Finite Group Databasebeta

← #17951782

Subgroups — #17951782

Order 4352.

710 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 136 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#14544510C211yesmaximal, characteristic, complemented
2C4×C22⋊#3624C211yesmaximal, characteristic, complemented
3C2×#11763653C212yesmaximal
4#14543622C212yesmaximal, complemented
5C2×#11762699C211yesmaximal, characteristic
6C2×C22⋊#3624C2212yes—
7C2×C22⋊#3624C2212yes—
8C2×#10587542—21no—
9C2×C4×#3624—21no—
10#11763675—21no—
11#11763675—21no—
12#11763653C418yescomplemented
13C22×#63227C411yescharacteristic
14C22×#63227C411yescharacteristic
15C22×#63227C2211yescharacteristic
16C22×C4×Dic17C2211yescharacteristic
17#11762699—21no—
18#11762699—21no—
19#11762699—21no—
20#11762699—21no—
21C2×#10587502C411yescharacteristic
22C2×#10587502C2211yescharacteristic
23C22×C4⋊C4×C17C411yescharacteristic
24C24×D17C2×C411yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17C2×C411yescharacteristic
27C23×C4×C17C2×C411yescharacteristic
28C23×Dic17C2×C411yescharacteristic
29C23×Dic17C2×C411yescharacteristic
30C23×Dic17C2311yescharacteristic
31C22×#3624—22no—
32C22×#3624—22no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—42no—
36C2×#63227—21no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×#63227—22no—
43C2×#63227—42no—
44C2×#63227—42no—
45C2×#63227C2×C414yes—
46C2×C4×Dic17—42no—
47C2×C4×Dic17D411yescharacteristic
48C2×C4×Dic17D411yescharacteristic
49C2×C4×Dic17D411yescharacteristic
50C2×C4×Dic17D411yescharacteristic
51C22⋊#3624—22no—
52C22⋊#3624—22no—
53C22⋊#3624C2×C414yes—
54C22⋊#3624C2×C414yes—
55#10587542—42no—
56C4×#3624—42no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60#10587502—21no—
61C24×C17C22×C411yescharacteristic
62C23×D17—21no—
63C23×D17—21no—
64C23×D17—22no—
65C23×D17—22no—
66C23×D17—41no—
67C23×D17—41no—
68C23×D17—41no—
69C23×D17—41no—
70C23×D17C4212yes—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17—21no—
76C22×C4×C17—22no—
77C22×C4×C17—41no—
78C22×C4×C17—41no—
79C22×C4×C17—41no—
80C22×C4×C17—41no—
81C22×C4×C17—41no—
82C22×C4×C17C4212yes—
83C22×C4×C17C22⋊C411yescharacteristic
84C22×C4×C17C22⋊C411yescharacteristic
85C22×Dic17—22no—
86C22×Dic17—22no—
87C22×Dic17—21no—
88C22×Dic17—22no—
89C22×Dic17—41no—
90C22×Dic17—41no—
91C22×Dic17—41no—
92C22×Dic17—41no—
93C22×Dic17—41no—
94C22×Dic17C22⋊C411yescharacteristic
95C22×Dic17C22⋊C411yescharacteristic
96C22×Dic17C22×C412yes—
97C22×Dic17C2×D411yescharacteristic
98C22×Dic17C2×D411yescharacteristic
99C22×Dic17#4112yes—
100C2×#3624—24no—
101C2×#3624—24no—
102C2×#3624—42no—
103C2×#3624—42no—
104C2×#3624—42no—
105C2×#3624—42no—
106C4⋊C4×C17—42no—
107C4⋊C4×C17—42no—
108#63227—24no—
109#63227—24no—
110#63227—42no—
111#63227—42no—
112#63227—42no—
113#63227—42no—
114C4×Dic17—22no—
115C4×Dic17—22no—
116C4×Dic17—41no—
117C4×Dic17—41no—
118#8155—171nomaximal, Sylow
119C23×C17—21no—
120C23×C17—21no—
121C23×C17—21no—
122C23×C17—21no—
123C23×C17C2×C4211yescharacteristic
124C23×C17C2×C22⋊C411yescharacteristic
125C23×C17#11711yescharacteristic
126C23×C17#13112yes—
127C23×C17#13112yes—
