Finite Group Databasebeta

← C17×#13070 #16870814

Subgroups — C17×#13070 #16870814

Order 4352.

456 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3407C211yesmaximal, characteristic, complemented
2C17×#3067C211yesmaximal, characteristic
3C2×C17×#360C211yesmaximal, characteristic, complemented
4C17×#2143C211yesmaximal, characteristic, complemented
5C17×#1815C211yesmaximal, characteristic, complemented
6C17×#1388C211yesmaximal, characteristic, complemented
7C17×#1383C211yesmaximal, characteristic, complemented
8C17×#575—21no—
9C2×C17×#135C2211yescharacteristic
10C2×C17×#131C2211yescharacteristic
11C17×#504—22no—
12C17×#504—22no—
13C2×C17×M32—22no—
14C2×C17×M32C2211yescharacteristic
15C2×C17×M32C2211yescharacteristic
16C17×#473C412yescomplemented
17C17×#472C412yescomplemented
18C17×#471—22no—
19C2×C17×#108C2211yescharacteristic
20C2×C17×#108C2211yescharacteristic
21C2×C17×#106C2211yescharacteristic
22C17×#417—21no—
23C17×#413—21no—
24C17×#360—22no—
25C17×#359—22no—
26C17×#343—21no—
27C2×#41×C17—21no—
28C2×#41×C17D411yescharacteristic
29C17×#137—22no—
30C17×#137—41no—
31C17×AGL(1,ℤ/8)—22no—
32C17×AGL(1,ℤ/8)—41no—
33C17×#135—21no—
34C17×#135—21no—
35C17×#135—21no—
36C17×#135—21no—
37C17×#135—42no—
38C2×Q16×C17C2×C411yescharacteristic
39C2×SD16×C17—21no—
40C2×D8×C17C2×C411yescharacteristic
41C17×#131—21no—
42C17×#131—21no—
43C17×#131C2×C412yes—
44C17×#131D412yes—
45C2×M16×C17—21no—
46C2×M16×C17—21no—
47C2×M16×C17—21no—
48C2×M16×C17D411yescharacteristic
49C22×C8×C17—21no—
50C22×C8×C17C2311yescharacteristic
51C2×C4⋊C4×C17—21no—
52C17×M32—24no—
53C17×M32—21no—
54C17×M32—21no—
55C17×M32—21no—
56C17×M32D412yes—
57C2×C16×C17—24no—
58C2×C16×C17—21no—
59C2×C16×C17D412yes—
60C17×#108—21no—
61C17×#108—21no—
62C17×#108—21no—
63C17×#108C2×C412yes—
64C17×#106—21no—
65C17×#106—21no—
66C17×#103—41no—
67C17×#102—41no—
68C17×#101—22no—
69C17×#100—22no—
70#41×C17—21no—
71#41×C17—21no—
72#41×C17—41no—
73#41×C17—41no—
74#41×C17—41no—
75#41×C17—41no—
76#41×C17C22⋊C412yes—
77#41×C17SD1612yes—
78C2×Q8×C17—21no—
79C2×Q8×C17D811yescharacteristic
80C2×D4×C17—21no—
81C2×D4×C17—21no—
82C2×D4×C17—41no—
83C2×D4×C17D811yescharacteristic
84C22×C4×C17—21no—
85C22×C4×C17—41no—
86C22×C4×C17—41no—
87C22×C4×C17C2×D411yescharacteristic
88Q16×C17—21no—
89Q16×C17—21no—
90Q16×C17—42no—
91SD16×C17—42no—
92SD16×C17—42no—
93SD16×C17—41no—
94SD16×C17—41no—
95D8×C17—21no—
96D8×C17—21no—
97D8×C17—42no—
98M16×C17—21no—
99M16×C17—22no—
100M16×C17—22no—
101M16×C17—22no—
102M16×C17—41no—
103M16×C17—41no—
104M16×C17—41no—
105M16×C17C22⋊C412yes—
106C2×C8×C17—21no—
107C2×C8×C17—22no—
108C2×C8×C17—22no—
109C2×C8×C17—41no—
110C2×C8×C17—41no—
111C2×C8×C17C22×C411yescharacteristic
112C2×C8×C17C2×D411yescharacteristic
113C2×C8×C17C2×D411yescharacteristic
114C2×C8×C17C2×D411yescharacteristic
115C2×C8×C17#4111yescharacteristic
116C2×C8×C17#4111yescharacteristic
117C4⋊C4×C17—41no—
118C4⋊C4×C17—41no—
