Finite Group Databasebeta

← #16932005

Subgroups — #16932005

Order 4352.

468 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 95 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#14537868C211yesmaximal, characteristic
2#14533773C211yesmaximal, characteristic
3#14527510C211yesmaximal, characteristic
4#14527127C211yesmaximal, characteristic
5#14526641C211yesmaximal, characteristic
6#14526639C211yesmaximal, characteristic
7C17×#1558C211yesmaximal, characteristic
8C2×#10587630C412yes—
9C2×C4×Dic34C2211yescharacteristic
10#11762938—22no—
11#11762937—22no—
12#11762866C2211yescharacteristic
13#11762866C2211yescharacteristic
14#11762863—21no—
15#11762859—21no—
16#11762740—21no—
17#11762740—21no—
18#11762716C412yes—
19#11762705—21no—
20#11762705—21no—
21C4×C4⋊Dic17C2211yescharacteristic
22#11762558—22no—
23#11762534—21no—
24#11762534—21no—
25C2×C17×C4⋊Q8C2211yescharacteristic
26C17×#432C2211yescharacteristic
27C17×#432C2211yescharacteristic
28C17×#333—22no—
29C22×Q8×C17C2×C412yes—
30C22×Dic34C2×C411yescharacteristic
31C17×C4⋊Q8—21no—
32C17×C4⋊Q8—21no—
33C17×C4⋊Q8—21no—
34C17×C4⋊Q8—21no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C42×C17C2311yescharacteristic
38#10587630—44no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—22no—
42C2×#63225—22no—
43C2×#63225D411yescharacteristic
44C2×#63225D411yescharacteristic
45C2×C4×Dic17C2×C412yes—
46C2×C4×Dic17D411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C4×Dic34—21no—
49C4×Dic34—21no—
50C4×Dic34—21no—
51C4×Dic34—21no—
52C17×C4⋊C8—21no—
53C17×C4⋊C8D412yes—
54C17×C22⋊C8—21no—
55C17×C22⋊C8—21no—
56C17×#97—21no—
57C17×#97—21no—
58#10587502—21no—
59#10587502—22no—
60#10587502—22no—
61C22⋊#3622—21no—
62C22⋊#3622—21no—
63#10587486—22no—
64#10587485—22no—
65#10587485—22no—
66C8×Dic17—22no—
67C4⋊#3622—21no—
68C4⋊#3622D412yes—
69#10587474—21no—
70#10587474—21no—
71C2×Q8×C17—22no—
72C2×Q8×C17—22no—
73C2×Q8×C17—42no—
74C2×Q8×C17—42no—
75C22×C4×C17—22no—
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×Dic17—22no—
80C22×Dic17C22⋊C411yescharacteristic
81C22×Dic17C22⋊C411yescharacteristic
82C22×Dic17C22⋊C411yescharacteristic
83C22×Dic17C22⋊C411yescharacteristic
84C2×Dic34—21no—
85C2×Dic34—21no—
86C2×Dic34—22no—
87C2×Dic34—41no—
88C2×C8×C17—22no—
89C2×C8×C17—41no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17—41no—
93C4⋊C4×C17—41no—
94C42×C17C2×D411yescharacteristic
95C42×C17C2×D411yescharacteristic
96C42×C17#4111yescharacteristic
97C42×C17#4111yescharacteristic
98C4⋊Dic17—24no—
99C4⋊Dic17—21no—
100C4⋊Dic17—21no—
101#63225—21no—
102#63225—21no—
103#63225—21no—
104#63225—21no—
105#63225—44no—
106#63225—44no—
107C4×Dic17—21no—
108C4×Dic17—21no—
109C4×Dic17—21no—
110C4×Dic17—21no—
111C4×Dic17—42no—
112C2×#3622—22no—
113C2×#3622—41no—
114#9452—171nomaximal, Sylow
115C23×C17C2×C22⋊C411yescharacteristic
116Q8×C17—82no—
117Q8×C17—82no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—22no—
121C2×C4×C17—41no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17C4×D411yescharacteristic
126C2×C4×C17C4×D411yescharacteristic
127C2×C4×C17#12011yescharacteristic
128C2×C4×C17#12111yescharacteristic
129C2×C4×C17#12111yescharacteristic
130C2×C4×C17#12311yescharacteristic
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—24no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17#10012yes—
144Dic34—42no—
145Dic34—42no—
146C8×C17—42no—
147C8×C17—81no—
148#3622—42no—
149#3622—81no—
150#2983—171no—
151#2546—171no—
152#1558—171no—
153#1558D1711yescharacteristic
154#1467—171no—
155#1285—171no—
156#1285—171no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17#38611yescharacteristic
160C22×C17C2×#10011yescharacteristic
