Finite Group Databasebeta

← #17897890

Subgroups — #17897890

Order 4352.

4874 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763645C211yesmaximal, characteristic
2#14541987C211yesmaximal, characteristic, complemented
3#14537318C218yesmaximal, complemented
4#14537311C218yesmaximal, complemented
5#14537304C216yesmaximal, complemented
6C2×#11762806C216yesmaximal, complemented
7C22×C17×#131C211yesmaximal, characteristic, complemented
8#11763645—28no—
9#11763645C4116yescomplemented
10C2×#10587694—21no—
11C2×#10587697C2214yes—
12C2×#10587696C2214yes—
13C2×#10587691—23no—
14C22×#63252C2216yes—
15C2×#41×D17—24no—
16#11763292C2218yescomplemented
17#11763287—21no—
18#11763284—23no—
19#41×Dic17—24no—
20C2×#10587612C2216yescomplemented
21#11762819C22112yescomplemented
22#11762815C22112yescomplemented
23#11762811C22124yescomplemented
24#11762806C2214yescomplemented
25#11762806C22112yescomplemented
26#11762806C22124yescomplemented
27C2×#10587487C22112yescomplemented
28C2×#10587492C22112yescomplemented
29C22×#41×C17C2211yescharacteristic
30C2×C17×#131—28no—
31C2×C17×#131C2218yescomplemented
32C22×M16×C17C2213yescomplemented
33C23×C8×C17C2213yescomplemented
34C2×#41×C17C2×C418yes—
35C2×#41×C17D418yes—
36C2×#41×C17D414yes—
37C2×#41×C17C2314yes—
38C22×Q8×C17D411yescharacteristic
39C22×D4×C17D413yes—
40C23×C4×C17C2313yes—
41C22×C22⋊D17—26no—
42#10587697—216no—
43#10587697C2×C4132yescomplemented
44#10587696—216no—
45#10587696C2×C4132yescomplemented
46#41×D17—264no—
47#10587694—216no—
48C2×#63256—24no—
49C2×Q8×D17—24no—
50#10587691—248no—
51C2×#63254—212no—
52C2×D4×D17—212no—
53C2×#63252—26no—
54C2×#63252—26no—
55C2×#63252—224no—
56C2×#63252C2×C4148yescomplemented
57C2×#63252C2314yes—
58C2×#63252C23112yes—
59C22×D68C2316yes—
60C22×C4×D17—26no—
61C22×Dic34C2316yes—
62C17×#131—232no—
63C2×M16×C17—212no—
64C2×M16×C17C23112yes—
65C22×C8×C17—212no—
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2313yes—
68C22×C8×C17C23112yes—
69C2×C22⋊Dic17—26no—
70Q8×Dic17—28no—
71D4×Dic17—224no—
72#10587612—26no—
73#10587612—26no—
74#10587612C2314yes—
75#10587612C23112yes—
76C2×C4⋊Dic17C23112yes—
77C2×C4×Dic17—26no—
78#10587492C2318yes—
79#10587492C23124yes—
80#10587487C2318yes—
81#10587487C23124yes—
82C24×C17C2×D413yes—
83C23×D17—26no—
84#41×C17C22⋊C4132yes—
85#41×C17C22×C4116yes—
86#41×C17C2×D4116yes—
87C2×Q8×C17C22⋊C418yes—
88C2×Q8×C17C2×D414yes—
89C2×D4×C17C22⋊C4124yes—
90C2×D4×C17C2×D4112yes—
91C22×C4×C17C22×C4112yes—
92C22×C4×C17C2×D416yes—
93C22×C4×C17C2×D413yes—
94C22×C4×C17C2×D4112yes—
95C22×C4×C17C2411yescharacteristic
96C22×C4×C17C2416yes—
97C2×C22⋊D17—248no—
98C2×C22⋊D17—24no—
99C2×C22⋊D17—212no—
100C22×Dic17—26no—
101C22×Dic17—212no—
102#63256—232no—
103Q8×D17—232no—
104#63254—296no—
105D4×D17—296no—
106#63252—248no—
107#63252—248no—
108#63252—28no—
109#63252—224no—
110#63252C22×C4116yes—
111#63252C22×C4148yes—
112C2×D68—26no—
113C2×D68—212no—
114C2×D68C22×C4124yes—
115C2×D68C2414yes—
116C2×C4×D17—248no—
117C2×C4×D17—24no—
118C2×C4×D17—212no—
119C2×Dic34—26no—
120C2×Dic34—212no—
121C2×Dic34C22×C4124yes—
122C2×Dic34C2414yes—
123M16×C17—224no—
124C2×C8×C17—22no—
125C2×C8×C17—26no—
126C2×C8×C17—224no—
127C2×C8×C17C2412yes—
