Finite Group Databasebeta

← #16936673

Subgroups — #16936673

Order 4352.

628 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763104C211yesmaximal, characteristic, complemented
2C2×#11763100C211yesmaximal, characteristic
3#14533702C211yesmaximal, characteristic
4#14529586C211yesmaximal, characteristic, complemented
5C2×#11762538C211yesmaximal, characteristic
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#10587555C2211yescharacteristic
9C2×#10587632—21no—
10C2×C4×D68C2211yescharacteristic
11#11763104—22no—
12#11763104—22no—
13#11763100—22no—
14#11763100—22no—
15C2×#10587478—21no—
16C2×#10587488—21no—
17C2×C4⋊#3622C2211yescharacteristic
18C2×#10587487—22no—
19C2×#10587506—21no—
20C2×#10587492—21no—
21#11762731—22no—
22#11762711—21no—
23#11762707—21no—
24#11762679C2211yescharacteristic
25#11762666—21no—
26#11762538—22no—
27C2×C4×Q8×C17C2211yescharacteristic
28C2×C17×C4⋊C8C2211yescharacteristic
29C2×C4×C8×C17C2211yescharacteristic
30C17×Q8⋊C8—22no—
31C22×Q8×C17D411yescharacteristic
32C22×D68D411yescharacteristic
33C22×C4×D17—21no—
34C22×Dic34—21no—
35C22×C8×C17—21no—
36C22×C8×C17—21no—
37C4×Q8×C17—21no—
38C4×Q8×C17—21no—
39C4×Q8×C17—21no—
40C4×Q8×C17—21no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17D411yescharacteristic
43C2×C42×C17—21no—
44C2×C42×C17C2311yescharacteristic
45#10587632—42no—
46#10587632—42no—
47C2×#63227—22no—
48C2×#63227—22no—
49C2×#63227—21no—
50C2×C4⋊Dic17—21no—
51C2×C4⋊Dic17—21no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17D411yescharacteristic
55C2×C4⋊Dic17D412yes—
56C2×#63225—22no—
57C22×#3622—21no—
58C4×D68—21no—
59C4×D68—21no—
60C4×D68—21no—
61C4×D68—21no—
62#10587555—21no—
63#10587555—21no—
64#10587555—21no—
65#10587555D412yes—
66C17×C4⋊C8—21no—
67C17×C4⋊C8D412yes—
68C4×C8×C17—21no—
69C4×C8×C17—21no—
70#10587506—24no—
71#10587502—22no—
72#10587492—24no—
73#10587488—24no—
74#10587487—44no—
75#10587478—24no—
76C4⋊#3622—21no—
77C4⋊#3622D412yes—
78C23×D17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—21no—
81C2×Q8×C17—21no—
82C2×Q8×C17—21no—
83C2×Q8×C17—21no—
84C2×Q8×C17SD1612yes—
85C22×C4×C17—21no—
86C22×C4×C17—21no—
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×Dic17—21no—
91C22×Dic17—22no—
92C22×Dic17—41no—
93C2×D68—21no—
94C2×D68—21no—
95C2×D68—21no—
96C2×D68—21no—
97C2×D68—21no—
98C2×D68Q1612yes—
99C2×C4×D17—41no—
100C2×C4×D17—41no—
101C2×C4×D17—41no—
102C2×C4×D17—41no—
103C2×C4×D17—41no—
104C2×C4×D17—41no—
105C2×Dic34—22no—
106C2×Dic34—42no—
107C2×Dic34—42no—
108C2×Dic34—41no—
109C2×C8×C17—22no—
110C2×C8×C17—22no—
111C2×C8×C17—41no—
112C2×C8×C17—41no—
113C2×C8×C17—41no—
114C2×C8×C17—41no—
115C4⋊C4×C17—21no—
116C4⋊C4×C17—21no—
117C4⋊C4×C17—41no—
118C4⋊C4×C17—41no—
119C42×C17—41no—
120C42×C17—41no—
121C42×C17C2×D411yescharacteristic
122C42×C17C2×D411yescharacteristic
123C42×C17C2×D411yescharacteristic
124C42×C17#4111yescharacteristic
125#63227—44no—
126#63227—41no—
127#63227—41no—
128#63227—82no—
129C4⋊Dic17—22no—
130C4⋊Dic17—21no—
131C4⋊Dic17—22no—
132C4⋊Dic17—21no—
133C4⋊Dic17—41no—
134C4⋊Dic17—41no—
135C4⋊Dic17—41no—
136C4⋊Dic17—41no—
137C4⋊Dic17—41no—
138C4⋊Dic17—41no—
139C4⋊Dic17—42no—
140#63225—44no—
141C2×#3622—22no—
142C2×#3622—41no—
143C2×#3622—41no—
144#10236—171nomaximal, Sylow
145C23×C17#12011yescharacteristic
146C22×D17—22no—
147C22×D17—41no—
148C22×D17—41no—
149C22×D17—41no—
150C22×D17—41no—
151C22×D17—41no—
152C22×D17—41no—
153Q8×C17—22no—
154Q8×C17—22no—
155Q8×C17—41no—
156Q8×C17—41no—
157C2×C4×C17—21no—
158C2×C4×C17—21no—
159C2×C4×C17—22no—
160C2×C4×C17—41no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—41no—
