Finite Group Databasebeta

← #17956561

Subgroups — #17956561

Order 4352.

684 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 120 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#14544641C211yesmaximal, characteristic
2C4×(C2×#595)⋊C4C211yesmaximal, characteristic
3#14543657C212yesmaximal
4#14543657C212yesmaximal
5#14537135C211yesmaximal, characteristic
6C22×C4×#595—21no—
7C2×#10587544—21no—
8C2×#10587543C2212yes—
9C2×(C2×#595)⋊C4—21no—
10C2×(C2×#595)⋊C4C2212yes—
11#11763675—21no—
12C2×#10587574C411yescharacteristic, complemented
13C2×#10587582—21no—
14C2×C42×D17C2211yescharacteristic
15#11762713—21no—
16#11762707C411yescharacteristic, complemented
17#11762704—21no—
18#11762678C411yescharacteristic, complemented
19#11762668—21no—
20#11762666C2211yescharacteristic
21#11762666C2211yescharacteristic
22C17×#389C411yescharacteristic, complemented
23C23×#595—22no—
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17C2311yescharacteristic
27C22×Dic34—21no—
28C22×#3624—22no—
29C2×C4×#595—24no—
30C2×C4×#595—42no—
31C2×C4×#595—42no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C2×#63227—21no—
37C2×#63227—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×C4×Dic17C2×C411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47#10587582—42no—
48#10587582—42no—
49#10587574—42no—
50C42×D17—42no—
51(C2×#595)⋊C4—24no—
52(C2×#595)⋊C4—22no—
53(C2×#595)⋊C4Q814yescomplemented
54#10587544—42no—
55#10587543—24no—
56C17×#95—21no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60C23×D17C2×D411yescharacteristic
61C22×#595—24no—
62C22×#595—24no—
63C22×#595—24no—
64C22×#595—42no—
65C22×#595—42no—
66C22×#595—42no—
67C22×#595—42no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17C22⋊C411yescharacteristic
76C22×Dic17C22⋊C411yescharacteristic
77C22×Dic17C2×D411yescharacteristic
78C2×C4×D17—22no—
79C2×C4×D17—22no—
80C2×C4×D17—22no—
81C2×C4×D17—22no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17—41no—
85C2×C4×D17D812yes—
86C2×C4×D17SD1612yes—
87C2×C4×D17C2×Q812yes—
88C2×C4×D17#4112yes—
89C2×Dic34—42no—
90C2×Dic34—42no—
91C2×Dic34—42no—
92C2×#3624—42no—
93C2×#3624—42no—
94C2×#3624—42no—
95C4×#595—48no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—41no—
98C4⋊C4×C17—42no—
99C42×C17—41no—
100#63227—41no—
101#63227—41no—
102#63227—41no—
103#63227—41no—
104C4⋊Dic17—41no—
105C4⋊Dic17—41no—
106#63225—42no—
107#63225—42no—
108#63225—42no—
109#63225—42no—
110C4×Dic17—22no—
111C4×Dic17—22no—
112C4×Dic17—41no—
113#9909—171nomaximal, Sylow
114C23×C17C2×C22⋊C411yescharacteristic
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17C22⋊Q812yes—
120C22×D17C2×D811yescharacteristic
121C22×D17C2×SD1611yescharacteristic
122C22×D17#13712yes—
123C2×#595—28no—
124C2×#595—44no—
125C2×#595—44no—
126C2×#595—44no—
127C2×#595—44no—
128C2×#595—44no—
129C2×#595—44no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17#10212yes—
144C2×C4×C17C4×Q812yes—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17#9912yes—
156C2×Dic17#10212yes—
157C2×Dic17#10412yes—
158C2×Dic17#10412yes—
159C2×Dic17#12411yescharacteristic
160C2×Dic17C4⋊Q811yescharacteristic
161C4×D17—24no—
162C4×D17—24no—
163C4×D17—42no—
164C4×D17—42no—
165C4×D17—44no—
166Dic34—84no—
167#3624—82no—
168#2857—171no—
169#1797—171no—
170C4×#95—171no—
171#1294—172no—
172#1294—172no—
173C22×C17#39111yescharacteristic
174C22×C17C2×#9911yescharacteristic
175C22×C17#41011yescharacteristic
176C22×C17C2×#10211yescharacteristic
177C22×C17#41911yescharacteristic
178C22×C17C2×#10411yescharacteristic
179C22×C17C2×#10411yescharacteristic
180C2×D17—22no—
181C2×D17—22no—
182C2×D17—21no—
183C2×D17—21no—
184C2×D17—21no—
185C2×D17—21no—
186C2×D17—22no—
187C2×D17—22no—
188C2×D17#47412yes—
189C2×D17#47512yes—
190#595—48no—
191C4×C17—22no—
192C4×C17—22no—
193C4×C17—41no—
194C4×C17—81no—
195C4×C17—81no—
196Dic17—21no—
197Dic17—21no—
198Dic17—22no—
199Dic17—81no—
200Dic17—81no—
201Dic17#32714yes—
202C2×C4⋊Q8—171no—
203C2×C22⋊Q8—341no—
204C22×C42—171no—
205C22×C42—341no—
206C2×C4⋊C8—341no—
207C2×C22⋊C8—172no—
