Finite Group Databasebeta

← #17051114

Subgroups — #17051114

Order 4352.

1430 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763481C211yesmaximal, characteristic
2Q8×C4⋊Dic17C211yesmaximal, characteristic, complemented
3#14529710C213yesmaximal, complemented
4#14529590C213yesmaximal, complemented
5#14528785C213yesmaximal, complemented
6C4×#10587492C213yesmaximal, complemented
7C2×C8×Q8×C17C211yesmaximal, characteristic, complemented
8C22×#63256—21no—
9#11763481—24no—
10#11763481C418yescomplemented
11C2×#10587560C2213yes—
12C2×#10587581—23no—
13C2×Q8×Dic17—21no—
14C2×C4×D68C2213yes—
15C2×#10587492C2213yes—
16C2×#10587492C2213yes—
17C2×#10587492C2216yes—
18#11762708—23no—
19#11762679C2213yes—
20C4×C4⋊Dic17C2213yes—
21#11762572C2213yes—
22#11762572C2211yescharacteristic
23C2×C4×Q8×C17C2211yescharacteristic
24C8×Q8×C17—24no—
25C2×C17×C4⋊C8C2213yes—
26C2×C4×C8×C17C2213yes—
27C22×Q8×C17D411yescharacteristic
28C2×#63256—28no—
29C2×#63256—44no—
30C2×#63256—44no—
31C22×D68—23no—
32C22×D68—23no—
33C22×D68C2313yes—
34C22×C4×D17—23no—
35C22×C8×C17C2313yes—
36C22×C8×C17C2311yescharacteristic
37C4×Q8×C17C2×C414yes—
38C4×Q8×C17D414yes—
39C2×C4⋊C4×C17D413yes—
40C2×C42×C17C2313yes—
41Q8×Dic17—44no—
42C2×#63227—23no—
43C2×C4⋊Dic17—23no—
44C2×C4⋊Dic17—23no—
45C2×C4⋊Dic17C2313yes—
46C2×C4⋊Dic17C2311yescharacteristic
47C2×C4⋊Dic17C2311yescharacteristic
48C2×C4×Dic17—23no—
49#10587581—224no—
50#10587560—26no—
51#10587560C2×C4112yes—
52C4×D68—26no—
53C4×D68C2×C4112yes—
54C17×C4⋊C8—26no—
55C4×C8×C17—26no—
56#10587502—23no—
57#10587492—26no—
58#10587492—26no—
59#10587492—46no—
60C23×D17—23no—
61C2×Q8×C17C22⋊C414yes—
62C2×Q8×C17D814yes—
63C2×Q8×C17SD1614yes—
64C22×C4×C17C2×D413yes—
65C22×C4×C17C2×D413yes—
66C22×C4×C17C2411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—23no—
70#63256—432no—
71C2×D68—23no—
72C2×D68—23no—
73C2×D68—23no—
74C2×D68—23no—
75C2×D68—212no—
76C2×D68—23no—
77C2×D68—212no—
78C2×D68—46no—
79C2×D68—46no—
80C2×D68—83no—
81C2×D68—83no—
82C2×D68C22×C416yes—
83C2×C4×D17—212no—
84C2×C4×D17—46no—
85C2×C4×D17—46no—
86C2×C8×C17—21no—
87C2×C8×C17—21no—
88C2×C8×C17—23no—
89C2×C8×C17—23no—
90C2×C8×C17—23no—
91C2×C8×C17#4112yescomplemented
92C4⋊C4×C17C22⋊C4112yes—
93C42×C17C22×C416yes—
94C42×C17C2×D416yes—
95#63227—212no—
96C4⋊Dic17—21no—
97C4⋊Dic17—21no—
98C4⋊Dic17—23no—
99C4⋊Dic17—23no—
100C4⋊Dic17—46no—
101C4⋊Dic17—83no—
102C4⋊Dic17C22×C412yes—
103C4⋊Dic17#4112yescomplemented
104C4×Dic17—46no—
105#22540—171nomaximal, Sylow
106C23×C17C22×D411yescharacteristic
107C22×D17—26no—
108C22×D17—43no—
109C22×D17—43no—
110C22×D17—43no—
111C22×D17—43no—
112C22×D17—43no—
113C22×D17—43no—
114Q8×C17#102116yescomplemented
115C2×C4×C17—23no—
116C2×C4×C17—23no—
117C2×C4×C17C2×C22⋊C416yes—
118C2×C4×C17C2×C22⋊C416yes—
119C2×C4×C17C2×D816yes—
120C2×C4×C17C2×SD1616yes—