164C2×C4×C17—41no—
165C2×C4×C17#12011yescharacteristic
166C2×C4×C17#12011yescharacteristic
167C2×C4×C17#12111yescharacteristic
168C2×C4×C17#12111yescharacteristic
169C2×C4×C17#12111yescharacteristic
170C2×C4×C17#12711yescharacteristic
171C2×C4×C17C2×SD1611yescharacteristic
172C2×C4×C17C2×Q1611yescharacteristic
173C2×C4×C17#13511yescharacteristic
174C2×C4×C17#13511yescharacteristic
175C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C17#13711yescharacteristic
177C2×C4×C17#13711yescharacteristic
178C2×C4×C17#13711yescharacteristic
179C2×Dic17—42no—
180C2×Dic17—41no—
181C2×Dic17—41no—
182C2×Dic17—42no—
183C2×Dic17—41no—
184C2×Dic17—42no—
185C2×Dic17—42no—
186C2×Dic17—81no—
187C2×Dic17—81no—
188D68—22no—
189D68—22no—
190D68—41no—
191D68—41no—
192C4×D17—81no—
193C4×D17—81no—
194C4×D17—81no—
195C4×D17—81no—
196Dic34—44no—
197Dic34—82no—
198C8×C17—42no—
199C8×C17—42no—
200#3622—42no—
201C2×#477—171no—
202C2×#476—171no—
203#2538—171no—
204#1956—171no—
205C2×Q8⋊C8D1711yescharacteristic, complemented
206C2×D4⋊C8—171no—
207#1296—171no—
208C22×C17#39211yescharacteristic
209C22×C17#45011yescharacteristic
210C22×C17#45111yescharacteristic
211C22×C17#45211yescharacteristic
212C22×C17#45211yescharacteristic
213C22×C17#45311yescharacteristic
214C22×C17#45411yescharacteristic
215C2×D17—42no—
216C2×D17—42no—
217C2×D17—42no—
218C2×D17—81no—
219C2×D17—81no—
220C2×D17—81no—
221C2×D17—81no—
222C4×C17—21no—
223C4×C17—21no—
224C4×C17—42no—
225C4×C17—81no—
226C4×C17#44811yescharacteristic
227C4×C17#44911yescharacteristic
228C4×C17#45111yescharacteristic
229C4×C17#45211yescharacteristic
230C4×C17#46511yescharacteristic
231C4×C17#46711yescharacteristic
232C4×C17#46811yescharacteristic
233C4×C17#47011yescharacteristic
234Dic17—82no—
235Dic17—81no—
236Dic17—82no—
237C2×C4⋊Q8—171no—
238C2×C22⋊Q8—341no—
239C2×C4×Q8C2×D1711yescharacteristic
240C2×C4×D4—171no—
241#477—342no—
242#477—342no—
243#476—342no—
244#476—342no—
245C2×#107—341no—
246C2×#106—341no—
247C2×C4⋊C8—171no—
248C2×C4⋊C8C2×D1711yescharacteristic
249C2×#103—341no—
250C2×#103—342no—
251C2×#102—341no—
252C2×C4×C8C2×D1711yescharacteristic
253#396—342no—
254#390—341no—
255#389—341no—
256#384—171no—
257#380—341no—
258Q8⋊C8—22no—
259D4⋊C8—342no—
260C2×C17#163111yescharacteristic
261C2×C17#163211yescharacteristic
262C2×C17#163711yescharacteristic
263C2×C17#163811yescharacteristic
264C2×C17#200011yescharacteristic
265C2×C17#200111yescharacteristic
266C2×C17#200611yescharacteristic
267D17—82no—
268C22×Q8—341no—
269C22×Q8D6811yescharacteristic
270C22×D4—171no—
271C23×C4—341no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8—341no—
275C4⋊Q8—172no—
276C4⋊Q8—341no—
277C4⋊Q8—341no—
278C4⋊Q8—341no—
279C22⋊Q8—682no—
280C22⋊Q8—682no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×D4—341no—
286C4×D4—341no—
287C4×D4—341no—
288C4×D4—341no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—171no—
291C2×C4⋊C4—172no—
292C2×C4⋊C4—341no—
293C2×C4⋊C4—341no—
294C2×C4⋊C4—342no—
295C2×C4⋊C4—341no—
296C2×C4⋊C4—341no—
297C2×C4⋊C4D6811yescharacteristic
298C2×C22⋊C4—342no—
299C2×C22⋊C4—341no—
300C2×C22⋊C4—342no—
301C2×C42—21no—
302C2×C42C22×D1711yescharacteristic
303#107—344no—
304#106—344no—
305C4⋊C8—21no—
306C4⋊C8—172no—
307C4⋊C8—341no—
308C4⋊C8C22⋊D1712yes—
309#103—344no—
310#103—684no—
311#102—344no—
312C4×C8—21no—
313C4×C8—21no—
314#95—342no—
315C17#1023611yescharacteristic, complemented, Sylow
316C24—341no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—342no—
323C2×Q8—682no—
324C2×Q8—682no—
325C2×Q8—681no—
