Finite Group Databasebeta

← #16952983

Subgroups — #16952983

Order 4352.

616 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#14532146C211yesmaximal, characteristic
2#14528774C211yesmaximal, characteristic
3#99×Dic17C211yesmaximal, characteristic
4#14526903C211yesmaximal, characteristic
5#14526902C211yesmaximal, characteristic
6#14526901C211yesmaximal, characteristic
7C17×#1390C211yesmaximal, characteristic
8C23×Dic34C411yescharacteristic, complemented
9C2×#10587564—21no—
10C2×D4×Dic17C2211yescharacteristic
11#11762832—21no—
12C2×#10587504C2211yescharacteristic
13C2×#10587505C2211yescharacteristic
14#11762730—21no—
15#11762721—21no—
16#11762713—21no—
17#11762689C2211yescharacteristic
18#11762684C411yescharacteristic, complemented
19#11762682C411yescharacteristic, complemented
20#11762667C411yescharacteristic, complemented
21C22⋊C4×Dic17—21no—
22#11762577—21no—
23#11762575—21no—
24C2×C17×#100C2211yescharacteristic
25C2×C17×#99C2211yescharacteristic
26C17×#386C2211yescharacteristic
27C22×D4×C17C2311yescharacteristic
28C23×C4×C17C2×C411yescharacteristic
29C23×Dic17C2×C411yescharacteristic
30C23×Dic17C2×C411yescharacteristic
31C23×Dic17D411yescharacteristic
32C22×Dic34—21no—
33C22×Dic34—21no—
34C22×Dic34—44no—
35C22×Dic34—41no—
36C22×Dic34—41no—
37C2×M16×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17C2×C411yescharacteristic
41C2×C22⋊Dic17—21no—
42C2×C22⋊Dic17—21no—
43C2×C22⋊Dic17C2×C411yescharacteristic
44C2×C22⋊Dic17C2×C411yescharacteristic
45C2×C22⋊Dic17D411yescharacteristic
46D4×Dic17—41no—
47D4×Dic17—41no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17D411yescharacteristic
50C2×#63225—21no—
51C2×#63225—21no—
52C2×C4×Dic17—41no—
53C2×C4×Dic17D411yescharacteristic
54C2×#63223—21no—
55#10587564—44no—
56C17×#100—21no—
57C17×#100—21no—
58C17×#99—21no—
59C17×#99Q812yescomplemented
60C17×#95—21no—
61#10587505—21no—
62#10587505Q812yescomplemented
63#10587504—21no—
64#10587504—21no—
65#10587502—21no—
66#10587502—21no—
67#10587502—21no—
68#10587502—21no—
69#10587502—41no—
70#10587485—41no—
71#10587485—41no—
72C24×C17C22×C411yescharacteristic
73C24×C17C2×D411yescharacteristic
74C2×D4×C17—41no—
75C2×D4×C17—41no—
76C2×D4×C17C2×Q811yescharacteristic
77C2×D4×C17C2×Q811yescharacteristic
78C2×D4×C17#4111yescharacteristic
79C2×D4×C17#4111yescharacteristic
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17—42no—
83C22×C4×C17—41no—
84C22×C4×C17—41no—
85C22×C4×C17C2×D411yescharacteristic
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—21no—
90C22×Dic17—21no—
91C22×Dic17—21no—
92C22×Dic17—42no—
93C22×Dic17—42no—
94C22×Dic17—42no—
95C22×Dic17—41no—
96C22×Dic17—41no—
97C22×Dic17C22⋊C411yescharacteristic
98C22×Dic17C22⋊C411yescharacteristic
99C22×Dic17C22⋊C411yescharacteristic
100C22×Dic17C22⋊C411yescharacteristic
101C2×Dic34—24no—
102C2×Dic34—42no—
103C2×Dic34—42no—
104C2×Dic34—42no—
105C2×Dic34—42no—
106C2×Dic34—44no—
107C2×Dic34—44no—
108C2×Dic34—44no—
109C2×Dic34—41no—
110C2×Dic34—41no—
111C2×Dic34—44no—
112M16×C17—41no—
113M16×C17—41no—
114C2×C8×C17—41no—
115C22⋊C4×C17—22no—
116C22⋊C4×C17—42no—
117C22⋊C4×C17—41no—
118C22⋊C4×C17—41no—
119C22⋊Dic17—22no—
120C22⋊Dic17—41no—
121C22⋊Dic17—41no—
122C22⋊Dic17—42no—
123C22⋊Dic17—41no—
124C22⋊Dic17—41no—
125C4⋊Dic17—41no—
126C4⋊Dic17—42no—
127#63225—42no—
128#63225—42no—
