Finite Group Databasebeta

← #16953019

Subgroups — #16953019

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587497C211yesmaximal, characteristic, complemented
2C2×#11763045C211yesmaximal, characteristic, complemented
3C2×#11762610C211yesmaximal, characteristic, complemented
4C2×#11762609C211yesmaximal, characteristic, complemented
5C2×C22⋊#63214C211yesmaximal, characteristic, complemented
6#14526875C211yesmaximal, characteristic, complemented
7C2×C17×#466C211yesmaximal, characteristic, complemented
8C22×D136C2211yescharacteristic
9C2×#10587616—21no—
10C2×#10587497—22no—
11C2×#10587497—22no—
12C2×#10587497—22no—
13C22×#63214C2211yescharacteristic
14C22×#63214C2211yescharacteristic
15#11763045—22no—
16#11763045—22no—
17C2×#10587489C2211yescharacteristic
18C2×#10587492—21no—
19#11762610—22no—
20#11762609—22no—
21C22⋊#63214—22no—
22C22×D8×C17C2211yescharacteristic
23C23×C8×C17C2211yescharacteristic
24C2×C17×#121—21no—
25C17×#466—22no—
26C17×#466—22no—
27C2×C17×#107C2211yescharacteristic
28C2×C17×#102—21no—
29C22×D4×C17—21no—
30C22×D4×C17—41no—
31C23×C4×C17D411yescharacteristic
32C22×C22⋊D17—41no—
33C22×D68—21no—
34C2×D8×C17—21no—
35C2×D8×C17—21no—
36C2×D8×C17—21no—
37C2×D8×C17—21no—
38C2×D8×C17—21no—
39C2×D8×C17D412yescomplemented
40C22×C8×C17—21no—
41C22×C8×C17—21no—
42C22×C8×C17—21no—
43C22×C8×C17—21no—
44C22×C8×C17—21no—
45C22×C8×C17—21no—
46C22×C8×C17D411yescharacteristic
47C22×C8×C17C2311yescharacteristic
48C17×#121—24no—
49C17×#121—42no—
50C2×C4⋊C4×C17—21no—
51C2×C22⋊C4×C17—41no—
52#10587616—24no—
53#10587616—42no—
54C2×#63227—41no—
55C2×C4⋊Dic17—21no—
56C2×D136—21no—
57C2×D136—21no—
58C2×D136—21no—
59C2×D136—21no—
60C2×D136—21no—
61C2×D136D412yescomplemented
62C17×#107—21no—
63C17×#107D412yescomplemented
64C17×#102—24no—
65#10587497—28no—
66#10587492—24no—
67#10587489—21no—
68#10587489D412yescomplemented
69C2×#63214—21no—
70C2×#63214—21no—
71C2×#63214—21no—
72C2×#63214—21no—
73C2×#63214—21no—
74C2×#63214D412yescomplemented
75C24×C17—41no—
76C24×C17D811yescharacteristic
77C23×D17—41no—
78C2×D4×C17—22no—
79C2×D4×C17—44no—
80C2×D4×C17—41no—
81C2×D4×C17—41no—
82C2×D4×C17—41no—
83C2×D4×C17—41no—
84C2×D4×C17—41no—
85C2×D4×C17—82no—
86C2×D4×C17—82no—
87C22×C4×C17—21no—
88C22×C4×C17—21no—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17—41no—
94C22×C4×C17D811yescharacteristic
95C22×C4×C17C2×D411yescharacteristic
96C2×C22⋊D17—44no—
97C2×C22⋊D17—82no—
98C2×C22⋊D17—82no—
99C22×Dic17—41no—
100C2×D68—22no—
101C2×D68—41no—
102C2×D68—41no—
103C2×D68—41no—
104C2×D68—41no—
105C2×D68—41no—
106D8×C17—22no—
107D8×C17—22no—
108D8×C17—41no—
109D8×C17—41no—
110C2×C8×C17—21no—
111C2×C8×C17—21no—
112C2×C8×C17—21no—
113C2×C8×C17—41no—
114C2×C8×C17—41no—
115C2×C8×C17—41no—
116C2×C8×C17—41no—
117C2×C8×C17C2×D411yescharacteristic
118C2×C8×C17C2×D411yescharacteristic
119C2×C8×C17C2×D411yescharacteristic
120C2×C8×C17C2×D411yescharacteristic
121C2×C8×C17C2×D411yescharacteristic
122C2×C8×C17#4111yescharacteristic
123C4⋊C4×C17—22no—
124C4⋊C4×C17—41no—
125C22⋊C4×C17—44no—
126#63227—44no—
127C4⋊Dic17—22no—
128C4⋊Dic17—41no—
129D136—22no—
130D136—22no—
131D136—41no—
132D136—41no—
133#63214—22no—
134#63214—41no—
135#13304—171nomaximal, Sylow
136C23×C17—21no—
137C23×C17—21no—
138C23×C17—21no—
139C23×C17—21no—
140C23×C17—21no—
141C23×C17—21no—
142C23×C17—42no—
143C23×C17—81no—
144C23×C17—81no—
145C23×C17—81no—
146C23×C17—81no—
147C23×C17—81no—
148C23×C17—81no—
149C23×C17D1612yescomplemented
150C23×C17C2×D811yescharacteristic
