Finite Group Databasebeta

← #17008246

Subgroups — #17008246

Order 4352.

940 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C2×#11763440C211yesmaximal, characteristic
2#14539556C212yesmaximal
3#14527825C214yesmaximal, complemented
4#14527757C214yesmaximal, complemented
5#14527355C212yesmaximal
6C2×#11762535C211yesmaximal, characteristic
7C17×#2627C211yesmaximal, characteristic
8#11763440—22no—
9#11763440—24no—
10#11763440C2214yes—
11C2×#10587574—22no—
12C2×#10587576—22no—
13C2×#10587610—21no—
14C2×#10587610C2212yes—
15C2×#10587612—21no—
16#11763144C2214yes—
17#11763142—22no—
18#11763048—24no—
19#11762965C2214yes—
20#11762964—22no—
21C2×C22⋊#3622C2212yes—
22#11762545C2218yes—
23#11762535C2214yes—
24C2×C17×C22⋊Q8C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#400C2212yes—
27C17×#398—22no—
28C17×#395C2212yes—
29C17×#385—21no—
30C17×#381—21no—
31C17×#380C2211yescharacteristic
32C22×Q8×C17C2311yescharacteristic
33C23×C4×C17D411yescharacteristic
34C23×C4×C17D411yescharacteristic
35C23×C4×C17C2311yescharacteristic
36C22×Dic34—21no—
37C22×Dic34D412yes—
38C17×C22⋊Q8—22no—
39C17×C22⋊Q8—22no—
40C2×C4⋊C4×C17—22no—
41C2×C4⋊C4×C17—22no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—22no—
45C2×C4⋊C4×C17D412yes—
46C2×C4⋊C4×C17C2311yescharacteristic
47C2×C4⋊C4×C17C2312yes—
48C2×C22⋊C4×C17—21no—
49C2×C22⋊C4×C17—21no—
50C2×C22⋊Dic17—22no—
51C2×#63231—24no—
52#10587612—24no—
53#10587612—24no—
54C2×C4⋊Dic17—21no—
55C2×C4⋊Dic17—21no—
56C2×C4⋊Dic17—21no—
57C2×C4⋊Dic17D412yes—
58#10587610—24no—
59#10587610—24no—
60#10587610—22no—
61#10587610—22no—
62#10587610D414yes—
63#10587610C2314yes—
64C2×#63225—22no—
65C2×#63225—22no—
66C2×C4×Dic17—22no—
67C22×#3622—22no—
68#10587576—28no—
69#10587576—28no—
70#10587574—28no—
71#10587574—28no—
72C17×#95—22no—
73C17×#95—22no—
74C17×#95—22no—
75C17×#95C2312yes—
76#10587506—24no—
77C22⋊#3622—22no—
78C22⋊#3622C2314yes—
79#10587481—24no—
80#10587478—24no—
81C24×C17C2×D411yescharacteristic
82C2×Q8×C17—22no—
83C2×Q8×C17—81no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—22no—
87C22×C4×C17—21no—
88C22×C4×C17—21no—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—22no—
92C22×C4×C17—81no—
93C22×C4×C17—81no—
94C22×C4×C17Q1614yescomplemented
95C22×C4×C17C2×D412yes—
96C22×C4×C17C2×D412yes—
97C22×C4×C17C2×D412yes—
98C22×C4×C17C2×D412yes—
99C22×C4×C17C2×D411yescharacteristic
100C22×C4×C17C2411yescharacteristic
101C22×Dic17—22no—
102C22×Dic17—22no—
103C2×Dic34—24no—
104C2×Dic34—22no—
105C2×Dic34—82no—
106C2×Dic34—82no—
107C2×Dic34C2×D414yes—
108C4⋊C4×C17—24no—
109C4⋊C4×C17—24no—
110C4⋊C4×C17—22no—
111C4⋊C4×C17—42no—
112C4⋊C4×C17—81no—
113C4⋊C4×C17—82no—
114C22⋊C4×C17—41no—
115C22⋊C4×C17—41no—
116C22⋊C4×C17—42no—
117C22⋊Dic17—24no—
118C22⋊Dic17—24no—
119#63231—84no—
120C4⋊Dic17—24no—
121C4⋊Dic17—24no—
122C4⋊Dic17—24no—
123C4⋊Dic17—22no—
124C4⋊Dic17C2×D414yes—
125#63225—28no—
126#63225—24no—
127#63225—24no—
128C4×Dic17—28no—
129C2×#3622—24no—
130C2×#3622—82no—
131#18295—171nomaximal, Sylow
132C23×C17—41no—
133C23×C17—41no—
134C23×C17#12011yescharacteristic
135C23×C17#12011yescharacteristic
136C23×C17C2×Q1612yes—
137C23×C17#13712yes—
138C23×C17C22×D411yescharacteristic
139Q8×C17—44no—
140C2×C4×C17—22no—
141C2×C4×C17—22no—
142C2×C4×C17—24no—
143C2×C4×C17—24no—
144C2×C4×C17—22no—
145C2×C4×C17—41no—
146C2×C4×C17—42no—
147C2×C4×C17—42no—
148C2×C4×C17—41no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×C4×C17—42no—
155C2×C4×C17—41no—
156C2×C4×C17—41no—
157C2×C4×C17—42no—
158C2×C4×C17—81no—
159C2×C4×C17#12012yes—
160C2×C4×C17C2×Q1614yes—
161C2×C4×C17C22×D412yes—
162C2×Dic17—24no—
163C2×Dic17—24no—
164C2×Dic17—44no—
165C2×Dic17—44no—
166Dic34—44no—
167Dic34—82no—
168Dic34—84no—
169#3622—44no—
170C2×#544—171no—
171#3189—172no—
172#2627D1711yescharacteristic
173#1617—174no—
174#1605—174no—
175#1520—172no—
176C2×#324—171no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—42no—
184C22×C17—42no—
185C22×C17#45112yes—
186C22×C17#45112yes—
187C22×C17C2×#12011yescharacteristic