129C4×Dic17—21no—
130C4×Dic17—21no—
131C4×Dic17—82no—
132#63223—41no—
133#63223—41no—
134C2×#3622—41no—
135#13296—171nomaximal, Sylow
136C23×C17—21no—
137C23×C17—21no—
138C23×C17—21no—
139C23×C17—21no—
140C23×C17—41no—
141C23×C17—41no—
142C23×C17—41no—
143C23×C17—41no—
144C23×C17C2×C22⋊C411yescharacteristic
145C23×C17C4×Q811yescharacteristic
146C23×C17C4×Q811yescharacteristic
147C23×C17C22⋊Q811yescharacteristic
148C23×C17C22⋊Q811yescharacteristic
149D4×C17—41no—
150D4×C17—41no—
151D4×C17—41no—
152D4×C17—41no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—22no—
156C2×C4×C17—41no—
157C2×C4×C17—42no—
158C2×C4×C17—41no—
159C2×C4×C17—42no—
160C2×C4×C17—42no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—41no—
164C2×C4×C17—81no—
165C2×C4×C17—81no—
166C2×C4×C17#12411yescharacteristic
167C2×C4×C17C4⋊Q811yescharacteristic
168C2×Dic17—22no—
169C2×Dic17—22no—
170C2×Dic17—21no—
171C2×Dic17—21no—
172C2×Dic17—21no—
173C2×Dic17—42no—
174C2×Dic17—41no—
175C2×Dic17—42no—
176C2×Dic17—41no—
177C2×Dic17—42no—
178C2×Dic17—42no—
179C2×Dic17—41no—
180C2×Dic17—41no—
181C2×Dic17—41no—
182C2×Dic17—42no—
183C2×Dic17—42no—
184C2×Dic17—41no—
185C2×Dic17—41no—
186C2×Dic17—41no—
187C2×Dic17—81no—
188C2×Dic17—81no—
189C2×Dic17—81no—
190C2×Dic17#9911yescharacteristic
191C2×Dic17#9911yescharacteristic
192Dic34—44no—
193Dic34—48no—
194Dic34—44no—
195C8×C17—81no—
196#3622—81no—
197#2409—171no—
198#1800—171no—
199C4×#99—171no—
200#1390—171no—
201#1390—171no—
202#1390—171no—
203#1390D1711yescharacteristic
204C22×C17—21no—
205C22×C17—21no—
206C22×C17—21no—
207C22×C17—21no—
208C22×C17—21no—
209C22×C17—21no—
210C22×C17—41no—
211C22×C17—41no—
212C22×C17—41no—
213C22×C17—41no—
214C22×C17—42no—
215C22×C17—41no—
216C22×C17—41no—
217C22×C17—41no—
218C22×C17—41no—
219C22×C17—81no—
220C22×C17#39111yescharacteristic
221C22×C17C2×#9911yescharacteristic
222C22×C17#41011yescharacteristic
223C4×C17—21no—
224C4×C17—21no—
225C4×C17—42no—
226C4×C17—81no—
227C4×C17—82no—
228Dic17—21no—
229Dic17—42no—
230Dic17—41no—
231Dic17—41no—
232Dic17—41no—
233Dic17—81no—
234Dic17—82no—
235Dic17AΣL(1,𝔽16)12yes—
236C23×Q8—171no—
237C2×C22⋊Q8—341no—
238C2×C4×D4—171no—
239#423—341no—
240C2×#100—171no—
241C2×#100C2×D1711yescharacteristic
242C2×#99—171no—
243C2×#99C2×D1711yescharacteristic
244#395—341no—
245#393—341no—
246#391—341no—
247#386—171no—
248#386C2×D1711yescharacteristic
249#385—171no—
250#385—171no—
251#380—171no—
252C4×C22⋊C4—341no—
253#343—341no—
254#342—341no—
255C2×C17—21no—
256C2×C17—21no—
257C2×C17—41no—
258C2×C17—41no—
259C2×C17—41no—
260C2×C17—41no—
261C2×C17#199611yescharacteristic
262C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
263C2×C17#212311yescharacteristic
264C22×Q8—341no—
265C22×Q8—341no—
266C22×Q8—681no—
267C22×Q8—681no—
268C22×Q8—684no—
269C22×D4C22×D1711yescharacteristic
270C23×C4—171no—
271C23×C4—171no—
272C23×C4—171no—
273C23×C4C4×D1711yescharacteristic
274C2×M16—21no—
275C2×M16—341no—
276C22⋊Q8—684no—
277C4×D4—681no—
278C4×D4—681no—
279C2×C4⋊C4—171no—
280C2×C4⋊C4—341no—
281C2×C4⋊C4—341no—
282C2×C4⋊C4—341no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—171no—
286C2×C22⋊C4—171no—
287C2×C22⋊C4—171no—
288C2×C22⋊C4—341no—
289C2×C22⋊C4—341no—
290C2×C22⋊C4C4×D1711yescharacteristic
291C2×C42—171no—
292C2×C42—681no—
293C4⋊C8—681no—
