Finite Group Databasebeta

← #16949821

Subgroups — #16949821

Order 4352.

1214 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1214 conjugacy classes of subgroups — too many to draw. Only its 139 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×#11762604C212yesmaximal, complemented
2C2×#11762602C212yesmaximal, complemented
3#14528699C212yesmaximal, complemented
4C2×C17×#535C211yesmaximal, characteristic
5C2×#10587505—22no—
6C2×#10587505C2214yes—
7#11762665—22no—
8#11762604—24no—
9#11762602—24no—
10C22×#41×C17C411yescharacteristic, complemented
11C23×D4×C17C2211yescharacteristic
12C17×#535—24no—
13C17×#535—24no—
14C2×C17×#127C411yescharacteristic, complemented
15C2×C17×#127C2211yescharacteristic
16C2×C17×#124C411yescharacteristic, complemented
17C2×C17×#124C2211yescharacteristic
18C2×C17×#121—22no—
19C2×C17×#120—22no—
20C2×C17×#117C411yescharacteristic, complemented
21C25×C17D412yes—
22C2×#41×C17—22no—
23C2×#41×C17—22no—
24C2×#41×C17—42no—
25C2×#41×C17—42no—
26C22×Q8×C17C2×C411yescharacteristic
27C22×D4×C17—21no—
28C22×D4×C17—22no—
29C22×D4×C17—22no—
30C22×D4×C17—22no—
31C22×D4×C17—22no—
32C22×D4×C17—42no—
33C22×D4×C17—42no—
34C22×D4×C17C2×C411yescharacteristic
35C22×D4×C17C2×C411yescharacteristic
36C22×D4×C17D412yes—
37C22×D4×C17C2311yescharacteristic
38C23×C4×C17—21no—
39C23×C4×C17C2×C411yescharacteristic
40C17×#127—22no—
41C17×#127—22no—
42C17×#127—22no—
43C17×#127D414yes—
44C17×#124—22no—
45C17×#124—22no—
46C17×#124—22no—
47C17×#124D414yes—
48C17×#121—216no—
49C17×#120—216no—
50C17×#117—22no—
51C17×#117—22no—
52C2×C4⋊C4×C17—21no—
53C2×C22⋊C4×C17—21no—
54C2×C22⋊C4×C17—22no—
55C2×C22⋊C4×C17—22no—
56C2×C42×C17C2×C411yescharacteristic
57C2×C42×C17C2×C411yescharacteristic
58C2×C22⋊Dic17—24no—
59C2×C22⋊Dic17—44no—
60#10587505—24no—
61#10587505—24no—
62#10587505—44no—
63C24×C17—21no—
64C24×C17—24no—
65C24×C17—24no—
66C24×C17—42no—
67C24×C17—84no—
68C24×C17C22⋊C412yes—
69C24×C17C22⋊C411yescharacteristic
70C24×C17C2×D412yes—
71#41×C17—48no—
72#41×C17—48no—
73C2×Q8×C17—21no—
74C2×Q8×C17—41no—
75C2×Q8×C17—41no—
76C2×Q8×C17C22⋊C412yescomplemented
77C2×D4×C17—24no—
78C2×D4×C17—24no—
79C2×D4×C17—28no—
80C2×D4×C17—28no—
81C2×D4×C17—24no—
82C2×D4×C17—21no—
83C2×D4×C17—21no—
84C2×D4×C17—21no—
85C2×D4×C17—41no—
86C2×D4×C17—41no—
87C2×D4×C17—42no—
88C2×D4×C17—41no—
89C2×D4×C17—41no—
90C2×D4×C17—41no—
91C2×D4×C17—41no—
92C2×D4×C17—48no—
93C2×D4×C17—48no—
94C2×D4×C17—42no—
95C2×D4×C17—42no—
96C2×D4×C17—84no—
97C2×D4×C17—84no—
98C2×D4×C17C22⋊C412yescomplemented
99C2×D4×C17C2×D412yes—
100C2×D4×C17C2×D412yes—
101C22×C4×C17—22no—
102C22×C4×C17—22no—
103C22×C4×C17—21no—
104C22×C4×C17—21no—
105C22×C4×C17—24no—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17—21no—
109C22×C4×C17—42no—
110C22×C4×C17—42no—
111C22×C4×C17C22⋊C411yescharacteristic
112C22×C4×C17C22×C411yescharacteristic
113C22×Dic17—44no—
114C4⋊C4×C17—24no—
115C22⋊C4×C17—28no—
116C22⋊C4×C17—24no—
117C22⋊C4×C17—28no—
118C42×C17—21no—
119C42×C17—21no—
120C42×C17C22⋊C412yescomplemented
