Finite Group Databasebeta

← #16925682

Subgroups — #16925682

Order 4352.

1174 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C22×#10587582C211yesmaximal, characteristic
2#14528693C212yesmaximal
3C2×#11762535C211yesmaximal, characteristic, complemented
4C2×#11762534C212yesmaximal, complemented
5C2×C17×#324C211yesmaximal, characteristic, complemented
6C23×Dic34C411yescharacteristic
7C23×C4×D17C2211yescharacteristic
8C2×#10587582—22no—
9C2×#10587582—22no—
10C2×#10587582—24no—
11C2×#10587582—24no—
12C22×#63225C411yescharacteristic
13C22×#63225C2211yescharacteristic
14C22×#63227C411yescharacteristic
15C22×#63227C411yescharacteristic
16C2×C22⋊#3622C2212yes—
17C2×#10587491—24no—
18#11762535—22no—
19#11762534—24no—
20C22×C4⋊C4×C17C2211yescharacteristic
21C2×C17×C22⋊C8C2212yes—
22C17×#324—22no—
23C24×D17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17C2311yescharacteristic
27C23×Dic17C2×C411yescharacteristic
28C23×Dic17C2×C411yescharacteristic
29C23×Dic17C2×C411yescharacteristic
30C22×C4×D17—21no—
31C22×C4×D17—22no—
32C22×C4×D17—21no—
33C22×C4×D17—44no—
34C22×C4×D17D414yes—
35C22×Dic34—22no—
36C22×Dic34—22no—
37C22×Dic34—22no—
38C22×Dic34—44no—
39C22×C8×C17—22no—
40C2×C4⋊C4×C17—22no—
41C2×C4⋊C4×C17—22no—
42C2×C4⋊C4×C17D412yes—
43C2×C4⋊C4×C17D412yes—
44C2×#63227—22no—
45C2×#63227—22no—
46C2×#63227—22no—
47C2×#63227—22no—
48C2×#63227—22no—
49C2×#63227—22no—
50C2×#63225—22no—
51C2×#63225—21no—
52C2×#63225—22no—
53C2×#63225—21no—
54C2×#63225—22no—
55C2×#63225—22no—
56C2×#63225D414yes—
57C22×#3622—22no—
58#10587582—216no—
59#10587582—216no—
60C17×C22⋊C8—22no—
61C17×C22⋊C8—22no—
62C22⋊#3622—22no—
63C22⋊#3622—22no—
64#10587491—48no—
65C24×C17C22×C411yescharacteristic
66C23×D17—21no—
67C23×D17—21no—
68C23×D17—24no—
69C23×D17—22no—
70C23×D17—22no—
71C23×D17—42no—
72C23×D17C22⋊C412yes—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—22no—
78C22×C4×C17—22no—
79C22×C4×C17—42no—
80C22×C4×C17—42no—
81C22×C4×C17C22⋊C412yes—
82C22×C4×C17C22⋊C412yes—
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D412yes—
85C22×C4×C17C2×D412yes—
86C22×C4×C17C2×D411yescharacteristic
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—24no—
90C22×Dic17—21no—
91C22×Dic17—21no—
92C22×Dic17—21no—
93C22×Dic17—21no—
94C22×Dic17—22no—
95C22×Dic17—22no—
96C22×Dic17—42no—
97C22×Dic17C22⋊C412yes—
98C22×Dic17C22⋊C412yes—
99C22×Dic17C22⋊C412yes—
100C2×C4×D17—24no—
101C2×C4×D17—24no—
102C2×C4×D17—28no—
103C2×C4×D17—44no—
104C2×C4×D17—44no—
105C2×C4×D17—48no—
106C2×C4×D17—44no—
107C2×Dic34—28no—
108C2×Dic34—48no—
109C2×Dic34—48no—
110C2×Dic34—44no—
111C2×Dic34—44no—
112C2×C8×C17—44no—
113C2×C8×C17—42no—
114C4⋊C4×C17—24no—
115C4⋊C4×C17—24no—
116#63227—28no—
117#63227—28no—
118#63225—28no—
119#63225—24no—
120#63225—24no—
121C2×#3622—44no—
122C2×#3622—42no—
123#8308—171nomaximal, Sylow
124C23×C17—22no—
125C23×C17—22no—
126C23×C17C2×C22⋊C411yescharacteristic
127C23×C17C2×C22⋊C411yescharacteristic
