Finite Group Databasebeta

← #17008239

Subgroups — #17008239

Order 4352.

1002 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1002 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×#11763440C211yesmaximal, characteristic, complemented
2#14539541C212yesmaximal
3#14527791C214yesmaximal, complemented
4#14527723C214yesmaximal, complemented
5#14527355C212yesmaximal, complemented
6C2×#11762535C211yesmaximal, characteristic, complemented
7C17×#2618C211yesmaximal, characteristic
8#11763440—22no—
9#11763440—24no—
10#11763440C2214yes—
11C2×#10587574—22no—
12C2×#10587576—22no—
13C2×#10587610—21no—
14C2×#10587610C2212yes—
15C2×#10587612—21no—
16#11763139C2214yes—
17#11763125—22no—
18#11763036—24no—
19#11762960C2214yes—
20#11762947—22no—
21C2×C22⋊#3622C2212yes—
22#11762545C2218yes—
23#11762535C2214yes—
24C2×C17×#123—21no—
25C2×C17×#121C2211yescharacteristic
26C22×C4⋊C4×C17C2211yescharacteristic
27C17×#398C2212yes—
28C17×#396—22no—
29C17×#394C2212yes—
30C17×#386—21no—
31C17×#380C2211yescharacteristic
32C22×D4×C17—21no—
33C22×D4×C17C2311yescharacteristic
34C23×C4×C17D411yescharacteristic
35C23×C4×C17D411yescharacteristic
36C23×C4×C17C2311yescharacteristic
37C22×Dic34—21no—
38C22×Dic34D412yes—
39C17×#123—44no—
40C17×#121—22no—
41C17×#121—22no—
42C2×C4⋊C4×C17—22no—
43C2×C4⋊C4×C17—22no—
44C2×C4⋊C4×C17—22no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17D412yes—
47C2×C4⋊C4×C17C2311yescharacteristic
48C2×C4⋊C4×C17C2312yes—
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—21no—
51C2×C22⋊C4×C17—22no—
52C2×C22⋊C4×C17—22no—
53C2×C22⋊Dic17—22no—
54C2×#63229—24no—
55#10587612—24no—
56#10587612—24no—
57C2×C4⋊Dic17—21no—
58C2×C4⋊Dic17—21no—
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—
69C2×C4×Dic17—22no—
70C22×#3622—22no—
71#10587576—28no—
72#10587576—28no—
73#10587574—28no—
74#10587574—28no—
75C17×#95—22no—
76C17×#95C2312yes—
77#10587503—24no—
78C22⋊#3622—22no—
79C22⋊#3622C2314yes—
80#10587481—24no—
81#10587479—24no—
82C24×C17—21no—
83C24×C17C2×D411yescharacteristic
84C2×D4×C17—22no—
85C2×D4×C17—41no—
86C2×D4×C17—41no—
87C2×D4×C17—42no—
88C2×D4×C17—42no—
89C2×D4×C17—81no—
90C22×C4×C17—21no—
91C22×C4×C17—21no—
92C22×C4×C17—22no—
93C22×C4×C17—21no—
94C22×C4×C17—21no—
95C22×C4×C17—21no—
96C22×C4×C17—22no—
97C22×C4×C17—81no—
98C22×C4×C17—81no—
99C22×C4×C17SD1614yescomplemented
100C22×C4×C17C2×D412yes—
101C22×C4×C17C2×D412yes—
102C22×C4×C17C2×D412yes—
103C22×C4×C17C2×D412yes—
104C22×C4×C17C2×D411yescharacteristic
105C22×C4×C17C2411yescharacteristic
106C22×Dic17—22no—
107C22×Dic17—22no—
108C2×Dic34—24no—
109C2×Dic34—22no—
110C2×Dic34—82no—
111C2×Dic34—82no—
112C2×Dic34C2×D414yes—
113C4⋊C4×C17—24no—
114C4⋊C4×C17—24no—
115C4⋊C4×C17—22no—
116C4⋊C4×C17—44no—
117C4⋊C4×C17—81no—
118C22⋊C4×C17—42no—
119C22⋊C4×C17—42no—
120C22⋊C4×C17—44no—
121C22⋊C4×C17—82no—
122C22⋊Dic17—24no—
123C22⋊Dic17—24no—
