Finite Group Databasebeta

← #16950304

Subgroups — #16950304

Order 4352.

680 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C4×#10587626C211yesmaximal, characteristic
2C2×#11762601C212yesmaximal
3#14528772C211yesmaximal, characteristic, complemented
4#14526908C212yesmaximal, complemented
5C17×#1799C211yesmaximal, characteristic, complemented
6C2×#10587626C2211yescharacteristic
7C2×C4×D68C411yescharacteristic, complemented
8C2×C4×C22⋊D17—21no—
9C2×C4×C22⋊D17—21no—
10C2×#10587505C2212yes—
11#11762712C2211yescharacteristic
12#11762710C411yescharacteristic, complemented
13#11762709—21no—
14#11762709—21no—
15#11762678C411yescharacteristic, complemented
16#11762669—21no—
17C42×Dic17C411yescharacteristic, complemented
18#11762601—24no—
19#11762576—21no—
20C2×C4×D4×C17C2211yescharacteristic
21C2×C17×#99C2212yes—
22C17×#380—21no—
23C17×#342—21no—
24C22×D4×C17C2311yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C22×C22⋊D17—21no—
28C22×C22⋊D17—21no—
29C22×D68C2×C411yescharacteristic
30C22×C4×D17—21no—
31C4×D4×C17—41no—
32C4×D4×C17—41no—
33C2×C4⋊C4×C17—41no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17D411yescharacteristic
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C42×C17C2×C411yescharacteristic
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41C2×C22⋊Dic17—22no—
42#10587626—21no—
43#10587626—21no—
44#10587626D412yes—
45#10587626Q812yes—
46C4×C22⋊D17—44no—
47C4×C22⋊D17—44no—
48C2×#63227—21no—
49C2×#63227—21no—
50C2×#63227—21no—
51C2×C4⋊Dic17—41no—
52C2×C4⋊Dic17C2×C411yescharacteristic
53C2×#63225—21no—
54C2×#63225—21no—
55C2×C4×Dic17—21no—
56C2×C4×Dic17—21no—
57C2×C4×Dic17C2×C411yescharacteristic
58C2×C4×Dic17C2×C411yescharacteristic
59C4×D68—41no—
60C4×D68—41no—
61C17×#99—22no—
62C17×#99—22no—
63C17×#95—41no—
64#10587505—22no—
65#10587505—22no—
66#10587502—41no—
67C24×C17C2×D411yescharacteristic
68C24×C17C2×D411yescharacteristic
69C23×D17—21no—
70C2×D4×C17—41no—
71C2×D4×C17—41no—
72C2×D4×C17C2×D411yescharacteristic
73C2×D4×C17C2×Q811yescharacteristic
74C2×D4×C17#4111yescharacteristic
75C2×D4×C17#4111yescharacteristic
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—42no—
81C22×C4×C17—42no—
82C22×C4×C17—42no—
83C22×C4×C17C22⋊C411yescharacteristic
84C22×C4×C17C22⋊C411yescharacteristic
85C22×C4×C17C22⋊C411yescharacteristic
86C22×C4×C17C22⋊C411yescharacteristic
87C22×C4×C17C22×C411yescharacteristic
88C2×C22⋊D17—22no—
89C2×C22⋊D17—22no—
90C2×C22⋊D17—22no—
91C2×C22⋊D17—22no—
92C2×C22⋊D17—42no—
93C2×C22⋊D17—42no—
94C2×C22⋊D17—42no—
95C2×C22⋊D17—42no—
96C22×Dic17—21no—
97C22×Dic17—21no—
98C22×Dic17—21no—
99C22×Dic17—42no—
100C2×D68—21no—
101C2×D68—41no—
102C2×D68—41no—
103C2×D68C4⋊C412yescomplemented
104C2×C4×D17—42no—
105C2×C4×D17—42no—
106C2×C4×D17—41no—
107C2×C4×D17—41no—
108C4⋊C4×C17—41no—
109C4⋊C4×C17—82no—
110C22⋊C4×C17—41no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—42no—
113C22⋊C4×C17—42no—
114C42×C17—21no—
115C42×C17—21no—
116C22⋊Dic17—42no—
117C22⋊Dic17—42no—
118C22⋊Dic17—42no—
119C22⋊Dic17—42no—
120#63227—42no—
121#63227—42no—
122#63227—42no—
123C4⋊Dic17—41no—
124C4⋊Dic17—82no—
125#63225—42no—
126#63225—42no—
127C4×Dic17—24no—
128C4×Dic17—21no—
129C4×Dic17—42no—
130C4×Dic17—41no—
131C4×Dic17C4⋊C412yescomplemented
132#12797—171nomaximal, Sylow
133C23×C17—21no—
134C23×C17—21no—
135C23×C17—21no—
136C23×C17—21no—
137C23×C17—41no—
138C23×C17—41no—
139C23×C17—41no—
