Finite Group Databasebeta

← #16931493

Subgroups — #16931493

Order 4352.

500 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 93 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#14538733C211yesmaximal, characteristic
2#14531675C211yesmaximal, characteristic
3#14529769C211yesmaximal, characteristic
4#14528884C211yesmaximal, characteristic
5#14527303C211yesmaximal, characteristic
6#14526689C211yesmaximal, characteristic
7C17×#1297C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9C2×#10587612C2211yescharacteristic
10#11763151—22no—
11#11763145—21no—
12#11763140—21no—
13#11762851C412yescomplemented
14#11762819C412yescomplemented
15#11762773C2211yescharacteristic
16#11762707—21no—
17#11762703—22no—
18#11762676—21no—
19#11762669C2211yescharacteristic
20C4×C4⋊Dic17—21no—
21#11762660—21no—
22#11762575—22no—
23#11762569—22no—
24#11762545—21no—
25#11762545—21no—
26C17×#407C2211yescharacteristic
27C4×C22⋊C4×C17C2211yescharacteristic
28C4×C22⋊C4×C17C2211yescharacteristic
29C23×C4×C17C2311yescharacteristic
30C22×Dic34C2×C411yescharacteristic
31C2×M16×C17C2×C412yes—
32C2×C22⋊C4×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C42×C17—21no—
35C2×C42×C17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17D412yes—
39#10587612—21no—
40#10587612—21no—
41#10587612C2×C412yes—
42#10587612D412yes—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17C2×C411yescharacteristic
45C2×C4⋊Dic17D411yescharacteristic
46C2×C4⋊Dic17D411yescharacteristic
47#10587610—21no—
48#10587610—21no—
49#10587610—21no—
50#10587610—21no—
51C2×#63225—22no—
52C2×#63225—21no—
53C2×C4×Dic17—22no—
54C2×C4×Dic17—21no—
55C2×#63223—21no—
56C17×C22⋊C8—21no—
57C17×C22⋊C8—21no—
58C17×#95—21no—
59C17×#95D411yescharacteristic
60C17×#95D411yescharacteristic
61#10587502—22no—
62#10587502—22no—
63#10587502—21no—
64C22⋊#3622—21no—
65C22⋊#3622—21no—
66#10587489—22no—
67#10587488—22no—
68#10587487—22no—
69#10587487—22no—
70C24×C17C2×D411yescharacteristic
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—42no—
74C22×C4×C17—41no—
75C22×C4×C17C22×C411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17#4111yescharacteristic
80C22×C4×C17#4111yescharacteristic
81C22×Dic17—21no—
82C22×Dic17—21no—
83C22×Dic17—22no—
84C22×Dic17—21no—
85C22×Dic17—21no—
86C2×Dic34—21no—
87C2×Dic34—21no—
88C2×Dic34—42no—
89M16×C17—42no—
90C2×C8×C17—22no—
91C22⋊C4×C17—22no—
92C22⋊C4×C17—42no—
93C22⋊C4×C17—41no—
94C42×C17—42no—
95C42×C17—42no—
96C22⋊Dic17—22no—
97C22⋊Dic17—22no—
98C22⋊Dic17—41no—
99C22⋊Dic17—41no—
100C22⋊Dic17—42no—
101C4⋊Dic17—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17—21no—
104C4⋊Dic17—21no—
105C4⋊Dic17—42no—
106C4⋊Dic17C22⋊C412yes—
107C4⋊Dic17C22⋊C412yes—
108#63225—44no—
109#63225—41no—
110#63225—41no—
111C4×Dic17—22no—
112C4×Dic17—22no—
113C4×Dic17—82no—
114#63223—42no—
115C2×#3622—41no—
116#9362—171nomaximal, Sylow
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—41no—
120C23×C17—41no—
121C23×C17C4×D411yescharacteristic
122C23×C17#12011yescharacteristic
123C23×C17#12111yescharacteristic
124C2×C4×C17—22no—
125C2×C4×C17—22no—
126C2×C4×C17—22no—
127C2×C4×C17—22no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—42no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—81no—
138C2×C4×C17—81no—
139C2×C4×C17C2×C22⋊C411yescharacteristic
140C2×C4×C17C4×D411yescharacteristic
141C2×C4×C17#12111yescharacteristic
142C2×C4×C17#12311yescharacteristic
143C2×Dic17—22no—
144C2×Dic17—22no—
145C2×Dic17—42no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—42no—
149C2×Dic17—42no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—42no—
153C2×Dic17—81no—
154C2×Dic17—81no—
155Dic34—42no—
156Dic34—81no—
157C8×C17—42no—
158#3622—81no—
159#3095—171no—
160#2347—171no—
161#1996—171no—
162#1826—171no—
163#1508—171no—
164#1305—171no—
165#1297D1711yescharacteristic
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—21no—
173C22×C17—21no—
174C22×C17—81no—
175C22×C17—81no—
