#117634782
#117634712
#117634662
#117633562
#117633432
#117633412
C4×#632562
C4×#632522
C4×Q8×D172
C4×#41×C172
#348017