326C2×Q8#6321912yes—
327C2×D4—172no—
328C2×D4—341no—
329C2×D4—341no—
330C2×D4—341no—
331C2×D4—341no—
332C2×D4—341no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—341no—
336C22×C4—342no—
337C22×C4—681no—
338C22×C4—681no—
339C22×C4—681no—
340C22×C4—681no—
341C22×C4—681no—
342C22×C4—681no—
343C22×C4—681no—
344C22×C4C2×D6811yescharacteristic
345C22×C4D4×D1711yescharacteristic
346C22×C4D4×D1711yescharacteristic
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—342no—
354C2×C8—681no—
355C2×C8—681no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—342no—
361C4⋊C4—341no—
362C4⋊C4—342no—
363C4⋊C4—341no—
364C4⋊C4—682no—
365C4⋊C4—684no—
366C4⋊C4—681no—
367C4⋊C4—681no—
368C4⋊C4—681no—
369C4⋊C4—681no—
370C4⋊C4—681no—
371C4⋊C4—681no—
372C22⋊C4—684no—
373C22⋊C4—681no—
374C22⋊C4—681no—
375C22⋊C4—1362no—
376C42—41no—
377C42—41no—
378C42#6325211yescharacteristic
379C42D4×D1711yescharacteristic
380C42D4×D1711yescharacteristic
381C42C2×C22⋊D1711yescharacteristic
382C23—342no—
383C23—681no—
384C23—681no—
385C23—681no—
386C23—681no—
387C23—681no—
388C23—681no—
389C23#1058756811yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—41no—
393Q8—41no—
394Q8—684no—
395Q8—1362no—
396D4—342no—
397D4—342no—
398D4—681no—
399D4—681no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—682no—
409C2×C4—681no—
410C2×C4—681no—
411C2×C4—681no—
412C2×C4—682no—
413C2×C4—682no—
414C2×C4—682no—
415C2×C4—1361no—
416C2×C4—1361no—
417C2×C4—1361no—
418C2×C4—1361no—
419C2×C4—1361no—
420C2×C4—1361no—
421C2×C4#1058756811yescharacteristic
422C2×C4#1058757011yescharacteristic
423C2×C4#1058757011yescharacteristic
424C2×C4C2×#6321911yescharacteristic
425C2×C4#1058759011yescharacteristic
426C2×C4#1058759411yescharacteristic
427C2×C4#1058759511yescharacteristic
428C2×C4#1058760111yescharacteristic
429C2×C4#1058760211yescharacteristic
430C2×C4Q16×D1711yescharacteristic
431C2×C4#1058760411yescharacteristic
432C2×C4#1058761611yescharacteristic
433C2×C4#1058762311yescharacteristic
434C2×C4#1058762611yescharacteristic
435C8—42no—
436C8—42no—
437C8—682no—
438C22—682no—
439C22—682no—
440C22—682no—
441C22—1361no—
442C22—1361no—
443C22—1361no—
444C22—1361no—
445C22#1176271811yescharacteristic
446C22#1176296211yescharacteristic
447C22#1176296611yescharacteristic
448C22#1176297311yescharacteristic
449C22#1176297411yescharacteristic
450C22#1176297811yescharacteristic
451C22#1176298311yescharacteristic
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—1362no—
457C4—1362no—
458C4—1361no—
459C4#1176295011yescharacteristic
460C4#1176295611yescharacteristic
461C4#1176296311yescharacteristic
462C4#1176296811yescharacteristic
463C4#1176303811yescharacteristic
464C4#1176304911yescharacteristic
465C4#1176305511yescharacteristic
466C4#1176306611yescharacteristic
467C2—1362no—
468C2#1452791311yescharacteristic
469C2#1452792111yescharacteristic
470C2#1452795111yescharacteristic
471C2#1452795911yescharacteristic
472C2#1452978911yescharacteristic
473C2#1452979311yescharacteristic
474C2#1452982711yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×Q8⋊C8C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenterC2×Q8⋊C8D17
hypocenterC17#10236
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#11763104C2semidirect
4. #14529586C2semidirect
6. #14526668C2semidirect
7. C2×C17×Q8⋊C8C2semidirect
205. C2×Q8⋊C8D17semidirect
315. C17#10236semidirect

Maximal subgroups

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

subgroupindex
C2×#117631042
C2×#117631002
#145337022
#145295862
C2×#117625382
#145266682
C2×C17×Q8⋊C82
#1023617