124#63229—84no—
125C4⋊Dic17—24no—
126C4⋊Dic17—24no—
127C4⋊Dic17—24no—
128C4⋊Dic17—22no—
129C4⋊Dic17C2×D414yes—
130#63225—28no—
131#63225—24no—
132#63225—24no—
133C4×Dic17—28no—
134C2×#3622—24no—
135C2×#3622—82no—
136#18294—171nomaximal, Sylow
137C23×C17—41no—
138C23×C17—41no—
139C23×C17—42no—
140C23×C17—42no—
141C23×C17—41no—
142C23×C17—41no—
143C23×C17—41no—
144C23×C17#12011yescharacteristic
145C23×C17#12011yescharacteristic
146C23×C17C2×SD1612yes—
147C23×C17#13712yes—
148C23×C17C22×D411yescharacteristic
149D4×C17—44no—
150D4×C17—82no—
151D4×C17—82no—
152C2×C4×C17—22no—
153C2×C4×C17—22no—
154C2×C4×C17—24no—
155C2×C4×C17—24no—
156C2×C4×C17—22no—
157C2×C4×C17—41no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17—41no—
161C2×C4×C17—42no—
162C2×C4×C17—42no—
163C2×C4×C17—42no—
164C2×C4×C17—42no—
165C2×C4×C17—41no—
166C2×C4×C17—42no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17—81no—
170C2×C4×C17#12012yes—
171C2×C4×C17C2×SD1614yes—
172C2×C4×C17C22×D412yes—
173C2×Dic17—24no—
174C2×Dic17—24no—
175C2×Dic17—44no—
176C2×Dic17—44no—
177Dic34—44no—
178Dic34—82no—
179Dic34—84no—
180#3622—44no—
181C2×#544—171no—
182#3187—172no—
183#2618D1711yescharacteristic
184#1611—174no—
185#1599—174no—
186#1520—172no—
187C2×#324—171no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—42no—
195C22×C17—42no—
196C22×C17—82no—
197C22×C17—82no—
198C22×C17—81no—
199C22×C17—81no—
200C22×C17—81no—
201C22×C17#44812yes—
202C22×C17#44812yes—
203C22×C17C2×#12011yescharacteristic
204C22×C17C22×SD1611yescharacteristic
205C22×C17C2×#13711yescharacteristic
206C4×C17—24no—
207C4×C17—22no—
208C4×C17—22no—
209C4×C17—42no—
210C4×C17—82no—
211C4×C17—81no—
212Dic17—44no—
213Dic17—44no—
214#544—174no—
215#544—344no—
216#544—342no—
217C2×C4⋊Q8—342no—
218C2×#125—342no—
219C2×#123—21no—
220C2×C22⋊Q8—172no—
221C2×C22⋊Q8—341no—
222C2×#121C2×D1711yescharacteristic
223C2×#117—341no—
224C22×C4⋊C4C2×D1711yescharacteristic
225#483—174no—
226#481—342no—
227#465—344no—
228#450—174no—
229#448—342no—
230C2×C22⋊C8—172no—
231#398C2×D1712yes—
232#396—22no—
233#394C2×D1712yes—
234#386—21no—
235#380C2×D1711yescharacteristic
236#328—178no—
237#324—174no—
238C2×C17—21no—
239C2×C17—21no—
240C2×C17—22no—
241C2×C17—22no—
242C2×C17—161no—
243C2×C17C2×#44811yescharacteristic
244C2×C17#302011yescharacteristic
245C2×C17#302611yescharacteristic
246C22×Q8—172no—
247C22×Q8—341no—
248C22×D4—21no—
249C22×D4C22×D1711yescharacteristic
250C23×C4C22⋊D1711yescharacteristic
251C23×C4C22⋊D1711yescharacteristic
252C23×C4C22×D1711yescharacteristic
253C2×SD16—344no—
254C22×C8—342no—
255C4⋊Q8—348no—
256C4⋊Q8—348no—
257#125—348no—
258#125—348no—
259#123—44no—
260C22⋊Q8—174no—
261C22⋊Q8—174no—
262C22⋊Q8—344no—
263C22⋊Q8—342no—
264C22⋊Q8—342no—
265C22⋊Q8—344no—
266#121—22no—
267#121—22no—
268#117—344no—
269#117—344no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—172no—
