Finite Group Databasebeta

← #17956510

Subgroups — #17956510

Order 4352.

598 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 108 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
2#14543960C211yesmaximal, characteristic
3#14543657C214yesmaximal
4#14537137C211yesmaximal, characteristic
5C2×#10587543C2212yes—
6C2×#10587542—21no—
7C2×(C2×#595)⋊C4—21no—
8C2×(C2×#595)⋊C4—21no—
9C2×(C2×#595)⋊C4C2212yes—
10#11763675—21no—
11C2×#10587555C411yescharacteristic, complemented
12C2×#10587583—21no—
13C2×C42×D17C2211yescharacteristic
14#11762714—21no—
15#11762708—21no—
16#11762708C412yescomplemented
17#11762666—21no—
18#11762666C2212yes—
19C17×#384C411yescharacteristic, complemented
20C23×#595—22no—
21C22×C4×D17D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×C4×D17C2311yescharacteristic
24C22×Dic34—21no—
25C22×#3624—22no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17C2×C412yes—
29C2×C42×C17C2×C411yescharacteristic
30C2×#63227—22no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—22no—
33C2×C4⋊Dic17C2×C412yes—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17D411yescharacteristic
36C2×C4×Dic17D411yescharacteristic
37#10587583—44no—
38C42×D17—42no—
39#10587555—42no—
40(C2×#595)⋊C4—24no—
41(C2×#595)⋊C4—24no—
42(C2×#595)⋊C4—42no—
43#10587543—22no—
44#10587543Q814yescomplemented
45#10587542—42no—
46#10587502—22no—
47#10587502—22no—
48C23×D17C2×D411yescharacteristic
49C22×#595—24no—
50C22×#595—42no—
51C22×#595—42no—
52C22×#595—42no—
53C22×#595—42no—
54C22×#595—42no—
55C22×#595—42no—
56C22×C4×C17—22no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22×C411yescharacteristic
60C22×Dic17—22no—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C2×D411yescharacteristic
64C2×C4×D17—22no—
65C2×C4×D17—22no—
66C2×C4×D17—22no—
67C2×C4×D17—22no—
68C2×C4×D17—41no—
69C2×C4×D17—41no—
70C2×C4×D17—41no—
71C2×C4×D17—41no—
72C2×C4×D17C2×Q812yes—
73C2×C4×D17#4112yes—
74C2×Dic34—42no—
75C2×Dic34—42no—
76C2×Dic34—42no—
77C2×#3624—24no—
78C2×#3624—42no—
79C2×#3624—42no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—42no—
83C42×C17—22no—
84#63227—44no—
85C4⋊Dic17—44no—
86C4⋊Dic17—42no—
87C4⋊Dic17—42no—
88C4×Dic17—22no—
89C4×Dic17—41no—
90C4×Dic17—41no—
91#9888—171nomaximal, Sylow
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—21no—
96C22×D17—21no—
97C22×D17#12411yescharacteristic
98C22×D17C4⋊Q811yescharacteristic
99C22×D17AGL(1,ℤ/8)12yes—
100C22×D17#13712yes—
101C2×#595—44no—
102C2×#595—44no—
103C2×#595—44no—
104C2×#595—82no—
105C2×#595—82no—
106C2×#595—82no—
107C2×#595—82no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17#9912yes—
118C2×C4×C17#10412yes—
119C2×C4×C17C4×Q812yes—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17C22⋊Q812yes—
132C4×D17—42no—
133C4×D17—42no—
134C4×D17—44no—
135C4×D17—44no—
136Dic34—84no—
137#3624—44no—
138#2857—171no—
139#1749—171no—
140#1434—171no—
141#1294—174no—
142C22×C17#39111yescharacteristic
143C22×C17C2×#9911yescharacteristic
144C22×C17#41011yescharacteristic
145C22×C17#41811yescharacteristic
146C22×C17#41911yescharacteristic
147C22×C17C2×#10411yescharacteristic
148C22×C17#42111yescharacteristic
149C2×D17—21no—
150C2×D17—21no—
151C2×D17—21no—
152C2×D17—21no—
153C2×D17—22no—
154C2×D17—22no—
155C2×D17—22no—
156C2×D17—22no—
157C2×D17#48912yes—
158C2×D17#50112yes—
159#595—84no—
160C4×C17—22no—
161C4×C17—22no—
162C4×C17—22no—
163C4×C17—82no—
164Dic17—22no—
165Dic17—41no—
166Dic17—41no—
167Dic17—82no—
168C2×C4⋊Q8—171no—
169C2×C22⋊Q8—341no—
170C22×C42—171no—
171C2×C22⋊C8—172no—
172C2×#97—341no—
173#391—341no—
174#389—172no—