128C23×C17C2×C22⋊C411yescharacteristic
129C23×C17#13112yes—
130C23×C17#13112yes—
131C22×D17—24no—
132C22×D17—24no—
133C22×D17—22no—
134C22×D17—24no—
135C22×D17—24no—
136C22×D17—22no—
137C22×D17—48no—
138C22×D17—48no—
139C22×D17—42no—
140C22×D17—42no—
141C22×D17—42no—
142C22×D17—41no—
143C22×D17—41no—
144C22×D17#10114yes—
145C2×C4×C17—24no—
146C2×C4×C17—22no—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—41no—
151C2×C4×C17—42no—
152C2×C4×C17—41no—
153C2×C4×C17—42no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—42no—
157C2×C4×C17—42no—
158C2×C4×C17#12012yes—
159C2×C4×C17#12012yes—
160C2×Dic17—24no—
161C2×Dic17—24no—
162C2×Dic17—22no—
163C2×Dic17—22no—
164C2×Dic17—22no—
165C2×Dic17—22no—
166C2×Dic17—24no—
167C2×Dic17—42no—
168C2×Dic17—42no—
169C2×Dic17—41no—
170C2×Dic17—41no—
171C2×Dic17—42no—
172C2×Dic17—42no—
173C2×Dic17#9914yes—
174C4×D17—48no—
175C4×D17—48no—
176Dic34—416no—
177C8×C17—82no—
178#3622—82no—
179C22×C22⋊Q8—171no—
180#1779—172no—
181C2×#324—172no—
182C2×#324—171no—
183C2×#324D1711yescharacteristic, complemented
184C22×C17—24no—
185C22×C17—24no—
186C22×C17—21no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17#37911yescharacteristic
191C22×C17#40812yes—
192C22×C17#40812yes—
193C22×C17#40811yescharacteristic
194C22×C17#40811yescharacteristic
195C22×C17C2×#9912yes—
196C22×C17C2×#10112yes—
197C2×D17—28no—
198C2×D17—24no—
199C2×D17—48no—
200C2×D17—44no—
201C2×D17—44no—
202C2×D17—44no—
203C2×D17—42no—
204C2×D17—42no—
205C4×C17—22no—
206C4×C17—22no—
207C4×C17—42no—
208C4×C17—42no—
209Dic17—24no—
210Dic17—42no—
211Dic17—42no—
212C23×Q8—171no—
213C24×C4—171no—
214C2×C22⋊Q8—344no—
215C2×C22⋊Q8—344no—
216C2×C22⋊Q8—342no—
217C2×C22⋊Q8—342no—
218C22×C4⋊C4—171no—
219C22×C4⋊C4—171no—
220C22×C4⋊C4C2×D1711yescharacteristic
221C22×C22⋊C4—171no—
222C22×C22⋊C4—171no—
223C2×C22⋊C8—172no—
224C2×C22⋊C8—344no—
225C2×C22⋊C8C2×D1712yes—
226#324—22no—
227#324—344no—
228#324—342no—
229C2×C17—22no—
230C2×C17—21no—
231C2×C17—21no—
232C2×C17#147214yes—
233C2×C17#178011yescharacteristic
234C2×C17#178111yescharacteristic
235C2×C17#178411yescharacteristic
236D17—44no—
237C25—171no—
238C22×Q8—342no—
239C22×Q8—342no—
240C22×Q8—342no—
241C22×Q8—684no—
242C23×C4—171no—
243C23×C4—171no—
244C23×C4—171no—
245C23×C4—174no—
246C23×C4—341no—
247C23×C4—342no—
248C23×C4—341no—
249C23×C4—684no—
250C23×C4C4×D1711yescharacteristic
251C23×C4C4×D1711yescharacteristic
252C23×C4C22×D1711yescharacteristic
253C22×C8—22no—
254C22×C8—342no—
255C22⋊Q8—3416no—
256C22⋊Q8—3416no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—174no—
260C2×C4⋊C4—342no—
261C2×C4⋊C4—342no—
262C2×C4⋊C4—341no—
263C2×C4⋊C4—342no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4—342no—
266C2×C4⋊C4D6812yes—
267C2×C4⋊C4C22⋊D1712yes—
268C2×C22⋊C4—342no—
269C2×C22⋊C4—342no—
270C2×C22⋊C4—342no—
271C2×C22⋊C4—342no—
272C2×C22⋊C4—342no—
273C2×C22⋊C4—342no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—342no—
277C22⋊C8—342no—
