Finite Group Databasebeta

← #17008512

Subgroups — #17008512

Order 4352.

976 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 183 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#14539586C212yesmaximal
2#14534264C211yesmaximal, characteristic
3#14527826C214yesmaximal, complemented
4#14527724C214yesmaximal, complemented
5#14527304C212yesmaximal, complemented
6#14527114C211yesmaximal, characteristic, complemented
7C2×C17×#544C211yesmaximal, characteristic, complemented
8C2×#10587616C2211yescharacteristic
9#11763142C2214yes—
10#11763122—22no—
11#11763057—24no—
12#11762959C2214yes—
13#11762948—22no—
14#11762773C2212yes—
15#11762748C2212yes—
16#11762726C2212yes—
17#11762699—22no—
18#11762688—21no—
19#11762688—21no—
20C4×C22⋊Dic17C2211yescharacteristic
21#11762546C2218yes—
22#11762535C2214yes—
23C17×#544—22no—
24C17×#544—24no—
25C17×#544C2214yes—
26C2×C17×C4⋊Q8—22no—
27C2×C17×#125—22no—
28C2×C17×C22⋊Q8—21no—
29C2×C17×C22⋊Q8C2212yes—
30C2×C17×#117—21no—
31C22×C4⋊C4×C17C2211yescharacteristic
32C22×Q8×C17—21no—
33C22×Q8×C17D412yes—
34C23×C4×C17D411yescharacteristic
35C23×C4×C17D411yescharacteristic
36C23×C4×C17C2311yescharacteristic
37C22×C22⋊D17—21no—
38C22×D68C2311yescharacteristic
39C17×C4⋊Q8—28no—
40C17×C4⋊Q8—28no—
41C17×#125—28no—
42C17×#125—28no—
43C17×C22⋊Q8—24no—
44C17×C22⋊Q8—24no—
45C17×C22⋊Q8—22no—
46C17×C22⋊Q8—22no—
47C17×C22⋊Q8D414yes—
48C17×C22⋊Q8C2314yes—
49C17×#117—24no—
50C17×#117—24no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17—21no—
53C2×C4⋊C4×C17—21no—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17—22no—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17D412yes—
59C2×C4⋊C4×C17D412yes—
60C2×C4⋊C4×C17C2312yes—
61C2×C22⋊C4×C17—22no—
62C2×C42×C17—22no—
63C2×C22⋊Dic17—21no—
64C2×#63230—24no—
65#10587616—22no—
66#10587616—22no—
67C2×#63227—22no—
68C2×#63227—21no—
69C2×#63227—22no—
70C2×C4⋊Dic17C2311yescharacteristic
71C2×C4×Dic17—21no—
72C2×#63223—22no—
73#10587506—24no—
74#10587502—22no—
75#10587502—22no—
76#10587502C2312yes—
77C22⋊#3622—22no—
78C22⋊#3622C2314yes—
79#10587480—24no—
80#10587478—24no—
81C24×C17C2×D411yescharacteristic
82C23×D17—21no—
83C2×Q8×C17—24no—
84C2×Q8×C17—22no—
85C2×Q8×C17—82no—
86C2×Q8×C17—82no—
87C2×Q8×C17C2×D414yes—
88C22×C4×C17—21no—
89C22×C4×C17—21no—
90C22×C4×C17—24no—
91C22×C4×C17—21no—
92C22×C4×C17—22no—
93C22×C4×C17—22no—
94C22×C4×C17—21no—
95C22×C4×C17—81no—
96C22×C4×C17—81no—
97C22×C4×C17C2×D412yes—
98C22×C4×C17C2×D412yes—
99C22×C4×C17C2×D411yescharacteristic
100C22×C4×C17C2×D412yes—
101C22×C4×C17C2×D412yes—
102C22×C4×C17C2411yescharacteristic
103C2×C22⋊D17—44no—
104C2×C22⋊D17—41no—
105C2×C22⋊D17—41no—
106C22×Dic17—21no—
107C22×Dic17—22no—
108C2×D68—22no—
109C2×D68—81no—
110C4⋊C4×C17—24no—
111C4⋊C4×C17—28no—
112C4⋊C4×C17—24no—
113C4⋊C4×C17—24no—
114C4⋊C4×C17—22no—
115C4⋊C4×C17—24no—
116C4⋊C4×C17—24no—
117C4⋊C4×C17—24no—
118C4⋊C4×C17—24no—
119C4⋊C4×C17C2×D414yes—
120C22⋊C4×C17—24no—
121C22⋊C4×C17—24no—
122C42×C17—28no—
123C22⋊Dic17—42no—
124#63230—84no—
125#63227—42no—
126#63227—44no—
127#63227—82no—
128C4⋊Dic17—22no—
129C4×Dic17—81no—
130#63223—82no—
131C2×#3622—24no—
132#18326—171nomaximal, Sylow
133C23×C17—22no—
134C23×C17—22no—
135C23×C17—41no—
136C23×C17#12011yescharacteristic
137C23×C17#12011yescharacteristic
138C23×C17C22×D411yescharacteristic
139C22×D17—42no—
140C22×D17—42no—
141C22×D17—41no—
142C22×D17—42no—
143Q8×C17—44no—
144Q8×C17—82no—
145Q8×C17—84no—
146C2×C4×C17—24no—
147C2×C4×C17—24no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—24no—
151C2×C4×C17—24no—
152C2×C4×C17—22no—
153C2×C4×C17—41no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—41no—
157C2×C4×C17—42no—
158C2×C4×C17—44no—
159C2×C4×C17—44no—
160C2×C4×C17—42no—
161C2×C4×C17—81no—
162C2×C4×C17#12012yes—
163C2×C4×C17#13714yes—
164C2×C4×C17C22×D412yes—
165C22⋊D17—82no—
166C22⋊D17—82no—
167C2×Dic17—42no—
168C2×Dic17—42no—
169C2×Dic17—42no—
170C2×Dic17—41no—
171C2×Dic17—42no—
172D68—44no—
173#3622—44no—
174C2×#544D1711yescharacteristic, complemented
175#3193—172no—
176#2591—171no—
177#1617—174no—
178#1599—174no—
179#1508—172no—
