Finite Group Databasebeta

← #16951856

Subgroups — #16951856

Order 4352.

724 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 103 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×#11762783C211yesmaximal, characteristic
2#14532408C211yesmaximal, characteristic
3C2×#11762603C212yesmaximal, complemented
4#14526908C212yesmaximal, complemented
5C17×#1782C211yesmaximal, characteristic
6C2×#10587630C411yescharacteristic, complemented
7C2×#10587622C2211yescharacteristic
8C2×#10587610C411yescharacteristic, complemented
9C2×#10587612—21no—
10#11762783—24no—
11C2×#10587505—21no—
12C2×#10587505C2212yes—
13#11762723—21no—
14#11762723—21no—
15#11762716C411yescharacteristic, complemented
16#11762712C2211yescharacteristic
17#11762682—21no—
18#11762682—21no—
19C4×C22⋊Dic17C411yescharacteristic, complemented
20#11762603—24no—
21C22×#41×C17C2211yescharacteristic
22C2×C17×#99—21no—
23C2×C17×#99C2212yes—
24C17×#381—21no—
25C2×#41×C17—22no—
26C2×#41×C17—22no—
27C2×#41×C17—41no—
28C2×#41×C17—41no—
29C2×#41×C17—42no—
30C22×Q8×C17C2×C411yescharacteristic
31C22×D4×C17D411yescharacteristic
32C22×D4×C17D411yescharacteristic
33C22×D4×C17C2311yescharacteristic
34C23×C4×C17C2×C411yescharacteristic
35C23×C4×C17D411yescharacteristic
36C23×C4×C17D411yescharacteristic
37C22×Dic34C2×C411yescharacteristic
38C2×C22⋊C4×C17—22no—
39C2×C22⋊Dic17—22no—
40C2×C22⋊Dic17—21no—
41C2×C22⋊Dic17—21no—
42C2×C22⋊Dic17—21no—
43C2×C22⋊Dic17—21no—
44C2×C22⋊Dic17—21no—
45C2×C22⋊Dic17—21no—
46#10587630—42no—
47#10587622—22no—
48#10587622—22no—
49#10587612—24no—
50#10587612—42no—
51C2×C4⋊Dic17—41no—
52C2×C4⋊Dic17C2×C411yescharacteristic
53#10587610—41no—
54#10587610—41no—
55C2×#63225—21no—
56C2×#63225—21no—
57C2×#63225—21no—
58C2×C4×Dic17—41no—
59C2×C4×Dic17C2×C411yescharacteristic
60C2×C4×Dic17C2×C411yescharacteristic
61C17×#99—22no—
62C17×#99—22no—
63C17×#99—42no—
64C17×#95—41no—
65C17×#95—41no—
66#10587505—24no—
67#10587505—22no—
68#10587505D414yes—
69#10587502—21no—
70#10587502—21no—
71#10587502—21no—
72C24×C17C22⋊C411yescharacteristic
73C24×C17C2×D411yescharacteristic
74C24×C17C2×D411yescharacteristic
75#41×C17—44no—
76#41×C17—44no—
77#41×C17—44no—
78#41×C17—44no—
79C2×Q8×C17—22no—
80C2×Q8×C17—22no—
81C2×Q8×C17—41no—
82C2×D4×C17—21no—
83C2×D4×C17—22no—
84C2×D4×C17—21no—
85C2×D4×C17—22no—
86C2×D4×C17—41no—
87C2×D4×C17—41no—
88C2×D4×C17—41no—
89C2×D4×C17—41no—
90C2×D4×C17—41no—
91C2×D4×C17C2×D412yes—
92C2×D4×C17#4112yes—
93C22×C4×C17—21no—
94C22×C4×C17—21no—
95C22×C4×C17—21no—
96C22×C4×C17—21no—
97C22×C4×C17—22no—
98C22×C4×C17—22no—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—22no—
104C22×C4×C17—41no—
105C22×C4×C17—41no—
106C22×C4×C17—42no—
107C22×C4×C17C22⋊C411yescharacteristic
108C22×C4×C17C22⋊C411yescharacteristic
109C22×C4×C17C22⋊C411yescharacteristic
110C22×C4×C17C22×C411yescharacteristic
111C22×Dic17—21no—
112C22×Dic17—21no—
113C22×Dic17—21no—
114C22×Dic17—21no—
115C22×Dic17—42no—
116C2×Dic34—22no—
117C2×Dic34—41no—
118C2×Dic34—41no—
119C22⋊C4×C17—42no—
120C22⋊C4×C17—42no—
121C22⋊Dic17—24no—
122C22⋊Dic17—42no—
123C22⋊Dic17—42no—
124C22⋊Dic17—42no—
125C22⋊Dic17—42no—
126C22⋊Dic17—42no—
127C22⋊Dic17—41no—
