Finite Group Databasebeta

← #16949806

Subgroups — #16949806

Order 4352.

976 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#11762599C214yesmaximal, complemented
2#14528712C211yesmaximal, characteristic
3#14528702C211yesmaximal, characteristic
4C2×C17×#536C211yesmaximal, characteristic
5C2×#10587507—21no—
6C2×#10587504C2212yes—
7C2×#10587505—21no—
8C2×#10587505C2212yes—
9#11762773—21no—
10#11762672—21no—
11#11762599—28no—
12C22×#41×C17C2211yescharacteristic
13C23×Q8×C17C411yescharacteristic, complemented
14C17×#536—28no—
15C2×C17×C4⋊Q8—21no—
16C2×C17×#124—21no—
17C2×C17×#123C412yescomplemented
18C2×C17×#123C2212yes—
19C2×C17×C22⋊Q8—21no—
20C2×C17×C22⋊Q8—21no—
21C2×C17×#117C411yescharacteristic, complemented
22C2×#41×C17—24no—
23C2×#41×C17—42no—
24C2×#41×C17—42no—
25C22×Q8×C17—21no—
26C22×Q8×C17—21no—
27C22×Q8×C17—24no—
28C22×Q8×C17—42no—
29C22×Q8×C17—42no—
30C22×Q8×C17D411yescharacteristic
31C22×D4×C17C2×C411yescharacteristic
32C22×D4×C17D411yescharacteristic
33C22×D4×C17C2311yescharacteristic
34C23×C4×C17C2×C411yescharacteristic
35C23×C4×C17C2×C412yes—
36C23×C4×C17D411yescharacteristic
37C23×C4×C17D411yescharacteristic
38C17×C4⋊Q8—28no—
39C17×#124—28no—
40C17×#123—24no—
41C17×#123—24no—
42C17×#123—24no—
43C17×#123D418yes—
44C17×C22⋊Q8—28no—
45C17×C22⋊Q8—28no—
46C17×#117—24no—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17—22no—
49C2×C4⋊C4×C17—22no—
50C2×C22⋊C4×C17—22no—
51C2×C22⋊C4×C17—22no—
52C2×C22⋊C4×C17C2×C412yes—
53C2×C42×C17—21no—
54C2×C22⋊Dic17—22no—
55C2×#63223—22no—
56#10587507—42no—
57#10587505—24no—
58#10587505—42no—
59#10587504—24no—
60#10587502—42no—
61C22⋊#3622—42no—
62C24×C17C22⋊C411yescharacteristic
63C24×C17C22×C411yescharacteristic
64C24×C17C2×D411yescharacteristic
65#41×C17—416no—
66C2×Q8×C17—24no—
67C2×Q8×C17—24no—
68C2×Q8×C17—22no—
69C2×Q8×C17—22no—
70C2×Q8×C17—41no—
71C2×Q8×C17—416no—
72C2×Q8×C17—42no—
73C2×Q8×C17—42no—
74C2×Q8×C17—42no—
75C2×Q8×C17—42no—
76C2×D4×C17—22no—
77C2×D4×C17—22no—
78C2×D4×C17—22no—
79C2×D4×C17—41no—
80C2×D4×C17—41no—
81C2×D4×C17—41no—
82C2×D4×C17—41no—
83C2×D4×C17—41no—
84C2×D4×C17C2×D414yes—
85C22×C4×C17—22no—
86C22×C4×C17—22no—
87C22×C4×C17—22no—
88C22×C4×C17—22no—
89C22×C4×C17—22no—
90C22×C4×C17—22no—
91C22×C4×C17—22no—
92C22×C4×C17—22no—
93C22×C4×C17—22no—
94C22×C4×C17—22no—
95C22×C4×C17—41no—
96C22×C4×C17—41no—
97C22×C4×C17—42no—
98C22×C4×C17—42no—
99C22×C4×C17C22⋊C411yescharacteristic
100C22×C4×C17C22⋊C411yescharacteristic
101C22×C4×C17C22⋊C414yes—
102C22×C4×C17C22⋊C411yescharacteristic
103C22×C4×C17C2×D411yescharacteristic
104C22×Dic17—42no—
105C4⋊C4×C17—24no—
106C4⋊C4×C17—28no—
107C4⋊C4×C17—28no—
108C22⋊C4×C17—28no—
109C22⋊C4×C17—22no—
110C22⋊C4×C17—28no—
111C22⋊C4×C17C22⋊C414yes—
112C42×C17—24no—
113C22⋊Dic17—42no—
114C22⋊Dic17—42no—
115#63223—42no—
116#63223—42no—
117C2×#3622—42no—
118#12692—171nomaximal, Sylow
119C23×C17—21no—
120C23×C17—21no—
121C23×C17—21no—
122C23×C17—21no—
123C23×C17—21no—
124C23×C17—21no—
125C23×C17—21no—
126C23×C17—42no—
127C23×C17—42no—
128C23×C17—42no—
129C23×C17C2×C22⋊C411yescharacteristic
130C23×C17C2×C22⋊C412yes—
131C23×C17#12012yes—
132Q8×C17—44no—
133Q8×C17—48no—
134Q8×C17—48no—
135D4×C17—44no—
136D4×C17—44no—
137D4×C17—44no—
138C2×C4×C17—28no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—24no—
146C2×C4×C17—24no—
147C2×C4×C17—21no—
148C2×C4×C17—21no—
149C2×C4×C17—21no—
150C2×C4×C17—44no—
151C2×C4×C17—44no—
152C2×C4×C17—44no—
153C2×C4×C17—44no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—44no—
157C2×C4×C17#9912yes—
158C2×C4×C17#9912yes—
159C2×C4×C17#12012yes—
160C2×Dic17—82no—
161C2×Dic17—84no—
162#3622—82no—
163C2×#536D1711yescharacteristic
164C2×#352—174no—
165#1785—171no—
166#1782—171no—
