Finite Group Databasebeta

← #16949807

Subgroups — #16949807

Order 4352.

976 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 115 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×#11763380C211yesmaximal, characteristic
2C2×#11762600C214yesmaximal, complemented
3#14528702C211yesmaximal, characteristic
4C17×#1785C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic, complemented
6#11763380—28no—
7C2×#10587630—21no—
8C2×#10587622—21no—
9C2×#10587621C412yescomplemented
10C2×#10587621C2212yes—
11C2×#10587610—21no—
12C2×#10587610—21no—
13C2×#10587612C411yescharacteristic, complemented
14C2×#10587505—21no—
15C2×#10587505C2212yes—
16#11762672—21no—
17#11762600—28no—
18C22×#41×C17C2211yescharacteristic
19C2×C17×#101—21no—
20C2×C17×#100C2212yes—
21C17×#407—21no—
22C2×#41×C17—24no—
23C2×#41×C17—42no—
24C2×#41×C17—42no—
25C22×Q8×C17D411yescharacteristic
26C22×D4×C17C2×C411yescharacteristic
27C22×D4×C17D411yescharacteristic
28C22×D4×C17C2311yescharacteristic
29C23×C4×C17C2×C411yescharacteristic
30C23×C4×C17D411yescharacteristic
31C23×C4×C17D411yescharacteristic
32C23×Dic17C2×C412yes—
33C22×Dic34—21no—
34C22×Dic34—21no—
35C22×Dic34—24no—
36C22×Dic34—42no—
37C22×Dic34—42no—
38C2×M16×C17—22no—
39C2×C22⋊Dic17—22no—
40C2×C22⋊Dic17—22no—
41C2×C22⋊Dic17—22no—
42C2×C22⋊Dic17C2×C412yes—
43#10587630—28no—
44#10587622—28no—
45#10587621—24no—
46#10587621—24no—
47#10587621—24no—
48#10587621D418yes—
49#10587612—24no—
50C2×C4⋊Dic17—21no—
51#10587610—28no—
52#10587610—28no—
53C2×#63225—22no—
54C2×#63225—22no—
55C2×C4×Dic17—21no—
56C17×#101—42no—
57C17×#100—24no—
58C17×C22⋊C8—42no—
59#10587505—24no—
60#10587505—42no—
61#10587502—42no—
62C24×C17C22⋊C411yescharacteristic
63C24×C17C22×C411yescharacteristic
64C24×C17C2×D411yescharacteristic
65#41×C17—416no—
66C2×Q8×C17—22no—
67C2×Q8×C17—22no—
68C2×Q8×C17—41no—
69C2×D4×C17—22no—
70C2×D4×C17—22no—
71C2×D4×C17—22no—
72C2×D4×C17—41no—
73C2×D4×C17—41no—
74C2×D4×C17—41no—
75C2×D4×C17—41no—
76C2×D4×C17—41no—
77C2×D4×C17C2×D414yes—
78C22×C4×C17—22no—
79C22×C4×C17—22no—
80C22×C4×C17—22no—
81C22×C4×C17—22no—
82C22×C4×C17—22no—
83C22×C4×C17—22no—
84C22×C4×C17—22no—
85C22×C4×C17—41no—
86C22×C4×C17—41no—
87C22×C4×C17C22⋊C411yescharacteristic
88C22×C4×C17C22⋊C411yescharacteristic
89C22×C4×C17C22⋊C411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×Dic17—22no—
92C22×Dic17—22no—
93C22×Dic17—22no—
94C22×Dic17—42no—
95C22×Dic17—42no—
96C22×Dic17—42no—
97C22×Dic17C22⋊C414yes—
98C2×Dic34—24no—
99C2×Dic34—24no—
100C2×Dic34—416no—
101C2×Dic34—42no—
102C2×Dic34—42no—
103C2×Dic34—42no—
104C2×Dic34—42no—
105M16×C17—42no—
106M16×C17—42no—
107C2×C8×C17—42no—
108C22⋊Dic17—28no—
109C22⋊Dic17—22no—
110C22⋊Dic17—28no—
111C22⋊Dic17—42no—
112C22⋊Dic17—42no—
113C22⋊Dic17C22⋊C414yes—
114C4⋊Dic17—24no—
115#63225—28no—
116#63225—28no—
117C4×Dic17—24no—
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⋊C412yes—
130C23×C17C2×C22⋊C411yescharacteristic
131C23×C17#12012yes—
132Q8×C17—44no—
133D4×C17—44no—
134D4×C17—44no—
135D4×C17—44no—
136C2×C4×C17—28no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
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—21no—
146C2×C4×C17—44no—
147C2×C4×C17—44no—
148C2×C4×C17#9912yes—
149C2×C4×C17#9912yes—
150C2×C4×C17#12012yes—
151C2×Dic17—24no—
152C2×Dic17—24no—
153C2×Dic17—44no—
154C2×Dic17—44no—
155C2×Dic17—42no—
156C2×Dic17—42no—
157C2×Dic17—44no—
158C2×Dic17—82no—
159C2×Dic17—84no—
160Dic34—48no—
161Dic34—48no—
162C8×C17—82no—
163C2×#536—171no—
164C2×#352—174no—
