Finite Group Databasebeta

← C17×#19423 #16877167

Subgroups — C17×#19423 #16877167

Order 4352.

630 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3209C211yesmaximal, characteristic, complemented
2C17×#3077C211yesmaximal, characteristic
3C2×C17×#474C211yesmaximal, characteristic
4C17×#2975C211yesmaximal, characteristic
5C17×#1703C214yesmaximal, complemented
6C17×#1689C214yesmaximal, complemented
7C17×#1570C211yesmaximal, characteristic, complemented
8C17×#1549C211yesmaximal, characteristic, complemented
9C17×#1542C211yesmaximal, characteristic, complemented
10C17×#533C2211yescharacteristic
11C2×C17×C4⋊Q8C2211yescharacteristic
12C2×C4×D4×C17C2211yescharacteristic
13C17×#494C2212yescomplemented
14C17×#487C2212yescomplemented
15C17×#478—22no—
16C17×#476C2212yes—
17C17×#474—22no—
18C17×#474C2212yes—
19C17×#474C2214yes—
20C17×#462—22no—
21C17×#451—22no—
22C17×#436—22no—
23C17×#436C2214yes—
24C17×#431C2211yescharacteristic
25C17×#431C2211yescharacteristic
26C17×#428—21no—
27C2×C17×#107—21no—
28C17×#423C2211yescharacteristic
29C2×C17×C4⋊C8C2211yescharacteristic
30C17×#418—21no—
31C2×C17×#102—21no—
32C17×#407—22no—
33C17×#334C2214yescomplemented
34C17×#333C2212yescomplemented
35C17×#332C2212yescomplemented
36C17×#332C2214yescomplemented
37C22×Q8×C17—21no—
38C22×D4×C17D411yescharacteristic
39C23×C4×C17—21no—
40C2×Q16×C17—24no—
41C2×M16×C17—21no—
42C2×M16×C17—24no—
43C22×C8×C17—21no—
44C17×C4⋊Q8—21no—
45C17×C4⋊Q8—21no—
46C17×C4⋊Q8—21no—
47C17×C4⋊Q8C2311yescharacteristic
48C17×C4⋊Q8C2312yes—
49C17×C4⋊Q8C2311yescharacteristic
50C17×#125—21no—
51C17×C22⋊Q8—21no—
52C17×C22⋊Q8—21no—
53C4×Q8×C17—21no—
54C4×Q8×C17—21no—
55C4×D4×C17—21no—
56C4×D4×C17—21no—
57C4×D4×C17D412yes—
58C4×D4×C17C2312yes—
59C17×#117—21no—
60C2×C4⋊C4×C17—21no—
61C2×C4⋊C4×C17—21no—
62C2×C4⋊C4×C17D411yescharacteristic
63C2×C22⋊C4×C17—21no—
64C2×C42×C17C2311yescharacteristic
65C17×#107—24no—
66C17×#107—22no—
67C17×#106—22no—
68C17×C4⋊C8—21no—
69C17×C4⋊C8—21no—
70C17×C4⋊C8D412yes—
71C17×C4⋊C8C2312yes—
72C17×C4⋊C8C2312yes—
73C17×C4⋊C8C2312yes—
74C17×#103—24no—
75C17×#102—24no—
76C17×#102—22no—
77C17×#102—22no—
78C17×C22⋊C8—21no—
79C17×C22⋊C8—24no—
80C17×C22⋊C8D412yes—
81C4×C8×C17—21no—
82C4×C8×C17C2312yes—
83C24×C17—21no—
84C2×Q8×C17—21no—
85C2×Q8×C17—22no—
86C2×Q8×C17—21no—
87C2×Q8×C17—41no—
88C2×Q8×C17—82no—
89C2×D4×C17—21no—
90C2×D4×C17—22no—
91C2×D4×C17—22no—
92C2×D4×C17C2×D412yes—
93C22×C4×C17—21no—
94C22×C4×C17—22no—
95C22×C4×C17—22no—
96C22×C4×C17—41no—
97C22×C4×C17—82no—
98C22×C4×C17C2×D411yescharacteristic
99C22×C4×C17C2×D411yescharacteristic
100C22×C4×C17#4111yescharacteristic
101Q16×C17—84no—
102M16×C17—42no—
103M16×C17—84no—
104C2×C8×C17—22no—
105C2×C8×C17—22no—
106C2×C8×C17—22no—
107C2×C8×C17—42no—
108C2×C8×C17C2×D414yes—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—21no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17—22no—
114C4⋊C4×C17—21no—
115C4⋊C4×C17—22no—
116C4⋊C4×C17—41no—
117C4⋊C4×C17—41no—
118C4⋊C4×C17—41no—
119C4⋊C4×C17—41no—
120C4⋊C4×C17C2×D412yes—
121C22⋊C4×C17—22no—
122C22⋊C4×C17—22no—