119C16×C17—22no—
120C16×C17—22no—
121#13070C1711yesmaximal, characteristic, direct product factor, Sylow
122C23×C17—21no—
123C23×C17—41no—
124C23×C17#13511yescharacteristic
125Q8×C17—21no—
126Q8×C17—41no—
127Q8×C17—41no—
128Q8×C17#10212yes—
129D4×C17—22no—
130D4×C17—22no—
131D4×C17—21no—
132D4×C17—41no—
133D4×C17—41no—
134D4×C17—81no—
135D4×C17—81no—
136D4×C17#10212yes—
137C2×C4×C17—21no—
138C2×C4×C17—22no—
139C2×C4×C17—22no—
140C2×C4×C17—41no—
141C2×C4×C17—81no—
142C2×C4×C17—81no—
143C2×C4×C17—81no—
144C2×C4×C17—161no—
145C2×C4×C17C2×C22⋊C411yescharacteristic
146C2×C4×C17#12111yescharacteristic
147C2×C4×C17#12111yescharacteristic
148C2×C4×C17C2×D811yescharacteristic
149C2×C4×C17C2×SD1611yescharacteristic
150C2×C4×C17#13511yescharacteristic
151C8×C17—22no—
152C8×C17—42no—
153C8×C17—81no—
154C8×C17C4×D411yescharacteristic
155C8×C17C4×D411yescharacteristic
156C8×C17#12011yescharacteristic
157C8×C17#12311yescharacteristic
158#3407C2×C1711yescharacteristic, complemented
159#3067C2×C1711yescharacteristic
160C2×#360C2×C1711yescharacteristic, complemented
161#2143C2×C1711yescharacteristic, complemented
162#1815C2×C1711yescharacteristic, complemented
163#1388C2×C1711yescharacteristic, complemented
164#1383C2×C1711yescharacteristic, complemented
165C22×C17—22no—
166C22×C17—41no—
167C22×C17—81no—
168C22×C17—81no—
169C22×C17—161no—
170C22×C17#41611yescharacteristic
171C22×C17#46511yescharacteristic
172C22×C17#46611yescharacteristic
173C4×C17—22no—
174C4×C17—81no—
175C4×C17—161no—
176C4×C17#38611yescharacteristic
177C4×C17C2×#10211yescharacteristic
178C4×C17#46511yescharacteristic
179C4×C17#46611yescharacteristic
180#575—21no—
181C2×#135C22×C1711yescharacteristic
182C2×#131C22×C1711yescharacteristic
183#504—22no—
184#504—22no—
185C2×M32—22no—
186C2×M32C22×C1711yescharacteristic
187C2×M32C22×C1711yescharacteristic
188#473C4×C1712yescomplemented
189#472C4×C1712yescomplemented
190#471—22no—
191C2×#108C22×C1711yescharacteristic
192C2×#108C22×C1711yescharacteristic
193C2×#106C22×C1711yescharacteristic
194#417—21no—
195#413—21no—
196#360—22no—
197#359—22no—
198#343—21no—
199C2×C17—21no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17—42no—
203C2×C17—161no—
204C2×C17#189311yescharacteristic
205C2×#41—21no—
206C2×#41D4×C1711yescharacteristic
207#137—22no—
208#137—41no—
209AGL(1,ℤ/8)—22no—
210AGL(1,ℤ/8)—41no—
211#135—21no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—42no—
216C2×Q16C2×C4×C1711yescharacteristic
217C2×SD16—21no—
218C2×D8C2×C4×C1711yescharacteristic
219#131—21no—
220#131—21no—
221#131C2×C4×C1712yes—
222#131D4×C1712yes—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16D4×C1711yescharacteristic
227C22×C8—21no—
228C22×C8C23×C1711yescharacteristic
229C2×C4⋊C4—21no—
230M32—24no—
231M32—21no—
232M32—21no—
233M32—21no—
234M32D4×C1712yes—
235C2×C16—24no—
236C2×C16—21no—
237C2×C16D4×C1712yes—
238#108—21no—
239#108—21no—
