Finite Group Databasebeta

← #17008233

Subgroups — #17008233

Order 4352.

1220 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539541C212yesmaximal, complemented
2#14534597C211yesmaximal, characteristic, complemented
3#14527821C214yesmaximal, complemented
4#14527753C214yesmaximal, complemented
5#14527352C212yesmaximal, complemented
6C2×#11762535C211yesmaximal, characteristic, complemented
7C2×C17×#537C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9#11763139—22no—
10#11763125C2214yescomplemented
11#11763036—24no—
12#11762960—22no—
13#11762947C2214yescomplemented
14C2×C22⋊#3622C2212yes—
15#11762748C2212yescomplemented
16#11762742—22no—
17#11762729C2212yescomplemented
18#11762682—21no—
19#11762672—21no—
20#11762669C2211yescharacteristic
21#11762543C2218yescomplemented
22#11762535C2214yescomplemented
23C22×#41×C17—21no—
24C17×#537—24no—
25C17×#537—22no—
26C17×#537C2214yes—
27C2×C17×C22⋊Q8—22no—
28C2×C17×#121—22no—
29C2×C17×#121—21no—
30C2×C17×#121C2212yes—
31C22×C4⋊C4×C17C2211yescharacteristic
32C2×#41×C17—24no—
33C2×#41×C17—24no—
34C2×#41×C17—44no—
35C2×#41×C17—44no—
36C22×Q8×C17—21no—
37C22×D4×C17—21no—
38C22×D4×C17—22no—
39C22×D4×C17—21no—
40C22×D4×C17D412yes—
41C23×C4×C17—22no—
42C23×C4×C17D411yescharacteristic
43C23×C4×C17D411yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C22×Dic34C2311yescharacteristic
46C17×C22⋊Q8—28no—
47C17×C22⋊Q8—28no—
48C17×#121—24no—
49C17×#121—28no—
50C17×#121—28no—
51C17×#121—24no—
52C17×#121—22no—
53C17×#121—22no—
54C17×#121D414yes—
55C17×#121C2314yes—
56C2×C4⋊C4×C17—21no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17—22no—
59C2×C4⋊C4×C17D412yes—
60C2×C4⋊C4×C17D412yes—
61C2×C4⋊C4×C17C2312yes—
62C2×C22⋊C4×C17—22no—
63C2×C22⋊C4×C17—22no—
64C2×C22⋊Dic17—21no—
65C2×C22⋊Dic17—21no—
66C2×#63229—24no—
67C2×C4⋊Dic17—21no—
68C2×C4⋊Dic17C2311yescharacteristic
69#10587610—22no—
70#10587610—22no—
71C2×#63225—21no—
72C2×#63225—22no—
73C22×#3622—22no—
74#10587503—24no—
75#10587502—22no—
76#10587502—22no—
77#10587502—22no—
78#10587502C2312yes—
79C22⋊#3622—22no—
80C22⋊#3622C2314yes—
81#10587481—24no—
82#10587479—24no—
83C24×C17—22no—
84C24×C17C2×D411yescharacteristic
85#41×C17—48no—
86#41×C17—416no—
87#41×C17—48no—
88C2×Q8×C17—24no—
89C2×Q8×C17—44no—
90C2×D4×C17—24no—
91C2×D4×C17—24no—
92C2×D4×C17—24no—
93C2×D4×C17—24no—
94C2×D4×C17—22no—
95C2×D4×C17—44no—
96C2×D4×C17—44no—
97C2×D4×C17—44no—
98C2×D4×C17—82no—
99C2×D4×C17—82no—
100C2×D4×C17C2×D414yes—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—22no—
104C22×C4×C17—28no—
105C22×C4×C17—22no—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17—48no—
109C22×C4×C17—81no—
110C22×C4×C17—81no—
111C22×C4×C17SD1614yescomplemented
112C22×C4×C17C2×D412yes—
113C22×C4×C17C2×D412yes—
114C22×C4×C17C2×D411yescharacteristic
115C22×C4×C17C2×D412yes—
116C22×C4×C17C2×D412yes—
117C22×C4×C17C2411yescharacteristic
118C22×Dic17—21no—
119C22×Dic17—21no—
120C22×Dic17—22no—
121C2×Dic34—22no—
122C2×Dic34—81no—
123C4⋊C4×C17—24no—
124C4⋊C4×C17—22no—
125C4⋊C4×C17—24no—
126C4⋊C4×C17—24no—
127C4⋊C4×C17C2×D414yes—
128C22⋊C4×C17—28no—