128C2×C8×C17C2416yes—
129C22⋊Dic17—24no—
130C22⋊Dic17—212no—
131C4⋊Dic17—212no—
132C4⋊Dic17C2418yes—
133C4×Dic17—24no—
134C4×Dic17—212no—
135#60086—171nomaximal, Sylow
136C23×C17—46no—
137C23×C17C2×C22⋊C4112yes—
138C23×C17C22×D411yescharacteristic
139C23×C17C22×D416yes—
140C22×D17—24no—
141C22×D17—224no—
142C22×D17—812no—
143Q8×C17C2×C22⋊C4116yes—
144D4×C17C2×C22⋊C4148yes—
145C2×C4×C17C2×C22⋊C4124yes—
146C2×C4×C17C2×C22⋊C4112yes—
147C2×C4×C17C23×C4112yes—
148C2×C4×C17C23×C412yes—
149C2×C4×C17C22×D416yes—
150C2×C4×C17C22×D411yescharacteristic
151C2×C4×C17C22×D412yes—
152C2×C4×C17C22×D4112yes—
153C2×C4×C17C2511yescharacteristic
154C22⋊D17—216no—
155C22⋊D17—248no—
156C2×Dic17—224no—
157C2×Dic17—24no—
158C2×Dic17—28no—
159C2×Dic17—812no—
160D68—224no—
161D68—24no—
162D68C23×C418yes—
163C4×D17—216no—
164C4×D17—248no—
165Dic34—224no—
166Dic34—24no—
167Dic34C23×C418yes—
168C8×C17—22no—
169C8×C17—26no—
170C2×#585—171no—
171C22×#131D1711yescharacteristic, complemented
172#3428—171no—
173#2898—178no—
174#2897—178no—
175#2896—176no—
176C2×#416—176no—
177C22×C17—212no—
178C22×C17—22no—
179C22×C17—412no—
180C22×C17C22×C22⋊C4112yes—
181C22×C17C22×C22⋊C412yes—
182C22×C17C23×D411yescharacteristic
183C2×D17—28no—
184C2×D17—48no—
185C2×D17—824no—
186C4×C17C22×C22⋊C4112yes—
187C4×C17C22×C22⋊C412yes—
188C4×C17C24×C411yescharacteristic
189C4×C17C23×D411yescharacteristic
190Dic17—28no—
191Dic17—48no—
192#585—1716no—
193#585—348no—
194C2×2−1+4—174no—
195C2×2−1+4—341no—
196C2×H(5,𝔽2)—174no—
197C2×H(5,𝔽2)—343no—
198C22×#41—176no—
199C22×#41—344no—
200C22×#41C2×D1711yescharacteristic
201C2×#131—28no—
202C2×#131C2×D1718yescomplemented
203C22×M16C2×D1713yescomplemented
204C23×C8C2×D1713yescomplemented
205#520—178no—
206#519—341no—
207#518—343no—
208C4×#41—344no—
209C2×#117—176no—
210#419—1712no—
211#418—1712no—
212#417—1724no—
213#416—174no—
214#416—1712no—
215#416—1724no—
216C2×#103—1712no—
217C2×#102—1712no—
218C2×C17—212no—
219C2×C17—22no—
220C2×C17C23×C22⋊C411yescharacteristic
221C2×C17#358412yes—
222D17—48no—
2232−1+4—1732no—
2242−1+4—3416no—
2252−1+4—3416no—
226H(5,𝔽2)—1732no—
227H(5,𝔽2)—3416no—
228H(5,𝔽2)—3448no—
229C2×#41—1748no—
230C2×#41—174no—
231C2×#41—1712no—
232C2×#41—3464no—
233C2×#41—344no—
234C2×#41—3412no—
235C2×#41—346no—
236C2×#41—346no—
237C2×#41—3424no—
238C2×#41C4×D1718yes—
239C2×#41D6814yes—
240C2×#41C22⋊D1718yes—
241C2×#41C22×D1714yes—
242C22×Q8—176no—
243C22×Q8—344no—
244C22×Q8D6811yescharacteristic
245C22×D4—176no—
246C22×D4—346no—
247C22×D4—3412no—
248C22×D4D6813yes—
249C23×C4—346no—
250C23×C4C22×D1713yes—
251#131—232no—
252C2×M16—212no—
253C2×M16C22×D17112yes—
254C22×C8—212no—
255C22×C8C22×D1711yescharacteristic
256C22×C8C22×D1713yes—
257C22×C8C22×D17112yes—
258C4×Q8—348no—
259C4×D4—3424no—
260#117—174no—
261#117—1712no—
262#117—346no—
263#117—346no—
264C2×C4⋊C4—1712no—
265C2×C22⋊C4—346no—
266C2×C42—346no—
267#103—178no—
268#103—1724no—
269#102—178no—
270#102—1724no—
271C17#6008611yescharacteristic, complemented, Sylow
272C24—346no—
273C24C2×D6813yes—
274#41—1716no—
275#41—1748no—
276#41—3432no—
277#41—3496no—
278#41—348no—
279#41—3424no—
280#41—3448no—
281#41—3448no—
282#41#63227132yes—
