Finite Group Databasebeta

← #17953561

Subgroups — #17953561

Order 4352.

940 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 126 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#14544607C211yesmaximal, characteristic
2#14543958C211yesmaximal, characteristic
3#14543657C214yesmaximal, complemented
4C2×#11762666C211yesmaximal, characteristic
5C22×C22⋊#595—21no—
6C2×#10587543—21no—
7C2×#10587543C2212yes—
8C2×C22⋊#3624—21no—
9C2×(C2×#595)⋊C4—21no—
10C2×(C2×#595)⋊C4C2212yes—
11C23×C4×D17C2211yescharacteristic
12C22×C4⋊Dic17C411yescharacteristic, complemented
13C22×#63227—21no—
14#11762666—21no—
15#11762666—21no—
16#11762666C412yescomplemented
17#11762666C2212yes—
18C2×#10587502—21no—
19C22×C4⋊C4×C17C411yescharacteristic, complemented
20C24×D17D411yescharacteristic
21C23×#595—22no—
22C23×C4×C17—21no—
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17—21no—
25C23×Dic17D411yescharacteristic
26C22×C4×D17—24no—
27C22×C4×D17—42no—
28C22×C4×D17—42no—
29C22×C4×D17C2×C411yescharacteristic
30C22×C4×D17D411yescharacteristic
31C22×C4×D17D411yescharacteristic
32C22×C4×D17C2311yescharacteristic
33C2×C22⋊#595—24no—
34C2×C22⋊#595—42no—
35C2×C22⋊#595—42no—
36C22×#3624—22no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—42no—
39C2×C4⋊C4×C17C2×C412yes—
40C2×#63227—22no—
41C2×#63227—22no—
42C2×#63227—44no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—42no—
45C2×C4⋊Dic17C2×C412yes—
46C22⋊#3624—42no—
47(C2×#595)⋊C4—24no—
48(C2×#595)⋊C4—42no—
49#10587543—24no—
50#10587543—42no—
51#10587502—22no—
52#10587502—22no—
53C24×C17C22⋊C411yescharacteristic
54C23×D17—22no—
55C23×D17—22no—
56C23×D17—22no—
57C23×D17—22no—
58C23×D17—41no—
59C23×D17—41no—
60C23×D17C22⋊C411yescharacteristic
61C23×D17SD1612yes—
62C23×D17Q1612yes—
63C23×D17C2×D411yescharacteristic
64C22×#595—24no—
65C22×#595—42no—
66C22×#595—42no—
67C22×#595—42no—
68C22×#595—42no—
69C22×#595—42no—
70C22×#595—42no—
71C22×C4×C17—22no—
72C22×C4×C17—22no—
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17—42no—
76C22×C4×C17—42no—
77C22×C4×C17—42no—
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22×C411yescharacteristic
80C22×Dic17—22no—
81C22×Dic17—22no—
82C22×Dic17—22no—
83C22×Dic17—41no—
84C22×Dic17—42no—
85C22×Dic17—42no—
86C22×Dic17—42no—
87C22×Dic17C22⋊C411yescharacteristic
88C22×Dic17C2×D411yescharacteristic
89C2×C4×D17—22no—
90C2×C4×D17—22no—
91C2×C4×D17—22no—
92C2×C4×D17—22no—
93C2×C4×D17—22no—
94C2×C4×D17—41no—
95C2×C4×D17—41no—
96C2×C4×D17—41no—
97C2×C4×D17—41no—
98C2×C4×D17—41no—
99C2×C4×D17—41no—
100C2×C4×D17—416no—
101C2×C4×D17#4114yes—
102C22⋊#595—48no—
103C2×#3624—42no—
104C2×#3624—42no—
105C2×#3624—42no—
106C4⋊C4×C17—42no—
107C4⋊C4×C17—42no—
108#63227—44no—
109#63227—44no—
110C4⋊Dic17—42no—
111C4⋊Dic17—42no—
112#8713—171nomaximal, Sylow
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17#9912yes—
118C23×C17#10312yes—
119C23×C17#10412yes—
120C23×C17C2×C22⋊C411yescharacteristic
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—24no—
126C22×D17—24no—
127C22×D17—21no—
128C22×D17—21no—
129C22×D17—21no—
130C22×D17—21no—
131C22×D17—21no—
132C22×D17—21no—
133C22×D17—24no—
134C22×D17—24no—
135C22×D17—42no—