129C22⋊C4×C17—24no—
130C22⋊C4×C17—24no—
131C22⋊Dic17—41no—
132C22⋊Dic17—41no—
133C22⋊Dic17—42no—
134#63229—84no—
135C4⋊Dic17—22no—
136C4⋊Dic17—81no—
137#63225—42no—
138#63225—82no—
139C2×#3622—24no—
140C2×#3622—82no—
141#18293—171nomaximal, Sylow
142C23×C17—24no—
143C23×C17—41no—
144C23×C17—41no—
145C23×C17—44no—
146C23×C17—44no—
147C23×C17—44no—
148C23×C17#12011yescharacteristic
149C23×C17#12011yescharacteristic
150C23×C17C2×SD1612yes—
151C23×C17AGL(1,ℤ/8)12yes—
152C23×C17C22×D411yescharacteristic
153Q8×C17—48no—
154D4×C17—44no—
155D4×C17—48no—
156D4×C17—44no—
157D4×C17—44no—
158D4×C17—48no—
159D4×C17—82no—
160D4×C17—84no—
161C2×C4×C17—22no—
162C2×C4×C17—22no—
163C2×C4×C17—24no—
164C2×C4×C17—24no—
165C2×C4×C17—22no—
166C2×C4×C17—24no—
167C2×C4×C17—41no—
168C2×C4×C17—42no—
169C2×C4×C17—42no—
170C2×C4×C17—41no—
171C2×C4×C17—416no—
172C2×C4×C17—42no—
173C2×C4×C17—44no—
174C2×C4×C17—42no—
175C2×C4×C17—81no—
176C2×C4×C17#12012yes—
177C2×C4×C17C2×SD1614yes—
178C2×C4×C17C22×D412yes—
179C2×Dic17—42no—
180C2×Dic17—42no—
181C2×Dic17—42no—
182C2×Dic17—42no—
183C2×Dic17—41no—
184C2×Dic17—41no—
185C2×Dic17—42no—
186Dic34—44no—
187#3622—44no—
188C2×#537D1711yescharacteristic
189#3187—172no—
190#2627—171no—
191#1616—174no—
192#1604—174no—
193#1519—172no—
194C2×#324—171no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—42no—
202C22×C17—42no—
203C22×C17—44no—
204C22×C17—44no—
205C22×C17—44no—
206C22×C17—88no—
207C22×C17#45012yes—
208C22×C17#45012yes—
209C22×C17C2×#12011yescharacteristic
210C22×C17C22×SD1611yescharacteristic
211C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
212C4×C17—24no—
213C4×C17—22no—
214C4×C17—22no—
215C4×C17—44no—
216C4×C17—42no—
217Dic17—82no—
218Dic17—81no—
219Dic17—81no—
220C22×#41—21no—
221#537—24no—
222#537—22no—
223#537C2×D1714yes—
224C2×C22⋊Q8—22no—
225C2×C22⋊Q8—171no—
226C2×#121—22no—
227C2×#121—21no—
228C2×#121C2×D1712yes—
229C22×C4⋊C4C2×D1711yescharacteristic
230#483—342no—
231#481—174no—
232#465—344no—
233#450—342no—
234#448—174no—
235C2×C22⋊C8—172no—
236#400—172no—
237#398—342no—
238#395—172no—
239#385—341no—
240#381—341no—
241#380—171no—
242#327—178no—
243#324—174no—
244C2×C17—21no—
245C2×C17—22no—
246C2×C17—21no—
247C2×C17—22no—
248C2×C17—84no—
249C2×C17C2×#45011yescharacteristic
250C2×C17#302011yescharacteristic
251C2×C17#302611yescharacteristic
252C2×#41—24no—
253C2×#41—24no—
254C2×#41—44no—
255C2×#41—44no—
256C22×Q8—21no—
257C22×Q8—171no—
258C22×D4—21no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4C22⋊D1712yes—
262C23×C4—22no—
263C23×C4C22⋊D1711yescharacteristic
264C23×C4C22⋊D1711yescharacteristic
265C23×C4C22×D1711yescharacteristic
266C2×SD16—344no—
267C22×C8—342no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—342no—
271C22⋊Q8—342no—
272#121—24no—
273#121—28no—
274#121—28no—
275#121—24no—
276#121—22no—
277#121—22no—
278#121C22⋊D1714yes—
279#121C22×D1714yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—171no—
284C2×C4⋊C4—341no—
285C2×C4⋊C4—341no—
286C2×C4⋊C4—342no—
287C2×C4⋊C4C22⋊D1712yes—
288C2×C4⋊C4C22⋊D1712yes—
