Finite Group Databasebeta

← C17×#13071 #16870815

Subgroups — C17×#13071 #16870815

Order 4352.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C17×#504C211yesmaximal, characteristic, complemented
2C17×#3067C211yesmaximal, characteristic
3C17×#2145C211yesmaximal, characteristic, complemented
4C2×C17×#359C211yesmaximal, characteristic, complemented
5C17×#1814C211yesmaximal, characteristic, complemented
6C17×#1388C211yesmaximal, characteristic, complemented
7C17×#1382C211yesmaximal, characteristic, complemented
8C17×#575—21no—
9C2×C17×#135C2211yescharacteristic
10C2×C17×#131C2211yescharacteristic
11C17×#504—22no—
12C17×#504—22no—
13C2×C17×M32—21no—
14C2×C17×M32C2211yescharacteristic
15C22×C16×C17—21no—
16C22×C16×C17C2211yescharacteristic
17C17×#473C412yescomplemented
18C17×#472C412yescomplemented
19C17×#471—22no—
20C2×C17×#108C2211yescharacteristic
21C2×C17×#108C2211yescharacteristic
22C17×#427C2211yescharacteristic
23C17×#421—21no—
24C17×#413—21no—
25C17×#360—22no—
26C17×#359—21no—
27C17×#359—21no—
28C17×#344—21no—
29C2×#41×C17—21no—
30C2×#41×C17D411yescharacteristic
31C17×#137—22no—
32C17×#137—41no—
33C17×AGL(1,ℤ/8)—22no—
34C17×AGL(1,ℤ/8)—41no—
35C17×#135—21no—
36C17×#135—21no—
37C17×#135—21no—
38C17×#135—21no—
39C17×#135—42no—
40C2×Q16×C17C2×C411yescharacteristic
41C2×SD16×C17—21no—
42C2×D8×C17C2×C411yescharacteristic
43C17×#131—21no—
44C17×#131—21no—
45C17×#131C2×C412yes—
46C17×#131D412yes—
47C2×M16×C17—21no—
48C2×M16×C17—21no—
49C2×M16×C17—21no—
50C2×M16×C17D411yescharacteristic
51C22×C8×C17—21no—
52C22×C8×C17C2311yescharacteristic
53C17×#117—21no—
54C17×M32—24no—
55C17×M32—21no—
56C17×M32D412yes—
57C2×C16×C17—24no—
58C2×C16×C17—21no—
59C2×C16×C17—21no—
60C2×C16×C17—21no—
61C2×C16×C17D411yescharacteristic
62C2×C16×C17D411yescharacteristic
63C17×#108—21no—
64C17×#108—21no—
65C17×#108—21no—
66C17×#108C2×C412yes—
67C17×#107—21no—
68C17×#106—21no—
69C17×#104—42no—
70C17×#101—22no—
71C17×#100—22no—
72#41×C17—21no—
73#41×C17—21no—
74#41×C17—41no—
75#41×C17—41no—
76#41×C17—41no—
77#41×C17—41no—
78#41×C17C22⋊C412yes—
79#41×C17D812yes—
80C2×Q8×C17—21no—
81C2×Q8×C17SD1611yescharacteristic
82C2×D4×C17—21no—
83C2×D4×C17—21no—
84C2×D4×C17—41no—
85C2×D4×C17SD1611yescharacteristic
86C22×C4×C17—21no—
87C22×C4×C17—41no—
88C22×C4×C17C2×D411yescharacteristic
89Q16×C17—21no—
90Q16×C17—21no—
91Q16×C17—42no—
92SD16×C17—42no—
93SD16×C17—42no—
94SD16×C17—41no—
95SD16×C17—41no—
96D8×C17—21no—
97D8×C17—21no—
98D8×C17—42no—
99M16×C17—21no—
100M16×C17—22no—
101M16×C17—22no—
102M16×C17—22no—
103M16×C17—41no—
104M16×C17—41no—
105M16×C17—41no—
106M16×C17C22⋊C412yes—
107C2×C8×C17—21no—
108C2×C8×C17—22no—
109C2×C8×C17—22no—
110C2×C8×C17—41no—
111C2×C8×C17—41no—
112C2×C8×C17C22×C411yescharacteristic
113C2×C8×C17C2×D411yescharacteristic
114C2×C8×C17C2×D411yescharacteristic
115C2×C8×C17C2×D411yescharacteristic
116C2×C8×C17#4111yescharacteristic
117C2×C8×C17#4111yescharacteristic
118C4⋊C4×C17—41no—