128C22⋊Dic17—42no—
129C22⋊Dic17—41no—
130C4⋊Dic17—44no—
131C4⋊Dic17—41no—
132#63225—42no—
133#63225—41no—
134#63225—41no—
135#63225—42no—
136C4×Dic17—22no—
137C4×Dic17—44no—
138C4×Dic17—41no—
139#13091—171nomaximal, Sylow
140C23×C17—21no—
141C23×C17—21no—
142C23×C17—21no—
143C23×C17—21no—
144C23×C17—21no—
145C23×C17—21no—
146C23×C17—21no—
147C23×C17—41no—
148C23×C17—41no—
149C23×C17—41no—
150C23×C17—41no—
151C23×C17—41no—
152C23×C17—41no—
153C23×C17C2×C22⋊C411yescharacteristic
154C23×C17#12011yescharacteristic
155C23×C17#12112yes—
156C23×C17#12311yescharacteristic
157Q8×C17—44no—
158D4×C17—42no—
159D4×C17—42no—
160D4×C17—44no—
161D4×C17—42no—
162D4×C17—42no—
163C2×C4×C17—22no—
164C2×C4×C17—22no—
165C2×C4×C17—22no—
166C2×C4×C17—22no—
167C2×C4×C17—21no—
168C2×C4×C17—21no—
169C2×C4×C17—21no—
170C2×C4×C17—21no—
171C2×C4×C17—21no—
172C2×C4×C17—21no—
173C2×C4×C17—21no—
174C2×C4×C17—21no—
175C2×C4×C17—21no—
176C2×C4×C17—42no—
177C2×C4×C17—42no—
178C2×C4×C17—42no—
179C2×C4×C17—42no—
180C2×C4×C17—82no—
181C2×C4×C17—82no—
182C2×C4×C17—82no—
183C2×C4×C17#9911yescharacteristic
184C2×C4×C17#9911yescharacteristic
185C2×C4×C17#9912yes—
186C2×C4×C17C4×D412yes—
187C2×Dic17—22no—
188C2×Dic17—41no—
189C2×Dic17—41no—
190C2×Dic17—44no—
191C2×Dic17—41no—
192C2×Dic17—41no—
193C2×Dic17—41no—
194C2×Dic17—41no—
195C2×Dic17—41no—
196C2×Dic17—41no—
197C2×Dic17—41no—
198C2×Dic17—41no—
199C2×Dic17—41no—
200C2×Dic17—82no—
201C2×Dic17—82no—
202Dic34—82no—
203C2×#410—171no—
204#2432—171no—
205C2×#354—172no—
206#1782D1711yescharacteristic
207#1393—172no—
208C22×C17—21no—
209C22×C17—21no—
210C22×C17—21no—
211C22×C17—21no—
212C22×C17—22no—
213C22×C17—41no—
214C22×C17—42no—
215C22×C17—41no—
216C22×C17—41no—
217C22×C17—41no—
218C22×C17—41no—
219C22×C17—41no—
220C22×C17—41no—
221C22×C17—41no—
222C22×C17—41no—
223C22×C17—81no—
224C22×C17—81no—
225C22×C17—81no—
226C22×C17—81no—
227C22×C17#38611yescharacteristic
228C22×C17C2×#9911yescharacteristic
229C22×C17C2×#9911yescharacteristic
230C4×C17—22no—
231C4×C17—21no—
232C4×C17—21no—
233C4×C17—22no—
234C4×C17—41no—
235C4×C17—162no—
236Dic17—42no—
237Dic17—81no—
238Dic17—84no—
239Dic17—81no—
240Dic17—81no—
241C22×#41C2×D1711yescharacteristic
242C2×C4⋊Q8—171no—
243C2×#124—171no—
244C2×C22⋊Q8—171no—
245C2×#117—341no—
246#410—344no—
247C2×#99—21no—
248C2×#99—172no—
249C2×#99—341no—
250C2×#99C2×D1712yes—
251#394—341no—
252#394—341no—
253#391—171no—
254#390—171no—
255#385—341no—
256#385—341no—
257#381—21no—
258C4×C22⋊C4—171no—
259#354—344no—
260C2×C17—21no—
261C2×C17—21no—
262C2×C17—41no—
263C2×C17—42no—
264C2×C17—41no—
265C2×C17—81no—
266C2×C17#189811yescharacteristic
267C2×C17#212611yescharacteristic
268C2×C17#212911yescharacteristic
269C2×#41—22no—
270C2×#41—22no—
271C2×#41—41no—
272C2×#41—41no—
273C2×#41—42no—
274C22×Q8—171no—
275C22×Q8C4×D1711yescharacteristic
276C22×D4D6811yescharacteristic
277C22×D4C22⋊D1711yescharacteristic
278C22×D4C22×D1711yescharacteristic
279C23×C4C4×D1711yescharacteristic
280C23×C4D6811yescharacteristic
281C23×C4C22⋊D1711yescharacteristic
282C4⋊Q8—682no—
283#124—342no—
284#124—342no—
285C22⋊Q8—681no—
286C22⋊Q8—681no—
287#117—344no—
288#117—682no—
289C2×C4⋊C4—171no—