289C2×C4⋊C4C22×D1712yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—341no—
293C2×C22⋊C4—341no—
294#106—344no—
295#103—344no—
296#102—344no—
297C22⋊C8—174no—
298C22⋊C8—342no—
299#95—172no—
300#95—342no—
301#95—342no—
302#95—342no—
303C17#1829311yescharacteristic, complemented, Sylow
304C24—22no—
305C24C2×C22⋊D1711yescharacteristic
306#41—416no—
307#41—48no—
308#41—48no—
309C2×Q8—24no—
310C2×Q8—44no—
311C2×Q8—342no—
312C2×Q8—1361no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4C2×C22⋊D1714yes—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—28no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—48no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—342no—
335C22×C4—341no—
336C22×C4—341no—
337C22×C4#6322914yescomplemented
338C22×C4C2×C22⋊D1712yes—
339C22×C4C2×C22⋊D1712yes—
340C22×C4C2×C22⋊D1712yes—
341C22×C4C2×C22⋊D1712yes—
342C22×C4C2×C22⋊D1711yescharacteristic
343C22×C4C23×D1711yescharacteristic
344SD16—1364no—
345C2×C8—344no—
346C2×C8—1362no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—342no—
352C4⋊C4—682no—
353C4⋊C4—1362no—
354C4⋊C4—1361no—
355C4⋊C4C2×C22⋊D1714yes—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—682no—
360C22⋊C4—681no—
361C22⋊C4—681no—
362C23—24no—
363C23—41no—
364C23—41no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23#1058761812yes—
369C23C2×#6322912yes—
370C23#1058763911yescharacteristic
371C23#1058763911yescharacteristic
372C23C22×C22⋊D1711yescharacteristic
373Q8—48no—
374Q8—684no—
375D4—44no—
376D4—48no—
377D4—48no—
378D4—44no—
379D4—44no—
380D4—82no—
381D4—84no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—416no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—81no—
397C2×C4—682no—
398C2×C4—682no—
399C2×C4—682no—
400C2×C4—682no—
401C2×C4—681no—
402C2×C4—681no—
403C2×C4—682no—
404C2×C4C2×#6322914yes—
405C2×C4#1058763912yes—
406C2×C4C22×C22⋊D1712yes—
407C8—684no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22#1176296012yes—
421C22#1176296012yes—
422C22C2×#1058763911yescharacteristic
423C22C22×#6322911yescharacteristic
424C22C2×#1058761811yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—44no—
430C4—1362no—
431C4—1361no—
432C4—1361no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2—21no—
437C2—84no—
438C2C2×#1176296011yescharacteristic
439C2#1453814811yescharacteristic
440C2#1453818111yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#537C2
socleC22×C17C2×#120
hypercenterC2×#537D17
hypocenterC17#18293
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #14539541C2semidirect
2. #14534597C2semidirect
3. #14527821C2semidirect
4. #14527753C2semidirect
5. #14527352C2semidirect
6. C2×#11762535C2semidirect
10. #11763125C22semidirect
13. #11762947C22semidirect
15. #11762748C22semidirect
17. #11762729C22semidirect
21. #11762543C22semidirect
22. #11762535C22semidirect
111. C22×C4×C17SD16semidirect
303. C17#18293semidirect
337. C22×C4#63229semidirect

Maximal subgroups

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

subgroupindex
#145395412
#145345972
#145278212
#145277532
#145273522
C2×#117625352
C2×C17×#5372
#1829317