208#391—341no—
209#389—171no—
210#389—341no—
211#389#59511yescharacteristic, complemented
212#384—171no—
213#380—171no—
214#380—171no—
215#380—341no—
216C2×#95—172no—
217C2×#95—341no—
218#336—341no—
219C2×C17C2×#32711yescharacteristic
220C2×C17#150211yescharacteristic
221C2×C17#150611yescharacteristic
222C2×C17#150811yescharacteristic
223C2×C17#198811yescharacteristic
224C2×C17#199411yescharacteristic
225C2×C17#199611yescharacteristic
226D17—22no—
227D17—22no—
228C22×Q8—341no—
229C23×C4—171no—
230C23×C4—171no—
231C23×C4—171no—
232C23×C4—342no—
233C22×C8—342no—
234C4⋊Q8—682no—
235C22⋊Q8—682no—
236C22⋊Q8—682no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—171no—
239C2×C4⋊C4—171no—
240C2×C4⋊C4—341no—
241C2×C4⋊C4—341no—
242C2×C4⋊C4—341no—
243C2×C4⋊C4—341no—
244C2×C4⋊C4C2×#59511yescharacteristic
245C2×C4⋊C4C2×#59511yescharacteristic
246C2×C22⋊C4—341no—
247C2×C22⋊C4—341no—
248C2×C42—171no—
249C2×C42—171no—
250C2×C42—171no—
251C2×C42—344no—
252C2×C42—682no—
253C2×C42—682no—
254C2×C42—682no—
255C2×C42C2×#59511yescharacteristic
256C4⋊C8—682no—
257C22⋊C8—344no—
258#95—21no—
259#95—174no—
260#95—341no—
261#95—341no—
262#95—342no—
263#95—344no—
264#95—341no—
265C17#990911yescharacteristic, complemented, Sylow
266C24—171no—
267C2×Q8—682no—
268C2×Q8—682no—
269C2×Q8—682no—
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—342no—
282C22×C4—342no—
283C22×C4—344no—
284C22×C4—344no—
285C22×C4—344no—
286C22×C4—341no—
287C22×C4—341no—
288C22×C4—681no—
289C22×C4—681no—
290C22×C4—681no—
291C22×C4—682no—
292C22×C4—682no—
293C22×C4—682no—
294C22×C4—682no—
295C22×C4C22⋊#59511yescharacteristic
296C22×C4C22⋊#59511yescharacteristic
297C22×C4C22×#59511yescharacteristic
298C2×C8—682no—
299C2×C8—682no—
300C2×C8—682no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—681no—
305C4⋊C4—681no—
306C4⋊C4—682no—
307C4⋊C4—682no—
308C4⋊C4—682no—
309C4⋊C4—682no—
310C22⋊C4—681no—
311C22⋊C4—681no—
312C22⋊C4—681no—
313C22⋊C4—681no—
314C42—41no—
315C42—342no—
316C42—342no—
317C42—681no—
318C42—688no—
319C23—172no—
320C23—171no—
321C23—171no—
322C23—172no—
323C23—341no—
324C23—341no—
325C23—341no—
326C23—341no—
327C23C2×C22⋊#59511yescharacteristic
328Q8—1364no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—172no—
343C2×C4—172no—
344C2×C4—172no—
345C2×C4—171no—
346C2×C4—171no—
347C2×C4—172no—
348C2×C4—344no—
349C2×C4—344no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—348no—
355C2×C4—682no—
356C2×C4—682no—
357C2×C4—684no—
358C2×C4—684no—
359C2×C4—684no—
360C2×C4—684no—
361C2×C4—684no—
362C2×C4—684no—
363C2×C4—684no—
364C2×C4—681no—
365C2×C4—681no—
366C2×C4—681no—
367C2×C4—681no—
368C2×C4—681no—
369C2×C4—681no—
370C2×C4#1058754712yes—
371C2×C4Q8×#59512yes—
372C8—1362no—
373C22—172no—
374C22—172no—
375C22—341no—
376C22—341no—
377C22—341no—
378C22—341no—
379C22—342no—
380C22—342no—
381C22—342no—
382C22—342no—
383C22(C2×#595)⋊Q811yescharacteristic
384C22D17⋊#9911yescharacteristic
385C22#1176378811yescharacteristic
386C22C2×#1058754711yescharacteristic
387C22#1176380211yescharacteristic
388C22D17⋊#10411yescharacteristic
389C22D17⋊#10411yescharacteristic
390C4—22no—
391C4—22no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—174no—
396C4—342no—
397C4—341no—
398C4—341no—
399C4—688no—
400C4—1361no—
401C4—1361no—
402C2—342no—
403C2—342no—
404C2D17⋊#32711yescharacteristic
405C2#1454416311yescharacteristic
406C2#1454417711yescharacteristic
407C2#1454418211yescharacteristic
408C2#1454496311yescharacteristic
409C2#1454497011yescharacteristic
410C2#1454497511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#389C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenter#389#595
hypocenterC17#9909
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
12. C2×#10587574C4semidirect
16. #11762707C4semidirect
18. #11762678C4semidirect
22. C17×#389C4semidirect
53. (C2×#595)⋊C4Q8semidirect
211. #389#595semidirect
265. C17#9909semidirect

Maximal subgroups

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

subgroupindex
#145446412
C4×(C2×#595)⋊C42
#145436572
#145371352
#990917