290C2×C4⋊C4—341no—
291C2×C4⋊C4—341no—
292C2×C4⋊C4—341no—
293C2×C4⋊C4—681no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—342no—
296C2×C22⋊C4—341no—
297C2×C22⋊C4—341no—
298C2×C22⋊C4—341no—
299C2×C22⋊C4—341no—
300C2×C22⋊C4—341no—
301C2×C22⋊C4—341no—
302C2×C42—171no—
303C2×C42—171no—
304C2×C42—681no—
305#99—22no—
306#99—22no—
307#99—42no—
308#99—174no—
309#99—344no—
310#99—342no—
311#95—41no—
312#95—41no—
313#95—341no—
314#95—341no—
315#95—341no—
316C17#1309111yescharacteristic, complemented, Sylow
317C24#6322711yescharacteristic
318C24C2×D6811yescharacteristic
319C24C2×C22⋊D1711yescharacteristic
320#41—44no—
321#41—44no—
322#41—44no—
323#41—44no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—342no—
328C2×Q8—681no—
329C2×Q8—681no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4D4×D1712yes—
340C2×D4#6325412yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—341no—
356C22×C4—341no—
357C22×C4—341no—
358C22×C4—341no—
359C22×C4—682no—
360C22×C4#6322711yescharacteristic
361C22×C4#6322711yescharacteristic
362C22×C4#6322711yescharacteristic
363C22×C4C2×C4×D1711yescharacteristic
364C4⋊C4—681no—
365C4⋊C4—682no—
366C4⋊C4—684no—
367C4⋊C4—681no—
368C4⋊C4—681no—
369C4⋊C4—682no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—344no—
373C22⋊C4—682no—
374C22⋊C4—682no—
375C22⋊C4—682no—
376C22⋊C4—682no—
377C22⋊C4—682no—
378C22⋊C4—681no—
379C22⋊C4—681no—
380C22⋊C4—682no—
381C42—342no—
382C42—681no—
383C42—684no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23#1058756811yescharacteristic
398C23#1058757211yescharacteristic
399C23C2×#6322711yescharacteristic
400C23#1058762512yes—
401Q8—44no—
402Q8—1362no—
403D4—42no—
404D4—42no—
405D4—44no—
406D4—42no—
407D4—42no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—342no—
429C2×C4—681no—
430C2×C4—681no—
431C2×C4—684no—
432C2×C4—681no—
433C2×C4—681no—
434C2×C4—681no—
435C2×C4—681no—
436C2×C4—681no—
437C2×C4—681no—
438C2×C4—681no—
439C2×C4—681no—
440C2×C4—681no—
441C2×C4—1362no—
442C2×C4—1362no—
443C2×C4#1058747712yes—
444C2×C4#1058747711yescharacteristic
445C2×C4#1058747711yescharacteristic
446C2×C4#1058756712yes—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22#1176268611yescharacteristic
467C22C2×#1058747711yescharacteristic
468C22C2×#1058747711yescharacteristic
469C4—21no—
470C4—21no—
471C4—22no—
472C4—22no—
473C4—41no—
474C4—162no—
475C4—682no—
476C4—1361no—
477C4—1364no—
478C4—1361no—
479C4—1361no—
480C2—21no—
481C2—21no—
482C2—42no—
483C2—41no—
484C2—41no—
485C2—81no—
486C2#1452923511yescharacteristic
487C2#1453044711yescharacteristic
488C2#1453045811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1782C2
socleC22×C17C2×#99
prime radicalC22×#41×C17C22
cosocleC22×D4×C17C23
hypercenter#1782D17
hypocenterC17#13091
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
3. C2×#11762603C2semidirect
4. #14526908C2semidirect
6. C2×#10587630C4semidirect
8. C2×#10587610C4semidirect
15. #11762716C4semidirect
19. C4×C22⋊Dic17C4semidirect
316. C17#13091semidirect

Maximal subgroups

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

subgroupindex
C2×#117627832
#145324082
C2×#117626032
#145269082
C17×#17822
#1309117