Finite Group Databasebeta

← #16944826

Subgroups — #16944826

Order 4352.

622 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 87 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#14532119C211yesmaximal, characteristic
2C2×#11762578C211yesmaximal, characteristic, complemented
3#14526815C212yesmaximal, complemented
4#14526788C212yesmaximal, complemented
5C17×#1463C211yesmaximal, characteristic, complemented
6C22×#63252C411yescharacteristic
7C22×#63223—21no—
8C2×#10587615—21no—
9#11762729C411yescharacteristic
10#11762729C2211yescharacteristic
11#11762726C411yescharacteristic
12#11762726C2211yescharacteristic
13#11762688—21no—
14#11762680—21no—
15#11762680—21no—
16#11762672C411yescharacteristic
17#11762578—21no—
18#11762578C2212yes—
19C22×C4⋊C4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#324—21no—
22C17×#324C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×C22⋊D17—21no—
27C2×#63252—41no—
28C2×#63252—41no—
29C2×#63252—44no—
30C22×D68C2×C411yescharacteristic
31C22×C4×D17—21no—
32C22×Dic34C2×C411yescharacteristic
33C2×M16×C17—41no—
34C2×C4⋊C4×C17—41no—
35C2×C4⋊C4×C17—41no—
36C2×C4⋊C4×C17C2×C411yescharacteristic
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C4⋊C4×C17D411yescharacteristic
39C2×C4⋊C4×C17C2311yescharacteristic
40C2×C22⋊Dic17—21no—
41#10587615—44no—
42C2×#63227—21no—
43C2×#63227—21no—
44C2×#63227—21no—
45C2×#63227—21no—
46C2×#63225—21no—
47C2×#63225—21no—
48C2×#63223—24no—
49C2×#63223—22no—
50C2×#63223—41no—
51C2×#63223—42no—
52C22×#3622—41no—
53C17×C22⋊C8—22no—
54C17×C22⋊C8—41no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587502C2×C411yescharacteristic
59#10587502C2×C411yescharacteristic
60C24×C17C22⋊C411yescharacteristic
61C23×D17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—22no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—42no—
72C22×C4×C17—42no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C2×C22⋊D17—24no—
80C2×C22⋊D17—42no—
81C2×C22⋊D17—41no—
82C2×C22⋊D17—41no—
83C22×Dic17—21no—
84C22×Dic17—21no—
85C22×Dic17—21no—
86#63252—44no—
87#63252—44no—
88#63252—44no—
89#63252—44no—
90C2×D68—41no—
91C2×D68—42no—
92C2×C4×D17—42no—
93C2×C4×D17—42no—
94C2×C4×D17—42no—
95C2×Dic34—41no—
96C2×Dic34—42no—
97M16×C17—82no—
98C2×C8×C17—42no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—41no—
102C4⋊C4×C17—41no—
103C22⋊Dic17—42no—
104#63227—42no—
105#63227—42no—
106#63227—42no—
107#63227—41no—
108#63227—41no—
109#63225—42no—
110#63225—42no—
111#63223—24no—
112#63223—44no—
113#63223—42no—
114C2×#3622—44no—
115C2×#3622—42no—
116#11762—171nomaximal, Sylow
117C23×C17—22no—
118C23×C17—21no—
119C23×C17—21no—
120C23×C17—41no—
121C23×C17#9911yescharacteristic
122C23×C17#9911yescharacteristic
123C23×C17C2×C22⋊C411yescharacteristic
124C22×D17—22no—
125C22×D17—41no—
126C22×D17—41no—
127C22×D17—41no—
128C22×D17—41no—
129C22×D17—41no—
130C22×D17—41no—
131C2×C4×C17—22no—
132C2×C4×C17—22no—
133C2×C4×C17—22no—
134C2×C4×C17—22no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—42no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—81no—
151C2×C4×C17#10412yes—
152C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C17#13711yescharacteristic
154C22⋊D17—44no—
155C22⋊D17—44no—
156C2×Dic17—22no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165D68—42no—
166D68—42no—
167C4×D17—48no—
168Dic34—42no—
169Dic34—42no—
170C8×C17—82no—
171#3622—44no—
172#2406—171no—
173C2×#343—171no—
174#1463D1711yescharacteristic, complemented
175#1354—172no—
176#1345—172no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—21no—
184C22×C17—42no—
185C22×C17—42no—
186C22×C17AΣL(1,𝔽16)12yes—
187C22×C17C2×#9911yescharacteristic
188C22×C17#41711yescharacteristic
189C22×C17C2×#10411yescharacteristic
190C2×D17—42no—
191C2×D17—42no—
192C2×D17—42no—
193C2×D17—81no—
194C2×D17—81no—
195C2×D17—81no—
196C2×D17—81no—
197C4×C17—21no—
198C4×C17—22no—
199C4×C17—21no—
200C4×C17—41no—
201C4×C17—41no—
202C4×C17—41no—
203C4×C17—41no—
204Dic17—42no—
205Dic17—81no—
