Finite Group Databasebeta

← #17064212

Subgroups — #17064212

Order 4352.

988 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762858C211yesmaximal, characteristic, complemented
2#14537347C212yesmaximal, complemented
3C2×#11762826C212yesmaximal
4#14530337C218yesmaximal, complemented
5C2×#11762566C211yesmaximal, characteristic, complemented
6C17×#2879C211yesmaximal, characteristic
7C22×#63223C2212yes—
8C2×#10587612C2212yes—
9#11762858—22no—
10#11762858C2214yescomplemented
11C2×#10587490—22no—
12#11762837—24no—
13#11762835C2218yescomplemented
14#11762832—24no—
15#11762826—24no—
16#11762826C2218yes—
17C2×#10587508—24no—
18#11762566C2214yescomplemented
19C22×#41×C17C2211yescharacteristic
20C22×M16×C17C2211yescharacteristic
21C22×M16×C17C2211yescharacteristic
22C17×#408—22no—
23C17×#408C2214yes—
24C17×#407—22no—
25C2×#41×C17—21no—
26C2×#41×C17—21no—
27C2×#41×C17—22no—
28C2×#41×C17—22no—
29C2×#41×C17—22no—
30C2×#41×C17—22no—
31C2×#41×C17Q812yes—
32C2×#41×C17C2312yes—
33C22×Q8×C17Q811yescharacteristic
34C22×D4×C17—21no—
35C22×D4×C17Q811yescharacteristic
36C23×C4×C17—21no—
37C23×C4×C17C2311yescharacteristic
38C2×M16×C17—22no—
39C2×M16×C17—22no—
40C2×M16×C17—22no—
41C2×M16×C17—22no—
42C2×M16×C17—21no—
43C2×M16×C17—21no—
44C2×M16×C17D412yes—
45C2×M16×C17C2312yes—
46C22×C8×C17D412yes—
47C22×C8×C17C2312yes—
48C2×C22⋊Dic17—22no—
49#10587612—22no—
50#10587612—22no—
51#10587612D414yes—
52#10587612C2314yes—
53C2×C4⋊Dic17—22no—
54C2×C4×Dic17D414yes—
55C2×#63223—22no—
56C2×#63223—22no—
57C2×#63223—22no—
58C2×#63223—22no—
59C2×#63223—22no—
60C2×#63223C2314yes—
61C22×#3622—22no—
62C17×C22⋊C8—28no—
63#10587508—216no—
64#10587490—28no—
65#10587485—216no—
66C24×C17—21no—
67C24×C17#4111yescharacteristic
68#41×C17—24no—
69#41×C17—24no—
70#41×C17—24no—
71#41×C17—24no—
72#41×C17—42no—
73#41×C17—42no—
74#41×C17—41no—
75#41×C17—41no—
76#41×C17—41no—
77#41×C17—41no—
78C2×Q8×C17—21no—
79C2×Q8×C17—22no—
80C2×Q8×C17—22no—
81C2×Q8×C17C2×Q812yes—
82C2×D4×C17—22no—
83C2×D4×C17—22no—
84C2×D4×C17—22no—
85C2×D4×C17—22no—
86C2×D4×C17—21no—
87C2×D4×C17—42no—
88C2×D4×C17—42no—
89C2×D4×C17C2×Q812yes—
90C22×C4×C17—21no—
91C22×C4×C17—22no—
92C22×C4×C17—22no—
93C22×C4×C17—42no—
94C22×C4×C17—42no—
95C22×C4×C17C2×D412yes—
96C22×C4×C17C2×D412yes—
97C22×C4×C17C2×D411yescharacteristic
98C22×C4×C17C2×D411yescharacteristic
99C22×C4×C17C2×Q811yescharacteristic
100C22×C4×C17C2×Q811yescharacteristic
101C22×C4×C17#4111yescharacteristic
102C22×C4×C17#4112yes—
103C22×C4×C17#4112yes—
104C22×C4×C17C2411yescharacteristic
105C22×Dic17—24no—
106M16×C17—24no—
107M16×C17—24no—
108M16×C17—42no—
109M16×C17—41no—
110M16×C17—41no—
111C2×C8×C17—22no—
112C2×C8×C17—22no—
113C2×C8×C17—24no—
114C2×C8×C17—22no—
115C2×C8×C17C2×D414yes—
116C22⋊Dic17—24no—
117C22⋊Dic17—24no—
118C4⋊Dic17—24no—
119C4⋊Dic17—24no—
120C4×Dic17—24no—
121C4×Dic17C2×D418yes—
122#63223—24no—
123#63223—24no—
124#63223—42no—
125#63223—42no—
126C2×#3622—24no—
127C2×#3622—42no—
128C2×#3622—42no—
129#23740—171nomaximal, Sylow
130C23×C17—22no—
131C23×C17—42no—
132C23×C17—81no—
133C23×C17—82no—