175#389—341no—
176#384#59511yescharacteristic, complemented
177#380—172no—
178#380—341no—
179C2×#95—172no—
180C2×#95—341no—
181C2×#95—341no—
182#336—341no—
183C2×C17#150611yescharacteristic
184C2×C17#150711yescharacteristic
185C2×C17#151111yescharacteristic
186C2×C17#151411yescharacteristic
187C2×C17#198611yescharacteristic
188C2×C17#199311yescharacteristic
189C2×C17#199611yescharacteristic
190D17—22no—
191D17—22no—
192C22×Q8—341no—
193C23×C4—171no—
194C23×C4—171no—
195C23×C4—171no—
196C23×C4—342no—
197C22×C8—342no—
198C4⋊Q8—682no—
199C22⋊Q8—684no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—172no—
203C2×C4⋊C4—342no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4C2×#59512yes—
206C2×C22⋊C4—342no—
207C2×C42—171no—
208C2×C42—171no—
209C2×C42—171no—
210C2×C42—682no—
211C2×C42C2×#59511yescharacteristic
212C22⋊C8—174no—
213C22⋊C8—342no—
214#97—682no—
215#95—342no—
216#95—342no—
217#95—344no—
218#95—344no—
219#95—682no—
220C17#988811yescharacteristic, complemented, Sylow
221C24—171no—
222C2×Q8—682no—
223C2×Q8—682no—
224C2×Q8—682no—
225C22×C4—22no—
226C22×C4—171no—
227C22×C4—172no—
228C22×C4—172no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—342no—
232C22×C4—342no—
233C22×C4—342no—
234C22×C4—342no—
235C22×C4—344no—
236C22×C4—342no—
237C22×C4—681no—
238C22×C4—681no—
239C22×C4—681no—
240C22×C4—681no—
241C22×C4—682no—
242C22×C4—682no—
243C22×C4—682no—
244C22×C4—682no—
245C22×C4—682no—
246C22×C4—682no—
247C22×C4C22⋊#59511yescharacteristic
248C22×C4C22⋊#59511yescharacteristic
249C22×C4C22×#59511yescharacteristic
250C2×C8—344no—
251C2×C8—682no—
252C2×C8—682no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—682no—
257C4⋊C4—684no—
258C4⋊C4—682no—
259C22⋊C4—684no—
260C42—22no—
261C42—342no—
262C42—681no—
263C42—681no—
264C23—171no—
265C23—171no—
266C23—172no—
267C23—172no—
268C23—341no—
269C23—341no—
270C23—341no—
271C23—341no—
272C23C2×C22⋊#59511yescharacteristic
273Q8—1364no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—172no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—682no—
293C2×C4—682no—
294C2×C4—684no—
295C2×C4—684no—
296C2×C4—684no—
297C2×C4—684no—
298C2×C4—684no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—682no—
302C2×C4—1362no—
303C2×C4—1362no—
304C2×C4—1362no—
305C2×C4—1362no—
306C2×C4#1058753912yes—
307C2×C4#1058755012yes—
308C2×C4Q8×#59512yes—
309C8—684no—
310C22—172no—
311C22—172no—
312C22—342no—
313C22—342no—
314C22—341no—
315C22—341no—
316C22—342no—
317C22—342no—
318C22—341no—
319C22—341no—
320C22(C2×#595)⋊Q811yescharacteristic
321C22C2×#1058753911yescharacteristic
322C22#1176378911yescharacteristic
323C22#1176380111yescharacteristic
324C22#1176380211yescharacteristic
325C22C2×#1058755011yescharacteristic
326C22#1176380711yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—82no—
331C4—342no—
332C4—681no—
333C4—681no—
334C4—1364no—
335C4—1362no—
336C2—342no—
337C2—342no—
338C2#1454417811yescharacteristic
339C2#1454418111yescharacteristic
340C2#1454419211yescharacteristic
341C2#1454420411yescharacteristic
342C2#1454495911yescharacteristic
343C2#1454496811yescharacteristic
344C2#1454497411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenter#384#595
hypocenterC17#9888
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
11. C2×#10587555C4semidirect
16. #11762708C4semidirect
19. C17×#384C4semidirect
44. #10587543Q8semidirect
176. #384#595semidirect
220. C17#9888semidirect

Maximal subgroups

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

subgroupindex
#145445112
#145439602
#145436572
#145371372
#988817