161C22×C17#41311yescharacteristic
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—82no—
168C4×C17#45612yes—
169Dic17—22no—
170Dic17—44no—
171Dic17—41no—
172Dic17—41no—
173Dic17—41no—
174C2×C4⋊Q8—172no—
175C2×C4⋊Q8C2×D1711yescharacteristic
176C2×C4×Q8—171no—
177#446—342no—
178#446—342no—
179#432—171no—
180#432—171no—
181#432—341no—
182#432C2×D1711yescharacteristic
183#432C2×D1711yescharacteristic
184#431—341no—
185#398—341no—
186#398—341no—
187#391—172no—
188#389—341no—
189#389—341no—
190C4×C4⋊C4—171no—
191#333—22no—
192#333—342no—
193#324—341no—
194#324—341no—
195C2×C17—21no—
196C2×C17—21no—
197C2×C17#188611yescharacteristic
198C2×C17#190011yescharacteristic
199C2×C17#190911yescharacteristic
200C22×Q8—171no—
201C22×Q8C4×D1712yes—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—684no—
207C4×Q8—341no—
208C4×Q8—341no—
209C4×Q8—341no—
210C4×Q8—341no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4—171no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—342no—
218C2×C4⋊C4—342no—
219C2×C42—171no—
220C2×C42—171no—
221C2×C42—172no—
222C2×C42C22×D1711yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—172no—
225C4⋊C8—342no—
226C4⋊C8—342no—
227C4⋊C8—341no—
228C4⋊C8C22⋊D1712yes—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—341no—
232C22⋊C8—341no—
233#97—21no—
234#97—21no—
235#97—342no—
236#97—341no—
237#97—341no—
238C4×C8—342no—
239#95—342no—
240#95—342no—
241#95—341no—
242C17#945211yescharacteristic, complemented, Sylow
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—342no—
248C2×Q8—341no—
249C2×Q8—341no—
250C2×Q8—681no—
251C22×C4—22no—
252C22×C4—171no—
253C22×C4—171no—
254C22×C4—171no—
255C22×C4—171no—
256C22×C4—342no—
257C22×C4C2×C4×D1711yescharacteristic
258C22×C4D4×D1711yescharacteristic
259C22×C4D4×D1711yescharacteristic
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—342no—
263C2×C8—681no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—344no—
269C4⋊C4—341no—
270C4⋊C4—341no—
271C4⋊C4—341no—
272C4⋊C4—341no—
273C4⋊C4—341no—
274C4⋊C4—341no—
275C4⋊C4—684no—
276C4⋊C4—684no—
277C42—341no—
278C42—341no—
279C42—341no—
280C42—341no—
281C42—682no—
282C42#6325211yescharacteristic
283C42D4×D1711yescharacteristic
284C42#6325411yescharacteristic
285C42C2×C22⋊D1711yescharacteristic
286C23C22⋊C4×D1711yescharacteristic
287Q8—82no—
288Q8—82no—
289Q8—682no—
290Q8—682no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—172no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—344no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—682no—
308C2×C4—682no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4#1058756711yescharacteristic
312C2×C4#1058756911yescharacteristic
313C2×C4#1058757011yescharacteristic
314C2×C4C4×C22⋊D1711yescharacteristic
315C2×C4#1058762311yescharacteristic
316C2×C4#1058762411yescharacteristic
317C8—42no—
318C8—81no—
319C8—682no—
320C8—1361no—
321C22—21no—
322C22—21no—
323C22#1176268511yescharacteristic
324C22#100×D1711yescharacteristic
325C22#1176279011yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—82no—
332C4—342no—
333C4—684no—
334C4—681no—
335C4—681no—
336C4—681no—
337C4#1176299012yes—
338C2—21no—
339C2—21no—
340C2#1452917411yescharacteristic
341C2#1452924411yescharacteristic
342C2#1452930311yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1558C2
socleC22×C17#386
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1558D17
hypocenterC17#9452
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
242. C17#9452semidirect

Maximal subgroups

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

subgroupindex
#145378682
#145337732
#145275102
#145271272
#145266412
#145266392
C17×#15582
#945217