151C22×D17—42no—
152C22×D17—81no—
153C22×D17—81no—
154C22×D17—81no—
155C22×D17—81no—
156C22×D17—81no—
157C22×D17—81no—
158D4×C17—42no—
159D4×C17—42no—
160D4×C17—88no—
161D4×C17—81no—
162D4×C17—81no—
163C2×C4×C17—21no—
164C2×C4×C17—21no—
165C2×C4×C17—21no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17—41no—
170C2×C4×C17—42no—
171C2×C4×C17—81no—
172C2×C4×C17—81no—
173C2×C4×C17#12011yescharacteristic
174C2×C4×C17#12011yescharacteristic
175C2×C4×C17#12111yescharacteristic
176C2×C4×C17#13511yescharacteristic
177C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
179C22⋊D17—88no—
180C2×Dic17—42no—
181C2×Dic17—81no—
182C2×Dic17—81no—
183D68—42no—
184D68—42no—
185D68—81no—
186D68—81no—
187C8×C17—41no—
188C8×C17#12011yescharacteristic
189C8×C17#12111yescharacteristic
190C8×C17#12111yescharacteristic
191C8×C17#12711yescharacteristic
192C22×D16—171no—
193C2×#466—171no—
194C2×#466D1711yescharacteristic, complemented
195C2×#357—171no—
196C2×#357—171no—
197C2×C22⋊C16—171no—
198#1380—171no—
199C22×C17—22no—
200C22×C17—22no—
201C22×C17—22no—
202C22×C17—41no—
203C22×C17—41no—
204C22×C17—41no—
205C22×C17—41no—
206C22×C17—82no—
207C22×C17—82no—
208C22×C17—82no—
209C22×C17—161no—
210C22×C17—161no—
211C22×C17—161no—
212C22×C17—161no—
213C22×C17#44711yescharacteristic
214C22×C17#44711yescharacteristic
215C22×C17#46611yescharacteristic
216C22×C17C2×D1611yescharacteristic
217C22×C17#50811yescharacteristic
218C22×C17#50911yescharacteristic
219C22×C17#50911yescharacteristic
220C2×D17—82no—
221C2×D17—82no—
222C2×D17—82no—
223C2×D17—161no—
224C2×D17—161no—
225C2×D17—161no—
226C2×D17—161no—
227C4×C17—41no—
228C4×C17—82no—
229C4×C17#39211yescharacteristic
230C4×C17#44911yescharacteristic
231C4×C17#44911yescharacteristic
232C4×C17#46911yescharacteristic
233Dic17—82no—
234C22×D8—171no—
235C22×D8C2×D1711yescharacteristic
236C23×C8C2×D1711yescharacteristic
237C2×#121—21no—
238C2×#121—341no—
239C2×D16—342no—
240C2×D16—342no—
241C2×D16—342no—
242C22×C16—171no—
243C22×C16—171no—
244#466—22no—
245#466—22no—
246#466—342no—
247#466—342no—
248C2×#107—171no—
249C2×#107C2×D1711yescharacteristic
250C2×#102—21no—
251C2×#102—341no—
252#357—342no—
253#357—342no—
254C22⋊C16—342no—
255C2×C17—42no—
256C2×C17—162no—
257C2×C17#199911yescharacteristic
258C2×C17#218411yescharacteristic
259C2×C17#218411yescharacteristic
260C2×C17#218611yescharacteristic
261C2×C17#218611yescharacteristic
262C2×C17#221511yescharacteristic
263C2×C17#221811yescharacteristic
264D17—162no—
265C22×D4—21no—
266C22×D4—41no—
267C22×D4—341no—
268C22×D4—681no—
269C23×C4C22⋊D1711yescharacteristic
270C2×D8—21no—
271C2×D8—21no—
272C2×D8—21no—
273C2×D8—21no—
274C2×D8—21no—
275C2×D8—172no—
276C2×D8—341no—
277C2×D8—341no—
278C2×D8—341no—
279C2×D8—341no—
280C2×D8—341no—
281C2×D8C22⋊D1712yescomplemented
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8C22⋊D1711yescharacteristic
289C22×C8C22×D1711yescharacteristic
290#121—24no—
291#121—42no—
292#121—344no—
293#121—682no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—341no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—681no—
298D16—348no—
299C2×C16—172no—
300C2×C16—341no—
301C2×C16—341no—
302C2×C16—341no—
303C2×C16—341no—
304C2×C16—341no—
305#107—21no—
306#107—172no—
307#107—341no—
308#107C22⋊D1712yescomplemented
309#102—24no—
310#102—344no—
311C17#1330411yescharacteristic, complemented, Sylow
312C24—41no—
313C24—681no—
314C24#6322811yescharacteristic
315C2×D4—22no—