176C22×C17#38611yescharacteristic
177C22×C17#45511yescharacteristic
178C22×C17#45611yescharacteristic
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—42no—
183C4×C17—81no—
184C4×C17—82no—
185C4×C17#41012yes—
186Dic17—42no—
187Dic17—42no—
188Dic17—82no—
189Dic17—81no—
190Dic17—81no—
191C2×C22⋊Q8—171no—
192C2×#117—171no—
193#485—342no—
194#484—341no—
195#483—341no—
196#428—172no—
197#419—172no—
198#407—171no—
199#407C2×D1711yescharacteristic
200#389—341no—
201#388—342no—
202#383—341no—
203#380—171no—
204C4×C4⋊C4—341no—
205C4×C4⋊C4—341no—
206C4×C22⋊C4C2×D1711yescharacteristic
207C4×C22⋊C4C2×D1711yescharacteristic
208#342—342no—
209#339—342no—
210#328—341no—
211#328—341no—
212C2×C17—21no—
213C2×C17—21no—
214C2×C17—41no—
215C2×C17—41no—
216C2×C17#188211yescharacteristic
217C2×C17#189911yescharacteristic
218C2×C17#190611yescharacteristic
219C22×Q8—171no—
220C23×C4C22×D1711yescharacteristic
221C2×M16—341no—
222C2×M16C4×D1712yes—
223C22⋊Q8—341no—
224C22⋊Q8—341no—
225C22⋊Q8—341no—
226C22⋊Q8—341no—
227#117—172no—
228#117—172no—
229#117—341no—
230#117—341no—
231C2×C4⋊C4—171no—
232C2×C4⋊C4—171no—
233C2×C4⋊C4—171no—
234C2×C4⋊C4—342no—
235C2×C4⋊C4—341no—
236C2×C4⋊C4—341no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—172no—
240C2×C22⋊C4—341no—
241C2×C22⋊C4—341no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—341no—
245C2×C42—342no—
246#107—342no—
247#106—342no—
248#103—342no—
249#103—342no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—341no—
253C22⋊C8—341no—
254#95—21no—
255#95—342no—
256#95—342no—
257#95—341no—
258#95D6811yescharacteristic
259#95D6811yescharacteristic
260C17#936211yescharacteristic, complemented, Sylow
261C24D4×D1711yescharacteristic
262C2×Q8—341no—
263C2×Q8—341no—
264C2×Q8—682no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4—342no—
272C22×C4—341no—
273C22×C4—341no—
274C22×C4C2×C4×D1711yescharacteristic
275C22×C4C2×D6811yescharacteristic
276C22×C4#6325211yescharacteristic
277C22×C4D4×D1711yescharacteristic
278C22×C4D4×D1711yescharacteristic
279C22×C4#6325611yescharacteristic
280M16—42no—
281M16—682no—
282C2×C8—22no—
283C2×C8—681no—
284C4⋊C4—172no—
285C4⋊C4—172no—
286C4⋊C4—341no—
287C4⋊C4—341no—
288C4⋊C4—341no—
289C4⋊C4—341no—
290C4⋊C4—684no—
291C4⋊C4—682no—
292C4⋊C4—681no—
293C4⋊C4—681no—
294C22⋊C4—22no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—342no—
298C22⋊C4—342no—
299C22⋊C4—681no—
300C22⋊C4—681no—
301C22⋊C4—682no—
302C42—42no—
303C42—42no—
304C42—342no—
305C42—342no—
306C42—1362no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23#1058756811yescharacteristic
312C23#1058757011yescharacteristic
313C23#1058757911yescharacteristic
314Q8—682no—
315Q8—1361no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—342no—
332C2×C4—342no—
333C2×C4—682no—
334C2×C4—681no—
335C2×C4—681no—
336C2×C4—682no—
337C2×C4—682no—
338C2×C4—682no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—1361no—
342C2×C4—1361no—
343C2×C4C4×D6811yescharacteristic
344C2×C4C22⋊C4×D1711yescharacteristic
345C2×C4#1058758011yescharacteristic
346C2×C4#1058758111yescharacteristic
347C8—42no—
348C8—1361no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1176268711yescharacteristic
360C22#1176298811yescharacteristic
361C22#1176299311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—82no—
368C4—682no—
369C4—682no—
370C4—1362no—
371C4—1361no—
372C4—1361no—
373C4#1176278112yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#1452915411yescharacteristic
379C2#1452924111yescharacteristic
380C2#1452928411yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1297C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1297D17
hypocenterC17#9362
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
13. #11762851C4semidirect
14. #11762819C4semidirect
260. C17#9362semidirect

Maximal subgroups

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

subgroupindex
#145387332
#145316752
#145297692
#145288842
#145273032
#145266892
C17×#12972
#936217