128C22×D17—22no—
129C22×D17—24no—
130C22×D17—42no—
131C22×D17—42no—
132C22×D17—41no—
133C22×D17—41no—
134C22×D17—41no—
135C22×D17—41no—
136C22×D17—41no—
137C22×D17—41no—
138C22×D17—42no—
139C22×D17—42no—
140C22×D17—42no—
141C22×D17—82no—
142C22×D17—82no—
143C22×D17—82no—
144C22×D17—82no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—22no—
148C2×C4×C17—21no—
149C2×C4×C17—21no—
150C2×C4×C17—21no—
151C2×C4×C17—44no—
152C2×C4×C17—42no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—42no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—82no—
160C2×C4×C17#10012yes—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17—21no—
166C2×Dic17—21no—
167C2×Dic17—21no—
168C2×Dic17—21no—
169C2×Dic17—21no—
170C2×Dic17—42no—
171C2×Dic17—42no—
172C2×Dic17—44no—
173C2×Dic17—41no—
174C2×Dic17—41no—
175C2×Dic17—41no—
176C2×Dic17—82no—
177C2×Dic17#9912yes—
178C2×Dic17C4×D412yes—
179C2×Dic17C4×D412yes—
180#3624—44no—
181#3624—44no—
182C2×#388—171no—
183#1749—171no—
184C4×C22⋊C8—171no—
185C2×#323—172no—
186#1280—172no—
187C22×C17—22no—
188C22×C17—22no—
189C22×C17—22no—
190C22×C17—22no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17C4×C22⋊C411yescharacteristic
196C22×C17#40512yes—
197C22×C17#40811yescharacteristic
198C22×C17#40811yescharacteristic
199C22×C17C2×#9911yescharacteristic
200C22×C17#41011yescharacteristic
201C22×C17C2×#10011yescharacteristic
202C22×C17#41311yescharacteristic
203C22×C17#43312yes—
204C2×D17—42no—
205C2×D17—42no—
206C2×D17—42no—
207C2×D17—84no—
208C2×D17—81no—
209C2×D17—81no—
210C2×D17—81no—
211C2×D17—81no—
212C2×D17—84no—
213C4×C17—22no—
214C4×C17—41no—
215C4×C17—42no—
216C4×C17—81no—
217Dic17—22no—
218Dic17—22no—
219Dic17—22no—
220Dic17—41no—
221Dic17—81no—
222C22×C4⋊C4#59511yescharacteristic
223C22×C22⋊C4—171no—
224C22×C22⋊C4—171no—
225C22×C22⋊C4—171no—
226C22×C42—171no—
227C2×C22⋊C8—172no—
228C2×C22⋊C8—172no—
229C2×#97—341no—
230C2×C4×C8—341no—
231#388—341no—
232#388—341no—
233#388—341no—
234#388—341no—
235C2×#95—171no—
236C2×#95—171no—
237#336—341no—
238#336—341no—
239#323—178no—
240C2×C17—22no—
241C2×C17—22no—
242C2×C17#146811yescharacteristic
243C2×C17#146911yescharacteristic
244C2×C17#147111yescharacteristic
245C2×C17#147311yescharacteristic
246C2×C17#175011yescharacteristic
247C2×C17C4×#9911yescharacteristic
248C2×C17C4×#10011yescharacteristic
249D17—82no—
250C25—171no—
251C23×C4—171no—
252C23×C4—171no—
253C23×C4—171no—
254C23×C4C2×#59511yescharacteristic
255C23×C4C2×#59511yescharacteristic
256C23×C4C2×#59511yescharacteristic
257C22×C8—342no—
258C22×C8—342no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—42no—
262C2×C22⋊C4—174no—
263C2×C22⋊C4—341no—
264C2×C22⋊C4—341no—
265C2×C22⋊C4—342no—
266C2×C22⋊C4—341no—
267C2×C22⋊C4—341no—
268C2×C22⋊C4—341no—
269C2×C22⋊C4—341no—
270C2×C22⋊C4—682no—
271C2×C22⋊C4—682no—
272C2×C42—171no—
273C2×C42—171no—
274C2×C42—171no—
275C2×C42—171no—
276C2×C42—682no—
277C22⋊C8—174no—
278C22⋊C8—174no—
279C22⋊C8—342no—
280C22⋊C8—342no—
281#97—682no—
282C4×C8—682no—
283#95—341no—
284#95—341no—
285#95—341no—
286#95—341no—
287C17#815511yescharacteristic, complemented, Sylow