134C23×C17C22⋊Q811yescharacteristic
135C23×C17C22⋊Q811yescharacteristic
136C23×C17#12312yes—
137C23×C17#12312yes—
138C23×C17C2×#4111yescharacteristic
139Q8×C17—24no—
140Q8×C17—41no—
141Q8×C17—41no—
142D4×C17—24no—
143D4×C17—44no—
144D4×C17—42no—
145D4×C17—42no—
146D4×C17—41no—
147D4×C17—41no—
148C2×C4×C17—22no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—44no—
153C2×C4×C17C22⋊Q812yes—
154C2×C4×C17C22⋊Q812yes—
155C2×C4×C17C22⋊Q812yes—
156C2×C4×C17C22⋊Q812yes—
157C2×C4×C17C22⋊Q811yescharacteristic
158C2×C4×C17C22⋊Q811yescharacteristic
159C2×C4×C17#12312yes—
160C2×C4×C17#12312yes—
161C2×C4×C17#12311yescharacteristic
162C2×C4×C17#12311yescharacteristic
163C2×C4×C17#12312yes—
164C2×C4×C17#12712yes—
165C2×C4×C17#12711yescharacteristic
166C2×C4×C17#12711yescharacteristic
167C2×C4×C17C22×D412yes—
168C2×C4×C17C22×D411yescharacteristic
169C2×C4×C17C22×Q811yescharacteristic
170C2×C4×C17C2×#4112yes—
171C2×Dic17—28no—
172C2×Dic17—84no—
173C8×C17—24no—
174C8×C17—42no—
175#3622—44no—
176C2×#430—171no—
177#2905—172no—
178C2×#421—172no—
179#2879D1711yescharacteristic
180#2098—178no—
181C2×H(3,ℤ/4)—171no—
182C22×C17—42no—
183C22×C17—41no—
184C22×C17—41no—
185C22×C17—42no—
186C22×C17—41no—
187C22×C17—44no—
188C22×C17—42no—
189C22×C17—84no—
190C22×C17#39912yes—
191C22×C17#39911yescharacteristic
192C22×C17#39911yescharacteristic
193C22×C17C2×C22⋊Q811yescharacteristic
194C22×C17C2×#12312yes—
195C4×C17—42no—
196C4×C17#39912yes—
197C4×C17#39911yescharacteristic
198C4×C17#39911yescharacteristic
199C4×C17C2×C22⋊Q812yes—
200C4×C17C2×#12311yescharacteristic
201C4×C17C2×#12711yescharacteristic
202Dic17—48no—
203C22×#41C2×D1711yescharacteristic
204C22×M16—172no—
205C22×M16C2×D1711yescharacteristic
206C22×M16C2×D1711yescharacteristic
207C2×#117—172no—
208#430—174no—
209#430—342no—
210C2×#108—342no—
211#424—178no—
212#424—344no—
213#423—344no—
214#421—178no—
215#421—344no—
216C2×#104—344no—
217#408—22no—
218#408C2×D1714yes—
219#407—22no—
220H(3,ℤ/4)—174no—
221C2×C17—22no—
222C2×C17—21no—
223C2×C17—22no—
224C2×C17—21no—
225C2×C17—21no—
226C2×C17—82no—
227C2×C17C2×#39911yescharacteristic
228C2×#41—21no—
229C2×#41—21no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—22no—
234C2×#41Dic3412yes—
235C2×#41C22×D1712yes—
236C22×Q8Dic3411yescharacteristic
237C22×D4—21no—
238C22×D4Dic3411yescharacteristic
239C23×C4—21no—
240C23×C4C22×D1711yescharacteristic
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—174no—
248C2×M16—342no—
249C2×M16—342no—
250C2×M16—342no—
251C2×M16—342no—
252C2×M16—342no—
253C2×M16C22⋊D1712yes—
254C2×M16C22×D1712yes—
255C22×C8—342no—
256C22×C8C22⋊D1712yes—
257C22×C8C22×D1712yes—
258#117—174no—
259#117—174no—
260#117—342no—
261#117—342no—
262C2×C4⋊C4—342no—
263C2×C22⋊C4—342no—
264C2×C42—174no—
265#108—348no—
266C4⋊C8—3416no—
267#104—3416no—
268C22⋊C8—28no—
269C17#2374011yescharacteristic, complemented, Sylow
270C24—21no—
271C24#6325211yescharacteristic
272#41—24no—
273#41—24no—
274#41—24no—
275#41—24no—
276#41—42no—
277#41—42no—
278#41—41no—
279#41—41no—
280#41—41no—
