Finite Group Databasebeta

← #17189829

Subgroups — #17189829

Order 4352.

474 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539321C211yesmaximal, characteristic
2#14539317C211yesmaximal, characteristic
3#14531070C211yesmaximal, characteristic, complemented
4#14531069C211yesmaximal, characteristic, complemented
5#14531068C211yesmaximal, characteristic, complemented
6#14531062C211yesmaximal, characteristic, complemented
7#14531060C211yesmaximal, characteristic, complemented
8#14531058C211yesmaximal, characteristic, complemented
9#14531057C211yesmaximal, characteristic, complemented
10#14530571C211yesmaximal, characteristic
11#14530569C211yesmaximal, characteristic
12#14530567C211yesmaximal, characteristic
13D17×#366C211yesmaximal, characteristic
14#14530382C211yesmaximal, characteristic
15C17×#2255C211yesmaximal, characteristic, complemented
16#11763439—21no—
17#11763436—21no—
18C2×#10587469C2211yescharacteristic
19C2×C16×D17C2211yescharacteristic
20#11763219—21no—
21#11763217—21no—
22#11763216—21no—
23#11763215—21no—
24#11763214C2211yescharacteristic
25#11763213C2211yescharacteristic
26#11763213C2211yescharacteristic
27#11763211C2211yescharacteristic
28#11763210C2211yescharacteristic
29#11763210C2211yescharacteristic
30#11762850—21no—
31#11762848—21no—
32#11762846C2211yescharacteristic
33#11762846C2211yescharacteristic
34#107×D17C2211yescharacteristic
35#107×D17C2211yescharacteristic
36#11762763C2211yescharacteristic
37#11762637C2211yescharacteristic
38#11762637C2211yescharacteristic
39#11762636C2211yescharacteristic
40#11762636C2211yescharacteristic
41#11762635C2211yescharacteristic
42#11762635C2211yescharacteristic
43#11762634C2211yescharacteristic
44#11762634C2211yescharacteristic
45#11762634C2211yescharacteristic
46#11762634C2211yescharacteristic
47#11762634C2211yescharacteristic
48#11762634C2211yescharacteristic
49#11762584C2211yescharacteristic
50C4×#63214C2211yescharacteristic
51C16×Dic17C2211yescharacteristic
52C17×#500C2211yescharacteristic
53C17×#499C2211yescharacteristic
54C17×#367C2211yescharacteristic
55C17×#367C2211yescharacteristic
56C17×#366C2211yescharacteristic
57C17×#366C2211yescharacteristic
58C17×#346C2211yescharacteristic
59C17×C4⋊Q8—21no—
60C17×#125—21no—
61#10587632—41no—
62#10587630—41no—
63C2×#63218D411yescharacteristic
64C2×C8×D17C2311yescharacteristic
65#10587583—41no—
66#10587582—41no—
67#10587582—41no—
68#10587578—21no—
69#10587578—21no—
70C4⋊C4×D17—21no—
71C4⋊C4×D17—21no—
72#10587576—21no—
73#10587576—21no—
74#10587575—41no—
75#10587575—41no—
76#10587574—21no—
77#10587574—21no—
78#10587574—41no—
79#10587558D411yescharacteristic
80#10587556—21no—
81#10587555—21no—
82C2×C16×C17C2311yescharacteristic
83C2×C16×C17C2311yescharacteristic
84C17×#107—21no—
85C17×#107C2311yescharacteristic
86C17×#107C2311yescharacteristic
87C17×#107C2311yescharacteristic
88C17×#107C2311yescharacteristic
89C17×#106—21no—
90C4×C8×C17C2311yescharacteristic
91#10587489—21no—
92#10587489C2311yescharacteristic
93#10587489C2311yescharacteristic
94#10587489C2311yescharacteristic
95#10587489C2311yescharacteristic
96#10587488—21no—
97#10587486D411yescharacteristic
98C8×Dic17C2311yescharacteristic
99C2×#63214C2311yescharacteristic
100C2×#63214C2311yescharacteristic
101#10587479—21no—
102#10587479—21no—
103#10587479—21no—
104#10587479—21no—
105#10587478—21no—
106#10587478—21no—
107#10587478—21no—
108#10587478—21no—
109#10587474D411yescharacteristic
