Finite Group Databasebeta

← #17956566

Subgroups — #17956566

Order 4352.

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

normal subgroups only

This group has 592 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#14544641C211yesmaximal, characteristic
2#14543960C211yesmaximal, characteristic
3#14543657C214yesmaximal
4#14537131C211yesmaximal, characteristic
5C2×#10587544—21no—
6C2×#10587543C2212yes—
7C2×(C2×#595)⋊C4—21no—
8C2×(C2×#595)⋊C4—21no—
9C2×(C2×#595)⋊C4C2212yes—
10#11763675—21no—
11C2×#10587632—21no—
12C2×C42×D17C2211yescharacteristic
13#11762716—21no—
14#11762705—21no—
15#11762705C412yescomplemented
16#11762679C411yescharacteristic, complemented
17#11762666—21no—
18#11762666C2212yes—
19C2×C17×C4⋊Q8C411yescharacteristic, complemented
20C23×#595—22no—
21C22×Q8×C17—21no—
22C22×C4×D17D411yescharacteristic
23C22×C4×D17D411yescharacteristic
24C22×C4×D17C2311yescharacteristic
25C22×#3624—22no—
26C17×C4⋊Q8—42no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17C2×C412yes—
29C2×C42×C17C2×C411yescharacteristic
30#10587632—44no—
31C2×#63227—22no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17C2×C412yes—
35C2×#63225—22no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17D411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C42×D17—42no—
40(C2×#595)⋊C4—22no—
41(C2×#595)⋊C4—42no—
42(C2×#595)⋊C4—42no—
43(C2×#595)⋊C4Q814yescomplemented
44#10587544—42no—
45#10587543—24no—
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—
56C2×Q8×C17—42no—
57C2×Q8×C17—42no—
58C2×Q8×C17—42no—
59C22×C4×C17—22no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×Dic17—22no—
64C22×Dic17C22⋊C411yescharacteristic
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C2×C4×D17—22no—
68C2×C4×D17—22no—
69C2×C4×D17—41no—
70C2×C4×D17—41no—
71C2×C4×D17—41no—
72C2×C4×D17—41no—
73C2×C4×D17—41no—
74C2×C4×D17SD1612yes—
75C2×C4×D17Q1612yes—
76C2×C4×D17C2×Q812yes—
77C2×C4×D17#4112yes—
78C2×#3624—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—42no—
83C42×C17—22no—
84#63227—44no—
85C4⋊Dic17—42no—
86C4⋊Dic17—42no—
87C4⋊Dic17—42no—
88#63225—44no—
89C4×Dic17—22no—
90C4×Dic17—41no—
91C4×Dic17—41no—
92#9911—171nomaximal, Sylow
93C23×C17C2×C22⋊C411yescharacteristic
94C22×D17—21no—
95C22×D17—21no—
96C22×D17—21no—
97C22×D17—21no—
98C22×D17C22⋊Q812yes—
99C22×D17C2×SD1611yescharacteristic
100C22×D17C2×Q1611yescharacteristic
101C22×D17AGL(1,ℤ/8)12yes—
102C2×#595—44no—
103C2×#595—44no—
104C2×#595—44no—
105C2×#595—82no—
106C2×#595—82no—
107C2×#595—82no—
108C2×#595—82no—
109Q8×C17—84no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17#9912yes—
120C2×C4×C17#10312yes—
121C2×C4×C17C4×Q812yes—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17#10312yes—
133C2×Dic17#12411yescharacteristic
134C2×Dic17C4⋊Q811yescharacteristic
135C4×D17—42no—
136C4×D17—42no—
137C4×D17—42no—
138C4×D17—42no—
139C4×D17—44no—
140#3624—82no—
141#2857—171no—
142#1797—171no—
143#1434—171no—
144#1294—174no—
145C22×C17#39111yescharacteristic
146C22×C17C2×#9911yescharacteristic
147C22×C17#41011yescharacteristic
148C22×C17C2×#10311yescharacteristic
149C22×C17#41811yescharacteristic
150C22×C17#42111yescharacteristic
151C22×C17#42111yescharacteristic
152C2×D17—22no—
153C2×D17—22no—
154C2×D17—22no—
155C2×D17—22no—
156C2×D17—21no—
157C2×D17—21no—
158C2×D17—21no—
159C2×D17—21no—
160C2×D17#47612yes—
161C2×D17#47712yes—
162#595—84no—
163C4×C17—22no—
164C4×C17—22no—
165C4×C17—22no—
166C4×C17—82no—
167Dic17—21no—
168Dic17—21no—
169Dic17—22no—
170Dic17—41no—
171Dic17—82no—
172C2×C4⋊Q8#59511yescharacteristic, complemented
173C2×C22⋊Q8—341no—
174C22×C42—171no—
