Finite Group Databasebeta

← #16934617

Subgroups — #16934617

Order 4352.

778 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11763142C211yesmaximal, characteristic, complemented
2#14534591C211yesmaximal, characteristic
3C2×#11762746C211yesmaximal, characteristic
4#14528875C211yesmaximal, characteristic, complemented
5C2×#11762532C211yesmaximal, characteristic
6#14526663C211yesmaximal, characteristic, complemented
7C2×C17×#328C211yesmaximal, characteristic, complemented
8C22×C4⋊Dic17—21no—
9C22×C4⋊Dic17C2211yescharacteristic
10C22×#63227—21no—
11C22×#63227C2211yescharacteristic
12#11763142—22no—
13#11763142—22no—
14C2×#10587478—21no—
15C2×#10587506—21no—
16C2×C22⋊#3622C2211yescharacteristic
17#11762746—22no—
18#11762746—22no—
19#11762744—21no—
20#11762731—22no—
21#11762720—21no—
22#11762681—21no—
23#11762669C2211yescharacteristic
24#11762664—21no—
25#11762572—22no—
26#11762532—22no—
27C2×C17×C22⋊Q8C2211yescharacteristic
28C2×C17×C22⋊C8C2211yescharacteristic
29C2×C17×#95C2211yescharacteristic
30C17×#328—22no—
31C24×D17D411yescharacteristic
32C22×Q8×C17D411yescharacteristic
33C23×C4×C17—21no—
34C23×C4×C17C2311yescharacteristic
35C23×Dic17—21no—
36C23×Dic17D411yescharacteristic
37C22×C8×C17—21no—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C17×C22⋊Q8—21no—
41C17×C22⋊Q8—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17D411yescharacteristic
44C2×C22⋊C4×C17—21no—
45C2×C22⋊Dic17—21no—
46C2×#63227—24no—
47C2×#63227—22no—
48C2×#63227—22no—
49C2×#63227—21no—
50C2×#63227—21no—
51C2×#63227—21no—
52C2×#63227—21no—
53C2×#63227—21no—
54C2×#63227—21no—
55C2×#63227—44no—
56C2×C4⋊Dic17—24no—
57C2×C4⋊Dic17—21no—
58C2×C4⋊Dic17—21no—
59C2×C4⋊Dic17—21no—
60C2×C4⋊Dic17—21no—
61C2×C4⋊Dic17—21no—
62C2×C4⋊Dic17—44no—
63C2×C4⋊Dic17D412yes—
64C2×C4⋊Dic17D412yes—
65C2×#63225—22no—
66C22×#3622—21no—
67C17×C22⋊C8—21no—
68C17×C22⋊C8—21no—
69C17×#95—21no—
70C17×#95D412yes—
71#10587506—24no—
72#10587502—22no—
73#10587502—21no—
74C22⋊#3622—21no—
75C22⋊#3622D412yes—
76#10587478—24no—
77C24×C17C2×D411yescharacteristic
78C23×D17—21no—
79C23×D17—21no—
80C23×D17—21no—
81C23×D17—21no—
82C23×D17—21no—
83C23×D17—21no—
84C23×D17—42no—
85C23×D17—42no—
86C23×D17Q1612yes—
87C2×Q8×C17—21no—
88C2×Q8×C17—21no—
89C2×Q8×C17—42no—
90C22×C4×C17—24no—
91C22×C4×C17—22no—
92C22×C4×C17—21no—
93C22×C4×C17—21no—
94C22×C4×C17—42no—
95C22×C4×C17—42no—
96C22×C4×C17—41no—
97C22×C4×C17—41no—
98C22×C4×C17C2×D411yescharacteristic
99C22×C4×C17C2×D411yescharacteristic
100C22×C4×C17C2×D411yescharacteristic
101C22×C4×C17C2×D411yescharacteristic
102C22×C4×C17C2×D411yescharacteristic
103C22×C4×C17#4111yescharacteristic
104C22×Dic17—22no—
105C22×Dic17—21no—
106C22×Dic17—21no—
107C22×Dic17—22no—
108C22×Dic17—21no—
109C22×Dic17—42no—
110C22×Dic17—42no—
111C22×Dic17—42no—
112C22×Dic17—41no—
113C22×Dic17—41no—
114C2×C8×C17—42no—
115C2×C8×C17—41no—
116C4⋊C4×C17—21no—
117C4⋊C4×C17—21no—
118C4⋊C4×C17—41no—
119C4⋊C4×C17—41no—
120C22⋊C4×C17—41no—
121C22⋊C4×C17—41no—
122C22⋊Dic17—41no—
123C22⋊Dic17—41no—
124#63227—48no—
125#63227—41no—
126#63227—41no—
127#63227—41no—
128#63227—41no—
129#63227—82no—
130#63227—82no—
131C4⋊Dic17—22no—
132C4⋊Dic17—22no—
133C4⋊Dic17—48no—
134C4⋊Dic17—41no—
135C4⋊Dic17—41no—
136C4⋊Dic17—42no—
137#63225—82no—
138C2×#3622—22no—
139C2×#3622—42no—
140#9928—171nomaximal, Sylow
141C23×C17—22no—
142C23×C17—22no—
143C23×C17#12011yescharacteristic
144C23×C17#12011yescharacteristic
145C23×C17#12111yescharacteristic
