Finite Group Databasebeta

← #16945406

Subgroups — #16945406

Order 4352.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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
1D17×#467C211yesmaximal, characteristic
2#14530393C211yesmaximal, characteristic
3#14528808C211yesmaximal, characteristic
4#14526797C211yesmaximal, characteristic, complemented
5#14526796C211yesmaximal, characteristic, complemented
6#14526795C211yesmaximal, characteristic, complemented
7C17×#1348C211yesmaximal, characteristic, complemented
8C2×Q16×D17C411yescharacteristic, complemented
9C22×C8×D17C2211yescharacteristic
10C22⋊Q8×D17—21no—
11C2×#10587469—21no—
12#11763051C411yescharacteristic, complemented
13#11763050C2211yescharacteristic
14#11763049—21no—
15#11763048—21no—
16#11763047C411yescharacteristic, complemented
17#107×D17C411yescharacteristic, complemented
18C2×#10587479—21no—
19#103×D17—21no—
20#11762666—21no—
21C22⋊#63214C2211yescharacteristic
22#11762588—21no—
23#11762572C2211yescharacteristic
24C17×#467C2211yescharacteristic
25C17×C22⋊C16C2211yescharacteristic
26C17×#340C2211yescharacteristic
27C2×Q8×D17—21no—
28C22×C4×D17D411yescharacteristic
29C2×Q16×C17C2×C411yescharacteristic
30C22×C8×C17C2311yescharacteristic
31C17×C22⋊Q8—21no—
32C2×C4⋊C4×C17—21no—
33#10587632—21no—
34C2×#63231—21no—
35C2×#63227—41no—
36C2×C4⋊Dic17—21no—
37#10587610—21no—
38C22×#3622D411yescharacteristic
39Q16×D17—42no—
40C2×Dic68C2×C411yescharacteristic
41C2×C8×D17—42no—
42C2×C8×D17C2×C411yescharacteristic
43C2×C8×D17C2×C411yescharacteristic
44C2×C8×D17D411yescharacteristic
45C2×C8×D17D411yescharacteristic
46#10587583—21no—
47#10587582—41no—
48C4⋊C4×D17—21no—
49C4⋊C4×D17—41no—
50C22⋊C4×D17—41no—
51#10587564—41no—
52C2×C16×C17—21no—
53C17×#107C2×C411yescharacteristic
54C17×#103—21no—
55#10587506—21no—
56#10587502—41no—
57#10587489C2×C411yescharacteristic
58#10587487—21no—
59C2×#63214—21no—
60#10587481—21no—
61#10587479—42no—
62#10587478—21no—
63#10587469—42no—
64C23×D17D811yescharacteristic
65C2×Q8×C17—21no—
66C22×C4×C17—41no—
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—41no—
69C22×Dic17D811yescharacteristic
70Q8×D17—42no—
71Q8×D17—42no—
72C2×C4×D17—41no—
73C2×C4×D17—41no—
74C2×C4×D17—41no—
75C2×C4×D17—41no—
76C2×C4×D17C22⋊C411yescharacteristic
77C2×C4×D17C22⋊C411yescharacteristic
78C2×C4×D17SD1611yescharacteristic
79C2×C4×D17SD1611yescharacteristic
80C2×Dic34—21no—
81C2×Dic34—41no—
82Q16×C17—41no—
83C2×C8×C17—41no—
84C2×C8×C17C22×C411yescharacteristic
85C2×C8×C17C2×D411yescharacteristic
86C4⋊C4×C17—21no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C22⋊C4×C17—41no—
91C22⋊Dic17—41no—
92#63231—42no—
93#63227—41no—
94#63227—41no—
95#63227—82no—
96C4⋊Dic17—21no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100#63225—41no—
101#63225—41no—
102#63225—41no—
103C2×#3622—22no—
104C2×#3622C22⋊C411yescharacteristic
105C2×#3622C22⋊C411yescharacteristic
106Dic68—41no—
107C8×D17—22no—
108C8×D17—22no—
109C8×D17—41no—
110C8×D17—41no—
111C16×C17—41no—
112#63214—41no—
113#11889—171nomaximal, Sylow
114C23×C17C2×D811yescharacteristic
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—22no—
118C22×D17—41no—
119C22×D17#10211yescharacteristic
120C22×D17#10211yescharacteristic
121Q8×C17—41no—
122Q8×C17—41no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—81no—
128C2×C4×C17—81no—
129C2×C4×C17—81no—
130C2×C4×C17C2×C22⋊C411yescharacteristic
131C2×C4×C17C2×SD1611yescharacteristic
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—81no—
136C2×Dic17—81no—
137C2×Dic17—81no—
138C2×Dic17#10211yescharacteristic
139C2×Dic17#10211yescharacteristic
140C4×D17—21no—
141C4×D17—21no—
142C4×D17—21no—
143C4×D17—21no—
144C4×D17—41no—
145C4×D17—41no—
146C4×D17—82no—
147C4×D17—82no—
148Dic34—41no—
149Dic34—41no—
150Dic34—82no—
151C8×C17—21no—
152C8×C17—21no—
153#3622—21no—
154#3622#9912yes—
155C2×#467—171no—
156#2112—171no—
157#1807—171no—
158#1348—171no—
159#1348—171no—
160#1348—171no—
161#1348D1711yescharacteristic, complemented
162C22×C17—21no—
163C22×C17—21no—
164C22×C17C2×#10211yescharacteristic
165C22×C17#51011yescharacteristic
166C22×C17#51011yescharacteristic
167C2×D17—21no—
168C2×D17—21no—
169C2×D17—21no—
