Finite Group Databasebeta

← #360×Dic17 #16949574

Subgroups — #360×Dic17 #16949574

Order 4352.

466 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×Dic17C211yesmaximal, characteristic
2C2×#11762619C211yesmaximal, characteristic
3M32×Dic17C211yesmaximal, characteristic, complemented
4#106×Dic17C211yesmaximal, characteristic, complemented
5#14526891C211yesmaximal, characteristic, complemented
6#14526890C211yesmaximal, characteristic, complemented
7C2×C17×#360C211yesmaximal, characteristic
8#41×Dic17—21no—
9C2×#10587483C2211yescharacteristic
10Q16×Dic17C411yescharacteristic, complemented
11Q16×Dic17C411yescharacteristic, complemented
12SD16×Dic17—21no—
13SD16×Dic17—21no—
14D8×Dic17C411yescharacteristic, complemented
15D8×Dic17C411yescharacteristic, complemented
16#11762847C2211yescharacteristic
17C2×#10587488C2211yescharacteristic
18#11762805—21no—
19C2×C8×Dic17C2211yescharacteristic
20C4⋊C4×Dic17—21no—
21#11762619C414yescomplemented
22#11762584—21no—
23C16×Dic17—21no—
24#11762571—21no—
25C2×C17×#135C2211yescharacteristic
26C2×C17×M32C2211yescharacteristic
27C2×C17×#106C2211yescharacteristic
28C17×#360C414yescomplemented
29C2×#41×C17—21no—
30C17×#135—21no—
31C17×#135—21no—
32C17×#135C2×C412yes—
33C17×#135C2×C412yes—
34C2×Q16×C17C2×C411yescharacteristic
35C2×SD16×C17—21no—
36C2×D8×C17C2×C411yescharacteristic
37C22×C8×C17C2311yescharacteristic
38C2×C4⋊C4×C17—21no—
39Q8×Dic17—21no—
40Q8×Dic17—21no—
41D4×Dic17—21no—
42D4×Dic17—21no—
43D4×Dic17—41no—
44D4×Dic17—41no—
45#10587612—21no—
46#10587612—41no—
47C2×C4⋊Dic17—21no—
48C2×C4×Dic17—41no—
49C2×C4×Dic17—41no—
50C2×C4×Dic17D411yescharacteristic
51C22×#3622D411yescharacteristic
52C17×M32—21no—
53C17×M32C2×C412yescomplemented
54C2×C16×C17—21no—
55C17×#106—21no—
56C17×#106C2×C412yescomplemented
57#10587506—21no—
58#10587506—21no—
59#10587503—21no—
60#10587503—21no—
61#10587502—41no—
62#10587489C2×C411yescharacteristic
63#10587489C2×C411yescharacteristic
64#10587488—21no—
65#10587488—21no—
66#10587488C2×C412yescomplemented
67C8×Dic17C2×C411yescharacteristic
68C8×Dic17C2×C411yescharacteristic
69C8×Dic17D411yescharacteristic
70C8×Dic17D411yescharacteristic
71#10587483—21no—
72#10587483C2×C412yescomplemented
73C2×#63214—21no—
74#41×C17—22no—
75#41×C17—22no—
76#41×C17—41no—
77#41×C17—41no—
78C2×Q8×C17—21no—
79C2×D4×C17—21no—
80C2×D4×C17—41no—
81C22×C4×C17—41no—
82C22×C4×C17—41no—
83C22×C4×C17C2×D411yescharacteristic
84C22×Dic17—41no—
85C22×Dic17—41no—
86C22×Dic17—41no—
87C22×Dic17D811yescharacteristic
88C22×Dic17D811yescharacteristic
89Q16×C17—21no—
90Q16×C17C4212yescomplemented
91SD16×C17—22no—
92SD16×C17—22no—
93D8×C17—21no—
94D8×C17C4212yescomplemented
95C2×C8×C17C22×C411yescharacteristic
96C2×C8×C17C22×C411yescharacteristic
97C2×C8×C17C22×C411yescharacteristic
98C2×C8×C17C2×D411yescharacteristic
99C2×C8×C17#4111yescharacteristic
100C2×C8×C17#4111yescharacteristic
101C4⋊C4×C17—22no—
102C4⋊C4×C17—41no—
103C22⋊Dic17—41no—
104C22⋊Dic17—41no—
105C22⋊Dic17—41no—
106C4⋊Dic17—21no—
107C4⋊Dic17—21no—
108C4⋊Dic17—22no—
109C4⋊Dic17—41no—
110C4⋊Dic17—41no—
111C4⋊Dic17—41no—
112C4×Dic17—41no—
113C4×Dic17—41no—
114C4×Dic17—41no—
115C4×Dic17—82no—
116C4×Dic17C22⋊C411yescharacteristic
117C4×Dic17C22⋊C411yescharacteristic
118C4×Dic17SD1611yescharacteristic
119C4×Dic17SD1611yescharacteristic
120C2×#3622—21no—
121C2×#3622—21no—
