Finite Group Databasebeta

← #16937678

Subgroups — #16937678

Order 4352.

576 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 113 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#14534898C211yesmaximal, characteristic
2#14529089C212yesmaximal
3C2×#11762547C212yesmaximal
4#14527040C211yesmaximal, characteristic
5C17×#1280C211yesmaximal, characteristic
6C2×#10587610C2212yes—
7C22×C4⋊Dic17C2211yescharacteristic
8C2×#10587487—22no—
9#11762741—22no—
10#11762730—22no—
11#11762669—21no—
12#11762669—21no—
13C2×#10587502—21no—
14C2×#10587502—21no—
15C2×#10587502C2211yescharacteristic
16#11762572—22no—
17#11762547—24no—
18C2×C17×C22⋊C8C2212yes—
19C2×C17×#95C2211yescharacteristic
20C23×C4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17C2311yescharacteristic
23C23×Dic17—21no—
24C23×Dic17—21no—
25C22×Dic34D412yes—
26C22×C8×C17—22no—
27C2×C22⋊Dic17—22no—
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—22no—
33C2×C4⋊Dic17—42no—
34C2×C4⋊Dic17D412yes—
35#10587610—22no—
36#10587610—22no—
37#10587610—22no—
38#10587610—22no—
39C2×#63225—22no—
40C17×C22⋊C8—22no—
41C17×C22⋊C8—22no—
42C17×#95—22no—
43#10587502—22no—
44#10587502—22no—
45#10587502—21no—
46#10587502—42no—
47#10587502—42no—
48#10587502D412yes—
49#10587487—44no—
50C24×C17C2×D411yescharacteristic
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—41no—
56C22×C4×C17—41no—
57C22×C4×C17—41no—
58C22×C4×C17—41no—
59C22×C4×C17—41no—
60C22×C4×C17—41no—
61C22×C4×C17SD1612yes—
62C22×C4×C17Q1612yes—
63C22×C4×C17C2×D412yes—
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17#4112yes—
66C22×C4×C17#4111yescharacteristic
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—41no—
72C22×Dic17—41no—
73C22×Dic17—41no—
74C22×Dic17—41no—
75C22×Dic17—41no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C22×Dic17—41no—
81C22×Dic17—41no—
82C22×Dic17—41no—
83C2×Dic34—22no—
84C2×Dic34—22no—
85C2×Dic34—42no—
86C2×Dic34—42no—
87C2×C8×C17—42no—
88C2×C8×C17—42no—
89C2×C8×C17—42no—
90C22⋊Dic17—42no—
91C22⋊Dic17—42no—
92C4⋊Dic17—24no—
93C4⋊Dic17—22no—
94C4⋊Dic17—22no—
95C4⋊Dic17—42no—
96C4⋊Dic17—81no—
97C4⋊Dic17—81no—
98C4⋊Dic17—81no—
99C4⋊Dic17—81no—
100#63225—42no—
101#63225—42no—
102#10407—171nomaximal, Sylow
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17#12011yescharacteristic
108C23×C17#12111yescharacteristic
109C23×C17#12411yescharacteristic
110C23×C17C2×SD1611yescharacteristic
111C23×C17C2×Q1611yescharacteristic
112C23×C17#13711yescharacteristic
113C23×C17#13711yescharacteristic
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—22no—
117C2×C4×C17—22no—
118C2×C4×C17—41no—
119C2×C4×C17—42no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—81no—
128C2×C4×C17—81no—
129C2×C4×C17#12112yes—
130C2×C4×C17#12312yes—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—81no—
144C2×Dic17—81no—
145C2×Dic17—81no—
146C2×Dic17—81no—
147C2×Dic17—81no—
148C2×Dic17—81no—
149C2×Dic17—81no—
150C2×Dic17—81no—
151Dic34—44no—
152Dic34—82no—
153C8×C17—82no—
154#2659—171no—
155#1872—172no—
156C2×#328—172no—
157#1442—171no—
158#1280D1711yescharacteristic
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—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17#39411yescharacteristic
172C22×C17#44811yescharacteristic
173C22×C17#45111yescharacteristic
174C22×C17#45411yescharacteristic
175C22×C17#45511yescharacteristic
176C22×C17B(4,𝔽2)11yescharacteristic
177C22×C17#45811yescharacteristic
178C22×C17#46111yescharacteristic
179C22×C17#46211yescharacteristic
180C22×C17#48711yescharacteristic
181C22×C17#49111yescharacteristic
182C4×C17—22no—
183C4×C17—22no—
184C4×C17—42no—
185C4×C17—42no—
186Dic17—82no—
187Dic17—82no—
188Dic17—82no—
189Dic17—82no—
190C2×C22⋊Q8—172no—
191C22×C4⋊C4—171no—
192C2×#103—342no—
193C2×C22⋊C8C2×D1712yes—
194#398—342no—
195#395—342no—
196#380—341no—