121C42×C17C22⋊C412yescomplemented
122C22⋊Dic17—44no—
123C22⋊Dic17—44no—
124C22⋊Dic17—88no—
125#12696—171nomaximal, Sylow
126C23×C17—21no—
127C23×C17—22no—
128C23×C17—21no—
129C23×C17—22no—
130C23×C17—22no—
131C23×C17—22no—
132C23×C17—22no—
133C23×C17—22no—
134C23×C17—42no—
135C23×C17—48no—
136C23×C17—44no—
137C23×C17—44no—
138C23×C17—48no—
139C23×C17—44no—
140C23×C17—41no—
141C23×C17—41no—
142C23×C17—84no—
143C23×C17—88no—
144C23×C17—88no—
145C23×C17#9912yes—
146C23×C17C2×C22⋊C411yescharacteristic
147C23×C17#12014yes—
148Q8×C17—42no—
149Q8×C17—42no—
150D4×C17—42no—
151D4×C17—42no—
152D4×C17—48no—
153D4×C17—42no—
154D4×C17—42no—
155D4×C17—42no—
156D4×C17—42no—
157D4×C17—48no—
158D4×C17—88no—
159D4×C17—88no—
160C2×C4×C17—24no—
161C2×C4×C17—24no—
162C2×C4×C17—21no—
163C2×C4×C17—21no—
164C2×C4×C17—21no—
165C2×C4×C17—21no—
166C2×C4×C17—22no—
167C2×C4×C17—22no—
168C2×C4×C17—22no—
169C2×C4×C17—48no—
170C2×C4×C17—42no—
171C2×C4×C17—42no—
172C2×C4×C17—41no—
173C2×C4×C17—41no—
174C2×C4×C17#9912yes—
175C2×C4×C17C2×C22⋊C411yescharacteristic
176C2×C4×C17C2×C22⋊C411yescharacteristic
177C2×Dic17—84no—
178C2×Dic17—88no—
179C2×#535D1711yescharacteristic
180C2×#354—172no—
181C2×#353—172no—
182#1781—172no—
183C22×C17—22no—
184C22×C17—24no—
185C22×C17—24no—
186C22×C17—22no—
187C22×C17—44no—
188C22×C17—42no—
189C22×C17—42no—
190C22×C17—44no—
191C22×C17—44no—
192C22×C17—44no—
193C22×C17—44no—
194C22×C17—42no—
195C22×C17—42no—
196C22×C17—82no—
197C22×C17—82no—
198C22×C17—82no—
199C22×C17—816no—
200C22×C17—82no—
201C22×C17#37911yescharacteristic
202C22×C17C2×#9911yescharacteristic
203C22×C17C2×#9911yescharacteristic
204C4×C17—22no—
205C4×C17—21no—
206C4×C17—21no—
207C4×C17—42no—
208C4×C17—42no—
209C4×C17—42no—
210Dic17—164no—
211C22×#41Dic1711yescharacteristic, complemented
212C23×D4C2×D1711yescharacteristic
213#535—24no—
214#535—24no—
215C2×#127Dic1711yescharacteristic, complemented
216C2×#127C2×D1711yescharacteristic
217C2×#124Dic1711yescharacteristic, complemented
218C2×#124C2×D1711yescharacteristic
219C2×#121—22no—
220C2×#120—22no—
221C2×#117Dic1711yescharacteristic, complemented
222C2×#99—174no—
223C2×#99—342no—
224#379—342no—
225#354—344no—
226#353—344no—
227C2×C17—21no—
228C2×C17—21no—
229C2×C17—44no—
230C2×C17—42no—
231C2×C17—82no—
232C2×C17—82no—
233C2×C17#178111yescharacteristic
234C2×C17#212712yes—
235C25C22⋊D1712yes—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—42no—
239C2×#41—42no—
240C22×Q8C2×Dic1711yescharacteristic
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4C2×Dic1711yescharacteristic
249C22×D4C2×Dic1711yescharacteristic
250C22×D4C22⋊D1712yes—
251C22×D4C22×D1711yescharacteristic
252C23×C4—21no—
253C23×C4C2×Dic1711yescharacteristic
254#127—22no—
255#127—22no—
256#127—22no—
257#127C22⋊D1714yes—
258#124—22no—
259#124—22no—
260#124—22no—
261#124C22⋊D1714yes—
262#121—216no—
263#120—216no—
264#117—22no—
265#117—22no—
266C2×C4⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—344no—
271C2×C22⋊C4—684no—
272C2×C42C2×Dic1711yescharacteristic