122C2×#3622C22⋊C411yescharacteristic
123C2×#3622C22⋊C411yescharacteristic
124C16×C17—22no—
125#63214—22no—
126C4×#360—171nomaximal, Sylow
127C23×C17—41no—
128C23×C17C2×D811yescharacteristic
129Q8×C17—22no—
130Q8×C17—41no—
131D4×C17—22no—
132D4×C17—42no—
133D4×C17—42no—
134D4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—42no—
137C2×C4×C17—42no—
138C2×C4×C17—42no—
139C2×C4×C17—81no—
140C2×C4×C17—81no—
141C2×C4×C17C2×C22⋊C411yescharacteristic
142C2×C4×C17C4×D411yescharacteristic
143C2×C4×C17C4×D411yescharacteristic
144C2×C4×C17C2×SD1611yescharacteristic
145C2×C4×C17#13511yescharacteristic
146C2×C4×C17#13511yescharacteristic
147C2×Dic17—22no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—42no—
152C2×Dic17—82no—
153C2×Dic17—81no—
154C2×Dic17—81no—
155C2×Dic17—82no—
156C2×Dic17#10211yescharacteristic
157C2×Dic17#10211yescharacteristic
158C2×Dic17#10211yescharacteristic
159C2×Dic17#10211yescharacteristic
160C8×C17C2×C4211yescharacteristic
161C8×C17#11711yescharacteristic
162C8×C17C4×D411yescharacteristic
163C8×C17C4×D411yescharacteristic
164#3622—21no—
165#3622—21no—
166C4×#135—171no—
167C2×#360—171no—
168C2×#360D1711yescharacteristic
169C4×M32—171no—
170C4×#106—171no—
171#1386—171no—
172#1386—171no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—42no—
176C22×C17—81no—
177C22×C17—81no—
178C22×C17C2×#10211yescharacteristic
179C22×C17C4×D811yescharacteristic
180C22×C17C4×D811yescharacteristic
181C4×C17—42no—
182C4×C17—82no—
183C4×C17C4×C22⋊C411yescharacteristic
184C4×C17#41611yescharacteristic
185C4×C17C4×SD1611yescharacteristic
186C4×C17C4×SD1611yescharacteristic
187Dic17—21no—
188Dic17—21no—
189Dic17—21no—
190Dic17—82no—
191Dic17—81no—
192Dic17—81no—
193Dic17#36012yesdirect product factor
194C2×#135C2×D1711yescharacteristic
195C4×#41—341no—
196C2×M32—171no—
197C2×M32C2×D1711yescharacteristic
198C4×Q16—171no—
199C4×Q16—171no—
200C4×SD16—341no—
201C4×SD16—341no—
202C4×D8—171no—
203C4×D8—171no—
204#427—171no—
205C2×#106—171no—
206C2×#106C2×D1711yescharacteristic
207#416—341no—
208C2×C4×C8—171no—
209C4×C4⋊C4—341no—
210#360—174no—
211#360Dic1714yesdirect product factor
212#346—341no—
213C4×C16—341no—
214#340—341no—
215C2×C17—21no—
216C2×C17—21no—
217C2×C17—82no—
218C2×C17C4×#10211yescharacteristic
219C2×C17C2×#36011yescharacteristic
220C2×C17#214511yescharacteristic
221C2×#41—21no—
222#135—21no—
223#135—21no—
224#135C4×D1712yes—
225#135C2×Dic1712yes—
226C2×Q16C4×D1711yescharacteristic
227C2×SD16—21no—
228C2×D8C4×D1711yescharacteristic
229C22×C8—171no—
230C22×C8C22×D1711yescharacteristic
231C4×Q8—341no—
232C4×Q8—341no—
233C4×D4—341no—
234C4×D4—341no—
235C4×D4—681no—
236C4×D4—681no—
237#117—341no—
238#117—681no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—341no—
241C2×C42—171no—
242C2×C42—681no—
243C2×C42—681no—
244M32—21no—
245M32—172no—
246M32—341no—
247M32C2×Dic1712yescomplemented
248C2×C16—21no—
249C2×C16—341no—
250#107—171no—
251#107—171no—
252#106—21no—
253#106—172no—
254#106—341no—
255#106—341no—
256#106C2×Dic1712yescomplemented
257#103—341no—
258#103—341no—
259#102—341no—
260#102—341no—
261C4×C8—171no—
262C4×C8—171no—
263C4×C8—171no—
264C4×C8—171no—
265#95—681no—
266C17C4×#36011yescharacteristic, complemented, Sylow
267#41—22no—
268#41—22no—
269#41—41no—
270#41—41no—
271C2×Q8—21no—
272C2×D4—21no—
273C2×D4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—171no—
