Finite Group Databasebeta

← #16943814

Subgroups — #16943814

Order 4352.

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

normal subgroups only

This group has 440 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
1#14531496C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526729C212yesmaximal, complemented
4#14526717C212yesmaximal, complemented
5C17×#1465C211yesmaximal, characteristic
6C2×#10587622C411yescharacteristic
7C2×C4×Dic34C411yescharacteristic
8#11762773—21no—
9#11762714C2211yescharacteristic
10#11762712C411yescharacteristic
11#11762697—21no—
12#11762697—21no—
13#11762682—21no—
14#11762682—21no—
15#11762678C2211yescharacteristic
16#11762655C411yescharacteristic
17#11762535C2212yes—
18#11762533—21no—
19C2×C4×D4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#324C2212yes—
22C17×#323—21no—
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C2×M16×C17—41no—
28C4×D4×C17—41no—
29C4×D4×C17—41no—
30C2×C4⋊C4×C17C2311yescharacteristic
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38#10587622—22no—
39#10587622—22no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×C4×Dic17—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17C2×C411yescharacteristic
47C2×#63223—41no—
48C4×Dic34—41no—
49C4×Dic34—41no—
50C17×C22⋊C8—22no—
51C17×C22⋊C8—41no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C22⋊#3622—22no—
57C22⋊#3622—41no—
58C24×C17C22⋊C411yescharacteristic
59C24×C17C22⋊C411yescharacteristic
60C2×D4×C17—21no—
61C2×D4×C17—21no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—81no—
69C22×C4×C17—81no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C2×Dic34—21no—
80C2×Dic34—41no—
81C2×Dic34—41no—
82C2×Dic34M1612yescomplemented
83M16×C17—82no—
84C2×C8×C17—42no—
85C4⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C42×C17—21no—
89C42×C17—21no—
90C22⋊Dic17—24no—
91C22⋊Dic17—24no—
92C22⋊Dic17—81no—
93C22⋊Dic17—81no—
94C4⋊Dic17—41no—
95#63225—41no—
96#63225—41no—
97#63225—81no—
98#63225—81no—
99C4×Dic17—21no—
100C4×Dic17—42no—
101C4×Dic17—41no—
102C4×Dic17M1612yescomplemented
103#63223—82no—
104C2×#3622—42no—
105#11535—171nomaximal, Sylow
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17C2×C22⋊C411yescharacteristic
117D4×C17—42no—
118D4×C17—42no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—81no—
131C2×C4×C17—81no—
132C2×C4×C17#9911yescharacteristic
133C2×C4×C17#9911yescharacteristic
134C2×C4×C17C2×M1611yescharacteristic
135C2×C4×C17C2×M1611yescharacteristic
136C2×Dic17—22no—
137C2×Dic17—22no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148Dic34—42no—
149Dic34—42no—
150C8×C17—82no—
151#3622—82no—
152#2326—171no—
153#1465—171no—
154#1465D1711yescharacteristic
155#1325—172no—
156#1319—172no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—42no—
160C22×C17—42no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—81no—
168C22×C17—81no—
169C22×C17#40711yescharacteristic
170C22×C17C2×#9911yescharacteristic
171C22×C17#41311yescharacteristic
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—41no—
179Dic17—42no—
180Dic17—42no—
181Dic17—81no—
182Dic17—81no—
183C2×#124—171no—
184C2×C4×Q8—171no—
185C2×C4×D4C2×D1711yescharacteristic
186#407—21no—
187#407—341no—
188#391—171no—
189#390—171no—
190#388—341no—
191#388—341no—
192#385—341no—
193#385—341no—
194#384—171no—
195#376—171no—
196#324—172no—
197#324C2×D1712yes—
198#323—21no—
199#323—341no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17—81no—
203C2×C17—81no—
204C2×C17#146511yescharacteristic
205C2×C17#212711yescharacteristic
206C2×C17#213511yescharacteristic
207C22×Q8—171no—
208C22×D4C4×D1711yescharacteristic
209C23×C4D6811yescharacteristic
210C23×C4C22⋊D1711yescharacteristic
211C2×M16—41no—
212C2×M16—681no—
213#124—342no—
214#124—342no—
215C4×Q8—681no—
216C4×Q8—681no—
217C4×D4—41no—
218C4×D4—41no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4C22×D1711yescharacteristic
224C2×C22⋊C4—341no—
225C2×C22⋊C4—341no—
226C2×C22⋊C4—341no—
227C2×C22⋊C4—341no—
228C2×C22⋊C4D6811yescharacteristic
229C2×C22⋊C4C22⋊D1711yescharacteristic
230C2×C42—171no—
231C2×C42—171no—
232C2×C42—341no—
233C2×C42C4×D1711yescharacteristic
234C22⋊C8—22no—
235C22⋊C8—41no—
236C22⋊C8—342no—
237C22⋊C8—681no—
238#95—341no—
239#95—341no—
240#95—341no—
241#95—341no—
242C17#1153511yescharacteristic, complemented, Sylow
243C24#6322711yescharacteristic
244C24#6322711yescharacteristic
245C2×Q8—172no—
246C2×Q8—341no—
247C2×Q8—681no—
248C2×Q8—681no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4#6322711yescharacteristic
264C22×C4#6322711yescharacteristic
265C22×C4C2×C4×D1711yescharacteristic
266C22×C4C2×D6811yescharacteristic
267C22×C4C2×C22⋊D1711yescharacteristic
268M16—82no—
269M16—1362no—
270C2×C8—42no—
271C2×C8—682no—
272C4⋊C4—41no—
273C4⋊C4—681no—
274C4⋊C4—681no—
275C4⋊C4—681no—
276C4⋊C4—1361no—
277C4⋊C4—1361no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—344no—
281C22⋊C4—344no—
282C22⋊C4—1361no—
283C22⋊C4—1361no—
284C42—21no—
285C42—21no—
286C42—172no—
287C42—341no—
288C42—681no—
289C42—682no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C2×#6322711yescharacteristic
301Q8—682no—
302Q8—682no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—342no—
319C2×C4—342no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4#1058747711yescharacteristic
331C2×C4#1058747711yescharacteristic
332C2×C4M16×D1711yescharacteristic
333C2×C4M16×D1711yescharacteristic
334C8—82no—
335C8—1362no—
336C22—21no—
337C22—21no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22#1176277211yescharacteristic
349C22C2×#1058747711yescharacteristic
350C22#1176279111yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—682no—
359C4—682no—
360C4—1361no—
361C4—1361no—
362C2—21no—
363C2—21no—
364C2—81no—
365C2—81no—
366C2#1452712011yescharacteristic
367C2#1453045111yescharacteristic
368C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11535
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #14526729C2semidirect
4. #14526717C2semidirect
82. C2×Dic34M16semidirect
102. C4×Dic17M16semidirect
242. C17#11535semidirect

Maximal subgroups

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

subgroupindex
#145314962
#145271222
#145267292
#145267172
C17×#14652
#1153517