Finite Group Databasebeta

← #16927806

Subgroups — #16927806

Order 4352.

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

normal subgroups only

This group has 418 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#14531280C211yesmaximal, characteristic
2#14528865C211yesmaximal, characteristic
3#14528859C211yesmaximal, characteristic
4#14526638C211yesmaximal, characteristic
5#14526637C211yesmaximal, characteristic
6#14526636C211yesmaximal, characteristic
7C17×#1284C211yesmaximal, characteristic
8C2×#10587612C2211yescharacteristic
9#11762773C2211yescharacteristic
10#11762703—21no—
11#11762697—21no—
12#11762694—21no—
13#11762692—21no—
14#11762672C2211yescharacteristic
15C4×#63225C411yescharacteristic, complemented
16C4×#63225C411yescharacteristic, complemented
17C4×C22⋊Dic17C2211yescharacteristic
18#11762655C411yescharacteristic, complemented
19C42×Dic17C411yescharacteristic, complemented
20#11762577—21no—
21#11762577—21no—
22#11762575—21no—
23#11762575—21no—
24C17×#407C2211yescharacteristic
25C17×#381C2211yescharacteristic
26C4×C22⋊C4×C17C2211yescharacteristic
27C23×C4×C17C2311yescharacteristic
28C2×M16×C17—21no—
29C2×C22⋊C4×C17—21no—
30C2×C22⋊C4×C17—21no—
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17C2×C411yescharacteristic
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17D411yescharacteristic
36C2×C22⋊Dic17D411yescharacteristic
37#10587612—21no—
38#10587612—21no—
39#10587612—21no—
40#10587612—21no—
41C2×C4⋊Dic17D411yescharacteristic
42C2×C4⋊Dic17D411yescharacteristic
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17—21no—
46C2×C4×Dic17—21no—
47C2×C4×Dic17C2×C411yescharacteristic
48C2×C4×Dic17C2×C411yescharacteristic
49C2×C4×Dic17C2×C411yescharacteristic
50C2×C4×Dic17C2×C411yescharacteristic
51C2×#63223—21no—
52C17×C22⋊C8—21no—
53C17×C22⋊C8—21no—
54C17×#95—21no—
55C17×#95—21no—
56C17×#95—21no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60#10587502—21no—
61#10587502—21no—
62#10587502—21no—
63#10587502—21no—
64C22⋊#3622—21no—
65C22⋊#3622—21no—
66C24×C17C2×D411yescharacteristic
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—41no—
70C22×C4×C17—41no—
71C22×C4×C17—41no—
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17#4111yescharacteristic
75C22×C4×C17#4111yescharacteristic
76C22×C4×C17#4111yescharacteristic
77C22×C4×C17#4111yescharacteristic
78C22×Dic17—21no—
79C22×Dic17—21no—
80C22×Dic17—41no—
81C22×Dic17C22⋊C411yescharacteristic
82C22×Dic17C22⋊C411yescharacteristic
83C22×Dic17C22⋊C411yescharacteristic
84C22×Dic17C22⋊C411yescharacteristic
85M16×C17—41no—
86M16×C17—41no—
87C2×C8×C17—41no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—41no—
90C22⋊C4×C17—41no—
91C42×C17—41no—
92C42×C17—41no—
93C22⋊Dic17—21no—
94C22⋊Dic17—21no—
95C22⋊Dic17—21no—
96C22⋊Dic17—21no—
97C22⋊Dic17—42no—
98C22⋊Dic17—41no—
99C22⋊Dic17—41no—
100C4⋊Dic17—21no—
101C4⋊Dic17—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17—21no—
104#63225—42no—
105#63225—42no—
106C4×Dic17—21no—
107C4×Dic17—21no—
108C4×Dic17—21no—
109C4×Dic17—21no—
110C4×Dic17—42no—
111C4×Dic17—42no—
112C4×Dic17—41no—
113C4×Dic17—41no—
114#63223—41no—
115#63223—41no—
116C2×#3622—41no—
117#8741—171nomaximal, Sylow
118C23×C17—21no—
119C23×C17—21no—
120C23×C17—41no—
121C23×C17—41no—
122C23×C17C2×C22⋊C411yescharacteristic
123C23×C17#12311yescharacteristic
124C23×C17#12311yescharacteristic
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—22no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—81no—
140C2×C4×C17—81no—
141C2×C4×C17—81no—
142C2×C4×C17#11711yescharacteristic
143C2×C4×C17#11711yescharacteristic
144C2×C4×C17#12311yescharacteristic
145C2×C4×C17#12311yescharacteristic
146C2×Dic17—22no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—81no—
161C2×Dic17—81no—
162C2×Dic17—81no—
163C8×C17—81no—
164#3622—81no—
165#2298—171no—
166#1822—171no—
167#1820—171no—
168#1284—171no—
169#1284—171no—
170#1284—171no—
171#1284D1711yescharacteristic
172C22×C17—21no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—21no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—81no—