288C24—172no—
289C24—341no—
290C24—341no—
291C24—342no—
292C24—342no—
293C24—681no—
294C24—681no—
295C24—681no—
296C24—681no—
297C24C22×#59511yescharacteristic
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—171no—
310C22×C4—171no—
311C22×C4—171no—
312C22×C4—172no—
313C22×C4—172no—
314C22×C4—171no—
315C22×C4—342no—
316C22×C4—342no—
317C22×C4—341no—
318C22×C4—342no—
319C22×C4—681no—
320C22×C4—681no—
321C22×C4—681no—
322C22×C4—681no—
323C22×C4—681no—
324C22×C4C4×#59512yes—
325C22×C4C22⋊#59511yescharacteristic
326C22×C4C22⋊#59511yescharacteristic
327C2×C8—344no—
328C2×C8—344no—
329C2×C8—682no—
330C2×C8—682no—
331C2×C8—682no—
332C2×C8—682no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C22⋊C4—344no—
336C22⋊C4—344no—
337C22⋊C4—682no—
338C22⋊C4—682no—
339C22⋊C4—682no—
340C22⋊C4—682no—
341C42—342no—
342C42—342no—
343C42—681no—
344C42—681no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—342no—
350C23—344no—
351C23—682no—
352C23—682no—
353C23—682no—
354C23—682no—
355C23—681no—
356C23—681no—
357C23—681no—
358C23—681no—
359C23—681no—
360C23—681no—
361C23—682no—
362C23—1362no—
363C23—1362no—
364C23—1362no—
365C23—1362no—
366C23C2×C4×#59511yescharacteristic
367C23#1058767711yescharacteristic
368C23#1058767812yes—
369C23#1058768012yes—
370C23C2×C22⋊#59511yescharacteristic
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—172no—
387C2×C4—172no—
388C2×C4—172no—
389C2×C4—341no—
390C2×C4—341no—
391C2×C4—341no—
392C2×C4—341no—
393C2×C4—341no—
394C2×C4—341no—
395C2×C4—341no—
396C2×C4—341no—
397C2×C4—341no—
398C2×C4—682no—
399C2×C4—682no—
400C2×C4—684no—
401C2×C4—681no—
402C2×C4—681no—
403C2×C4—681no—
404C2×C4—1362no—
405C2×C4#1058755312yes—
406C8—684no—
407C8—684no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—682no—
417C22—682no—
418C22—682no—
419C22—1364no—
420C22—1364no—
421C22—1361no—
422C22—1361no—
423C22—1361no—
424C22—1361no—
425C22C4×C22⋊#59511yescharacteristic
426C22#1176377412yes—
427C22#1176378211yescharacteristic
428C22#1176378411yescharacteristic
429C22D17⋊#9911yescharacteristic
430C22#1176379011yescharacteristic
431C22C2×#1058755311yescharacteristic
432C22#1176379511yescharacteristic
433C22#1176383412yes—
434C4—22no—
435C4—42no—
436C4—41no—
437C4—81no—
438C4—342no—
439C4—342no—
440C4—342no—
441C4—681no—
442C4—1361no—
443C2—22no—
444C2—22no—
445C2—1362no—
446C2#1454405211yescharacteristic
447C2#1454405511yescharacteristic
448C2#1454406011yescharacteristic
449C2#1454406811yescharacteristic
450C2#1454451311yescharacteristic
451C2#1454452911yescharacteristic
452C2C4×#1058755311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC23×C17C2×C42
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×D17C2×C4
cosocleC23×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8155
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #14544510C2semidirect
2. C4×C22⋊#3624C2semidirect
4. #14543622C2semidirect
12. #11763653C4semidirect
287. C17#8155semidirect

Maximal subgroups

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

subgroupindex
#145445102
C4×C22⋊#36242
C2×#117636532
#145436222
C2×#117626992
#815517