316C2×D4—44no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—342no—
325C2×D4—684no—
326C2×D4—681no—
327C2×D4—681no—
328C2×D4—681no—
329C2×D4—681no—
330C2×D4—681no—
331C2×D4—1362no—
332C2×D4—1362no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—681no—
341C22×C4#6322811yescharacteristic
342C22×C4C2×C22⋊D1711yescharacteristic
343D8—22no—
344D8—22no—
345D8—41no—
346D8—41no—
347D8—342no—
348D8—342no—
349D8—681no—
350D8—681no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8D4×D1711yescharacteristic
359C2×C8D4×D1711yescharacteristic
360C2×C8D4×D1711yescharacteristic
361C2×C8#6325411yescharacteristic
362C2×C8C2×C22⋊D1711yescharacteristic
363C2×C8C2×C22⋊D1711yescharacteristic
364C4⋊C4—22no—
365C4⋊C4—41no—
366C4⋊C4—342no—
367C4⋊C4—681no—
368C22⋊C4—44no—
369C22⋊C4—684no—
370C16—342no—
371C16—681no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—42no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—682no—
386C23—1361no—
387C23—1361no—
388C23—1361no—
389C23—1361no—
390C23—1361no—
391C23—1361no—
392C23#1058749712yescomplemented
393C23C2×#6322811yescharacteristic
394D4—42no—
395D4—42no—
396D4—88no—
397D4—81no—
398D4—81no—
399D4—682no—
400D4—682no—
401D4—1368no—
402D4—1361no—
403D4—1361no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—682no—
415C2×C4—1361no—
416C2×C4—1361no—
417C2×C4#1058761811yescharacteristic
418C2×C4#1058762311yescharacteristic
419C2×C4#1058762511yescharacteristic
420C2×C4#1058763511yescharacteristic
421C2×C4#1058763611yescharacteristic
422C2×C4#1058763911yescharacteristic
423C8—41no—
424C8#1058762311yescharacteristic
425C8#1058762411yescharacteristic
426C8#1058762511yescharacteristic
427C8#1058762611yescharacteristic
428C22—22no—
429C22—22no—
430C22—22no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—2721no—
443C22—2721no—
444C22—2721no—
445C22—2721no—
446C22—1362no—
447C22—1362no—
448C22—1362no—
449C22#1176294211yescharacteristic
450C22#1176294311yescharacteristic
451C22#1176304311yescharacteristic
452C22C2×#1058749711yescharacteristic
453C22#1176323911yescharacteristic
454C22#1176324411yescharacteristic
455C22#1176324511yescharacteristic
456C4—41no—
457C4—82no—
458C4—1362no—
459C4#1176271711yescharacteristic
460C4#1176295211yescharacteristic
461C4#1176295411yescharacteristic
462C4#1176306011yescharacteristic
463C2—42no—
464C2—162no—
465C2—2722no—
466C2#1452978011yescharacteristic
467C2#1453069511yescharacteristic
468C2#1453069611yescharacteristic
469C2#1453070511yescharacteristic
470C2#1453070711yescharacteristic
471C2#1453086811yescharacteristic
472C2#1453088511yescharacteristic
subgroupG/H
centerC23C2×#63228
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
FittingC2×C17×#466C2
socleC23×C17C2×D8
hypercenterC2×#466D17
hypocenterC17#13304
LCSG ▷ C22×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C22×#10587497C2semidirect
2. C2×#11763045C2semidirect
3. C2×#11762610C2semidirect
4. C2×#11762609C2semidirect
5. C2×C22⋊#63214C2semidirect
6. #14526875C2semidirect
7. C2×C17×#466C2semidirect
39. C2×D8×C17D4semidirect
61. C2×D136D4semidirect
63. C17×#107D4semidirect
68. #10587489D4semidirect
74. C2×#63214D4semidirect
149. C23×C17D16semidirect
194. C2×#466D17semidirect
281. C2×D8C22⋊D17semidirect
308. #107C22⋊D17semidirect
311. C17#13304semidirect
392. C23#10587497semidirect

Maximal subgroups

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

subgroupindex
C22×#105874972
C2×#117630452
C2×#117626102
C2×#117626092
C2×C22⋊#632142
#145268752
C2×C17×#4662
#1330417