140C23×C17—41no—
141C23×C17C2×C22⋊C411yescharacteristic
142C23×C17#12111yescharacteristic
143C23×C17#12111yescharacteristic
144C23×C17C22⋊Q811yescharacteristic
145C23×C17C22⋊Q811yescharacteristic
146C22×D17—22no—
147C22×D17—41no—
148C22×D17—41no—
149C22×D17—41no—
150C22×D17—41no—
151C22×D17—41no—
152C22×D17—41no—
153D4×C17—41no—
154D4×C17—41no—
155D4×C17—41no—
156D4×C17—41no—
157C2×C4×C17—21no—
158C2×C4×C17—21no—
159C2×C4×C17—22no—
160C2×C4×C17—21no—
161C2×C4×C17—21no—
162C2×C4×C17—21no—
163C2×C4×C17—42no—
164C2×C4×C17—42no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—81no—
169C2×C4×C17—81no—
170C2×C4×C17—82no—
171C2×C4×C17—82no—
172C2×C4×C17—82no—
173C2×C4×C17#9911yescharacteristic
174C2×C4×C17#9911yescharacteristic
175C2×C4×C17C2×C4⋊C411yescharacteristic
176C2×C4×C17#11711yescharacteristic
177C22⋊D17—44no—
178C22⋊D17—44no—
179C22⋊D17—44no—
180C22⋊D17—44no—
181C2×Dic17—22no—
182C2×Dic17—22no—
183C2×Dic17—22no—
184C2×Dic17—22no—
185C2×Dic17—41no—
186C2×Dic17—41no—
187C2×Dic17—41no—
188C2×Dic17—41no—
189C2×Dic17—41no—
190C2×Dic17—82no—
191C2×Dic17—82no—
192C2×Dic17—82no—
193D68—42no—
194D68—42no—
195C4×D17—82no—
196C4×D17—82no—
197C4×#127—171no—
198C2×#353—172no—
199#1799—171no—
200#1799D1711yescharacteristic, complemented
201#1393—172no—
202C22×C17—21no—
203C22×C17—21no—
204C22×C17—21no—
205C22×C17—21no—
206C22×C17—21no—
207C22×C17—21no—
208C22×C17—41no—
209C22×C17—41no—
210C22×C17—41no—
211C22×C17—41no—
212C22×C17—41no—
213C22×C17—41no—
214C22×C17—41no—
215C22×C17—41no—
216C22×C17—81no—
217C22×C17—81no—
218C22×C17#38011yescharacteristic
219C22×C17C2×#9911yescharacteristic
220C22×C17#41011yescharacteristic
221C2×D17—42no—
222C2×D17—42no—
223C2×D17—42no—
224C2×D17—81no—
225C2×D17—81no—
226C2×D17—81no—
227C2×D17—81no—
228C4×C17—21no—
229C4×C17—21no—
230C4×C17—21no—
231C4×C17—41no—
232C4×C17—41no—
233C4×C17—41no—
234C4×C17—162no—
235C4×C17#35312yescomplemented
236Dic17—24no—
237Dic17—42no—
238Dic17—81no—
239Dic17—162no—
240C2×#127—171no—
241C2×C4×D4—171no—
242C2×C4×D4—341no—
243C2×C4×D4—341no—
244C2×C4×D4C2×D1711yescharacteristic
245C2×#99—172no—
246C2×#99C2×D1712yes—
247#390—171no—
248#390—171no—
249#390—341no—
250#390—341no—
251#384—171no—
252#380—21no—
253#380—341no—
254C43—171no—
255#353—344no—
256#342—21no—
257#342—341no—
258C2×C17—21no—
259C2×C17—21no—
260C2×C17—41no—
261C2×C17—41no—
262C2×C17—41no—
263C2×C17—41no—
264C2×C17#179911yescharacteristic
265C2×C17C2×#35311yescharacteristic
266C2×C17#212611yescharacteristic
267D17—82no—
268C22×D4—171no—
269C22×D4—341no—
270C22×D4—341no—
271C22×D4C22×D1711yescharacteristic
272C23×C4—341no—
273C23×C4D6811yescharacteristic
274C23×C4C22⋊D1711yescharacteristic
275#127—172no—
276#127—172no—
277#127—341no—
278#127—341no—
279C4×D4—41no—
280C4×D4—41no—
281C4×D4—681no—
282C4×D4—681no—
283C4×D4—684no—
284C4×D4—684no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4—171no—
287C2×C4⋊C4—341no—
288C2×C4⋊C4—341no—
289C2×C4⋊C4—681no—
290C2×C4⋊C4C4×D1711yescharacteristic
291C2×C22⋊C4—22no—
292C2×C22⋊C4—341no—
293C2×C22⋊C4—341no—
294C2×C22⋊C4—341no—
295C2×C22⋊C4—341no—
296C2×C22⋊C4—341no—
297C2×C22⋊C4—342no—
298C2×C22⋊C4D6811yescharacteristic
299C2×C22⋊C4C22⋊D1711yescharacteristic
300C2×C42—171no—
301C2×C42—171no—
302C2×C42—341no—
303C2×C42—341no—
304C2×C42C4×D1711yescharacteristic
305#99—22no—
306#99—22no—
307#99—342no—
308#99—342no—
309#95—41no—
310#95—681no—