181C22×C17—81no—
182C22×C17#38111yescharacteristic
183C22×C17#41011yescharacteristic
184C22×C17#41011yescharacteristic
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—42no—
190C4×C17—81no—
191C4×C17—161no—
192Dic17—42no—
193Dic17—41no—
194Dic17—41no—
195Dic17—41no—
196Dic17—41no—
197Dic17—81no—
198Dic17—161no—
199C2×#117—171no—
200#407—171no—
201#407C2×D1711yescharacteristic
202#388—341no—
203#388—341no—
204#387—341no—
205#387—341no—
206#381—171no—
207#381C2×D1711yescharacteristic
208C4×C4⋊C4—171no—
209C4×C4⋊C4—171no—
210C4×C22⋊C4—171no—
211C4×C22⋊C4C2×D1711yescharacteristic
212#376—171no—
213C43—171no—
214#343—341no—
215#343—341no—
216#342—341no—
217#342—341no—
218C2×C17—21no—
219C2×C17—21no—
220C2×C17—41no—
221C2×C17—41no—
222C2×C17#182011yescharacteristic
223C2×C17#182711yescharacteristic
224C2×C17#182711yescharacteristic
225C23×C4C22×D1711yescharacteristic
226C2×M16—21no—
227C2×M16—341no—
228#117—341no—
229#117—341no—
230#117—341no—
231#117—341no—
232C2×C4⋊C4—171no—
233C2×C4⋊C4—171no—
234C2×C4⋊C4—341no—
235C2×C4⋊C4—341no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—171no—
239C2×C22⋊C4—171no—
240C2×C22⋊C4—341no—
241C2×C22⋊C4—341no—
242C2×C42—171no—
243C2×C42—171no—
244C2×C42—171no—
245C2×C42—171no—
246C2×C42—341no—
247C2×C42—341no—
248C2×C42C4×D1711yescharacteristic
249C2×C42C4×D1711yescharacteristic
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—341no—
253C22⋊C8—341no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—341no—
258#95—341no—
259#95—341no—
260#95—341no—
261#95—341no—
262#95—341no—
263#95—341no—
264C17#874111yescharacteristic, complemented, Sylow
265C24D4×D1711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—171no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—341no—
276C22×C4—341no—
277C22×C4—681no—
278C22×C4C2×C4×D1711yescharacteristic
279C22×C4#6325211yescharacteristic
280C22×C4#6325211yescharacteristic
281C22×C4D4×D1711yescharacteristic
282C22×C4#6325411yescharacteristic
283C22×C4#6325611yescharacteristic
284M16—41no—
285M16—41no—
286M16—681no—
287M16—681no—
288C2×C8—41no—
289C2×C8—681no—
290C4⋊C4—341no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4—682no—
295C4⋊C4—682no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—682no—
304C22⋊C4—681no—
305C22⋊C4—681no—
306C42—41no—
307C42—41no—
308C42—341no—
309C42—341no—
310C42—341no—
311C42—341no—
312C42—681no—
313C42—681no—
314C42—682no—
315C42—682no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23C22⋊C4×D1711yescharacteristic
321C23#1058756911yescharacteristic
322C23#1058758011yescharacteristic
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—342no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—681no—
346C2×C4—681no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4—681no—
351C2×C4—681no—
352C2×C4—681no—
353C2×C4—681no—
354C2×C4—1361no—
355C2×C4—1361no—
356C2×C4—1361no—
357C2×C4#1058755811yescharacteristic
358C2×C4#1058756911yescharacteristic
359C2×C4#1058757811yescharacteristic
360C2×C4#1058758011yescharacteristic
361C8—81no—
362C8—1361no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—81no—
372C22—81no—
373C22#1176267111yescharacteristic
374C22#1176278111yescharacteristic
375C22#1176278111yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—81no—
382C4—161no—
383C4—2721no—
384C4—682no—
385C4—681no—
386C4—681no—
387C4—681no—
388C4—681no—
389C4—1361no—
390C2—21no—
391C2—21no—
392C2—41no—
393C2—41no—
394C2#1452886011yescharacteristic
395C2#1452888611yescharacteristic
396C2#1452888711yescharacteristic
subgroupG/H
centerC22#11762671
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1284C2
socleC22×C17#381
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1284D17
hypocenterC17#8741
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
15. C4×#63225C4semidirect
16. C4×#63225C4semidirect
18. #11762655C4semidirect
19. C42×Dic17C4semidirect
264. C17#8741semidirect

Maximal subgroups

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

subgroupindex
#145312802
#145288652
#145288592
#145266382
#145266372
#145266362
C17×#12842
#874117