167C22×C17—22no—
168C22×C17—22no—
169C22×C17—22no—
170C22×C17—42no—
171C22×C17—42no—
172C22×C17—42no—
173C22×C17—42no—
174C22×C17—42no—
175C22×C17—42no—
176C22×C17—42no—
177C22×C17—41no—
178C22×C17—81no—
179C22×C17—81no—
180C22×C17#37911yescharacteristic
181C22×C17C2×#9911yescharacteristic
182C22×C17C2×#9911yescharacteristic
183C4×C17—22no—
184C4×C17—22no—
185C4×C17—22no—
186C4×C17—44no—
187C4×C17—41no—
188C4×C17—44no—
189Dic17—162no—
190C22×#41C2×D1711yescharacteristic
191C23×Q8Dic1711yescharacteristic, complemented
192#536—28no—
193C2×C4⋊Q8—21no—
194C2×#124—21no—
195C2×#123Dic1712yescomplemented
196C2×#123C2×D1712yes—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8—21no—
199C2×#117Dic1711yescharacteristic, complemented
200C2×#101—341no—
201C2×#100—172no—
202C2×#99—172no—
203C2×#99—341no—
204#407—341no—
205#381—341no—
206#352—348no—
207C2×C17—21no—
208C2×C17—21no—
209C2×C17—42no—
210C2×C17—42no—
211C2×C17—81no—
212C2×C17#178111yescharacteristic
213C2×C17#212312yes—
214C2×#41—24no—
215C2×#41—42no—
216C2×#41—42no—
217C22×Q8—21no—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8C22⋊D1711yescharacteristic
223C22×D4C2×Dic1711yescharacteristic
224C22×D4C22⋊D1711yescharacteristic
225C22×D4C22×D1711yescharacteristic
226C23×C4C2×Dic1711yescharacteristic
227C23×C4C2×Dic1712yes—
228C23×C4C22⋊D1711yescharacteristic
229C23×C4C22⋊D1711yescharacteristic
230C2×M16—342no—
231C4⋊Q8—28no—
232#124—28no—
233#123—24no—
234#123—24no—
235#123—24no—
236#123C22⋊D1718yes—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239#117—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—342no—
246C2×C22⋊C4C2×Dic1712yes—
247C2×C42—21no—
248#101—682no—
249#100—344no—
250#99—344no—
251#99—682no—
252C22⋊C8—682no—
253#95—682no—
254C17#1269211yescharacteristic, complemented, Sylow
255C24C22⋊Dic1711yescharacteristic
256C24C22×Dic1711yescharacteristic
257C24C2×C22⋊D1711yescharacteristic
258#41—416no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4C2×C22⋊D1714yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—682no—
293C22×C4C22⋊Dic1711yescharacteristic
294C22×C4C22⋊Dic1711yescharacteristic
295C22×C4C22⋊Dic1714yes—
296C22×C4C22⋊Dic1711yescharacteristic
297C22×C4C2×C22⋊D1711yescharacteristic
298M16—682no—
299M16—682no—
300C2×C8—682no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—22no—
306C22⋊C4—28no—
307C22⋊C4—682no—
308C22⋊C4—682no—
309C22⋊C4C22⋊Dic1714yes—
310C42—24no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23C2×C22⋊Dic1711yescharacteristic
322C23C2×C22⋊Dic1712yes—
323C23#1058763912yes—
324Q8—44no—
325Q8—48no—
326Q8—48no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—1362no—
350C2×C4—1364no—
351C2×C4#1058750512yes—
352C2×C4#1058750512yes—
353C2×C4#1058763912yes—
354C8—1362no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—81no—
367C22—81no—
368C22#1176266511yescharacteristic
369C22C2×#1058750511yescharacteristic
370C22C2×#1058750511yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—44no—
375C4—41no—
376C4—44no—
377C4—2722no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2#1452869911yescharacteristic
384C2#1453043812yes—
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC2×C17×#536C2
socleC22×C17C2×#99
prime radicalC22×#41×C17C22
cosocleC22×D4×C17C23
hypercenterC2×#536D17
hypocenterC17#12692
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#11762599C2semidirect
13. C23×Q8×C17C4semidirect
17. C2×C17×#123C4semidirect
21. C2×C17×#117C4semidirect
191. C23×Q8Dic17semidirect
195. C2×#123Dic17semidirect
199. C2×#117Dic17semidirect
254. C17#12692semidirect

Maximal subgroups

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

subgroupindex
C2×#117625992
#145287122
#145287022
C2×C17×#5362
#1269217