165#1785D1711yescharacteristic
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—41no—
187Dic17—44no—
188Dic17—44no—
189Dic17—162no—
190C22×#41C2×D1711yescharacteristic
191C23×Q8—171no—
192#536—348no—
193C2×C4⋊Q8—341no—
194C2×#124—341no—
195C2×#123—172no—
196C2×#123—172no—
197C2×C22⋊Q8—341no—
198C2×C22⋊Q8—341no—
199C2×#117—171no—
200C2×#101—21no—
201C2×#100C2×D1712yes—
202C2×#99—172no—
203C2×#99—341no—
204#407—21no—
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—344no—
218C22×Q8—341no—
219C22×Q8—341no—
220C22×Q8—682no—
221C22×Q8—682no—
222C22×Q8C22⋊D1711yescharacteristic
223C22×D4C4×D1711yescharacteristic
224C22×D4D6811yescharacteristic
225C22×D4C22×D1711yescharacteristic
226C23×C4—172no—
227C23×C4C4×D1711yescharacteristic
228C23×C4D6811yescharacteristic
229C23×C4C22⋊D1711yescharacteristic
230C2×M16—22no—
231C4⋊Q8—348no—
232#124—348no—
233#123—178no—
234#123—344no—
235#123—344no—
236#123—344no—
237C22⋊Q8—348no—
238C22⋊Q8—348no—
239#117—344no—
240C2×C4⋊C4—342no—
241C2×C4⋊C4—342no—
242C2×C4⋊C4—341no—
243C2×C22⋊C4—172no—
244C2×C22⋊C4—342no—
245C2×C22⋊C4—342no—
246C2×C22⋊C4—342no—
247C2×C42—341no—
248#101—42no—
249#100—24no—
250#99—344no—
251#99—682no—
252C22⋊C8—42no—
253#95—682no—
254C17#1269211yescharacteristic, complemented, Sylow
255C24#6322711yescharacteristic
256C24C2×C4×D1711yescharacteristic
257C24C2×D6811yescharacteristic
258#41—416no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—344no—
263C2×Q8—344no—
264C2×Q8—6816no—
265C2×Q8—682no—
266C2×Q8—682no—
267C2×Q8—682no—
268C2×Q8—682no—
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×D4D4×D1714yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—174no—
288C22×C4—342no—
289C22×C4—342no—
290C22×C4—342no—
291C22×C4—682no—
292C22×C4—682no—
293C22×C4—682no—
294C22×C4#6322711yescharacteristic
295C22×C4#6322711yescharacteristic
296C22×C4#6322711yescharacteristic
297C22×C4C2×C22⋊D1711yescharacteristic
298M16—42no—
299M16—42no—
300C2×C8—42no—
301C4⋊C4—344no—
302C4⋊C4—348no—
303C4⋊C4—348no—
304C22⋊C4—174no—
305C22⋊C4—348no—
306C22⋊C4—342no—
307C22⋊C4—348no—
308C22⋊C4—682no—
309C22⋊C4—682no—
310C42—344no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23C22⋊C4×D1712yes—
322C23#1058756812yes—
323C23C2×#6322711yescharacteristic
324Q8—44no—
325Q8—688no—
326Q8—688no—
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—21no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—344no—
343C2×C4—344no—
344C2×C4—684no—
345C2×C4—684no—
346C2×C4—682no—
347C2×C4—682no—
348C2×C4—684no—
349C2×C4—1362no—
350C2×C4—1364no—
351C2×C4#1058747712yes—
352C2×C4#1058747712yes—
353C2×C4#1058762312yes—
354C8—82no—
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#1176266411yescharacteristic
369C22C2×#1058747711yescharacteristic
370C22C2×#1058747711yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—41no—
375C4—2722no—
376C4—684no—
377C4—684no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2#1452869811yescharacteristic
384C2#1453043912yes—
subgroupG/H
centerC22C2×#10587477
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1785C2
socleC22×C17C2×#99
prime radicalC22×#41×C17C22
cosocleC22×D4×C17C23
hypercenter#1785D17
hypocenterC17#12692
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. C2×#11762600C2semidirect
5. C23×Dic34C4semidirect
9. C2×#10587621C4semidirect
13. C2×#10587612C4semidirect
254. C17#12692semidirect

Maximal subgroups

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

subgroupindex
C2×#117633802
C2×#117626002
#145287022
C17×#17852
#1269217