277C22×C4—171no—
278C22×C4—681no—
279C22×C4—681no—
280C22×C4—681no—
281C22×C4D4×D1711yescharacteristic
282Q16—21no—
283Q16C4×Dic1712yescomplemented
284SD16—22no—
285SD16—22no—
286D8—21no—
287D8C4×Dic1712yescomplemented
288C2×C8—171no—
289C2×C8—171no—
290C2×C8—341no—
291C2×C8—341no—
292C2×C8C2×C4×D1711yescharacteristic
293C2×C8C2×C4×D1711yescharacteristic
294C2×C8D4×D1711yescharacteristic
295C2×C8#6325411yescharacteristic
296C2×C8#6325411yescharacteristic
297C2×C8C22×Dic1711yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—341no—
301C4⋊C4—341no—
302C4⋊C4—342no—
303C4⋊C4—681no—
304C4⋊C4—681no—
305C4⋊C4—681no—
306C22⋊C4—681no—
307C22⋊C4—681no—
308C22⋊C4—681no—
309C42—171no—
310C42—171no—
311C42—171no—
312C42—171no—
313C42—681no—
314C42—681no—
315C42—681no—
316C42—1362no—
317C16—22no—
318C16—342no—
319C23—41no—
320C23D8×D1711yescharacteristic
321Q8—22no—
322Q8—41no—
323D4—22no—
324D4—42no—
325D4—42no—
326D4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—171no—
334C2×C4—171no—
335C2×C4—171no—
336C2×C4—171no—
337C2×C4—342no—
338C2×C4—681no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—682no—
342C2×C4—1362no—
343C2×C4—1362no—
344C2×C4—1361no—
345C2×C4—1361no—
346C2×C4C22⋊C4×D1711yescharacteristic
347C2×C4#1058756711yescharacteristic
348C2×C4#1058759811yescharacteristic
349C2×C4SD16×D1711yescharacteristic
350C2×C4#1058760211yescharacteristic
351C2×C4D4×Dic1711yescharacteristic
352C8—341no—
353C8—341no—
354C8#1058756311yescharacteristic
355C8#1058756711yescharacteristic
356C8C2×C4×Dic1711yescharacteristic
357C8D4×Dic1711yescharacteristic
358C22—21no—
359C22—21no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22#102×D1711yescharacteristic
364C22#1176288911yescharacteristic
365C22D8×Dic1711yescharacteristic
366C4—42no—
367C4—82no—
368C4—172no—
369C4—341no—
370C4—341no—
371C4—341no—
372C4—1362no—
373C4—1361no—
374C4—1361no—
375C4C22⋊C4×Dic1711yescharacteristic
376C4#1176280211yescharacteristic
377C4SD16×Dic1711yescharacteristic
378C4#1176290011yescharacteristic
379C2—21no—
380C2—21no—
381C2—82no—
382C2#102×Dic1711yescharacteristic
383C2D17×#36011yescharacteristic
384C2#1453051811yescharacteristic
subgroupG/H
centerC22#102×D17
commutatorC8×C17C2×C42
FrattiniC22×C8C22×D17
FittingC2×C17×#360C2
socleC22×C17C2×#102
prime radicalC2×C17×#135C22
cosocleC22×C8×C17C23
hypercenterC2×#360D17
hypocenterC17C4×#360
LCSG ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#360

Complements

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

NH ≅ G/Ntype
3. M32×Dic17C2semidirect
4. #106×Dic17C2semidirect
5. #14526891C2semidirect
6. #14526890C2semidirect
10. Q16×Dic17C4semidirect
11. Q16×Dic17C4semidirect
14. D8×Dic17C4semidirect
15. D8×Dic17C4semidirect
21. #11762619C4semidirect
28. C17×#360C4semidirect
53. C17×M32C2×C4semidirect
56. C17×#106C2×C4semidirect
66. #10587488C2×C4semidirect
72. #10587483C2×C4semidirect
90. Q16×C17C42semidirect
94. D8×C17C42semidirect
193. Dic17#360direct (N × H)
211. #360Dic17direct (N × H)
247. M32C2×Dic17semidirect
256. #106C2×Dic17semidirect
266. C17C4×#360semidirect
283. Q16C4×Dic17semidirect
287. D8C4×Dic17semidirect

Maximal subgroups

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

subgroupindex
#135×Dic172
C2×#117626192
M32×Dic172
#106×Dic172
#145268912
#145268902
C2×C17×#3602
C4×#36017