Finite Group Databasebeta

← #17952070

Subgroups — #17952070

Order 4352.

720 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 130 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#14544511C211yesmaximal, characteristic
2C2×#11763663C212yesmaximal
3C4×(C2×#595)⋊C4C211yesmaximal, characteristic
4#14543657C212yesmaximal
5#14531322C211yesmaximal, characteristic
6C22×C4×#595—21no—
7C2×#10587543C2212yes—
8C2×#10587542—21no—
9C2×(C2×#595)⋊C4—21no—
10C2×(C2×#595)⋊C4C2212yes—
11#11763675—21no—
12#11763663C418yescomplemented
13C2×#10587632C2211yescharacteristic
14C2×Q8×Dic17C411yescharacteristic, complemented
15C2×C42×D17C2211yescharacteristic
16#11762707C411yescharacteristic, complemented
17#11762696—21no—
18#11762683—21no—
19#11762666C2211yescharacteristic
20C4⋊C4×Dic17C411yescharacteristic, complemented
21#11762660—21no—
22#11762656—21no—
23C17×#391C411yescharacteristic, complemented
24C23×#595—22no—
25C22×Q8×C17C2×C411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×C4×D17D411yescharacteristic
28C22×C4×D17C2311yescharacteristic
29C22×#3624—22no—
30C2×C4×#595—24no—
31C2×C4×#595—42no—
32C2×C4×#595—42no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35#10587632—22no—
36#10587632C2×C414yes—
37Q8×Dic17—42no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17—21no—
46C2×C4×Dic17—21no—
47C2×C4×Dic17C2×C411yescharacteristic
48C2×C4×Dic17D411yescharacteristic
49C2×C4×Dic17D411yescharacteristic
50C42×D17—42no—
51(C2×#595)⋊C4—24no—
52(C2×#595)⋊C4—22no—
53(C2×#595)⋊C4C2×C414yes—
54#10587543—22no—
55#10587543C2×C414yes—
56#10587542—42no—
57C17×#95—21no—
58C17×#95—21no—
59#10587502—21no—
60#10587502—21no—
61#10587502—21no—
62C23×D17C2×D411yescharacteristic
63C22×#595—24no—
64C22×#595—24no—
65C22×#595—24no—
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×C17C4212yescomplemented
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—21no—
85C2×C4×D17—21no—
86C2×C4×D17—22no—
87C2×C4×D17—22no—
88C2×C4×D17—22no—
89C2×C4×D17—22no—
90C2×C4×D17—41no—
91C2×C4×D17—41no—
92C2×C4×D17—41no—
93C2×C4×D17C22×C412yes—
94C2×C4×D17#4112yes—
95C2×#3624—24no—
96C2×#3624—42no—
97C2×#3624—42no—
98C4×#595—48no—
99C4⋊C4×C17—41no—
100C42×C17—41no—
101#63227—24no—
102#63227—41no—
103#63227—41no—
104C4⋊Dic17—21no—
105C4⋊Dic17—42no—
106C4⋊Dic17—41no—
107C4⋊Dic17C4212yescomplemented
108#63225—24no—
109#63225—42no—
110C4×Dic17—22no—
111C4×Dic17—22no—
112C4×Dic17—41no—
113C4×Dic17—42no—
114C4×Dic17—42no—
115#8219—171nomaximal, Sylow
116C23×C17C2×C22⋊C411yescharacteristic
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17—21no—
121C22×D17C4×D412yes—
122C22×D17AGL(1,ℤ/8)11yescharacteristic
123C22×D17#13712yes—
124C22×D17#13711yescharacteristic
125C2×#595—28no—
126C2×#595—44no—
127C2×#595—44no—
128C2×#595—44no—
129C2×#595—44no—
130C2×#595—44no—
131C2×#595—44no—
132Q8×C17—42no—
133Q8×C17—42no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—22no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17#10412yes—
148C2×C4×C17C2×C4211yescharacteristic
149C2×C4×C17#11711yescharacteristic
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—22no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17#9912yes—
163C2×Dic17#10412yes—
164C2×Dic17C4×D412yes—
165C4×D17—24no—
166C4×D17—24no—
167C4×D17—42no—
168C4×D17—42no—
169C4×D17—44no—
170#3624—44no—
171C4×C22⋊Q8—171no—
172#1749—171no—
173C2×#328—172no—
174C4×#95—171no—
175#1294—172no—
176C22×C17C4×C22⋊C411yescharacteristic
177C22×C17C2×#9911yescharacteristic
178C22×C17#41011yescharacteristic
179C22×C17#41711yescharacteristic
180C22×C17#41911yescharacteristic
181C22×C17C2×#10411yescharacteristic
182C22×C17C2×#10411yescharacteristic
183C2×D17—22no—
184C2×D17—22no—
185C2×D17—21no—
186C2×D17—21no—
187C2×D17—22no—
188C2×D17—22no—
189C2×D17—21no—
190C2×D17—21no—
191C2×D17#44012yes—
192C2×D17#44112yes—
193#595—48no—
194C4×C17—21no—
195C4×C17—21no—
196C4×C17—22no—
197C4×C17—42no—
198C4×C17—41no—
199C4×C17—81no—
200Dic17—22no—
201Dic17—22no—
202Dic17—22no—
203Dic17—42no—
204Dic17—81no—
205C2×C22⋊Q8—171no—
206C2×C4×Q8—171no—
207C22×C42—171no—
208C22×C42—341no—
209C2×C22⋊C8—172no—
210C2×#97—341no—
211#391#59511yescharacteristic, complemented
212#389—171no—
213#387—341no—