273C2×C42C2×Dic1711yescharacteristic
274#99—344no—
275#99—344no—
276#99—684no—
277C17#1269611yescharacteristic, complemented, Sylow
278C24—21no—
279C24—24no—
280C24—24no—
281C24—42no—
282C24—84no—
283C24C22⋊Dic1712yes—
284C24C22⋊Dic1711yescharacteristic
285C24C2×C22⋊D1712yes—
286#41—48no—
287#41—48no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8C22⋊Dic1712yescomplemented
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—24no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4C22⋊Dic1712yescomplemented
314C2×D4C2×C22⋊D1712yes—
315C2×D4C2×C22⋊D1712yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—684no—
327C22×C4C22⋊Dic1711yescharacteristic
328C22×C4C22×Dic1711yescharacteristic
329C4⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—684no—
334C22⋊C4—684no—
335C22⋊C4—1368no—
336C42—21no—
337C42—21no—
338C42C22⋊Dic1712yescomplemented
339C42C22⋊Dic1712yescomplemented
340C23—21no—
341C23—22no—
342C23—21no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—48no—
350C23—44no—
351C23—44no—
352C23—48no—
353C23—44no—
354C23—41no—
355C23—41no—
356C23—84no—
357C23—88no—
358C23—88no—
359C23#1058750512yes—
360C23C2×C22⋊Dic1711yescharacteristic
361C23#1058763914yes—
362Q8—42no—
363Q8—42no—
364D4—42no—
365D4—42no—
366D4—48no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—48no—
372D4—88no—
373D4—88no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—48no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—1364no—
389C2×C4—1368no—
390C2×C4#1058750512yes—
391C2×C4C2×C22⋊Dic1711yescharacteristic
392C2×C4C2×C22⋊Dic1711yescharacteristic
393C22—22no—
394C22—24no—
395C22—22no—
396C22—24no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—42no—
405C22—42no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—816no—
410C22—82no—
411C22#1176266511yescharacteristic
412C22C2×#1058750511yescharacteristic
413C22C2×#1058750511yescharacteristic
414C4—22no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—2724no—
421C2—21no—
422C2—21no—
423C2—44no—
424C2—42no—
425C2—82no—
426C2—82no—
427C2#1452869911yescharacteristic
428C2#1453045312yes—
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
Fitting, prime radicalC2×C17×#535C2
socleC22×C17C2×#99
cosocleC22×D4×C17C23
hypercenterC2×#535D17
hypocenterC17#12696
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#11762604C2semidirect
2. C2×#11762602C2semidirect
3. #14528699C2semidirect
10. C22×#41×C17C4semidirect
14. C2×C17×#127C4semidirect
16. C2×C17×#124C4semidirect
20. C2×C17×#117C4semidirect
76. C2×Q8×C17C22⋊C4semidirect
98. C2×D4×C17C22⋊C4semidirect
120. C42×C17C22⋊C4semidirect
121. C42×C17C22⋊C4semidirect
211. C22×#41Dic17semidirect
215. C2×#127Dic17semidirect
217. C2×#124Dic17semidirect
221. C2×#117Dic17semidirect
277. C17#12696semidirect
291. C2×Q8C22⋊Dic17semidirect
313. C2×D4C22⋊Dic17semidirect
338. C42C22⋊Dic17semidirect
339. C42C22⋊Dic17semidirect

Maximal subgroups

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

subgroupindex
C2×#117626042
C2×#117626022
#145286992
C2×C17×#5352
#1269617