119C22⋊C4×C17—41no—
120C42×C17—41no—
121C16×C17—21no—
122C16×C17—22no—
123C16×C17—21no—
124#13071C1711yesmaximal, characteristic, direct product factor, Sylow
125C23×C17—21no—
126C23×C17—41no—
127C23×C17#13511yescharacteristic
128Q8×C17—21no—
129Q8×C17—41no—
130Q8×C17—41no—
131Q8×C17#10212yes—
132D4×C17—22no—
133D4×C17—22no—
134D4×C17—21no—
135D4×C17—41no—
136D4×C17—41no—
137D4×C17—81no—
138D4×C17—81no—
139D4×C17#10212yes—
140C2×C4×C17—21no—
141C2×C4×C17—22no—
142C2×C4×C17—22no—
143C2×C4×C17—41no—
144C2×C4×C17—81no—
145C2×C4×C17—81no—
146C2×C4×C17—81no—
147C2×C4×C17C2×C22⋊C411yescharacteristic
148C2×C4×C17#12111yescharacteristic
149C2×C4×C17#12111yescharacteristic
150C2×C4×C17C2×D811yescharacteristic
151C2×C4×C17C2×SD1611yescharacteristic
152C2×C4×C17#13511yescharacteristic
153C8×C17—22no—
154C8×C17—42no—
155C8×C17—81no—
156C8×C17C4×D411yescharacteristic
157C8×C17C4×D411yescharacteristic
158C8×C17#12011yescharacteristic
159C8×C17#12311yescharacteristic
160C2×#504C2×C1711yescharacteristic, complemented
161#3067C2×C1711yescharacteristic
162#2145C2×C1711yescharacteristic, complemented
163C2×#359C2×C1711yescharacteristic, complemented
164#1814C2×C1711yescharacteristic, complemented
165#1388C2×C1711yescharacteristic, complemented
166#1382C2×C1711yescharacteristic, complemented
167C22×C17—22no—
168C22×C17—41no—
169C22×C17—42no—
170C22×C17—81no—
171C22×C17—161no—
172C22×C17#41611yescharacteristic
173C22×C17#46511yescharacteristic
174C22×C17#46611yescharacteristic
175C4×C17—22no—
176C4×C17—81no—
177C4×C17—161no—
178C4×C17#38611yescharacteristic
179C4×C17C2×#10211yescharacteristic
180C4×C17#46511yescharacteristic
181C4×C17#46611yescharacteristic
182#575—21no—
183C2×#135C22×C1711yescharacteristic
184C2×#131C22×C1711yescharacteristic
185#504—22no—
186#504—22no—
187C2×M32—21no—
188C2×M32C22×C1711yescharacteristic
189C22×C16—21no—
190C22×C16C22×C1711yescharacteristic
191#473C4×C1712yescomplemented
192#472C4×C1712yescomplemented
193#471—22no—
194C2×#108C22×C1711yescharacteristic
195C2×#108C22×C1711yescharacteristic
196#427C22×C1711yescharacteristic
197#421—21no—
198#413—21no—
199#360—22no—
200#359—21no—
201#359—21no—
202#344—21no—
203C2×C17—21no—
204C2×C17—21no—
205C2×C17—21no—
206C2×C17—42no—
207C2×C17—161no—
208C2×C17#189311yescharacteristic
209C2×#41—21no—
210C2×#41D4×C1711yescharacteristic
211#137—22no—
212#137—41no—
213AGL(1,ℤ/8)—22no—
214AGL(1,ℤ/8)—41no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135—21no—
219#135—42no—
220C2×Q16C2×C4×C1711yescharacteristic
221C2×SD16—21no—
222C2×D8C2×C4×C1711yescharacteristic
223#131—21no—
224#131—21no—
225#131C2×C4×C1712yes—
226#131D4×C1712yes—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16D4×C1711yescharacteristic
231C22×C8—21no—
232C22×C8C23×C1711yescharacteristic
233#117—21no—
234M32—24no—
235M32—21no—
236M32D4×C1712yes—
237C2×C16—24no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16D4×C1711yescharacteristic