214#385—341no—
215#380—171no—
216C4×C4⋊C4—171no—
217C4×C4⋊C4—341no—
218C4×C22⋊C4—341no—
219C2×#95—172no—
220C2×#95—341no—
221#336—341no—
222#328—178no—
223C2×C17#150411yescharacteristic
224C2×C17#150611yescharacteristic
225C2×C17#150811yescharacteristic
226C2×C17#150911yescharacteristic
227C2×C17C4×#9911yescharacteristic
228C2×C17C4×#10411yescharacteristic
229C2×C17#176311yescharacteristic
230D17—22no—
231D17—22no—
232C22×Q8C2×#59511yescharacteristic
233C23×C4—171no—
234C23×C4—171no—
235C23×C4—171no—
236C23×C4—342no—
237C22×C8—342no—
238C22⋊Q8—174no—
239C22⋊Q8—342no—
240C4×Q8—682no—
241C2×C4⋊C4—171no—
242C2×C4⋊C4—171no—
243C2×C4⋊C4—341no—
244C2×C4⋊C4—341no—
245C2×C4⋊C4—341no—
246C2×C4⋊C4C2×#59511yescharacteristic
247C2×C22⋊C4—341no—
248C2×C22⋊C4—341no—
249C2×C42—171no—
250C2×C42—171no—
251C2×C42—171no—
252C2×C42—341no—
253C2×C42—341no—
254C2×C42—344no—
255C2×C42—682no—
256C2×C42—682no—
257C2×C42—682no—
258C2×C42C2×#59511yescharacteristic
259C22⋊C8—174no—
260C22⋊C8—342no—
261#97—682no—
262#95—21no—
263#95—21no—
264#95—174no—
265#95—341no—
266#95—341no—
267#95—342no—
268#95—344no—
269#95—341no—
270C17#821911yescharacteristic, complemented, Sylow
271C24—171no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8C4×#59512yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—171no—
279C22×C4—171no—
280C22×C4—171no—
281C22×C4—172no—
282C22×C4—172no—
283C22×C4—342no—
284C22×C4—342no—
285C22×C4—341no—
286C22×C4—341no—
287C22×C4—342no—
288C22×C4—342no—
289C22×C4—344no—
290C22×C4—344no—
291C22×C4—344no—
292C22×C4—341no—
293C22×C4—341no—
294C22×C4—681no—
295C22×C4—681no—
296C22×C4—681no—
297C22×C4—682no—
298C22×C4—682no—
299C22×C4—682no—
300C22×C4—682no—
301C22×C4C22⋊#59511yescharacteristic
302C22×C4C22⋊#59511yescharacteristic
303C22×C4C22×#59511yescharacteristic
304C2×C8—344no—
305C2×C8—682no—
306C2×C8—682no—
307C4⋊C4—41no—
308C4⋊C4—172no—
309C4⋊C4—344no—
310C4⋊C4—341no—
311C4⋊C4—681no—
312C4⋊C4—682no—
313C4⋊C4—682no—
314C22⋊C4—344no—
315C22⋊C4—681no—
316C22⋊C4—681no—
317C42—41no—
318C42—342no—
319C42—342no—
320C42—681no—
321C42—682no—
322C42—682no—
323C42—688no—
324C23—172no—
325C23—171no—
326C23—171no—
327C23—172no—
328C23—341no—
329C23—341no—
330C23—341no—
331C23—341no—
332C23C2×C22⋊#59511yescharacteristic
333Q8—42no—
334Q8—42no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—172no—
349C2×C4—172no—
350C2×C4—172no—
351C2×C4—344no—
352C2×C4—344no—
353C2×C4—341no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—348no—
360C2×C4—342no—
361C2×C4—682no—
362C2×C4—682no—
363C2×C4—684no—
364C2×C4—684no—
365C2×C4—684no—
366C2×C4—684no—
367C2×C4—684no—
368C2×C4—684no—
369C2×C4—684no—
370C2×C4—681no—
371C2×C4—681no—
372C2×C4—681no—
373C2×C4—681no—
374C2×C4—681no—
375C2×C4#1058755012yes—
376C2×C4C2×C4×#59511yescharacteristic
377C2×C4#1058767711yescharacteristic
378C8—684no—
379C22—172no—
380C22—172no—
381C22—341no—
382C22—341no—
383C22—342no—
384C22—342no—
385C22—341no—
386C22—341no—
387C22—342no—
388C22—342no—
389C22C4×C22⋊#59511yescharacteristic
390C22D17⋊#9911yescharacteristic
391C22#1176378911yescharacteristic
392C22#1176380011yescharacteristic
393C22#1176380211yescharacteristic
394C22D17⋊#10411yescharacteristic
395C22C2×#1058755011yescharacteristic
396C4—22no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—42no—
401C4—81no—
402C4—342no—
403C4—342no—
404C4—342no—
405C4—688no—
406C4—682no—
407C4—1361no—
408C2—342no—
409C2—342no—
410C2#1454417111yescharacteristic
411C2#1454417811yescharacteristic
412C2#1454418211yescharacteristic
413C2#1454418511yescharacteristic
414C2#1454453111yescharacteristic
415C2C4×#1058755011yescharacteristic
416C2#1454455611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenter#391#595
hypocenterC17#8219
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
12. #11763663C4semidirect
14. C2×Q8×Dic17C4semidirect
16. #11762707C4semidirect
20. C4⋊C4×Dic17C4semidirect
23. C17×#391C4semidirect
73. C2×Q8×C17C42semidirect
107. C4⋊Dic17C42semidirect
211. #391#595semidirect
270. C17#8219semidirect
275. C2×Q8C4×#595semidirect

Maximal subgroups

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

subgroupindex
#145445112
C2×#117636632
C4×(C2×#595)⋊C42
#145436572
#145313222
#821917