Finite Group Databasebeta

← #17008178

Subgroups — #17008178

Order 4352.

1064 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763435C211yesmaximal, characteristic
2#14539556C212yesmaximal, complemented
3#14527808C214yesmaximal, complemented
4#14527740C214yesmaximal, complemented
5#14527355C212yesmaximal, complemented
6C2×#11762533C211yesmaximal, characteristic
7C17×#2626C211yesmaximal, characteristic
8#11763435—22no—
9#11763435—24no—
10#11763435C2214yes—
11C2×#10587610—21no—
12C2×#10587610C2212yes—
13C2×#10587564—22no—
14C2×#10587564—22no—
15C22×C22⋊Dic17—21no—
16#11763144C2214yescomplemented
17#11763142—22no—
18#11763048—24no—
19#11762965C2214yescomplemented
20#11762964—22no—
21C2×C22⋊#3622C2212yes—
22#11762545C2218yescomplemented
23#11762533C2214yes—
24C2×C17×C22⋊Q8C2211yescharacteristic
25C22×C22⋊C4×C17C2211yescharacteristic
26C17×#399C2212yes—
27C17×#396—22no—
28C17×#393C2212yes—
29C17×#385—21no—
30C17×#380C2211yescharacteristic
31C17×#379—21no—
32C25×C17D411yescharacteristic
33C22×Q8×C17C2311yescharacteristic
34C23×C4×C17D411yescharacteristic
35C23×C4×C17C2311yescharacteristic
36C23×Dic17—21no—
37C23×Dic17—21no—
38C22×Dic34—21no—
39C22×Dic34D412yes—
40C17×C22⋊Q8—22no—
41C17×C22⋊Q8—22no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—22no—
45C2×C4⋊C4×C17C2311yescharacteristic
46C2×C22⋊C4×C17—22no—
47C2×C22⋊C4×C17—22no—
48C2×C22⋊C4×C17—21no—
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—22no—
51C2×C22⋊C4×C17—22no—
52C2×C22⋊C4×C17D412yes—
53C2×C22⋊C4×C17C2312yes—
54C2×C22⋊Dic17—22no—
55C2×C22⋊Dic17—24no—
56C2×C22⋊Dic17—22no—
57C2×C22⋊Dic17—24no—
58C2×#63231—24no—
59C2×C4⋊Dic17—21no—
60C2×C4⋊Dic17D412yes—
61#10587610—24no—
62#10587610—24no—
63#10587610—22no—
64#10587610—22no—
65#10587610D414yes—
66#10587610C2314yes—
67C2×#63225—22no—
68C2×#63225—22no—
69C22×#3622—22no—
70#10587564—28no—
71#10587564—28no—
72#10587564—28no—
73#10587564—28no—
74C17×#95—22no—
75C17×#95C2312yes—
76#10587506—24no—
77C22⋊#3622—22no—
78C22⋊#3622C2314yes—
79#10587481—24no—
80#10587478—24no—
81C24×C17—21no—
82C24×C17—21no—
83C24×C17—22no—
84C24×C17—81no—
85C24×C17—81no—
86C24×C17Q1614yescomplemented
87C24×C17C2×D412yes—
88C24×C17C2×D411yescharacteristic
89C2×Q8×C17—22no—
90C2×Q8×C17—81no—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17—21no—
94C22×C4×C17—22no—
95C22×C4×C17—21no—
96C22×C4×C17—81no—
97C22×C4×C17—81no—
98C22×C4×C17C2×D412yes—
99C22×C4×C17C2×D412yes—
100C22×C4×C17C2×D412yes—
101C22×C4×C17C2×D411yescharacteristic
102C22×C4×C17C2411yescharacteristic
103C22×Dic17—22no—
104C22×Dic17—22no—
105C22×Dic17—24no—
106C22×Dic17—24no—
107C22×Dic17—82no—
108C22×Dic17—82no—
109C2×Dic34—24no—
110C2×Dic34—22no—
111C2×Dic34—82no—
112C2×Dic34—82no—
113C2×Dic34C2×D414yes—
114C4⋊C4×C17—22no—
115C4⋊C4×C17—42no—
116C4⋊C4×C17—81no—
117C4⋊C4×C17—82no—
118C22⋊C4×C17—24no—
119C22⋊C4×C17—24no—
120C22⋊C4×C17—41no—
121C22⋊C4×C17—41no—
122C22⋊C4×C17—42no—
123C22⋊C4×C17—82no—
124C22⋊C4×C17—82no—
125C22⋊Dic17—28no—