197#380—341no—
198C2×#95—171no—
199C2×#95—341no—
200C2×#95—341no—
201C2×#95C2×D1711yescharacteristic
202#340—342no—
203#328—344no—
204C2×C17—21no—
205C2×C17—21no—
206C2×C17—21no—
207C2×C17—21no—
208C2×C17#159911yescharacteristic
209C2×C17#160211yescharacteristic
210C2×C17#160811yescharacteristic
211C2×C17#161711yescharacteristic
212C2×C17#204011yescharacteristic
213C2×C17#204311yescharacteristic
214C2×C17#204411yescharacteristic
215C22×Q8—172no—
216C23×C4—341no—
217C23×C4—341no—
218C23×C4D6811yescharacteristic
219C23×C4D6811yescharacteristic
220C23×C4C22×D1711yescharacteristic
221C22×C8—22no—
222C22⋊Q8—342no—
223C22⋊Q8—342no—
224C22⋊Q8—342no—
225C22⋊Q8—342no—
226C2×C4⋊C4—172no—
227C2×C4⋊C4—342no—
228C2×C4⋊C4—342no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—341no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4—341no—
233C2×C4⋊C4—682no—
234C2×C22⋊C4—342no—
235#103—684no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238#95—22no—
239#95—172no—
240#95—342no—
241#95—342no—
242#95—341no—
243#95—682no—
244#95—682no—
245C17#1040711yescharacteristic, complemented, Sylow
246C24C2×D6811yescharacteristic
247C2×Q8—342no—
248C2×Q8—342no—
249C2×Q8—682no—
250C2×Q8—682no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—342no—
262C22×C4—342no—
263C22×C4—342no—
264C22×C4—342no—
265C22×C4—681no—
266C22×C4—681no—
267C22×C4—681no—
268C22×C4—681no—
269C22×C4—681no—
270C22×C4—681no—
271C22×C4—681no—
272C22×C4—681no—
273C22×C4—681no—
274C22×C4—681no—
275C22×C4—681no—
276C22×C4—681no—
277C22×C4#6321912yes—
278C22×C4Dic6812yes—
279C22×C4#6325212yes—
280C22×C4D4×D1712yes—
281C22×C4D4×D1711yescharacteristic
282C22×C4#6325611yescharacteristic
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C4⋊C4—344no—
287C4⋊C4—342no—
288C4⋊C4—342no—
289C4⋊C4—682no—
290C4⋊C4—682no—
291C4⋊C4—682no—
292C4⋊C4—1361no—
293C4⋊C4—1361no—
294C4⋊C4—1361no—
295C4⋊C4—1361no—
296C22⋊C4—682no—
297C22⋊C4—682no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#1058756111yescharacteristic
303C23#1058756811yescharacteristic
304C23#1058758111yescharacteristic
305C23C2×#6321911yescharacteristic
306C23C2×Dic6811yescharacteristic
307C23#1058759511yescharacteristic
308C23#1058759511yescharacteristic
309Q8—684no—
310Q8—1362no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—81no—
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—682no—
335C2×C4—682no—
336C2×C4—682no—
337C2×C4—682no—
338C2×C4—1361no—
339C2×C4—1361no—
340C2×C4—1361no—
341C2×C4—1361no—
342C2×C4—1361no—
343C2×C4—1361no—
344C2×C4—1361no—
345C2×C4—1361no—
346C2×C4#1058757012yes—
347C2×C4#1058758012yes—
348C8—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22#1176272711yescharacteristic
362C22#1176295011yescharacteristic
363C22#1176296711yescharacteristic
364C22#1176298411yescharacteristic
365C22#1176298911yescharacteristic
366C22#1176299711yescharacteristic
367C22#1176300011yescharacteristic
368C22#1176301511yescharacteristic
369C22#1176302011yescharacteristic
370C22#1176316011yescharacteristic
371C22#1176317811yescharacteristic
372C4—22no—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—1362no—
377C4—1362no—
378C4—1362no—
379C4—1362no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#1452772611yescharacteristic
385C2#1452774311yescharacteristic
386C2#1452777711yescharacteristic
387C2#1452782811yescharacteristic
388C2#1453001411yescharacteristic
389C2#1453003111yescharacteristic
390C2#1453003611yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1280C2
socleC23×C17#120
prime radicalC24×C17C2×D4
hypercenter#1280D17
hypocenterC17#10407
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
245. C17#10407semidirect

Maximal subgroups

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

subgroupindex
#145348982
#145290892
C2×#117625472
#145270402
C17×#12802
#1040717