175C2×C4⋊C8—341no—
176C2×C22⋊C8—172no—
177#391—341no—
178#389—172no—
179#389—341no—
180#384—171no—
181#380—172no—
182#380—341no—
183C2×#95—172no—
184C2×#95—341no—
185C2×#95—341no—
186#336—341no—
187C2×C17#151011yescharacteristic
188C2×C17#151411yescharacteristic
189C2×C17#151811yescharacteristic
190C2×C17#152011yescharacteristic
191C2×C17#198711yescharacteristic
192C2×C17#199511yescharacteristic
193C2×C17#199611yescharacteristic
194D17—22no—
195D17—22no—
196C22×Q8—21no—
197C23×C4—171no—
198C23×C4—171no—
199C23×C4—171no—
200C23×C4—342no—
201C22×C8—342no—
202C4⋊Q8—42no—
203C22⋊Q8—684no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—172no—
206C2×C4⋊C4—342no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4C2×#59512yes—
210C2×C22⋊C4—342no—
211C2×C42—171no—
212C2×C42—171no—
213C2×C42—171no—
214C2×C42—682no—
215C2×C42C2×#59511yescharacteristic
216C4⋊C8—682no—
217C22⋊C8—344no—
218#95—174no—
219#95—342no—
220#95—342no—
221#95—342no—
222#95—682no—
223#95—682no—
224C17#991111yescharacteristic, complemented, Sylow
225C24—171no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C22×C4—22no—
230C22×C4—171no—
231C22×C4—171no—
232C22×C4—172no—
233C22×C4—172no—
234C22×C4—172no—
235C22×C4—172no—
236C22×C4—171no—
237C22×C4—342no—
238C22×C4—342no—
239C22×C4—344no—
240C22×C4—342no—
241C22×C4—681no—
242C22×C4—681no—
243C22×C4—681no—
244C22×C4—681no—
245C22×C4—681no—
246C22×C4—682no—
247C22×C4—682no—
248C22×C4—682no—
249C22×C4—682no—
250C22×C4—682no—
251C22×C4—682no—
252C22×C4C22⋊#59511yescharacteristic
253C22×C4C22⋊#59511yescharacteristic
254C22×C4C22×#59511yescharacteristic
255C2×C8—682no—
256C2×C8—682no—
257C2×C8—682no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—682no—
261C4⋊C4—682no—
262C4⋊C4—682no—
263C4⋊C4—684no—
264C22⋊C4—684no—
265C42—22no—
266C42—342no—
267C42—681no—
268C42—681no—
269C23—172no—
270C23—171no—
271C23—171no—
272C23—172no—
273C23—341no—
274C23—341no—
275C23—341no—
276C23—341no—
277C23C2×C22⋊#59511yescharacteristic
278Q8—84no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—171no—
289C2×C4—171no—
290C2×C4—172no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—682no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—682no—
302C2×C4—684no—
303C2×C4—684no—
304C2×C4—684no—
305C2×C4—684no—
306C2×C4—682no—
307C2×C4—682no—
308C2×C4—682no—
309C2×C4—1362no—
310C2×C4—1362no—
311C2×C4—1362no—
312C2×C4—1362no—
313C2×C4#1058754912yes—
314C2×C4#1058755112yes—
315C2×C4Q8×#59512yes—
316C8—1362no—
317C22—172no—
318C22—172no—
319C22—342no—
320C22—342no—
321C22—342no—
322C22—342no—
323C22—341no—
324C22—341no—
325C22—341no—
326C22—341no—
327C22(C2×#595)⋊Q811yescharacteristic
328C22C2×#1058755111yescharacteristic
329C22#1176379011yescharacteristic
330C22C2×#1058754911yescharacteristic
331C22#1176380111yescharacteristic
332C22#1176380711yescharacteristic
333C22#1176380811yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—82no—
338C4—341no—
339C4—341no—
340C4—342no—
341C4—681no—
342C4—1364no—
343C4—1362no—
344C2—342no—
345C2—342no—
346C2#1454418911yescharacteristic
347C2#1454420511yescharacteristic
348C2#1454421611yescharacteristic
349C2#1454422411yescharacteristic
350C2#1454496211yescharacteristic
351C2#1454497211yescharacteristic
352C2#1454497311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#9911
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. #11762705C4semidirect
16. #11762679C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
43. (C2×#595)⋊C4Q8semidirect
172. C2×C4⋊Q8#595semidirect
224. C17#9911semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145439602
#145436572
#145371312
#991117