294C4⋊C8—681no—
295#100—21no—
296#100—21no—
297#100—341no—
298#100—341no—
299#99—21no—
300#99—172no—
301#99—341no—
302#99Dic3412yescomplemented
303#95—21no—
304#95—341no—
305#95—341no—
306#95—341no—
307#95—341no—
308#95—681no—
309C17#1329611yescharacteristic, complemented, Sylow
310C24C2×C4×D1711yescharacteristic
311C24D4×D1711yescharacteristic
312C2×Q8—344no—
313C2×Q8—682no—
314C2×Q8—681no—
315C2×Q8—681no—
316C2×Q8—682no—
317C2×Q8—682no—
318C2×Q8—682no—
319C2×Q8—684no—
320C2×Q8—684no—
321C2×Q8—684no—
322C2×Q8—684no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4C2×Dic3411yescharacteristic
326C2×D4#6325211yescharacteristic
327C2×D4#6325411yescharacteristic
328C2×D4Q8×D1711yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—171no—
335C22×C4—171no—
336C22×C4—171no—
337C22×C4—171no—
338C22×C4—341no—
339C22×C4—341no—
340C22×C4—341no—
341C22×C4—341no—
342C22×C4—341no—
343C22×C4—341no—
344C22×C4—682no—
345C22×C4—682no—
346C22×C4—682no—
347C22×C4—681no—
348C22×C4—681no—
349C22×C4D4×D1711yescharacteristic
350M16—41no—
351M16—41no—
352M16—681no—
353M16—681no—
354C2×C8—41no—
355C2×C8—681no—
356C4⋊C4—681no—
357C4⋊C4—682no—
358C4⋊C4—682no—
359C4⋊C4—682no—
360C22⋊C4—22no—
361C22⋊C4—42no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—342no—
365C22⋊C4—681no—
366C22⋊C4—681no—
367C22⋊C4—682no—
368C22⋊C4—681no—
369C22⋊C4—681no—
370C42—341no—
371C42—341no—
372C42—1362no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23C4×Dic3411yescharacteristic
382C23#1058756411yescharacteristic
383C23C22⋊C4×D1711yescharacteristic
384C23#1058757311yescharacteristic
385C23#1058758211yescharacteristic
386Q8—684no—
387Q8—684no—
388Q8—688no—
389D4—41no—
390D4—41no—
391D4—41no—
392D4—41no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—171no—
407C2×C4—171no—
408C2×C4—342no—
409C2×C4—342no—
410C2×C4—341no—
411C2×C4—341no—
412C2×C4—341no—
413C2×C4—682no—
414C2×C4—682no—
415C2×C4—682no—
416C2×C4—681no—
417C2×C4—682no—
418C2×C4—681no—
419C2×C4—681no—
420C2×C4—681no—
421C2×C4—681no—
422C2×C4—682no—
423C2×C4—682no—
424C2×C4—681no—
425C2×C4—681no—
426C2×C4—681no—
427C2×C4—1361no—
428C2×C4—1361no—
429C2×C4—1361no—
430C2×C4#1058757111yescharacteristic
431C2×C4#1058757411yescharacteristic
432C8—81no—
433C8—1361no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—81no—
450C22#1176271311yescharacteristic
451C22#99×D1711yescharacteristic
452C22#1176278111yescharacteristic
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—82no—
458C4—172no—
459C4—341no—
460C4—682no—
461C4—681no—
462C4—681no—
463C4—681no—
464C4—1361no—
465C4—1362no—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—41no—
471C2—41no—
472C2#1452976711yescharacteristic
473C2D17×AΣL(1,𝔽16)11yescharacteristic
474C2#1453043611yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1390C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#1390D17
hypocenterC17#13296
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
8. C23×Dic34C4semidirect
18. #11762684C4semidirect
19. #11762682C4semidirect
20. #11762667C4semidirect
59. C17×#99Q8semidirect
62. #10587505Q8semidirect
302. #99Dic34semidirect
309. C17#13296semidirect

Maximal subgroups

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

subgroupindex
#145321462
#145287742
#99×Dic172
#145269032
#145269022
#145269012
C17×#13902
#1329617