242C2×C16D4×C1711yescharacteristic
243#108—21no—
244#108—21no—
245#108—21no—
246#108C2×C4×C1712yes—
247#107—21no—
248#106—21no—
249#104—42no—
250#101—22no—
251#100—22no—
252C17#1307111yescharacteristic, direct product factor, Sylow
253#41—21no—
254#41—21no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41C22⋊C4×C1712yes—
260#41D8×C1712yes—
261C2×Q8—21no—
262C2×Q8SD16×C1711yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4SD16×C1711yescharacteristic
267C22×C4—21no—
268C22×C4—41no—
269C22×C4C2×D4×C1711yescharacteristic
270Q16—21no—
271Q16—21no—
272Q16—42no—
273SD16—42no—
274SD16—42no—
275SD16—41no—
276SD16—41no—
277D8—21no—
278D8—21no—
279D8—42no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—22no—
284M16—41no—
285M16—41no—
286M16—41no—
287M16C22⋊C4×C1712yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8C22×C4×C1711yescharacteristic
294C2×C8C2×D4×C1711yescharacteristic
295C2×C8C2×D4×C1711yescharacteristic
296C2×C8C2×D4×C1711yescharacteristic
297C2×C8#41×C1711yescharacteristic
298C2×C8#41×C1711yescharacteristic
299C4⋊C4—41no—
300C22⋊C4—41no—
301C42—41no—
302C16—21no—
303C16—22no—
304C16—21no—
305C23—21no—
306C23—41no—
307C23C17×#13511yescharacteristic
308Q8—21no—
309Q8—41no—
310Q8—41no—
311Q8C17×#10212yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319D4C17×#10212yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4C2×C22⋊C4×C1711yescharacteristic
328C2×C4C17×#12111yescharacteristic
329C2×C4C17×#12111yescharacteristic
330C2×C4C2×D8×C1711yescharacteristic
331C2×C4C2×SD16×C1711yescharacteristic
332C2×C4C17×#13511yescharacteristic
333C8—22no—
334C8—42no—
335C8—81no—
336C8C4×D4×C1711yescharacteristic
337C8C4×D4×C1711yescharacteristic
338C8C17×#12011yescharacteristic
339C8C17×#12311yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—81no—
344C22—161no—
345C22C17×#41611yescharacteristic
346C22C17×#46511yescharacteristic
347C22C17×#46611yescharacteristic
348C4—22no—
349C4—81no—
350C4—161no—
351C4C17×#38611yescharacteristic
352C4C2×C17×#10211yescharacteristic
353C4C17×#46511yescharacteristic
354C4C17×#46611yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—161no—
360C2C17×#189311yescharacteristic
subgroupG/H
centerC4×C17#466
commutatorC2×C8C22×C4×C17
Frattini, cosocleC22×C8C23×C17
socleC2×C17#1893
prime radicalC17×#3067C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C17×#504C2semidirect
3. C17×#2145C2semidirect
4. C2×C17×#359C2semidirect
5. C17×#1814C2semidirect
6. C17×#1388C2semidirect
7. C17×#1382C2semidirect
17. C17×#473C4semidirect
18. C17×#472C4semidirect
124. #13071C17direct (N × H)
160. C2×#504C2×C17semidirect
162. #2145C2×C17semidirect
163. C2×#359C2×C17semidirect
164. #1814C2×C17semidirect
165. #1388C2×C17semidirect
166. #1382C2×C17semidirect
191. #473C4×C17semidirect
192. #472C4×C17semidirect
252. C17#13071direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C17×#5042
C17×#30672
C17×#21452
C2×C17×#3592
C17×#18142
C17×#13882
C17×#13822
#1307117