Finite Group Databasebeta

← #17952080

Subgroups — #17952080

Order 4352.

640 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C4×#10587543C211yesmaximal, characteristic
2C2×#11763663C212yesmaximal
3#14543960C211yesmaximal, characteristic
4#14543657C212yesmaximal
5#14531322C211yesmaximal, characteristic
6C2×#10587543C2212yes—
7C2×C4×#3624—21no—
8C2×(C2×#595)⋊C4—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⋊C4×C17C2×C411yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32#10587632—22no—
33#10587632C2×C414yes—
34Q8×Dic17—42no—
35C2×#63227—21no—
36C2×#63227—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17C2×C411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47C42×D17—42no—
48(C2×#595)⋊C4—22no—
49(C2×#595)⋊C4—42no—
50(C2×#595)⋊C4—42no—
51(C2×#595)⋊C4C2×C414yes—
52#10587543—22no—
53#10587543C2×C414yes—
54C4×#3624—42no—
55C17×#95—21no—
56C17×#95—21no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60C23×D17C2×D411yescharacteristic
61C22×#595—24no—
62C22×#595—42no—
63C22×#595—42no—
64C22×#595—42no—
65C22×#595—42no—
66C22×#595—42no—
67C22×#595—42no—
68C2×Q8×C17—21no—
69C2×Q8×C17—41no—
70C2×Q8×C17—41no—
71C2×Q8×C17C4212yes—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17C22⋊C411yescharacteristic
80C22×Dic17C22⋊C411yescharacteristic
81C22×Dic17C2×D411yescharacteristic
82C2×C4×D17—22no—
83C2×C4×D17—22no—
84C2×C4×D17—41no—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×C4×D17—41no—
88C2×C4×D17—41no—
89C2×C4×D17SD1612yes—
90C2×C4×D17Q1612yes—
91C2×C4×D17C22×C412yes—
92C2×C4×D17#4112yes—
93C2×#3624—24no—
94C2×#3624—42no—
95C2×#3624—42no—
96C4⋊C4×C17—41no—
97C42×C17—41no—
98#63227—24no—
99#63227—42no—
100C4⋊Dic17—21no—
101C4⋊Dic17—42no—
102C4⋊Dic17—41no—
103C4⋊Dic17C4212yes—
104#63225—24no—
105#63225—42no—
106C4×Dic17—22no—
107C4×Dic17—22no—
108C4×Dic17—41no—
109C4×Dic17—42no—
110C4×Dic17—42no—
111#8221—171nomaximal, Sylow
112C23×C17C2×C22⋊C411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17C4×D412yes—
118C22×D17C2×SD1611yescharacteristic
119C22×D17C2×Q1611yescharacteristic
120C22×D17#13512yes—
121C2×#595—44no—
122C2×#595—44no—
123C2×#595—44no—
124C2×#595—82no—
125C2×#595—82no—
126C2×#595—82no—
127C2×#595—82no—
128Q8×C17—42no—
129Q8×C17—42no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—22no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17#10312yes—
144C2×C4×C17C2×C4211yescharacteristic
145C2×C4×C17#11711yescharacteristic
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—22no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17#9912yes—
159C2×Dic17#10312yes—
160C2×Dic17C4×D412yes—
161C4×D17—42no—
162C4×D17—42no—
163C4×D17—42no—
164C4×D17—42no—
165C4×D17—44no—
166#3624—44no—
167C4×C22⋊Q8—171no—
168C4×C22⋊C8—171no—
169C2×#328—172no—
170#1434—171no—
171#1294—172no—
172C22×C17C4×C22⋊C411yescharacteristic
173C22×C17C2×#9911yescharacteristic
174C22×C17#41011yescharacteristic
175C22×C17C2×#10311yescharacteristic
176C22×C17#41611yescharacteristic
177C22×C17#42111yescharacteristic
178C22×C17#42111yescharacteristic
179C2×D17—22no—
180C2×D17—22no—
181C2×D17—22no—
182C2×D17—21no—
183C2×D17—21no—
184C2×D17—21no—
185C2×D17—21no—
186C2×D17—22no—
187C2×D17C4×SD1612yes—
188C2×D17C4×Q1612yes—
189#595—84no—
190C4×C17—21no—
191C4×C17—21no—
192C4×C17—22no—
193C4×C17—42no—
194C4×C17—41no—
195C4×C17—81no—
196Dic17—22no—
197Dic17—22no—
198Dic17—22no—
199Dic17—42no—
200Dic17—81no—
201C2×C22⋊Q8—171no—
202C2×C4×Q8—171no—
203C22×C42—171no—
204C2×C22⋊C8—172no—
205C2×C4×C8—341no—
206#391#59511yescharacteristic, complemented
207#389—171no—
208#387—341no—
209#385—341no—
210#380—171no—
211C4×C4⋊C4—171no—
212C4×C4⋊C4—341no—
213C4×C22⋊C4—341no—
214C2×#95—172no—
215C2×#95—341no—
216C2×#95—341no—