146C23×C17C2×Q1611yescharacteristic
147C23×C17#13511yescharacteristic
148C23×C17#13711yescharacteristic
149C23×C17#13711yescharacteristic
150C22×D17—22no—
151C22×D17—22no—
152C22×D17—22no—
153C22×D17—44no—
154C22×D17—44no—
155C22×D17—42no—
156C22×D17—42no—
157C22×D17—41no—
158C22×D17—41no—
159C22×D17—41no—
160C22×D17—42no—
161C22×D17—42no—
162C22×D17—41no—
163C22×D17—82no—
164C22×D17—82no—
165C22×D17—81no—
166C22×D17—81no—
167Q8×C17—42no—
168Q8×C17—81no—
169C2×C4×C17—22no—
170C2×C4×C17—22no—
171C2×C4×C17—24no—
172C2×C4×C17—41no—
173C2×C4×C17—41no—
174C2×C4×C17—41no—
175C2×C4×C17—41no—
176C2×C4×C17—44no—
177C2×C4×C17—44no—
178C2×C4×C17—42no—
179C2×C4×C17—42no—
180C2×C4×C17—41no—
181C2×C4×C17—41no—
182C2×C4×C17—42no—
183C2×C4×C17—42no—
184C2×C4×C17#12011yescharacteristic
185C2×C4×C17#12111yescharacteristic
186C2×C4×C17#12111yescharacteristic
187C2×C4×C17#12711yescharacteristic
188C2×Dic17—42no—
189C2×Dic17—42no—
190C2×Dic17—42no—
191C2×Dic17—41no—
192C2×Dic17—42no—
193C2×Dic17—42no—
194C2×Dic17—42no—
195C2×Dic17—42no—
196C2×Dic17—42no—
197C2×Dic17—82no—
198C2×Dic17—82no—
199C2×Dic17—81no—
200C8×C17—81no—
201#3622—42no—
202C2×#484—171no—
203#2626—171no—
204C2×#399—171no—
205#1824—171no—
206C2×#328D1711yescharacteristic, complemented
207C2×#323—171no—
208#1294—171no—
209C22×C17—22no—
210C22×C17—22no—
211C22×C17—22no—
212C22×C17—22no—
213C22×C17—21no—
214C22×C17—21no—
215C22×C17—21no—
216C22×C17—21no—
217C22×C17#39211yescharacteristic
218C22×C17#45111yescharacteristic
219C22×C17#45111yescharacteristic
220C22×C17#45211yescharacteristic
221C22×C17#45211yescharacteristic
222C22×C17#45312yes—
223C22×C17#45812yes—
224C22×C17#46711yescharacteristic
225C22×C17#47011yescharacteristic
226C2×D17—44no—
227C2×D17—44no—
228C2×D17—42no—
229C2×D17—82no—
230C2×D17—82no—
231C2×D17—82no—
232C2×D17—81no—
233C2×D17—81no—
234C2×D17—82no—
235C4×C17—21no—
236C4×C17—21no—
237C4×C17—21no—
238C4×C17—21no—
239C4×C17—44no—
240C4×C17—81no—
241C4×C17—81no—
242Dic17—82no—
243Dic17—81no—
244Dic17—82no—
245C2×C22⋊Q8C2×D1711yescharacteristic
246C22×C4⋊C4—171no—
247C22×C4⋊C4—341no—
248C22×C22⋊C4—171no—
249C22×C22⋊C4—341no—
250#484—342no—
251#484—342no—
252C2×#107—341no—
253C2×#103—341no—
254C2×C22⋊C8—171no—
255C2×C22⋊C8C2×D1711yescharacteristic
256#399—342no—
257#399—341no—
258#399—342no—
259#396—342no—
260#393—341no—
261#385—341no—
262#380—171no—
263#379—341no—
264C2×#95C2×D1711yescharacteristic
265#340—342no—
266#328—22no—
267#323—342no—
268C2×C17—21no—
269C2×C17—21no—
270C2×C17—21no—
271C2×C17—21no—
272C2×C17#161412yes—
273C2×C17#161812yes—
274C2×C17#200111yescharacteristic
275C2×C17#200211yescharacteristic
276C2×C17#202211yescharacteristic
277D17—82no—
278C25—171no—
279C22×Q8D6811yescharacteristic
280C23×C4—21no—
281C23×C4—171no—
282C23×C4—341no—
283C23×C4C22×D1711yescharacteristic
284C22×C8—21no—
285C22×C8—341no—
286C22⋊Q8—21no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—172no—
292C2×C4⋊C4—172no—
293C2×C4⋊C4—341no—
294C2×C4⋊C4—341no—
295C2×C4⋊C4—341no—
296C2×C4⋊C4—341no—
297C2×C4⋊C4—344no—
298C2×C4⋊C4—341no—
299C2×C4⋊C4—342no—
300C2×C4⋊C4—684no—
301C2×C4⋊C4D6811yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—344no—
304C2×C22⋊C4—341no—
305C2×C22⋊C4—341no—
306C2×C22⋊C4—341no—
307C2×C22⋊C4—341no—
308C2×C22⋊C4—341no—
309C2×C22⋊C4—341no—
310C2×C22⋊C4—341no—
311C2×C22⋊C4—342no—
312C2×C22⋊C4—342no—
313C2×C22⋊C4—684no—
314#107—344no—
315#103—344no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—172no—
319C22⋊C8—341no—
320#95—21no—