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—42no—
173C2×D17—42no—
174C2×D17#36012yes—
175C4×C17—21no—
176C4×C17—81no—
177C4×C17—81no—
178C4×C17—161no—
179C4×C17C2×#9911yescharacteristic
180C4×C17#42111yescharacteristic
181Dic17—21no—
182Dic17—21no—
183Dic17—81no—
184Dic17—81no—
185Dic17—161no—
186C22×Q16—171no—
187C23×C8—171no—
188C2×C22⋊Q8—341no—
189C2×M32—341no—
190#467—171no—
191#467—171no—
192#467—171no—
193#467—341no—
194#467—341no—
195#467C2×D1711yescharacteristic
196C2×#107—171no—
197C2×#106—341no—
198C2×#103—341no—
199#380—341no—
200C22⋊C16—171no—
201C22⋊C16—341no—
202C22⋊C16C2×D1711yescharacteristic
203#340—171no—
204#340C2×D1711yescharacteristic
205C2×C17—21no—
206C2×C17—21no—
207C2×C17#151911yescharacteristic
208C2×C17#214111yescharacteristic
209C2×C17C2×#36011yescharacteristic
210D17—22no—
211D17—41no—
212C22×Q8—341no—
213C23×C4—171no—
214C2×Q16—171no—
215C2×Q16—341no—
216C2×Q16—682no—
217C2×Q16C4×D1711yescharacteristic
218C22×C8—171no—
219C22×C8—171no—
220C22×C8—171no—
221C22×C8—171no—
222C22×C8—171no—
223C22×C8—682no—
224C22×C8C22×D1711yescharacteristic
225C22⋊Q8—21no—
226C22⋊Q8—341no—
227C22⋊Q8—341no—
228C22⋊Q8—341no—
229C22⋊Q8—681no—
230C22⋊Q8—681no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—341no—
233C2×C4⋊C4—341no—
234C2×C4⋊C4—681no—
235C2×C22⋊C4—681no—
236C2×C22⋊C4—681no—
237M32—682no—
238C2×C16—21no—
239C2×C16—341no—
240#107—171no—
241#107—341no—
242#107C4×D1711yescharacteristic
243#106—682no—
244#103—21no—
245#103—341no—
246#103—341no—
247#103—341no—
248#95—681no—
249C17#1188911yescharacteristic, complemented, Sylow
250C24—171no—
251C2×Q8—21no—
252C2×Q8—341no—
253C2×Q8—681no—
254C2×Q8—682no—
255C2×Q8—682no—
256C22×C4—41no—
257C22×C4—171no—
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—171no—
261C22×C4—171no—
262C22×C4—681no—
263C22×C4—681no—
264C22×C4—681no—
265C22×C4—681no—
266C22×C4—681no—
267C22×C4D4×D1711yescharacteristic
268Q16—41no—
269Q16—681no—
270Q16—682no—
271C2×C8—41no—
272C2×C8—171no—
273C2×C8—171no—
274C2×C8—342no—
275C2×C8—342no—
276C2×C8—342no—
277C2×C8—681no—
278C2×C8—681no—
279C2×C8C2×C4×D1711yescharacteristic
280C2×C8D4×D1711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—341no—
286C4⋊C4—681no—
287C4⋊C4—681no—
288C4⋊C4—681no—
289C4⋊C4—681no—
290C4⋊C4—681no—
291C4⋊C4—681no—
292C22⋊C4—41no—
293C22⋊C4—681no—
294C22⋊C4—681no—
295C22⋊C4—681no—
296C22⋊C4—1362no—
297C16—41no—
298C16—681no—
299C23—171no—
300C23—171no—
301C23—342no—
302C23—341no—
303C23—341no—
304C23—681no—
305C23D8×D1711yescharacteristic
306Q8—41no—
307Q8—41no—
308Q8—681no—
309Q8—681no—
310Q8—1362no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—171no—
319C2×C4—171no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—1362no—
330C2×C4—1362no—
331C2×C4—1361no—
332C2×C4—1361no—
333C2×C4—1361no—
334C2×C4C22⋊C4×D1711yescharacteristic
335C2×C4SD16×D1711yescharacteristic
336C8—21no—
337C8—21no—
338C8—172no—
339C8—341no—
340C22—21no—
341C22—21no—
342C22—172no—
343C22—341no—
344C22—341no—
345C22—341no—
346C22—341no—
347C22—341no—
348C22—682no—
349C22—682no—
350C22#102×D1711yescharacteristic
351C22#1176324811yescharacteristic
352C22#1176325111yescharacteristic
353C4—21no—
354C4—81no—
355C4—81no—
356C4—161no—
357C4—2721no—
358C4—341no—
359C4—341no—
360C4—1361no—
361C4—1361no—
362C4#99×D1711yescharacteristic
363C4#1176282411yescharacteristic
364C2—21no—
365C2—21no—
366C2—342no—
367C2—681no—
368C2#1452735011yescharacteristic
369C2#1453050211yescharacteristic
370C2D17×#36011yescharacteristic
subgroupG/H
centerC22#102×D17
commutatorC2×C8×C17C22×C4
FrattiniC22×C8C22×D17
FittingC17×#1348C2
socleC22×C17C2×#102
prime radicalC23×D17D8
cosocleC22×C8×C17C23
hypercenter#1348D17
hypocenterC17#11889
LCSG ▷ C2×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
4. #14526797C2semidirect
5. #14526796C2semidirect
6. #14526795C2semidirect
7. C17×#1348C2semidirect
8. C2×Q16×D17C4semidirect
12. #11763051C4semidirect
16. #11763047C4semidirect
17. #107×D17C4semidirect
161. #1348D17semidirect
249. C17#11889semidirect

Maximal subgroups

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

subgroupindex
D17×#4672
#145303932
#145288082
#145267972
#145267962
#145267952
C17×#13482
#1188917