240#108—21no—
241#108C2×C4×C1712yes—
242#106—21no—
243#106—21no—
244#103—41no—
245#102—41no—
246#101—22no—
247#100—22no—
248C17#1307011yescharacteristic, direct product factor, Sylow
249#41—21no—
250#41—21no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41C22⋊C4×C1712yes—
256#41SD16×C1712yes—
257C2×Q8—21no—
258C2×Q8D8×C1711yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4D8×C1711yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C2×D4×C1711yescharacteristic
267Q16—21no—
268Q16—21no—
269Q16—42no—
270SD16—42no—
271SD16—42no—
272SD16—41no—
273SD16—41no—
274D8—21no—
275D8—21no—
276D8—42no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—22no—
281M16—41no—
282M16—41no—
283M16—41no—
284M16C22⋊C4×C1712yes—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8C22×C4×C1711yescharacteristic
291C2×C8C2×D4×C1711yescharacteristic
292C2×C8C2×D4×C1711yescharacteristic
293C2×C8C2×D4×C1711yescharacteristic
294C2×C8#41×C1711yescharacteristic
295C2×C8#41×C1711yescharacteristic
296C4⋊C4—41no—
297C4⋊C4—41no—
298C16—22no—
299C16—22no—
300C23—21no—
301C23—41no—
302C23C17×#13511yescharacteristic
303Q8—21no—
304Q8—41no—
305Q8—41no—
306Q8C17×#10212yes—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4—81no—
313D4—81no—
314D4C17×#10212yes—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—161no—
323C2×C4C2×C22⋊C4×C1711yescharacteristic
324C2×C4C17×#12111yescharacteristic
325C2×C4C17×#12111yescharacteristic
326C2×C4C2×D8×C1711yescharacteristic
327C2×C4C2×SD16×C1711yescharacteristic
328C2×C4C17×#13511yescharacteristic
329C8—22no—
330C8—42no—
331C8—81no—
332C8C4×D4×C1711yescharacteristic
333C8C4×D4×C1711yescharacteristic
334C8C17×#12011yescharacteristic
335C8C17×#12311yescharacteristic
336C22—22no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—161no—
341C22C17×#41611yescharacteristic
342C22C17×#46511yescharacteristic
343C22C17×#46611yescharacteristic
344C4—22no—
345C4—81no—
346C4—161no—
347C4C17×#38611yescharacteristic
348C4C2×C17×#10211yescharacteristic
349C4C17×#46511yescharacteristic
350C4C17×#46611yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2C17×#189311yescharacteristic
subgroupG/H
center, socleC2×C17#1893
commutatorC2×C8C22×C4×C17
Frattini, cosocleC22×C8C23×C17
prime radicalC17×#3067C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#3407C2semidirect
3. C2×C17×#360C2semidirect
4. C17×#2143C2semidirect
5. C17×#1815C2semidirect
6. C17×#1388C2semidirect
7. C17×#1383C2semidirect
16. C17×#473C4semidirect
17. C17×#472C4semidirect
121. #13070C17direct (N × H)
158. #3407C2×C17semidirect
160. C2×#360C2×C17semidirect
161. #2143C2×C17semidirect
162. #1815C2×C17semidirect
163. #1388C2×C17semidirect
164. #1383C2×C17semidirect
188. #473C4×C17semidirect
189. #472C4×C17semidirect
248. C17#13070direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#34072
C17×#30672
C2×C17×#3602
C17×#21432
C17×#18152
C17×#13882
C17×#13832
#1307017