110#10587469Q814yescomplemented
111C16×D17Q814yescomplemented
112C2×Q8×C17—41no—
113C2×C4×D17—41no—
114C2×C4×D17—41no—
115C2×C4×D17C2×D411yescharacteristic
116C2×Dic34—41no—
117C2×Dic34—41no—
118C2×Dic34—41no—
119C2×C8×C17C2×D411yescharacteristic
120C2×C8×C17C2411yescharacteristic
121C4⋊C4×C17—21no—
122C4⋊C4×C17—21no—
123C4⋊C4×C17—21no—
124C4⋊C4×C17—21no—
125C4⋊C4×C17—41no—
126C4⋊C4×C17—41no—
127C4⋊C4×C17—41no—
128C42×C17C2×D411yescharacteristic
129#63227—41no—
130#63227—41no—
131#63227D811yescharacteristic
132#63227D811yescharacteristic
133C4⋊Dic17—21no—
134C4⋊Dic17—21no—
135C4⋊Dic17—21no—
136C4⋊Dic17—21no—
137C4⋊Dic17—41no—
138C4⋊Dic17—41no—
139C4⋊Dic17—41no—
140#63225—41no—
141#63225—41no—
142#63225—41no—
143#63225—41no—
144#63225—41no—
145#63225—41no—
146#63225—41no—
147#63225—41no—
148#63225D811yescharacteristic
149#63225D811yescharacteristic
150C4×Dic17—41no—
151C4×Dic17—41no—
152C4×Dic17C2×D411yescharacteristic
153C2×#3622C2×D411yescharacteristic
154C2×#3622C2×D411yescharacteristic
155#63218Q1614yes—
156C8×D17C2×Q812yes—
157C8×D17C2×Q812yes—
158C16×C17C2×Q812yes—
159C16×C17C2×Q812yes—
160#63214C2×Q812yes—
161#63214C2×Q812yes—
162#33729—171nomaximal, Sylow
163C22×D17C2×D811yescharacteristic
164Q8×C17—161no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17C2×D811yescharacteristic
170C2×C4×C17C2×D811yescharacteristic
171C2×C4×C17C22×D411yescharacteristic
172C2×Dic17—41no—
173C2×Dic17—41no—
174C2×Dic17—41no—
175C2×Dic17—41no—
176C2×Dic17C2×D811yescharacteristic
177C2×Dic17C2×D811yescharacteristic
178C2×Dic17C2×D811yescharacteristic
179C4×D17—161no—
180C4×D17—161no—
181C4×D17C4⋊Q812yes—
182C4×D17C2×Q1612yes—
183Dic34—161no—
184Dic34—161no—
185Dic34—161no—
186C8×C17C2×Q1612yes—
187C8×C17C22×Q811yescharacteristic
188C8×C17C22×Q811yescharacteristic
189#3622C4⋊Q812yes—
190#3622C2×Q1612yes—
191#3164—171no—
192#3164—171no—
193#2255—171no—
194#2255—171no—
195#2255—171no—
196#2255D1711yescharacteristic, complemented
197#2253—171no—
198#2253—171no—
199#2252—171no—
200#2252—171no—
201#2159—171no—
202#2159—171no—
203#2158—171no—
204C2×#366—171no—
205#2108—171no—
206C22×C17C22×D811yescharacteristic
207C2×D17—21no—
208C2×D17—21no—
209C2×D17#50012yes—
210C4×C17—21no—
211C4×C17—21no—
212C4×C17—81no—
213C4×C17—81no—
214C4×C17—81no—
215C4×C17—81no—
216C4×C17C2×C4⋊Q811yescharacteristic
217C4×C17C22×Q1611yescharacteristic
218Dic17—21no—
219Dic17—21no—
220Dic17—81no—
221Dic17—81no—
222Dic17—81no—
223Dic17—81no—
224Dic17#50012yes—
225#544—341no—
226#544—341no—
227C2×M32—171no—
228C22×C16—171no—
229#501—341no—
230#501—341no—
231#501—341no—
232#501—341no—
233#500—171no—
234#500—171no—
235#500—171no—
236#500C2×D1711yescharacteristic
237#499—171no—
238#499—171no—
239#499—171no—
240#499C2×D1711yescharacteristic
241#428—341no—
242#428—341no—
243#427—171no—
244#427—171no—
245C2×#107—171no—
246C2×#107—171no—
247#405—171no—
248#367—171no—
249#367—171no—
250#367—171no—
251#367—171no—
252#367C2×D1711yescharacteristic
253#367C2×D1711yescharacteristic
254#366—171no—
255#366—171no—
256#366—171no—
257#366—171no—
258#366—171no—
259#366—171no—
260#366—171no—
261#366—171no—
262#366C2×D1711yescharacteristic
263#366C2×D1711yescharacteristic
264#346—171no—
265#346C2×D1711yescharacteristic
266C4×C16—171no—
267C4×C16—171no—
268C2×C17C2×#50011yescharacteristic
269C2×C17#341511yescharacteristic
270C2×C17#341711yescharacteristic
271D17—22no—
272C2×M16—171no—
273C22×C8—171no—
274C4⋊Q8—21no—
275C4⋊Q8—341no—