121C2×C4×C17C23×C411yescharacteristic
122C2×C4×C17C22×D411yescharacteristic
123C2×C4×C17C2×#4111yescharacteristic
124C2×C4×C17C2×#4111yescharacteristic
125C2×C4×C17H(5,𝔽2)11yescharacteristic
126C2×C4×C172−1+411yescharacteristic
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—43no—
134C2×Dic17—43no—
135C2×Dic17—43no—
136D68—26no—
137D68—26no—
138D68—43no—
139D68—43no—
140D68—424no—
141D68—812no—
142C4×D17—424no—
143C8×C17—22no—
144C8×C17—43no—
145C2×#550—171no—
146C2×C8×Q8D1711yescharacteristic, complemented
147Q8×C4⋊C4—171no—
148#1982—173no—
149#1957—173no—
150#1802—173no—
151C4×#102—173no—
152C22×C17C22×C22⋊C411yescharacteristic
153C22×C17#54811yescharacteristic
154C22×C17#55011yescharacteristic
155C22×C17C22×D811yescharacteristic
156C22×C17C22×SD1611yescharacteristic
157C22×C17#57611yescharacteristic
158C22×C17#57711yescharacteristic
159C2×D17—46no—
160C2×D17—46no—
161C2×D17—46no—
162C2×D17—83no—
163C2×D17—83no—
164C2×D17—83no—
165C2×D17—83no—
166C4×C17—26no—
167C4×C17C2×#102112yes—
168C4×C17C4×#4111yescharacteristic
169C4×C17#52011yescharacteristic
170C4×C17#54811yescharacteristic
171C4×C17#55011yescharacteristic
172Dic17—42no—
173Dic17—42no—
174Dic17—83no—
175C22×#41—341no—
176#550—178no—
177#550—344no—
178C2×#127—173no—
179C2×#121—343no—
180C2×C4×Q8—341no—
181C2×C4×Q8C2×D1711yescharacteristic
182C2×C4×D4—173no—
183C8×Q8—24no—
184C2×C4⋊C8C2×D1713yes—
185C2×#102—173no—
186C2×#102—173no—
187C2×#102—176no—
188C2×C4×C8C2×D1713yes—
189#389—343no—
190#384—173no—
191C4×C4⋊C4—173no—
192#340—173no—
193#340—171no—
194C2×C17#234111yescharacteristic
195C2×C17C22×#10211yescharacteristic
196C2×C17#289711yescharacteristic
197C2×C17#335811yescharacteristic
198C2×C17#336211yescharacteristic
199C2×C17#339111yescharacteristic
200C2×C17#339211yescharacteristic
201D17—86no—
202C2×#41—348no—
203C2×#41—684no—
204C2×#41—684no—
205C22×Q8D6811yescharacteristic
206C22×D4—173no—
207C22×D4—343no—
208C22×D4—343no—
209C23×C4—343no—
210C22×C8C22×D1713yes—
211C22×C8C22×D1711yescharacteristic
212#127—1712no—
213#127—346no—
214#121—3424no—
215C4×Q8—684no—
216C4×Q8C4×D1714yes—
217C4×Q8C22⋊D1714yes—
218C4×D4—1712no—
219C4×D4—346no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—173no—
223C2×C4⋊C4—343no—
224C2×C4⋊C4—343no—
225C2×C4⋊C4D6813yes—
226C2×C22⋊C4—343no—
227C2×C42—343no—
228C2×C42C22×D1713yes—
229C4⋊C8—26no—
230#102—346no—
231#102—346no—
232#102—686no—
233C4×C8—26no—
234#95—343no—
235C17#2254011yescharacteristic, complemented, Sylow
236C24—343no—
237#41—6832no—
238C2×Q8#6321914yes—
239C2×Q8D13614yes—
240C2×Q8#6322714yes—
241C2×D4—176no—
242C2×D4—343no—
243C2×D4—343no—
244C2×D4—343no—
245C2×D4—343no—
246C2×D4—343no—
247C2×D4—3412no—
248C2×D4—3412no—
249C2×D4—686no—
250C2×D4—686no—
251C2×D4—1363no—
252C2×D4—1363no—
253C22×C4—341no—
254C22×C4—341no—
255C22×C4—3412no—