217#336—341no—
218#328—178no—
219C2×C17#151611yescharacteristic
220C2×C17#151811yescharacteristic
221C2×C17#152011yescharacteristic
222C2×C17#152111yescharacteristic
223C2×C17C4×#9911yescharacteristic
224C2×C17#175911yescharacteristic
225C2×C17C4×#10311yescharacteristic
226D17—22no—
227D17—22no—
228C22×Q8C2×#59511yescharacteristic
229C23×C4—171no—
230C23×C4—171no—
231C23×C4—171no—
232C23×C4—342no—
233C22×C8—342no—
234C22⋊Q8—174no—
235C22⋊Q8—342no—
236C4×Q8—682no—
237C2×C4⋊C4—171no—
238C2×C4⋊C4—171no—
239C2×C4⋊C4—341no—
240C2×C4⋊C4—341no—
241C2×C4⋊C4—341no—
242C2×C4⋊C4C2×#59511yescharacteristic
243C2×C22⋊C4—341no—
244C2×C22⋊C4—341no—
245C2×C42—171no—
246C2×C42—171no—
247C2×C42—171no—
248C2×C42—341no—
249C2×C42—341no—
250C2×C42—682no—
251C2×C42C2×#59511yescharacteristic
252C22⋊C8—174no—
253C22⋊C8—342no—
254C4×C8—682no—
255#95—21no—
256#95—21no—
257#95—174no—
258#95—341no—
259#95—341no—
260#95—342no—
261#95—341no—
262#95—682no—
263#95—682no—
264C17#822111yescharacteristic, complemented, Sylow
265C24—171no—
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8C4×#59512yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—172no—
276C22×C4—172no—
277C22×C4—172no—
278C22×C4—172no—
279C22×C4—342no—
280C22×C4—342no—
281C22×C4—341no—
282C22×C4—344no—
283C22×C4—341no—
284C22×C4—681no—
285C22×C4—681no—
286C22×C4—681no—
287C22×C4—681no—
288C22×C4—681no—
289C22×C4—682no—
290C22×C4—682no—
291C22×C4—682no—
292C22×C4—682no—
293C22×C4—682no—
294C22×C4—682no—
295C22×C4C22⋊#59511yescharacteristic
296C22×C4C22⋊#59511yescharacteristic
297C22×C4C22×#59511yescharacteristic
298C2×C8—344no—
299C2×C8—682no—
300C2×C8—682no—
301C4⋊C4—41no—
302C4⋊C4—172no—
303C4⋊C4—344no—
304C4⋊C4—341no—
305C4⋊C4—681no—
306C4⋊C4—682no—
307C4⋊C4—682no—
308C22⋊C4—344no—
309C22⋊C4—682no—
310C42—41no—
311C42—342no—
312C42—342no—
313C42—681no—
314C42—682no—
315C42—682no—
316C23—172no—
317C23—171no—
318C23—171no—
319C23—172no—
320C23—341no—
321C23—341no—
322C23—341no—
323C23—341no—
324C23C2×C22⋊#59511yescharacteristic
325Q8—42no—
326Q8—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—172no—
341C2×C4—172no—
342C2×C4—172no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—342no—
350C2×C4—682no—
351C2×C4—682no—
352C2×C4—682no—
353C2×C4—682no—
354C2×C4—684no—
355C2×C4—681no—
356C2×C4—681no—
357C2×C4—684no—
358C2×C4—684no—
359C2×C4—684no—
360C2×C4—681no—
361C2×C4—681no—
362C2×C4—681no—
363C2×C4—1362no—
364C2×C4—1362no—
365C2×C4—1362no—
366C2×C4—1362no—
367C2×C4#1058754912yes—
368C2×C4C2×C4×#59511yescharacteristic
369C2×C4#1058767711yescharacteristic
370C8—684no—
371C22—172no—
372C22—172no—
373C22—342no—
374C22—341no—
375C22—341no—
376C22—341no—
377C22—341no—
378C22—342no—
379C22—342no—
380C22—342no—
381C22C4×C22⋊#59511yescharacteristic
382C22D17⋊#9911yescharacteristic
383C22#1176378911yescharacteristic
384C22C2×#1058754911yescharacteristic
385C22#1176379911yescharacteristic
386C22#1176380611yescharacteristic
387C22#1176380811yescharacteristic
388C4—22no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—42no—
393C4—81no—
394C4—342no—
395C4—342no—
396C4—342no—
397C4—682no—
398C4—1364no—
399C4—1361no—
400C2—342no—
401C2—342no—
402C2#1454421111yescharacteristic
403C2#1454421811yescharacteristic
404C2#1454422211yescharacteristic
405C2#1454422511yescharacteristic
406C2#1454453111yescharacteristic
407C2#1454454811yescharacteristic
408C2C4×#1058754911yescharacteristic
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#8221
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
206. #391#595semidirect
264. C17#8221semidirect

Maximal subgroups

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

subgroupindex
C4×#105875432
C2×#117636632
#145439602
#145436572
#145313222
#822117