Finite Group Databasebeta

← #17953195

Subgroups — #17953195

Order 4352.

614 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 114 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#14544639C211yesmaximal, characteristic
2C2×#11763663C212yesmaximal
3#14543960C211yesmaximal, characteristic
4#14543657C212yesmaximal
5C4×#10587632C211yesmaximal, characteristic
6C2×C4⋊#3624—21no—
7C2×#10587543C2212yes—
8C2×(C2×#595)⋊C4—21no—
9C2×(C2×#595)⋊C4—21no—
10C2×(C2×#595)⋊C4C2212yes—
11#11763675—21no—
12#11763663—24no—
13C2×#10587632C2211yescharacteristic
14C2×C42×D17C2211yescharacteristic
15#11762716C411yescharacteristic, complemented
16#11762705C411yescharacteristic, complemented
17#11762690—21no—
18#11762680—21no—
19#11762666C2211yescharacteristic
20C4×#63225—21no—
21C4×C4⋊Dic17C411yescharacteristic, complemented
22C4×#63227—21no—
23C2×C4×Q8×C17C411yescharacteristic, complemented
24C23×#595—22no—
25C22×Q8×C17C2×C411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×C4×D17D411yescharacteristic
28C22×C4×D17C2311yescharacteristic
29C22×#3624—22no—
30C4×Q8×C17—42no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C42×C17—21no—
34C2×C42×C17C2×C411yescharacteristic
35#10587632—21no—
36#10587632—21no—
37#10587632D412yes—
38#10587632Q812yes—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17—21no—
46C2×C4×Dic17C2×C411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C2×C4×Dic17D411yescharacteristic
49C42×D17—42no—
50(C2×#595)⋊C4—24no—
51(C2×#595)⋊C4—22no—
52(C2×#595)⋊C4—22no—
53(C2×#595)⋊C4—42no—
54#10587543—22no—
55#10587543—22no—
56C4⋊#3624—42no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60#10587502—21no—
61#10587502—21no—
62C23×D17C2×D411yescharacteristic
63C22×#595—24no—
64C22×#595—42no—
65C22×#595—42no—
66C22×#595—42no—
67C22×#595—42no—
68C22×#595—42no—
69C22×#595—42no—
70C2×Q8×C17—21no—
71C2×Q8×C17—41no—
72C2×Q8×C17—41no—
73C2×Q8×C17C4⋊C412yes—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22×C411yescharacteristic
79C22×Dic17—21no—
80C22×Dic17—21no—
81C22×Dic17C22⋊C411yescharacteristic
82C22×Dic17C22⋊C411yescharacteristic
83C22×Dic17C2×D411yescharacteristic
84C2×C4×D17—22no—
85C2×C4×D17—22no—
86C2×C4×D17—22no—
87C2×C4×D17—22no—
88C2×C4×D17—41no—
89C2×C4×D17—41no—
90C2×C4×D17—41no—
91C2×C4×D17—41no—
92C2×C4×D17C2×D411yescharacteristic
93C2×C4×D17C2×Q811yescharacteristic
94C2×C4×D17#4111yescharacteristic
95C2×C4×D17#4111yescharacteristic
96C2×#3624—42no—
97C2×#3624—42no—
98C2×#3624—42no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—42no—
101C42×C17—22no—
102C42×C17—42no—
103#63227—22no—
104#63227—22no—
105#63227—42no—
106C4⋊Dic17—21no—
107C4⋊Dic17—41no—
108C4⋊Dic17C4⋊C412yes—
109#63225—22no—
110#63225—22no—
111#63225—42no—
112C4×Dic17—22no—
113C4×Dic17—41no—
114C4×Dic17—41no—
115C4×Dic17—42no—
116#8574—171nomaximal, Sylow
117C23×C17C2×C22⋊C411yescharacteristic
118C22×D17—21no—
119C22×D17—21no—
120C22×D17—21no—
121C22×D17—21no—
122C22×D17#12111yescharacteristic
123C22×D17C22⋊Q811yescharacteristic
124C22×D17#13512yes—