311C17#1279711yescharacteristic, complemented, Sylow
312C24—341no—
313C24C2×D6811yescharacteristic
314C24C2×C22⋊D1711yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—172no—
318C2×D4—342no—
319C2×D4—342no—
320C2×D4—342no—
321C2×D4—342no—
322C2×D4—341no—
323C2×D4—681no—
324C2×D4—681no—
325C2×D4—682no—
326C2×D4—682no—
327C2×D4—682no—
328C2×D4—682no—
329C2×D4D4×D1711yescharacteristic
330C2×D4#6325411yescharacteristic
331C2×D4Q8×D1711yescharacteristic
332C2×D4#6325611yescharacteristic
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—341no—
341C22×C4—341no—
342C22×C4—341no—
343C22×C4—682no—
344C22×C4—682no—
345C22×C4—681no—
346C22×C4—681no—
347C22×C4—682no—
348C22×C4#6322711yescharacteristic
349C22×C4#6322711yescharacteristic
350C22×C4#6322711yescharacteristic
351C22×C4#6322711yescharacteristic
352C22×C4C2×C4×D1711yescharacteristic
353C4⋊C4—41no—
354C4⋊C4—82no—
355C4⋊C4—681no—
356C4⋊C4—682no—
357C4⋊C4—682no—
358C4⋊C4—1362no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—682no—
364C22⋊C4—682no—
365C22⋊C4—682no—
366C22⋊C4—682no—
367C22⋊C4—682no—
368C22⋊C4—682no—
369C22⋊C4—682no—
370C42—21no—
371C42—21no—
372C42—172no—
373C42—344no—
374C42—341no—
375C42—681no—
376C42—682no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—342no—
386C23—681no—
387C23—681no—
388C23—681no—
389C23—681no—
390C23—681no—
391C23—681no—
392C23#1058758111yescharacteristic
393C23#1058758311yescharacteristic
394C23C2×#6322711yescharacteristic
395C23#1058762411yescharacteristic
396C23#1058763211yescharacteristic
397D4—41no—
398D4—41no—
399D4—41no—
400D4—41no—
401D4—682no—
402D4—682no—
403D4—684no—
404D4—684no—
405D4—684no—
406D4—684no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—342no—
424C2×C4—342no—
425C2×C4—342no—
426C2×C4—342no—
427C2×C4—681no—
428C2×C4—681no—
429C2×C4—681no—
430C2×C4—681no—
431C2×C4—681no—
432C2×C4—1362no—
433C2×C4—1362no—
434C2×C4—1362no—
435C2×C4—1362no—
436C2×C4—1362no—
437C2×C4#1058747711yescharacteristic
438C2×C4#1058747711yescharacteristic
439C2×C4C4⋊C4×D1711yescharacteristic
440C2×C4#1058757811yescharacteristic
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—81no—
456C22—81no—
457C22—682no—
458C22—682no—
459C22—682no—
460C22—1361no—
461C22—1361no—
462C22—1361no—
463C22—1361no—
464C22#1176266611yescharacteristic
465C22C2×#1058747711yescharacteristic
466C22#1176278211yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—162no—
474C4—2722no—
475C4—344no—
476C4—682no—
477C4—1361no—
478C4#1176260112yescomplemented
479C2—21no—
480C2—21no—
481C2—41no—
482C2—41no—
483C2—41no—
484C2—41no—
485C2—1362no—
486C2#1452876911yescharacteristic
487C2C2×#1176260111yescharacteristic
488C2#1453044711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1799C2
socleC22×C17C2×#99
prime radicalC2×#10587626C22
cosocleC22×D4×C17C23
hypercenter#1799D17
hypocenterC17#12797
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #14528772C2semidirect
4. #14526908C2semidirect
5. C17×#1799C2semidirect
7. C2×C4×D68C4semidirect
12. #11762710C4semidirect
15. #11762678C4semidirect
17. C42×Dic17C4semidirect
103. C2×D68C4⋊C4semidirect
131. C4×Dic17C4⋊C4semidirect
200. #1799D17semidirect
235. C4×C17#353semidirect
311. C17#12797semidirect
478. C4#11762601semidirect

Maximal subgroups

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

subgroupindex
C4×#105876262
C2×#117626012
#145287722
#145269082
C17×#17992
#1279717