126C22⋊Dic17—24no—
127C22⋊Dic17—24no—
128#63231—84no—
129C4⋊Dic17—24no—
130C4⋊Dic17—22no—
131C4⋊Dic17C2×D414yes—
132#63225—28no—
133#63225—24no—
134#63225—24no—
135C2×#3622—24no—
136C2×#3622—82no—
137#18287—171nomaximal, Sylow
138C23×C17—22no—
139C23×C17—24no—
140C23×C17—22no—
141C23×C17—24no—
142C23×C17—41no—
143C23×C17—41no—
144C23×C17—42no—
145C23×C17—42no—
146C23×C17—42no—
147C23×C17—42no—
148C23×C17—81no—
149C23×C17—81no—
150C23×C17—84no—
151C23×C17—84no—
152C23×C17#12011yescharacteristic
153C23×C17#12011yescharacteristic
154C23×C17C2×Q1614yes—
155C23×C17C2×Q1612yes—
156C23×C17#13712yes—
157C23×C17C22×D411yescharacteristic
158Q8×C17—44no—
159C2×C4×C17—22no—
160C2×C4×C17—41no—
161C2×C4×C17—42no—
162C2×C4×C17—42no—
163C2×C4×C17—41no—
164C2×C4×C17—42no—
165C2×C4×C17—42no—
166C2×C4×C17—42no—
167C2×C4×C17—42no—
168C2×C4×C17—42no—
169C2×C4×C17—42no—
170C2×C4×C17—41no—
171C2×C4×C17—41no—
172C2×C4×C17—42no—
173C2×C4×C17—81no—
174C2×C4×C17#12012yes—
175C2×C4×C17C22×D412yes—
176C2×Dic17—24no—
177C2×Dic17—24no—
178C2×Dic17—44no—
179C2×Dic17—44no—
180C2×Dic17—84no—
181C2×Dic17—84no—
182Dic34—44no—
183Dic34—82no—
184Dic34—84no—
185#3622—44no—
186C2×#543—171no—
187#3189—172no—
188#2626D1711yescharacteristic
189#1614—174no—
190#1602—174no—
191#1520—172no—
192C2×#323—171no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—24no—
200C22×C17—42no—
201C22×C17—42no—
202C22×C17—44no—
203C22×C17—48no—
204C22×C17—42no—
205C22×C17—42no—
206C22×C17—82no—
207C22×C17—84no—
208C22×C17#45112yes—
209C22×C17#45112yes—
210C22×C17C2×#12011yescharacteristic
211C22×C17C22×Q1611yescharacteristic
212C22×C17C2×#13711yescharacteristic
213C4×C17—22no—
214C4×C17—22no—
215C4×C17—42no—
216C4×C17—82no—
217C4×C17—81no—
218C4×C17—81no—
219Dic17—44no—
220Dic17—44no—
221#543—174no—
222#543—344no—
223#543—342no—
224C2×C22⋊Q8—172no—
225C2×C22⋊Q8—342no—
226C2×C22⋊Q8—342no—
227C2×C22⋊Q8—341no—
228C2×C22⋊Q8C2×D1711yescharacteristic
229C22×C22⋊C4—341no—
230C22×C22⋊C4C2×D1711yescharacteristic
231#484—174no—
232#484—342no—
233#467—344no—
234#451—174no—
235#451—342no—
236C2×C22⋊C8—172no—
237#399C2×D1712yes—
238#396—22no—
239#393C2×D1712yes—
240#385—21no—
241#380C2×D1711yescharacteristic
242#379—21no—
243#328—178no—
244#323—174no—
245C2×C17—21no—
246C2×C17—21no—
247C2×C17—22no—
248C2×C17—22no—
249C2×C17—44no—
250C2×C17C2×#45111yescharacteristic
251C2×C17#301911yescharacteristic
252C2×C17#302611yescharacteristic
253C25C22⋊D1711yescharacteristic
254C22×Q8—172no—
255C22×Q8—341no—
256C22×Q8C22×D1711yescharacteristic
257C23×C4—341no—
258C23×C4—341no—
259C23×C4C22⋊D1711yescharacteristic
260C23×C4C22×D1711yescharacteristic
261C2×Q16—344no—
262C22×C8—342no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—174no—
266C22⋊Q8—174no—
267C22⋊Q8—348no—
268C22⋊Q8—348no—
269C22⋊Q8—348no—
270C22⋊Q8—348no—
271C22⋊Q8—344no—
272C22⋊Q8—342no—
273C22⋊Q8—342no—
274C22⋊Q8—344no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—172no—
279C2×C4⋊C4—342no—