136C22×D17—42no—
137C22×D17—44no—
138C22×D17—44no—
139C22×D17—44no—
140C22×D17—44no—
141C22×D17#10312yes—
142C22×D17#12312yes—
143C22×D17C2×SD1611yescharacteristic
144C22×D17C2×Q1611yescharacteristic
145C22×D17AGL(1,ℤ/8)12yes—
146C2×#595—44no—
147C2×#595—44no—
148C2×#595—44no—
149C2×#595—82no—
150C2×#595—82no—
151C2×#595—82no—
152C2×#595—82no—
153C2×C4×C17—28no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17—42no—
161C2×C4×C17—84no—
162C2×C4×C17#9912yes—
163C2×C4×C17#11712yes—
164C2×Dic17—21no—
165C2×Dic17—21no—
166C2×Dic17—21no—
167C2×Dic17—21no—
168C2×Dic17—21no—
169C2×Dic17—44no—
170C2×Dic17—42no—
171C2×Dic17—42no—
172C2×Dic17—42no—
173C2×Dic17—84no—
174C2×Dic17#12312yes—
175C4×D17—44no—
176C4×D17—44no—
177C4×D17—44no—
178C4×D17—44no—
179#3624—82no—
180C2×#380—171no—
181#1779—171no—
182C22⋊#95—171no—
183#1294—174no—
184C22×C17—22no—
185C22×C17—22no—
186C22×C17—22no—
187C22×C17—22no—
188C22×C17—22no—
189C22×C17—22no—
190C22×C17#32814yes—
191C22×C17#38111yescharacteristic
192C22×C17C2×#9911yescharacteristic
193C22×C17C2×#9911yescharacteristic
194C22×C17C2×#10311yescharacteristic
195C22×C17#41811yescharacteristic
196C22×C17C2×#10411yescharacteristic
197C22×C17#42111yescharacteristic
198C2×D17—22no—
199C2×D17—22no—
200C2×D17—21no—
201C2×D17—21no—
202C2×D17—22no—
203C2×D17—22no—
204C2×D17—21no—
205C2×D17—21no—
206C2×D17—22no—
207C2×D17—22no—
208C2×D17—22no—
209C2×D17—48no—
210C2×D17—48no—
211C2×D17—41no—
212C2×D17—41no—
213C2×D17—41no—
214C2×D17—41no—
215C2×D17#48112yes—
216C2×D17#48412yes—
217#595—84no—
218C4×C17—22no—
219C4×C17—22no—
220C4×C17—82no—
221Dic17—22no—
222Dic17—41no—
223Dic17—82no—
224C24×C4—171no—
225C22×C4⋊C4—171no—
226C22×C4⋊C4#59511yescharacteristic, complemented
227C22×C22⋊C4—341no—
228C22×C22⋊C4—341no—
229C2×C22⋊C8—172no—
230C2×C22⋊C8—341no—
231C2×C22⋊C8—341no—
232#380—172no—
233#380—172no—
234#380—341no—
235#380—341no—
236C2×#95—172no—
237C2×#95—341no—
238C2×#95—341no—
239C2×C17—24no—
240C2×C17C2×#32811yescharacteristic
241C2×C17#150711yescharacteristic
242C2×C17#151011yescharacteristic
243C2×C17#152011yescharacteristic
244C2×C17#181911yescharacteristic
245C2×C17#182411yescharacteristic
246C2×C17#182611yescharacteristic
247D17—22no—
248D17—22no—
249D17—42no—
250C25—171no—
251C23×C4—21no—
252C23×C4—171no—
253C23×C4—171no—
254C23×C4—171no—
255C23×C4—171no—
256C23×C4—171no—
257C23×C4—344no—
258C23×C4—342no—
259C23×C4—341no—
260C23×C4—682no—
261C23×C4—682no—
262C23×C4C2×#59511yescharacteristic
263C22×C8—342no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—172no—
267C2×C4⋊C4—341no—
268C2×C4⋊C4—682no—
269C2×C4⋊C4C2×#59512yes—
270C2×C22⋊C4—342no—
271C2×C22⋊C4—342no—
272C2×C22⋊C4—344no—
273C2×C22⋊C4—684no—
274C2×C22⋊C4—682no—
275C2×C22⋊C4—682no—
276C22⋊C8—344no—
277C22⋊C8—682no—
278C22⋊C8—682no—
279#95—342no—
280#95—342no—
281#95—344no—
282#95—682no—
283C17#871311yescharacteristic, complemented, Sylow
284C24—171no—
285C24—171no—
286C24—172no—
287C24—172no—
288C24—342no—
289C24—342no—
290C24—342no—
291C24—342no—
292C24—681no—
293C24—681no—
294C24C22⋊#59511yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—171no—
303C22×C4—174no—
304C22×C4—171no—