321#95—342no—
322#95—341no—
323#95C22⋊D1712yes—
324C17#992811yescharacteristic, complemented, Sylow
325C24—172no—
326C24—341no—
327C24—341no—
328C24—341no—
329C24—341no—
330C24—341no—
331C24—341no—
332C24—682no—
333C24—682no—
334C24D4×D1711yescharacteristic
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—42no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—342no—
347C22×C4—341no—
348C22×C4—342no—
349C22×C4—341no—
350C22×C4—341no—
351C22×C4—682no—
352C22×C4—682no—
353C22×C4—682no—
354C22×C4—681no—
355C22×C4—681no—
356C22×C4C2×D6811yescharacteristic
357C22×C4#6325211yescharacteristic
358C22×C4D4×D1711yescharacteristic
359C22×C4D4×D1711yescharacteristic
360C22×C4D4×D1711yescharacteristic
361C22×C4C2×C22⋊D1711yescharacteristic
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—342no—
365C2×C8—682no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—342no—
371C4⋊C4—342no—
372C4⋊C4—682no—
373C4⋊C4—688no—
374C4⋊C4—681no—
375C4⋊C4—681no—
376C4⋊C4—1362no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—688no—
380C22⋊C4—681no—
381C22⋊C4—681no—
382C22⋊C4—681no—
383C22⋊C4—681no—
384C22⋊C4—681no—
385C22⋊C4—681no—
386C22⋊C4—1362no—
387C22⋊C4—1362no—
388C23—22no—
389C23—22no—
390C23—342no—
391C23—342no—
392C23—342no—
393C23—684no—
394C23—682no—
395C23—684no—
396C23—682no—
397C23—681no—
398C23—681no—
399C23—681no—
400C23—682no—
401C23—682no—
402C23—681no—
403C23—1362no—
404C23—1361no—
405C23—1362no—
406C23—1361no—
407C23#1058756811yescharacteristic
408C23#1058757011yescharacteristic
409C23#1058760111yescharacteristic
410C23#1058760211yescharacteristic
411C23Q16×D1711yescharacteristic
412C23#1058760411yescharacteristic
413C23#1058762311yescharacteristic
414Q8—42no—
415Q8—81no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—682no—
432C2×C4—682no—
433C2×C4—682no—
434C2×C4—681no—
435C2×C4—682no—
436C2×C4—682no—
437C2×C4—682no—
438C2×C4—682no—
439C2×C4—682no—
440C2×C4—1362no—
441C2×C4—1362no—
442C2×C4—1361no—
443C2×C4#1058756811yescharacteristic
444C2×C4#1058757011yescharacteristic
445C2×C4#1058761611yescharacteristic
446C2×C4#1058762611yescharacteristic
447C8—81no—
448C8—682no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—21no—
456C22—21no—
457C22—684no—
458C22—682no—
459C22—684no—
460C22—1362no—
461C22—1362no—
462C22—1362no—
463C22—1361no—
464C22—1362no—
465C22—1361no—
466C22#1176271811yescharacteristic
467C22#1176296311yescharacteristic
468C22#1176296611yescharacteristic
469C22#1176296811yescharacteristic
470C22#1176297411yescharacteristic
471C22#1176297812yes—
472C22#1176300012yes—
473C22#1176304911yescharacteristic
474C22#1176306611yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—44no—
480C4—81no—
481C4—81no—
482C4—1361no—
483C4—1362no—
484C4—1362no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—1362no—
490C2#1452781012yes—
491C2#1452783512yes—
492C2#1452979311yescharacteristic
493C2#1452979811yescharacteristic
494C2#1452991411yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC2×C17×#328C2
socleC23×C17#120
prime radicalC24×D17D4
hypercenterC2×#328D17
hypocenterC17#9928
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
1. C2×#11763142C2semidirect
4. #14528875C2semidirect
6. #14526663C2semidirect
7. C2×C17×#328C2semidirect
206. C2×#328D17semidirect
324. C17#9928semidirect

Maximal subgroups

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

subgroupindex
C2×#117631422
#145345912
C2×#117627462
#145288752
C2×#117625322
#145266632
C2×C17×#3282
#992817