276C4⋊Q8—341no—
277C4⋊Q8—341no—
278C4⋊Q8—681no—
279C4⋊Q8—681no—
280#125—21no—
281#125—341no—
282#125—341no—
283#125—341no—
284#125—681no—
285#125—681no—
286C22⋊Q8—681no—
287C22⋊Q8—681no—
288C22⋊Q8—681no—
289C22⋊Q8—681no—
290#117—171no—
291#117—341no—
292#117—341no—
293C2×C4⋊C4—341no—
294C2×C4⋊C4—341no—
295M32—174no—
296C2×C16—171no—
297C2×C16—171no—
298C2×C16—174no—
299C2×C16C22×D1711yescharacteristic
300C2×C16C22×D1711yescharacteristic
301#107—21no—
302#107—171no—
303#107—171no—
304#107—171no—
305#107—171no—
306#107—341no—
307#107—341no—
308#107—341no—
309#107—341no—
310#107—341no—
311#107C22×D1711yescharacteristic
312#107C22×D1711yescharacteristic
313#107C22×D1711yescharacteristic
314#107C22×D1711yescharacteristic
315#106—21no—
316#106—341no—
317#106—341no—
318#106—341no—
319#106—341no—
320#106—341no—
321#97—171no—
322#97—171no—
323C4×C8—171no—
324C4×C8C22×D1711yescharacteristic
325C17#3372911yescharacteristic, complemented, Sylow
326C2×Q8—41no—
327C2×Q8—681no—
328C2×Q8—681no—
329C2×Q8—681no—
330C22×C4—171no—
331C22×C4—681no—
332C22×C4—681no—
333M16—174no—
334C2×C8—171no—
335C2×C8—171no—
336C2×C8—172no—
337C2×C8—172no—
338C2×C8D4×D1711yescharacteristic
339C2×C8C23×D1711yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—171no—
348C4⋊C4—171no—
349C4⋊C4—341no—
350C4⋊C4—341no—
351C4⋊C4—341no—
352C4⋊C4—341no—
353C4⋊C4—681no—
354C4⋊C4—681no—
355C4⋊C4—681no—
356C4⋊C4—681no—
357C4⋊C4—681no—
358C4⋊C4—681no—
359C4⋊C4—681no—
360C4⋊C4—681no—
361C4⋊C4—681no—
362C4⋊C4—681no—
363C4⋊C4—681no—
364C22⋊C4—171no—
365C22⋊C4—171no—
366C22⋊C4—681no—
367C22⋊C4—681no—
368C42—171no—
369C42—681no—
370C42—681no—
371C42D4×D1711yescharacteristic
372C16—172no—
373C16—172no—
374C16Q8×D1712yes—
375C16Q8×D1712yes—
376C23—171no—
377Q8—161no—
378Q8—2721no—
379Q8—2721no—
380Q8—2721no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—2721no—
386C2×C4—2721no—
387C2×C4—171no—
388C2×C4—172no—
389C2×C4—172no—
390C2×C4—171no—
391C2×C4—171no—
392C2×C4—681no—
393C2×C4—681no—
394C2×C4—681no—
395C2×C4—681no—
396C2×C4D8×D1711yescharacteristic
397C2×C4D8×D1711yescharacteristic
398C2×C4C2×D4×D1711yescharacteristic
399C8—172no—
400C8—172no—
401C8Q16×D1712yes—
402C8C2×Q8×D1711yescharacteristic
403C8C2×Q8×D1711yescharacteristic
404C22—172no—
405C22—341no—
406C22—341no—
407C22C2×D8×D1711yescharacteristic
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—172no—
415C4—341no—
416C4—341no—
417C4—1361no—
418C4—1361no—
419C4—1361no—
420C4—1361no—
421C4C4⋊Q8×D1711yescharacteristic
422C4C2×Q16×D1711yescharacteristic
423C2—342no—
424C2D17×#50011yescharacteristic
425C2#1454192411yescharacteristic
426C2#1454193611yescharacteristic
subgroupG/H
centerC22C2×D8×D17
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#2255C2
socleC22×C17C22×D8
prime radicalC22×D17C2×D8
hypercenter#2255D17
hypocenterC17#33729
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #14531070C2semidirect
4. #14531069C2semidirect
5. #14531068C2semidirect
6. #14531062C2semidirect
7. #14531060C2semidirect
8. #14531058C2semidirect
9. #14531057C2semidirect
15. C17×#2255C2semidirect
110. #10587469Q8semidirect
111. C16×D17Q8semidirect
196. #2255D17semidirect
325. C17#33729semidirect

Maximal subgroups

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

subgroupindex
#145393212
#145393172
#145310702
#145310692
#145310682
#145310622
#145310602
#145310582
#145310572
#145305712
#145305692
#145305672
D17×#3662
#145303822
C17×#22552
#3372917