Finite Group Databasebeta

← #16938739

Subgroups — #16938739

Order 4352.

604 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 121 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#14528976C212yesmaximal
2#14527048C211yesmaximal, characteristic
3#14526662C212yesmaximal
4#14526627C211yesmaximal, characteristic
5C17×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic17C2211yescharacteristic
7C22×#63225—21no—
8C2×#10587479—22no—
9C2×C22⋊#3622C2212yes—
10C2×#10587502—21no—
11C2×#10587502C2211yescharacteristic
12#11762571—22no—
13C2×C17×C22⋊Q8—21no—
14C17×#398—22no—
15C17×#395—22no—
16C17×#380—21no—
17C17×#380C2212yes—
18C2×C17×#95C2211yescharacteristic
19C22×Q8×C17—21no—
20C23×C4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17C2311yescharacteristic
23C23×Dic17—21no—
24C23×Dic17—21no—
25C17×C22⋊Q8—42no—
26C17×C22⋊Q8—42no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D412yes—
32C2×C4⋊C4×C17D412yes—
33C2×C22⋊C4×C17—22no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17—42no—
36C2×#63225—42no—
37C2×#63225—42no—
38C2×#63225—42no—
39C22×#3622—22no—
40C17×#95—22no—
41C17×#95—22no—
42C17×#95—22no—
43#10587502—21no—
44#10587502—42no—
45#10587502Q812yescomplemented
46C22⋊#3622—22no—
47C22⋊#3622Q814yescomplemented
48#10587479—44no—
49C24×C17C2×D411yescharacteristic
50C2×Q8×C17—42no—
51C2×Q8×C17—81no—
52C2×Q8×C17—81no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17—41no—
60C22×C4×C17—41no—
61C22×C4×C17—41no—
62C22×C4×C17—41no—
63C22×C4×C17—41no—
64C22×C4×C17—41no—
65C22×C4×C17SD1612yes—
66C22×C4×C17SD1612yes—
67C22×C4×C17C2×D412yes—
68C22×C4×C17C2×Q812yes—
69C22×C4×C17C2×Q811yescharacteristic
70C22×C4×C17#4111yescharacteristic
71C22×Dic17—22no—
72C22×Dic17—22no—
73C22×Dic17—22no—
74C22×Dic17—22no—
75C22×Dic17—41no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C22×Dic17—41no—
81C22×Dic17—41no—
82C22×Dic17—41no—
83C22×Dic17—41no—
84C22×Dic17—41no—
85C4⋊C4×C17—42no—
86C4⋊C4×C17—42no—
87C4⋊C4×C17—42no—
88C4⋊C4×C17—44no—
89C4⋊C4×C17—81no—
90C4⋊C4×C17—81no—
91C22⋊C4×C17—42no—
92C22⋊C4×C17—42no—
93C4⋊Dic17—24no—
94C4⋊Dic17—42no—
95#63225—48no—
96C2×#3622—24no—
97C2×#3622—42no—
98C2×#3622—42no—
99#10570—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17#12011yescharacteristic
105C23×C17C22⋊Q811yescharacteristic
106C23×C17C4⋊Q811yescharacteristic
107C23×C17C2×SD1611yescharacteristic
108C23×C17C2×SD1611yescharacteristic
109C23×C17AGL(1,ℤ/8)11yescharacteristic
110C23×C17#13711yescharacteristic
111Q8×C17—84no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—42no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—81no—
132C2×C4×C17—81no—
133C2×C4×C17C22⋊Q812yes—
134C2×C4×C17C4⋊Q812yes—
135C2×Dic17—24no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—42no—
144C2×Dic17—42no—
145C2×Dic17—81no—
146C2×Dic17—81no—
147C2×Dic17—81no—
148C2×Dic17—81no—
149#3622—44no—
150#2659D1711yescharacteristic
151#1853—172no—
152#1444—171no—
153#1294—172no—
154#1280—171no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17#39511yescharacteristic
168C22×C17#44811yescharacteristic
169C22×C17#45011yescharacteristic
170C22×C17#45311yescharacteristic
171C22×C17#45611yescharacteristic
172C22×C17B(4,𝔽2)11yescharacteristic
173C22×C17#45811yescharacteristic
174C22×C17#47511yescharacteristic
175C22×C17#47611yescharacteristic
176C22×C17#49811yescharacteristic
177C22×C17#50111yescharacteristic
178C4×C17—22no—
179C4×C17—22no—
180C4×C17—42no—
181C4×C17—42no—
182C4×C17—82no—
183C4×C17—82no—
184Dic17—44no—
185Dic17—82no—
186C2×C22⋊Q8—21no—
187C22×C4⋊C4—171no—
188C22×C4⋊C4—341no—
189C2×#106—342no—