123C22⋊C4×C17—41no—
124C42×C17—41no—
125C42×C17C2×D411yescharacteristic
126C42×C17C2×D411yescharacteristic
127C42×C17#4111yescharacteristic
128C42×C17C2411yescharacteristic
129#19423C1711yesmaximal, characteristic, direct product factor, Sylow
130C23×C17—22no—
131C23×C17—42no—
132C23×C17—81no—
133C23×C17—81no—
134C23×C17#12111yescharacteristic
135Q8×C17—42no—
136Q8×C17—44no—
137Q8×C17—81no—
138Q8×C17—82no—
139D4×C17—22no—
140D4×C17—22no—
141D4×C17—42no—
142C2×C4×C17—22no—
143C2×C4×C17—22no—
144C2×C4×C17—41no—
145C2×C4×C17—44no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—42no—
149C2×C4×C17—82no—
150C2×C4×C17—81no—
151C2×C4×C17—82no—
152C2×C4×C17#12111yescharacteristic
153C2×C4×C17#12111yescharacteristic
154C2×C4×C17#12111yescharacteristic
155C2×C4×C17#13712yes—
156C2×C4×C17#13712yes—
157C2×C4×C17C22×D411yescharacteristic
158C2×C4×C17C22×D411yescharacteristic
159C2×C4×C17C2×#4111yescharacteristic
160C8×C17—24no—
161C8×C17—42no—
162C8×C17—42no—
163C8×C17—42no—
164#3209C2×C1711yescharacteristic, complemented
165#3077C2×C1711yescharacteristic
166C2×#474C2×C1711yescharacteristic
167#2975C2×C1711yescharacteristic
168#1703C2×C1714yescomplemented
169#1689C2×C1714yescomplemented
170#1570C2×C1711yescharacteristic, complemented
171#1549C2×C1711yescharacteristic, complemented
172#1542C2×C1711yescharacteristic, complemented
173C22×C17—21no—
174C22×C17—21no—
175C22×C17—42no—
176C22×C17—42no—
177C22×C17—42no—
178C22×C17—84no—
179C22×C17#47012yes—
180C22×C17C2×#12111yescharacteristic
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—21no—
184C4×C17—42no—
185C4×C17—82no—
186C4×C17—81no—
187C4×C17—81no—
188C4×C17#47012yes—
189C4×C17C2×#13711yescharacteristic
190C4×C17C2×#13711yescharacteristic
191C4×C17#57612yes—
192#533C22×C1711yescharacteristic
193C2×C4⋊Q8C22×C1711yescharacteristic
194C2×C4×D4C22×C1711yescharacteristic
195#494C22×C1712yescomplemented
196#487C22×C1712yescomplemented
197#478—22no—
198#476C22×C1712yes—
199#474—22no—
200#474C22×C1712yes—
201#474C22×C1714yes—
202#462—22no—
203#451—22no—
204#436—22no—
205#436C22×C1714yes—
206#431C22×C1711yescharacteristic
207#431C22×C1711yescharacteristic
208#428—21no—
209C2×#107—21no—
210#423C22×C1711yescharacteristic
211C2×C4⋊C8C22×C1711yescharacteristic
212#418—21no—
213C2×#102—21no—
214#407—22no—
215#334C22×C1714yescomplemented
216#333C22×C1712yescomplemented
217#332C22×C1712yescomplemented
218#332C22×C1714yescomplemented
219C2×C17—22no—
220C2×C17—82no—
221C2×C17#304111yescharacteristic
222C2×C17C2×#47011yescharacteristic
223C2×C17#306911yescharacteristic
224C22×Q8—21no—
225C22×D4D4×C1711yescharacteristic
226C23×C4—21no—
227C2×Q16—24no—
228C2×M16—21no—
229C2×M16—24no—
230C22×C8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8C23×C1712yes—
235C4⋊Q8C23×C1711yescharacteristic
236C4⋊Q8C23×C1711yescharacteristic
237#125—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4D4×C1712yes—
245C4×D4C23×C1712yes—
246#117—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4D4×C1711yescharacteristic
250C2×C22⋊C4—21no—
251C2×C42C23×C1711yescharacteristic
252#107—24no—