206Dic17—81no—
207C22×#41—171no—
208C22×M16—341no—
209C2×#123—341no—
210C22×C4⋊C4C2×D1711yescharacteristic
211#407—21no—
212#395—171no—
213#395—171no—
214#394—171no—
215#394—171no—
216#386—341no—
217#385—341no—
218#385—341no—
219#381—171no—
220#343—172no—
221#343—341no—
222#324—21no—
223#324C2×D1712yes—
224C2×C17—21no—
225C2×C17—21no—
226C2×C17—22no—
227C2×C17—41no—
228C2×C17#150411yescharacteristic
229C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
230C2×C17#213311yescharacteristic
231D17—82no—
232C2×#41—681no—
233C2×#41—681no—
234C2×#41—684no—
235C22×Q8—171no—
236C22×D4—171no—
237C22×D4—341no—
238C23×C4—341no—
239C23×C4C4×D1711yescharacteristic
240C23×C4D6811yescharacteristic
241C23×C4C22⋊D1711yescharacteristic
242C2×M16—41no—
243C2×M16—344no—
244C2×M16—342no—
245C2×M16—681no—
246C2×M16—682no—
247C22×C8—681no—
248#123—684no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—341no—
252C2×C4⋊C4—341no—
253C2×C4⋊C4C4×D1711yescharacteristic
254C2×C4⋊C4D6811yescharacteristic
255C2×C4⋊C4C22⋊D1711yescharacteristic
256C2×C4⋊C4C22×D1711yescharacteristic
257C2×C22⋊C4—341no—
258C2×C22⋊C4—341no—
259C2×C22⋊C4—341no—
260C2×C22⋊C4—341no—
261C2×C22⋊C4—341no—
262C22⋊C8—22no—
263C22⋊C8—41no—
264#95—171no—
265#95—171no—
266#95—341no—
267#95—341no—
268#95—341no—
269C17#1176211yescharacteristic, complemented, Sylow
270C24—341no—
271C24#6322711yescharacteristic
272#41—684no—
273#41—684no—
274#41—684no—
275#41—684no—
276C2×Q8—681no—
277C2×Q8—682no—
278C2×D4—344no—
279C2×D4—681no—
280C2×D4—682no—
281C2×D4—682no—
282C2×D4—681no—
283C2×D4—681no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—341no—
296C22×C4—341no—
297C22×C4—341no—
298C22×C4—682no—
299C22×C4—682no—
300C22×C4—682no—
301C22×C4#6322711yescharacteristic
302C22×C4#6322711yescharacteristic
303C22×C4#6322711yescharacteristic
304C22×C4C2×C4×D1711yescharacteristic
305C22×C4C2×D6811yescharacteristic
306C22×C4C2×C22⋊D1711yescharacteristic
307M16—82no—
308M16—344no—
309M16—684no—
310M16—682no—
311C2×C8—42no—
312C2×C8—684no—
313C2×C8—682no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—682no—
319C4⋊C4—682no—
320C22⋊C4—682no—
321C22⋊C4—682no—
322C22⋊C4—682no—
323C22⋊C4—682no—
324C22⋊C4—681no—
325C22⋊C4—681no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—41no—
330C23—342no—
331C23—681no—
332C23—681no—
333C23—681no—
334C23—681no—
335C23—681no—
336C23—681no—
337C23#1058747711yescharacteristic
338C23#1058747711yescharacteristic
339C23C2×#6322711yescharacteristic
340Q8—682no—
341Q8—682no—
342D4—682no—
343D4—682no—
344D4—684no—
345D4—684no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—342no—
367C2×C4—688no—
368C2×C4—681no—
369C2×C4—681no—
370C2×C4—681no—
371C2×C4—681no—
372C2×C4—681no—
373C2×C4—681no—
374C2×C4—681no—
375C2×C4—681no—
376C2×C4#1058749612yes—
377C2×C4#1058759411yescharacteristic
378C2×C4#1058759511yescharacteristic
379C8—82no—
380C8—684no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—682no—
391C22—682no—
392C22—682no—
393C22—1361no—
394C22—1361no—
395C22—1361no—
396C22—1361no—
397C22#1176259512yes—
398C22C2×#1058747711yescharacteristic
399C22#1176281111yescharacteristic
400C22C2×#1058749611yescharacteristic
401C4—21no—
402C4—22no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—682no—
409C4—1361no—
410C4—1361no—
411C2—21no—
412C2—21no—
413C2—22no—
414C2—41no—
415C2—1362no—
416C2#1452728911yescharacteristic
417C2C2×#1176259511yescharacteristic
418C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1463C2
socleC22×C17C2×#99
prime radicalC22×#63252C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1463D17
hypocenterC17#11762
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. C2×#11762578C2semidirect
3. #14526815C2semidirect
4. #14526788C2semidirect
5. C17×#1463C2semidirect
174. #1463D17semidirect
269. C17#11762semidirect

Maximal subgroups

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

subgroupindex
#145321192
C2×#117625782
#145268152
#145267882
C17×#14632
#1176217