190C2×C22⋊C8—172no—
191#398—22no—
192#395—22no—
193#380—21no—
194#380C2×D1712yes—
195C2×#95—171no—
196C2×#95—341no—
197C2×#95C2×D1711yescharacteristic
198#340—342no—
199C2×C17—21no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17—21no—
203C2×C17#159611yescharacteristic
204C2×C17#160411yescharacteristic
205C2×C17#161111yescharacteristic
206C2×C17#161311yescharacteristic
207C2×C17#205811yescharacteristic
208C2×C17#206011yescharacteristic
209C2×C17#206211yescharacteristic
210C22×Q8—21no—
211C23×C4—341no—
212C23×C4—341no—
213C23×C4C22⋊D1711yescharacteristic
214C23×C4C22⋊D1711yescharacteristic
215C23×C4C22×D1711yescharacteristic
216C22×C8—342no—
217C22⋊Q8—42no—
218C22⋊Q8—42no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—342no—
224C2×C4⋊C4—682no—
225C2×C4⋊C4—682no—
226C2×C4⋊C4—682no—
227C2×C4⋊C4—682no—
228C2×C4⋊C4C22⋊D1712yes—
229C2×C4⋊C4C22⋊D1712yes—
230C2×C22⋊C4—22no—
231#106—684no—
232C22⋊C8—174no—
233C22⋊C8—342no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—172no—
238#95—341no—
239#95—682no—
240C17#1057011yescharacteristic, complemented, Sylow
241C24C2×C22⋊D1711yescharacteristic
242C2×Q8—42no—
243C2×Q8—81no—
244C2×Q8—81no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—342no—
258C22×C4—342no—
259C22×C4—342no—
260C22×C4—342no—
261C22×C4—681no—
262C22×C4—681no—
263C22×C4—681no—
264C22×C4—681no—
265C22×C4—681no—
266C22×C4—681no—
267C22×C4—681no—
268C22×C4—681no—
269C22×C4—681no—
270C22×C4—681no—
271C22×C4#6322912yes—
272C22×C4#6323012yes—
273C22×C4Q8×D1711yescharacteristic
274C22×C4Q8×D1712yes—
275C22×C4#6325611yescharacteristic
276C22×C4C2×C22⋊D1712yes—
277C2×C8—344no—
278C2×C8—682no—
279C2×C8—682no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—344no—
287C4⋊C4—682no—
288C4⋊C4—688no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1058761811yescharacteristic
296C23C2×#6322911yescharacteristic
297C23C2×#6323011yescharacteristic
298C23#1058762811yescharacteristic
299C23#1058763011yescharacteristic
300C23#1058763211yescharacteristic
301C23#1058763911yescharacteristic
302Q8—84no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—344no—
325C2×C4—682no—
326C2×C4—682no—
327C2×C4—682no—
328C2×C4—682no—
329C2×C4—682no—
330C2×C4—682no—
331C2×C4—682no—
332C2×C4—682no—
333C2×C4—682no—
334C2×C4—1361no—
335C2×C4—1361no—
336C2×C4—1361no—
337C2×C4—1361no—
338C2×C4#1058763012yes—
339C2×C4#1058763212yes—
340C8—684no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1176272811yescharacteristic
354C22#1176294811yescharacteristic
355C22#1176296011yescharacteristic
356C22#1176297711yescharacteristic
357C22#1176299211yescharacteristic
358C22#1176299611yescharacteristic
359C22#1176299911yescharacteristic
360C22#1176309111yescharacteristic
361C22#1176309611yescharacteristic
362C22#1176320511yescharacteristic
363C22#1176321511yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—82no—
370C4—684no—
371C4—1362no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#1452770711yescharacteristic
377C2#1452775311yescharacteristic
378C2#1452779211yescharacteristic
379C2#1452780411yescharacteristic
380C2#1453011411yescharacteristic
381C2#1453012611yescharacteristic
382C2#1453013611yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#2659C2
socleC23×C17#120
prime radicalC24×C17C2×D4
hypercenter#2659D17
hypocenterC17#10570
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
45. #10587502Q8semidirect
47. C22⋊#3622Q8semidirect
240. C17#10570semidirect

Maximal subgroups

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

subgroupindex
#145289762
#145270482
#145266622
#145266272
C17×#26592
#1057017