253#107—22no—
254#106—22no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8D4×C1712yes—
258C4⋊C8C23×C1712yes—
259C4⋊C8C23×C1712yes—
260C4⋊C8C23×C1712yes—
261#103—24no—
262#102—24no—
263#102—22no—
264#102—22no—
265C22⋊C8—21no—
266C22⋊C8—24no—
267C22⋊C8D4×C1712yes—
268C4×C8—21no—
269C4×C8C23×C1712yes—
270C17#1942311yescharacteristic, direct product factor, Sylow
271C24—21no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—82no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—21no—
280C2×D4C2×D4×C1712yes—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—82no—
286C22×C4C2×D4×C1711yescharacteristic
287C22×C4C2×D4×C1711yescharacteristic
288C22×C4#41×C1711yescharacteristic
289Q16—84no—
290M16—42no—
291M16—84no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—42no—
296C2×C8C2×D4×C1714yes—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4C2×D4×C1712yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C42—41no—
313C42C2×D4×C1711yescharacteristic
314C42C2×D4×C1711yescharacteristic
315C42#41×C1711yescharacteristic
316C42C24×C1711yescharacteristic
317C23—22no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23C17×#12111yescharacteristic
322Q8—42no—
323Q8—44no—
324Q8—81no—
325Q8—82no—
326D4—22no—
327D4—22no—
328D4—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4C17×#12111yescharacteristic
340C2×C4C17×#12111yescharacteristic
341C2×C4C17×#12111yescharacteristic
342C2×C4C17×#13712yes—
343C2×C4C17×#13712yes—
344C2×C4C22×D4×C1711yescharacteristic
345C2×C4C22×D4×C1711yescharacteristic
346C2×C4C2×#41×C1711yescharacteristic
347C8—24no—
348C8—42no—
349C8—42no—
350C8—42no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—84no—
357C22C17×#47012yes—
358C22C2×C17×#12111yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—42no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4C17×#47012yes—
367C4C2×C17×#13711yescharacteristic
368C4C2×C17×#13711yescharacteristic
369C4C17×#57612yes—
370C2—22no—
371C2—82no—
372C2C17×#304111yescharacteristic
373C2C2×C17×#47011yescharacteristic
374C2C17×#306911yescharacteristic
subgroupG/H
center, socleC22×C17C2×#121
commutator, Frattini, cosocleC42C24×C17
prime radicalC22×D4×C17D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#3209C2semidirect
5. C17×#1703C2semidirect
6. C17×#1689C2semidirect
7. C17×#1570C2semidirect
8. C17×#1549C2semidirect
9. C17×#1542C2semidirect
13. C17×#494C22semidirect
14. C17×#487C22semidirect
33. C17×#334C22semidirect
34. C17×#333C22semidirect
35. C17×#332C22semidirect
36. C17×#332C22semidirect
129. #19423C17direct (N × H)
164. #3209C2×C17semidirect
168. #1703C2×C17semidirect
169. #1689C2×C17semidirect
170. #1570C2×C17semidirect
171. #1549C2×C17semidirect
172. #1542C2×C17semidirect
195. #494C22×C17semidirect
196. #487C22×C17semidirect
215. #334C22×C17semidirect
216. #333C22×C17semidirect
217. #332C22×C17semidirect
218. #332C22×C17semidirect
270. C17#19423direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#32092
C17×#30772
C2×C17×#4742
C17×#29752
C17×#17032
C17×#16892
C17×#15702
C17×#15492
C17×#15422
#1942317