283#41C2×C4×D17116yes—
284#41C2×C22⋊D17116yes—
285C2×Q8—1724no—
286C2×Q8—174no—
287C2×Q8—3432no—
288C2×Q8—346no—
289C2×Q8—3412no—
290C2×Q8#6322718yes—
291C2×Q8C2×D6814yes—
292C2×D4—1724no—
293C2×D4—174no—
294C2×D4—344no—
295C2×D4—3412no—
296C2×D4—3448no—
297C2×D4—3496no—
298C2×D4—346no—
299C2×D4—3412no—
300C2×D4#63227124yes—
301C2×D4C2×D68112yes—
302C22×C4—346no—
303C22×C4—3412no—
304C22×C4—344no—
305C22×C4—3448no—
306C22×C4—3412no—
307C22×C4C2×C4×D17112yes—
308C22×C4C2×D6813yes—
309C22×C4C2×D6816yes—
310C22×C4C2×C22⋊D17112yes—
311C22×C4C23×D1716yes—
312C22×C4C23×D1711yescharacteristic
313M16—224no—
314C2×C8—22no—
315C2×C8—26no—
316C2×C8—224no—
317C2×C8C23×D1712yes—
318C2×C8C23×D1716yes—
319C4⋊C4—178no—
320C4⋊C4—3412no—
321C22⋊C4—344no—
322C22⋊C4—3412no—
323C42—3412no—
324C42—344no—
325C23—46no—
326C23—344no—
327C23—3424no—
328C23—13612no—
329C23C2×#63227112yes—
330C23C22×D6811yescharacteristic
331C23C22×D6816yes—
332Q8—178no—
333Q8—3424no—
334Q8—344no—
335Q8C2×#63227116yes—
336D4—178no—
337D4—3448no—
338D4—3416no—
339D4—3424no—
340D4—344no—
341D4C2×#63227148yes—
342C2×C4—3424no—
343C2×C4—344no—
344C2×C4—3416no—
345C2×C4—3448no—
346C2×C4—348no—
347C2×C4—13612no—
348C2×C4C2×#63227124yes—
349C2×C4C2×#63227112yes—
350C2×C4C22×C4×D1712yes—
351C2×C4C22×C4×D17112yes—
352C2×C4C22×D6816yes—
353C2×C4C22×D6811yescharacteristic
354C2×C4C22×C22⋊D1712yes—
355C2×C4C22×C22⋊D17112yes—
356C2×C4C24×D1711yescharacteristic
357C8—22no—
358C8—26no—
359C22—212no—
360C22—22no—
361C22—412no—
362C22—348no—
363C22—688no—
364C22—13624no—
365C22C22×#63227112yes—
366C22C22×#6322712yes—
367C22C23×D6811yescharacteristic
368C4—348no—
369C4—688no—
370C4C22×#6322712yes—
371C4C22×#63227112yes—
372C4C23×C4×D1711yescharacteristic
373C4C23×C22⋊D1711yescharacteristic
374C2—22no—
375C2—212no—
376C2—688no—
377C2C23×#6322711yescharacteristic
378C2#1454355812yes—
subgroupG/H
centerC2×C4C22×D68
commutatorC4×C17C24×C4
FrattiniC2×C4C24×D17
FittingC22×C17×#131C2
socleC22×C17C23×D4
prime radicalC2×#11763645C2
cosocleC2×C4×C17C25
hypercenterC22×#131D17
hypocenterC17#60086
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ C22×#131

Complements

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

NH ≅ G/Ntype
2. #14541987C2semidirect
3. #14537318C2semidirect
4. #14537311C2semidirect
5. #14537304C2semidirect
6. C2×#11762806C2semidirect
7. C22×C17×#131C2semidirect
9. #11763645C4semidirect
16. #11763292C22semidirect
20. C2×#10587612C22semidirect
21. #11762819C22semidirect
22. #11762815C22semidirect
23. #11762811C22semidirect
24. #11762806C22semidirect
25. #11762806C22semidirect
26. #11762806C22semidirect
27. C2×#10587487C22semidirect
28. C2×#10587492C22semidirect
31. C2×C17×#131C22semidirect
32. C22×M16×C17C22semidirect
33. C23×C8×C17C22semidirect
43. #10587697C2×C4semidirect
45. #10587696C2×C4semidirect
56. C2×#63252C2×C4semidirect
171. C22×#131D17semidirect
202. C2×#131C2×D17semidirect
203. C22×M16C2×D17semidirect
204. C23×C8C2×D17semidirect
271. C17#60086semidirect

Maximal subgroups

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

subgroupindex
C2×#117636452
#145419872
#145373182
#145373112
#145373042
C2×#117628062
C22×C17×#1312
#6008617