280C2×C4⋊C4—341no—
281C2×C4⋊C4—342no—
282C2×C4⋊C4C22×D1711yescharacteristic
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—342no—
290C2×C22⋊C4—344no—
291C2×C22⋊C4—342no—
292C2×C22⋊C4—344no—
293C2×C22⋊C4C22⋊D1712yes—
294C2×C22⋊C4C22×D1712yes—
295#107—344no—
296#103—344no—
297#103—344no—
298C22⋊C8—174no—
299C22⋊C8—342no—
300#95—22no—
301#95C22×D1712yes—
302C17#1828711yescharacteristic, complemented, Sylow
303C24—21no—
304C24—21no—
305C24—22no—
306C24—81no—
307C24—81no—
308C24#6323114yescomplemented
309C24C2×C22⋊D1712yes—
310C24C2×C22⋊D1711yescharacteristic
311C2×Q8—22no—
312C2×Q8—81no—
313C2×Q8—174no—
314C2×Q8—344no—
315C2×Q8—342no—
316C2×Q8—1362no—
317C2×Q8—1362no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—342no—
326C22×C4—342no—
327C22×C4—344no—
328C22×C4—344no—
329C22×C4—1362no—
330C22×C4—1362no—
331C22×C4D4×D1712yes—
332C22×C4D4×D1712yes—
333C22×C4C2×C22⋊D1712yes—
334C22×C4C2×C22⋊D1711yescharacteristic
335C22×C4C23×D1711yescharacteristic
336Q16—1364no—
337C2×C8—344no—
338C2×C8—1362no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—81no—
342C4⋊C4—82no—
343C4⋊C4—174no—
344C4⋊C4—348no—
345C4⋊C4—344no—
346C4⋊C4—344no—
347C4⋊C4—342no—
348C4⋊C4—344no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—348no—
357C22⋊C4—344no—
358C22⋊C4—344no—
359C23—24no—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—84no—
372C23—84no—
373C23#1058762311yescharacteristic
374C23#1058762311yescharacteristic
375C23C2×#6323114yes—
376C23C2×#6323112yes—
377C23#1058763712yes—
378C23C22×C22⋊D1711yescharacteristic
379Q8—44no—
380Q8—684no—
381Q8—1362no—
382Q8—1364no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—81no—
398C2×C4—344no—
399C2×C4—344no—
400C2×C4—684no—
401C2×C4—684no—
402C2×C4—1364no—
403C2×C4—1364no—
404C2×C4#1058762312yes—
405C2×C4C2×D4×D1712yes—
406C8—684no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—48no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—84no—
422C22#1176296412yes—
423C22#1176296412yes—
424C22C2×#1058762311yescharacteristic
425C22C22×#6323111yescharacteristic
426C22C2×#1058763711yescharacteristic
427C4—22no—
428C4—22no—
429C4—42no—
430C4—82no—
431C4—81no—
432C4—81no—
433C4—684no—
434C4—684no—
435C2—21no—
436C2—22no—
437C2—21no—
438C2—22no—
439C2—44no—
440C2C2×#1176296411yescharacteristic
441C2#1453814011yescharacteristic
442C2#1453818011yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2626C2
socleC22×C17C2×#120
prime radicalC25×C17D4
hypercenter#2626D17
hypocenterC17#18287
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #14539556C2semidirect
3. #14527808C2semidirect
4. #14527740C2semidirect
5. #14527355C2semidirect
16. #11763144C22semidirect
19. #11762965C22semidirect
22. #11762545C22semidirect
86. C24×C17Q16semidirect
302. C17#18287semidirect
308. C24#63231semidirect

Maximal subgroups

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

subgroupindex
C2×#117634352
#145395562
#145278082
#145277402
#145273552
C2×#117625332
C17×#26262
#1828717