125C22×D17#13712yes—
126C2×#595—44no—
127C2×#595—44no—
128C2×#595—44no—
129C2×#595—82no—
130C2×#595—82no—
131C2×#595—82no—
132C2×#595—82no—
133Q8×C17—42no—
134Q8×C17—42no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—22no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17#9912yes—
148C2×C4×C17#10412yes—
149C2×C4×C17C2×C4⋊C411yescharacteristic
150C2×C4×C17#11711yescharacteristic
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—22no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—21no—
157C2×Dic17—21no—
158C2×Dic17—21no—
159C2×Dic17—21no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165C2×Dic17#12111yescharacteristic
166C2×Dic17C22⋊Q811yescharacteristic
167C4×D17—41no—
168C4×D17—41no—
169C4×D17—41no—
170C4×D17—41no—
171C4×D17—44no—
172C4×D17—44no—
173#3624—82no—
174C4×C22⋊Q8—171no—
175#1797—171no—
176C2×#328—172no—
177#1434—171no—
178#1294—172no—
179C22×C17#38011yescharacteristic
180C22×C17C2×#9911yescharacteristic
181C22×C17#41011yescharacteristic
182C22×C17#41611yescharacteristic
183C22×C17#41911yescharacteristic
184C22×C17C2×#10411yescharacteristic
185C22×C17#42111yescharacteristic
186C2×D17—21no—
187C2×D17—21no—
188C2×D17—21no—
189C2×D17—21no—
190C2×D17—22no—
191C2×D17—22no—
192C2×D17—22no—
193C2×D17—22no—
194C2×D17#46412yes—
195C2×D17#47912yes—
196#595—84no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—22no—
200C4×C17—22no—
201C4×C17—42no—
202C4×C17—81no—
203Dic17—21no—
204Dic17—21no—
205Dic17—42no—
206Dic17—41no—
207Dic17—41no—
208Dic17—81no—
209C2×C22⋊Q8—171no—
210C2×C4×Q8#59511yescharacteristic, complemented
211C22×C42—171no—
212C2×C4⋊C8—341no—
213C2×C22⋊C8—172no—
214#391—171no—
215#389—171no—
216#387—341no—
217#385—341no—
218#380—171no—
219C4×C4⋊C4—171no—
220C4×C4⋊C4—341no—
221C4×C22⋊C4—341no—
222C2×#95—172no—
223C2×#95—341no—
224C2×#95—341no—
225#336—341no—
226#328—344no—
227C2×C17#150011yescharacteristic
228C2×C17#150811yescharacteristic
229C2×C17#151511yescharacteristic
230C2×C17#151611yescharacteristic
231C2×C17#179911yescharacteristic
232C2×C17#180511yescharacteristic
233C2×C17#180611yescharacteristic
234D17—22no—
235D17—22no—
236C22×Q8C2×#59511yescharacteristic
237C23×C4—171no—
238C23×C4—171no—
239C23×C4—171no—
240C23×C4—342no—
241C22×C8—342no—
242C22⋊Q8—172no—
243C22⋊Q8—172no—
244C22⋊Q8—341no—
245C22⋊Q8—341no—
246C4×Q8—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—171no—
249C2×C4⋊C4—171no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—341no—
252C2×C4⋊C4C2×#59511yescharacteristic
253C2×C22⋊C4—341no—
254C2×C22⋊C4—341no—
255C2×C42—21no—
256C2×C42—171no—
257C2×C42—171no—
258C2×C42—171no—
259C2×C42—341no—
260C2×C42—682no—
261C2×C42C2×#59511yescharacteristic
262C4⋊C8—682no—
263C22⋊C8—342no—
264C22⋊C8—342no—
265#95—341no—
266#95—341no—
267#95—341no—
268#95—341no—
269#95—342no—
270#95—342no—
271#95—344no—
272#95—341no—
273#95—682no—
274C17#857411yescharacteristic, complemented, Sylow
275C24—171no—
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8C4⋊#59512yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—171no—