180#1463—171no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—21no—
184C22×C17—21no—
185C22×C17—21no—
186C22×C17—21no—
187C22×C17—21no—
188C22×C17—44no—
189C22×C17#45412yes—
190C22×C17C2×#12011yescharacteristic
191C22×C17C2×#13712yes—
192C2×D17—84no—
193C2×D17—81no—
194C2×D17—81no—
195C2×D17—81no—
196C4×C17—24no—
197C4×C17—22no—
198C4×C17—22no—
199C4×C17—44no—
200C4×C17—42no—
201C4×C17—44no—
202Dic17—82no—
203Dic17—81no—
204#544—24no—
205#544—22no—
206#544C2×D1714yes—
207C2×C4⋊Q8—22no—
208C2×#125—22no—
209C2×C22⋊Q8—21no—
210C2×C22⋊Q8C2×D1712yes—
211C2×#121—171no—
212C2×#117—21no—
213C22×C4⋊C4C2×D1711yescharacteristic
214#484—174no—
215#480—342no—
216#468—344no—
217#450—174no—
218#448—342no—
219#407—172no—
220#400—172no—
221#394—172no—
222#388—342no—
223#386—341no—
224#386—341no—
225C4×C22⋊C4—171no—
226#328—178no—
227#324—174no—
228C2×C17—22no—
229C2×C17—22no—
230C2×C17—41no—
231C2×C17#300411yescharacteristic
232C2×C17C2×#45411yescharacteristic
233C2×C17#302611yescharacteristic
234D17—161no—
235C22×Q8—21no—
236C22×Q8C22⋊D1712yes—
237C22×D4—171no—
238C22×D4—341no—
239C23×C4C22⋊D1711yescharacteristic
240C23×C4C22⋊D1711yescharacteristic
241C23×C4C22×D1711yescharacteristic
242C2×SD16—344no—
243C2×M16—342no—
244C4⋊Q8—28no—
245C4⋊Q8—28no—
246#125—28no—
247#125—28no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8C22⋊D1714yes—
253C22⋊Q8C22×D1714yes—
254#121—342no—
255#121—342no—
256#117—24no—
257#117—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—171no—
266C2×C4⋊C4C22⋊D1712yes—
267C2×C4⋊C4C22⋊D1712yes—
268C2×C4⋊C4C22×D1712yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—341no—
271C2×C22⋊C4—341no—
272C2×C22⋊C4—342no—
273C2×C22⋊C4—342no—
274C2×C42—22no—
275C2×C42—341no—
276#107—344no—
277#103—344no—
278#102—344no—
279C22⋊C8—174no—
280C22⋊C8—342no—
281#95—172no—
282#95—342no—
283#95—342no—
284C17#1832611yescharacteristic, complemented, Sylow
285C24—341no—
286C24C2×C22⋊D1711yescharacteristic
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8C2×C22⋊D1714yes—
292C2×D4—342no—
293C2×D4—684no—
294C2×D4—681no—
295C2×D4—681no—
296C2×D4—1361no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—342no—
307C22×C4—341no—
308C22×C4C2×C22⋊D1712yes—
309C22×C4C2×C22⋊D1712yes—
310C22×C4C2×C22⋊D1712yes—
311C22×C4C2×C22⋊D1712yes—
312C22×C4C2×C22⋊D1711yescharacteristic
313C22×C4C23×D1711yescharacteristic
314SD16—1364no—
315M16—1362no—
316C2×C8—344no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—342no—
327C4⋊C4C2×C22⋊D1714yes—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—682no—
331C22⋊C4—684no—
332C22⋊C4—682no—
333C22⋊C4—1362no—
334C42—28no—
335C42—1361no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—682no—
340C23—682no—
341C23—681no—
342C23—682no—
343C23#1058763911yescharacteristic
344C23#1058763911yescharacteristic
345C23C22×C22⋊D1711yescharacteristic
346Q8—44no—
347Q8—82no—
348Q8—84no—
349D4—684no—
350D4—1362no—
351D4—1362no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4—682no—
369C2×C4—682no—
370C2×C4—682no—
371C2×C4—681no—
372C2×C4—682no—
373C2×C4#1058762814yes—
374C2×C4#1058763912yes—
375C2×C4C22×C22⋊D1712yes—
376C8—684no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—44no—
385C22—1364no—
386C22—1361no—
387C22—1361no—
388C22—1361no—
389C22#1176298212yes—
390C22C2×#1058763911yescharacteristic
391C22C2×#1058762812yes—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—44no—
396C4—42no—
397C4—44no—
398C4—1362no—
399C4—1361no—
400C2—22no—
401C2—22no—
402C2—41no—
403C2—2721no—
404C2#1453801811yescharacteristic
405C2C2×#1176298211yescharacteristic
406C2#1453818111yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#544C2
socleC22×C17C2×#120
prime radicalC2×#10587616C22
hypercenterC2×#544D17
hypocenterC17#18326
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #14527826C2semidirect
4. #14527724C2semidirect
5. #14527304C2semidirect
6. #14527114C2semidirect
7. C2×C17×#544C2semidirect
174. C2×#544D17semidirect
284. C17#18326semidirect

Maximal subgroups

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

subgroupindex
#145395862
#145342642
#145278262
#145277242
#145273042
#145271142
C2×C17×#5442
#1832617