188C22×C17C22×Q1611yescharacteristic
189C22×C17C2×#13711yescharacteristic
190C4×C17—24no—
191C4×C17—22no—
192C4×C17—22no—
193C4×C17—42no—
194C4×C17—82no—
195C4×C17—81no—
196C4×C17—81no—
197Dic17—44no—
198Dic17—44no—
199#544—174no—
200#544—344no—
201#544—342no—
202C2×C4⋊Q8—342no—
203C2×#125—342no—
204C2×C22⋊Q8—172no—
205C2×C22⋊Q8—341no—
206C2×C22⋊Q8C2×D1711yescharacteristic
207C2×#117—341no—
208C22×C4⋊C4C2×D1711yescharacteristic
209#484—174no—
210#484—342no—
211#467—344no—
212#451—174no—
213#451—342no—
214C2×C22⋊C8—172no—
215#400C2×D1712yes—
216#398—22no—
217#395C2×D1712yes—
218#385—21no—
219#381—21no—
220#380C2×D1711yescharacteristic
221#328—178no—
222#324—174no—
223C2×C17—21no—
224C2×C17—21no—
225C2×C17—22no—
226C2×C17—22no—
227C2×C17C2×#45111yescharacteristic
228C2×C17#302011yescharacteristic
229C2×C17#302611yescharacteristic
230C22×Q8—172no—
231C22×Q8—341no—
232C22×Q8C22×D1711yescharacteristic
233C23×C4C22⋊D1711yescharacteristic
234C23×C4C22⋊D1711yescharacteristic
235C23×C4C22×D1711yescharacteristic
236C2×Q16—344no—
237C22×C8—342no—
238C4⋊Q8—348no—
239C4⋊Q8—348no—
240#125—348no—
241#125—348no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—174no—
245C22⋊Q8—174no—
246C22⋊Q8—344no—
247C22⋊Q8—342no—
248C22⋊Q8—342no—
249C22⋊Q8—344no—
250#117—344no—
251#117—344no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—172no—
258C2×C4⋊C4—341no—
259C2×C4⋊C4—341no—
260C2×C4⋊C4—341no—
261C2×C4⋊C4—342no—
262C2×C4⋊C4—342no—
263C2×C4⋊C4C22⋊D1712yes—
264C2×C4⋊C4C22×D1711yescharacteristic
265C2×C4⋊C4C22×D1712yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—342no—
269C2×C42—342no—
270#107—344no—
271#103—344no—
272#103—344no—
273C22⋊C8—174no—
274C22⋊C8—342no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95C22×D1712yes—
279C17#1829511yescharacteristic, complemented, Sylow
280C24C2×C22⋊D1711yescharacteristic
281C2×Q8—22no—
282C2×Q8—81no—
283C2×Q8—174no—
284C2×Q8—344no—
285C2×Q8—342no—
286C2×Q8—1362no—
287C2×Q8—1362no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—342no—
299C22×C4—342no—
300C22×C4#6323114yescomplemented
301C22×C4D4×D1712yes—
302C22×C4D4×D1712yes—
303C22×C4C2×C22⋊D1712yes—
304C22×C4C2×C22⋊D1712yes—
305C22×C4C2×C22⋊D1711yescharacteristic
306C22×C4C23×D1711yescharacteristic
307Q16—1364no—
308C2×C8—344no—
309C2×C8—1362no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—81no—
315C4⋊C4—82no—
316C4⋊C4—174no—
317C4⋊C4—344no—
318C4⋊C4—344no—
319C4⋊C4—344no—
320C4⋊C4—348no—
321C4⋊C4—344no—
322C4⋊C4—342no—
323C4⋊C4—344no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—344no—
328C22⋊C4—344no—
329C42—348no—
330C23—41no—
331C23—41no—
332C23#1058762311yescharacteristic
333C23#1058762311yescharacteristic
334C23C2×#6323112yes—
335C23#1058763712yes—
336C23C22×C22⋊D1711yescharacteristic
337Q8—44no—
338Q8—684no—
339Q8—1362no—
340Q8—1364no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—344no—
361C2×C4—344no—
362C2×C4—684no—
363C2×C4—684no—
364C2×C4#1058762312yes—
365C2×C4C2×#6323114yes—
366C2×C4C2×D4×D1712yes—
367C8—684no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1176296412yes—
377C22#1176296412yes—
378C22C2×#1058762311yescharacteristic
379C22C22×#6323111yescharacteristic
380C22C2×#1058763711yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—42no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4—684no—
389C4—684no—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2C2×#1176296411yescharacteristic
395C2#1453814711yescharacteristic
396C2#1453818011yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2627C2
socleC22×C17C2×#120
prime radicalC24×C17C2×D4
hypercenter#2627D17
hypocenterC17#18295
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
3. #14527825C2semidirect
4. #14527757C2semidirect
94. C22×C4×C17Q16semidirect
279. C17#18295semidirect
300. C22×C4#63231semidirect

Maximal subgroups

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

subgroupindex
C2×#117634402
#145395562
#145278252
#145277572
#145273552
C2×#117625352
C17×#26272
#1829517