275C2×C4⋊C4—341no—
276C2×C4⋊C4—341no—
277C2×C4⋊C4—341no—
278C2×C4⋊C4—342no—
279C2×C4⋊C4—342no—
280C2×C4⋊C4C22⋊D1712yes—
281C2×C4⋊C4C22×D1711yescharacteristic
282C2×C4⋊C4C22×D1712yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—342no—
288C2×C42—342no—
289#106—344no—
290#103—344no—
291#102—344no—
292C22⋊C8—174no—
293C22⋊C8—342no—
294#95—22no—
295#95C22×D1712yes—
296C17#1829411yescharacteristic, complemented, Sylow
297C24—21no—
298C24C2×C22⋊D1711yescharacteristic
299C2×Q8—174no—
300C2×Q8—344no—
301C2×Q8—342no—
302C2×Q8—1362no—
303C2×Q8—1362no—
304C2×D4—22no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—42no—
309C2×D4—81no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—342no—
320C22×C4—342no—
321C22×C4#6322914yescomplemented
322C22×C4D4×D1712yes—
323C22×C4D4×D1712yes—
324C22×C4C2×C22⋊D1712yes—
325C22×C4C2×C22⋊D1712yes—
326C22×C4C2×C22⋊D1711yescharacteristic
327C22×C4C23×D1711yescharacteristic
328SD16—1364no—
329C2×C8—344no—
330C2×C8—1362no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—44no—
335C4⋊C4—81no—
336C4⋊C4—174no—
337C4⋊C4—344no—
338C4⋊C4—344no—
339C4⋊C4—344no—
340C4⋊C4—348no—
341C4⋊C4—344no—
342C4⋊C4—342no—
343C4⋊C4—344no—
344C22⋊C4—42no—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—82no—
348C22⋊C4—344no—
349C22⋊C4—344no—
350C42—348no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23C2×#6322912yes—
359C23#1058762311yescharacteristic
360C23#1058762311yescharacteristic
361C23#1058763712yes—
362C23C22×C22⋊D1711yescharacteristic
363Q8—684no—
364Q8—1362no—
365Q8—1364no—
366D4—44no—
367D4—82no—
368D4—82no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—344no—
388C2×C4—344no—
389C2×C4—684no—
390C2×C4—684no—
391C2×C4C2×#6322914yes—
392C2×C4#1058762312yes—
393C2×C4C2×D4×D1712yes—
394C8—684no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#1176294712yes—
409C22#1176294712yes—
410C22C2×#1058762311yescharacteristic
411C22C22×#6322911yescharacteristic
412C22C2×#1058763711yescharacteristic
413C4—22no—
414C4—22no—
415C4—24no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—684no—
420C4—684no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—161no—
426C2C2×#1176294711yescharacteristic
427C2#1453814711yescharacteristic
428C2#1453818011yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2618C2
socleC22×C17C2×#120
prime radicalC2×C17×#121C22
hypercenter#2618D17
hypocenterC17#18294
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#11763440C2semidirect
3. #14527791C2semidirect
4. #14527723C2semidirect
5. #14527355C2semidirect
6. C2×#11762535C2semidirect
99. C22×C4×C17SD16semidirect
296. C17#18294semidirect
321. C22×C4#63229semidirect

Maximal subgroups

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

subgroupindex
C2×#117634402
#145395412
#145277912
#145277232
#145273552
C2×#117625352
C17×#26182
#1829417