283C22×C4—171no—
284C22×C4—171no—
285C22×C4—171no—
286C22×C4—171no—
287C22×C4—171no—
288C22×C4—171no—
289C22×C4—342no—
290C22×C4—342no—
291C22×C4—341no—
292C22×C4—342no—
293C22×C4—342no—
294C22×C4—344no—
295C22×C4—341no—
296C22×C4—681no—
297C22×C4—681no—
298C22×C4—681no—
299C22×C4—681no—
300C22×C4—682no—
301C22×C4—682no—
302C22×C4—682no—
303C22×C4—682no—
304C22×C4—682no—
305C22×C4—682no—
306C22×C4C22⋊#59511yescharacteristic
307C22×C4C22⋊#59511yescharacteristic
308C22×C4C22×#59511yescharacteristic
309C2×C8—682no—
310C2×C8—682no—
311C2×C8—682no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—172no—
315C4⋊C4—342no—
316C4⋊C4—342no—
317C4⋊C4—341no—
318C4⋊C4—681no—
319C4⋊C4—682no—
320C22⋊C4—342no—
321C22⋊C4—342no—
322C22⋊C4—682no—
323C42—22no—
324C42—42no—
325C42—342no—
326C42—681no—
327C42—681no—
328C42—682no—
329C23—171no—
330C23—171no—
331C23—172no—
332C23—172no—
333C23—341no—
334C23—341no—
335C23—341no—
336C23—341no—
337C23C2×C22⋊#59511yescharacteristic
338Q8—42no—
339Q8—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—171no—
353C2×C4—171no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—342no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—341no—
362C2×C4—341no—
363C2×C4—681no—
364C2×C4—681no—
365C2×C4—681no—
366C2×C4—681no—
367C2×C4—684no—
368C2×C4—684no—
369C2×C4—681no—
370C2×C4—681no—
371C2×C4—684no—
372C2×C4—684no—
373C2×C4—684no—
374C2×C4—681no—
375C2×C4—681no—
376C2×C4—681no—
377C2×C4—1362no—
378C2×C4—1362no—
379C2×C4—1362no—
380C2×C4—1362no—
381C2×C4#1058755012yes—
382C2×C4#1058755112yes—
383C2×C4C2×C4⋊#59511yescharacteristic
384C2×C4#1058767711yescharacteristic
385C8—1362no—
386C22—172no—
387C22—172no—
388C22—341no—
389C22—341no—
390C22—341no—
391C22—341no—
392C22—342no—
393C22—342no—
394C22—342no—
395C22—342no—
396C22#1176375111yescharacteristic
397C22C2×#1058755111yescharacteristic
398C22#1176378811yescharacteristic
399C22#1176379911yescharacteristic
400C22#1176380211yescharacteristic
401C22C2×#1058755011yescharacteristic
402C22#1176380611yescharacteristic
403C4—21no—
404C4—21no—
405C4—22no—
406C4—22no—
407C4—42no—
408C4—81no—
409C4—341no—
410C4—341no—
411C4—682no—
412C4—681no—
413C4—681no—
414C4—1364no—
415C4—1361no—
416C2—342no—
417C2—342no—
418C2#1454415711yescharacteristic
419C2#1454418411yescharacteristic
420C2#1454420611yescharacteristic
421C2#1454421011yescharacteristic
422C2#1454464411yescharacteristic
423C2#1454465511yescharacteristic
424C2#1454465711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C4×Q8×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenterC2×C4×Q8#595
hypocenterC17#8574
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
15. #11762716C4semidirect
16. #11762705C4semidirect
21. C4×C4⋊Dic17C4semidirect
23. C2×C4×Q8×C17C4semidirect
210. C2×C4×Q8#595semidirect
274. C17#8574semidirect

Maximal subgroups

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

subgroupindex
#145446392
C2×#117636632
#145439602
#145436572
C4×#105876322
#857417