281#41—41no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8C2×Dic3412yes—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4C2×Dic3412yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—344no—
300C22×C4C2×Dic3411yescharacteristic
301C22×C4C2×Dic3411yescharacteristic
302C22×C4#6325211yescharacteristic
303C22×C4D4×D1712yes—
304C22×C4D4×D1712yes—
305C22×C4#6325412yes—
306C22×C4#6325412yes—
307C22×C4C2×C22⋊D1711yescharacteristic
308C22×C4C2×C22⋊D1711yescharacteristic
309C22×C4C23×D1711yescharacteristic
310M16—24no—
311M16—24no—
312M16—41no—
313M16—42no—
314M16—41no—
315M16—344no—
316M16—344no—
317M16—682no—
318M16—682no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—22no—
323C2×C8—344no—
324C2×C8—682no—
325C2×C8—682no—
326C2×C8C2×C22⋊D1714yes—
327C4⋊C4—344no—
328C4⋊C4—344no—
329C22⋊C4—344no—
330C22⋊C4—344no—
331C42—178no—
332C42—344no—
333C23—22no—
334C23—42no—
335C23—81no—
336C23—82no—
337C23#1058756912yes—
338C23#1058756912yes—
339C23#1058761011yescharacteristic
340C23#1058761011yescharacteristic
341C23C2×#6325211yescharacteristic
342Q8—24no—
343Q8—41no—
344Q8—41no—
345D4—24no—
346D4—42no—
347D4—42no—
348D4—44no—
349D4—41no—
350D4—41no—
351C2×C4—22no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—348no—
357C2×C4—1364no—
358C2×C4#1058756412yes—
359C2×C4#1058756412yes—
360C2×C4#1058756412yes—
361C2×C4#1058756412yes—
362C2×C4#1058756912yes—
363C2×C4#1058756912yes—
364C2×C4#1058761011yescharacteristic
365C2×C4#1058761011yescharacteristic
366C2×C4#1058762111yescharacteristic
367C2×C4#1058762111yescharacteristic
368C2×C4#1058762112yes—
369C2×C4#1058762611yescharacteristic
370C2×C4#1058762612yes—
371C2×C4#1058762611yescharacteristic
372C2×C4C22×Dic3411yescharacteristic
373C2×C4C2×D4×D1712yes—
374C2×C4C2×#6325412yes—
375C2×C4C22×C22⋊D1711yescharacteristic
376C8—24no—
377C8—42no—
378C8—684no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—41no—
384C22—44no—
385C22—42no—
386C22—84no—
387C22#1176274512yes—
388C22#1176274511yescharacteristic
389C22#1176274511yescharacteristic
390C22C2×#1058761011yescharacteristic
391C22C2×#1058756912yes—
392C4—42no—
393C4—688no—
394C4#1176274512yes—
395C4#1176274511yescharacteristic
396C4#1176274511yescharacteristic
397C4C2×#1058756412yes—
398C4C2×#1058762111yescharacteristic
399C4C2×#1058762611yescharacteristic
400C2—22no—
401C2—21no—
402C2—22no—
403C2—21no—
404C2—21no—
405C2—82no—
406C2C2×#1176274511yescharacteristic
subgroupG/H
centerC4C2×#10587626
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2879C2
socleC2×C17C2×#399
prime radicalC22×#41×C17C22
hypercenter#2879D17
hypocenterC17#23740
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C23×C4 ◁ #2879

Complements

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

NH ≅ G/Ntype
1. C2×#11762858C2semidirect
2. #14537347C2semidirect
4. #14530337C2semidirect
5. C2×#11762566C2semidirect
10. #11762858C22semidirect
13. #11762835C22semidirect
18. #11762566C22semidirect
269. C17#23740semidirect

Maximal subgroups

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

subgroupindex
C2×#117628582
#145373472
C2×#117628262
#145303372
C2×#117625662
C17×#28792
#2374017