278C22⋊C8—688no—
279C17#830811yescharacteristic, complemented, Sylow
280C24—172no—
281C24—344no—
282C24—341no—
283C24—341no—
284C24—342no—
285C24—342no—
286C24—682no—
287C24C2×C4×D1711yescharacteristic
288C2×Q8—348no—
289C2×Q8—684no—
290C2×Q8—688no—
291C2×Q8—684no—
292C2×Q8—688no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—172no—
302C22×C4—172no—
303C22×C4—172no—
304C22×C4—348no—
305C22×C4—344no—
306C22×C4—341no—
307C22×C4—341no—
308C22×C4—341no—
309C22×C4—341no—
310C22×C4—342no—
311C22×C4—342no—
312C22×C4—341no—
313C22×C4—341no—
314C22×C4—344no—
315C22×C4—344no—
316C22×C4—684no—
317C22×C4—684no—
318C22×C4—688no—
319C22×C4—684no—
320C22×C4—682no—
321C22×C4#6322712yes—
322C22×C4#6322712yes—
323C22×C4C2×D6811yescharacteristic
324C22×C4D4×D1712yes—
325C22×C4D4×D1712yes—
326C22×C4C2×C22⋊D1711yescharacteristic
327C2×C8—44no—
328C2×C8—42no—
329C2×C8—684no—
330C2×C8—682no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—348no—
334C4⋊C4—344no—
335C4⋊C4—344no—
336C22⋊C4—348no—
337C22⋊C4—348no—
338C23—22no—
339C23—22no—
340C23—174no—
341C23—344no—
342C23—344no—
343C23—344no—
344C23—344no—
345C23—342no—
346C23—342no—
347C23—688no—
348C23—688no—
349C23—682no—
350C23—682no—
351C23—682no—
352C23—681no—
353C23—681no—
354C23C22⋊C4×D1711yescharacteristic
355C23C22⋊C4×D1711yescharacteristic
356C23#1058758712yes—
357C23#1058759312yes—
358C23C2×#6322711yescharacteristic
359Q8—6816no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—174no—
374C2×C4—344no—
375C2×C4—344no—
376C2×C4—342no—
377C2×C4—342no—
378C2×C4—344no—
379C2×C4—342no—
380C2×C4—342no—
381C2×C4—688no—
382C2×C4—688no—
383C2×C4—682no—
384C2×C4—682no—
385C2×C4—681no—
386C2×C4—681no—
387C2×C4—682no—
388C2×C4—682no—
389C2×C4#1058756812yes—
390C2×C4#1058762312yes—
391C8—82no—
392C8—1362no—
393C22—24no—
394C22—24no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—348no—
400C22—344no—
401C22—688no—
402C22—684no—
403C22—684no—
404C22—684no—
405C22—682no—
406C22—682no—
407C22#1176266411yescharacteristic
408C22#1176277412yes—
409C22#1176277412yes—
410C22#1176277511yescharacteristic
411C22#1176277711yescharacteristic
412C22#99×D1712yes—
413C22#101×D1712yes—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—42no—
418C4—344no—
419C4—682no—
420C4—682no—
421C2—22no—
422C2—21no—
423C2—21no—
424C2—684no—
425C2#1452714614yes—
426C2#1452869511yescharacteristic
427C2#1452869711yescharacteristic
428C2#1452870611yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#324C2
socleC23×C17C2×C22⋊C4
prime radicalC24×D17C2×C4
cosocleC23×C4×C17C23
hypercenterC2×#324D17
hypocenterC17#8308
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. C2×#11762535C2semidirect
4. C2×#11762534C2semidirect
5. C2×C17×#324C2semidirect
183. C2×#324D17semidirect
279. C17#8308semidirect

Maximal subgroups

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

subgroupindex
C22×#105875822
#145286932
C2×#117625352
C2×#117625342
C2×C17×#3242
#830817