305C22×C4—342no—
306C22×C4—342no—
307C22×C4—342no—
308C22×C4—342no—
309C22×C4—342no—
310C22×C4—342no—
311C22×C4—342no—
312C22×C4—344no—
313C22×C4—342no—
314C22×C4—681no—
315C22×C4—681no—
316C22×C4—681no—
317C22×C4—681no—
318C22×C4—681no—
319C22×C4—681no—
320C22×C4—6816no—
321C22×C4—681no—
322C22×C4—682no—
323C22×C4—682no—
324C22×C4—682no—
325C22×C4—682no—
326C22×C4—682no—
327C22×C4—682no—
328C22×C4—682no—
329C22×C4—682no—
330C22×C4—682no—
331C22×C4C22⋊#59511yescharacteristic
332C22×C4C22×#59511yescharacteristic
333C2×C8—682no—
334C2×C8—682no—
335C2×C8—682no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—682no—
339C4⋊C4—682no—
340C22⋊C4—684no—
341C22⋊C4—684no—
342C22⋊C4—688no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—172no—
348C23—171no—
349C23—171no—
350C23—172no—
351C23—172no—
352C23—344no—
353C23—344no—
354C23—341no—
355C23—341no—
356C23—341no—
357C23—341no—
358C23—341no—
359C23—341no—
360C23—341no—
361C23—341no—
362C23—341no—
363C23—341no—
364C23—344no—
365C23—344no—
366C23—682no—
367C23—682no—
368C23—684no—
369C23—684no—
370C23—684no—
371C23—684no—
372C23#1058753912yes—
373C23#1058754812yes—
374C23#1058754912yes—
375C23C2×C22⋊#59511yescharacteristic
376C2×C4—28no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—84no—
385C2×C4—172no—
386C2×C4—341no—
387C2×C4—341no—
388C2×C4—341no—
389C2×C4—341no—
390C2×C4—341no—
391C2×C4—684no—
392C2×C4—684no—
393C2×C4—684no—
394C2×C4—684no—
395C2×C4—684no—
396C2×C4—684no—
397C2×C4—684no—
398C2×C4—684no—
399C2×C4—682no—
400C2×C4—682no—
401C2×C4—682no—
402C2×C4—1364no—
403C2×C4—1362no—
404C2×C4—1362no—
405C2×C4—1362no—
406C2×C4—1362no—
407C2×C4#1058755112yes—
408C2×C4#1058767712yes—
409C8—1362no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—172no—
417C22—172no—
418C22—341no—
419C22—341no—
420C22—342no—
421C22—342no—
422C22—342no—
423C22—342no—
424C22—341no—
425C22—341no—
426C22—342no—
427C22—342no—
428C22—342no—
429C22—688no—
430C22—688no—
431C22—681no—
432C22—681no—
433C22—681no—
434C22—681no—
435C22#1176366214yes—
436C22#1176375211yescharacteristic
437C22C2×#1058755111yescharacteristic
438C22C2×#1058753911yescharacteristic
439C22C2×#1058754911yescharacteristic
440C22#1176380111yescharacteristic
441C22C2×#1058754811yescharacteristic
442C22#1176380811yescharacteristic
443C4—22no—
444C4—22no—
445C4—82no—
446C4—342no—
447C4—681no—
448C4—1364no—
449C4—1362no—
450C2—24no—
451C2—342no—
452C2—342no—
453C2—682no—
454C2C2×#1176366211yescharacteristic
455C2#1454418011yescharacteristic
456C2#1454419111yescharacteristic
457C2#1454422411yescharacteristic
458C2#1454467911yescharacteristic
459C2#1454468811yescharacteristic
460C2#1454469011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×D17D4
cosocleC22×C4×D17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8713
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #14543657C2semidirect
12. C22×C4⋊Dic17C4semidirect
16. #11762666C4semidirect
19. C22×C4⋊C4×C17C4semidirect
226. C22×C4⋊C4#595semidirect
283. C17#8713semidirect

Maximal subgroups

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

subgroupindex
#145446072
#145439582
#145436572
C2×#117626662
#871317