256C22×C4—343no—
257C22×C4—686no—
258C22×C4—686no—
259C22×C4C2×D6813yes—
260C22×C4C2×D6813yes—
261C22×C4C23×D1711yescharacteristic
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—23no—
265C2×C8—23no—
266C2×C8—23no—
267C2×C8#6325612yescomplemented
268C4⋊C4—172no—
269C4⋊C4—172no—
270C4⋊C4—341no—
271C4⋊C4—341no—
272C4⋊C4—343no—
273C4⋊C4—343no—
274C4⋊C4—686no—
275C4⋊C4—1363no—
276C4⋊C4#63227112yes—
277C22⋊C4—3412no—
278C42—686no—
279C42C2×C4×D1716yes—
280C42C2×C22⋊D1716yes—
281C23—346no—
282C23—683no—
283C23—683no—
284C23—683no—
285C23—683no—
286C23—683no—
287C23—683no—
288C23C22×D6811yescharacteristic
289Q8#10587492116yescomplemented
290D4—346no—
291D4—346no—
292D4—683no—
293D4—683no—
294D4—6824no—
295D4—13612no—
296C2×C4—23no—
297C2×C4—23no—
298C2×C4—342no—
299C2×C4—342no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—6824no—
305C2×C4—683no—
306C2×C4—683no—
307C2×C4—683no—
308C2×C4C2×#6321916yes—
309C2×C4C2×D13616yes—
310C2×C4C2×#6322716yes—
311C2×C4C2×#6322716yes—
312C2×C4C22×C4×D1711yescharacteristic
313C2×C4C2×#6325611yescharacteristic
314C2×C4#1058769411yescharacteristic
315C2×C4#41×D1711yescharacteristic
316C2×C4#1058769611yescharacteristic
317C2×C4C22×C22⋊D1711yescharacteristic
318C8—22no—
319C8—43no—
320C22—686no—
321C22—686no—
322C22—686no—
323C22—1363no—
324C22—1363no—
325C22—1363no—
326C22—1363no—
327C22C22×#6322711yescharacteristic
328C22#1176347111yescharacteristic
329C22#1176348111yescharacteristic
330C22C22×D13611yescharacteristic
331C22C22×#6321911yescharacteristic
332C22#1176361411yescharacteristic
333C22#1176362111yescharacteristic
334C4—26no—
335C4—682no—
336C4—682no—
337C4—1363no—
338C4C2×#10587492112yes—
339C4C4×#6325611yescharacteristic
340C4#1176329111yescharacteristic
341C4#1176347011yescharacteristic
342C4#1176347811yescharacteristic
343C2—1366no—
344C2#1453162211yescharacteristic
345C2C22×#1058749211yescharacteristic
346C2#1453731111yescharacteristic
347C2#1454142011yescharacteristic
348C2#1454145611yescharacteristic
349C2#1454175911yescharacteristic
350C2#1454176611yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C8×Q8×C17C2
socleC23×C17C22×D4
prime radicalC2×#11763481C2
cosocleC22×C4×C17C24
hypercenterC2×C8×Q8D17
hypocenterC17#22540
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic17C2semidirect
3. #14529710C2semidirect
4. #14529590C2semidirect
5. #14528785C2semidirect
6. C4×#10587492C2semidirect
7. C2×C8×Q8×C17C2semidirect
10. #11763481C4semidirect
91. C2×C8×C17#41semidirect
103. C4⋊Dic17#41semidirect
114. Q8×C17#102semidirect
146. C2×C8×Q8D17semidirect
235. C17#22540semidirect
267. C2×C8#63256semidirect
289. Q8#10587492semidirect

Maximal subgroups

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

subgroupindex
C2×#117634812
Q8×C4⋊Dic172
#145297102
#145295902
#145287852
C4×#105874922
C2×C8×Q8×C172
#2254017