Finite Group Databasebeta

← C17×#7750 #16865494

Subgroups — C17×#7750 #16865494

Order 4352.

740 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#2679C211yesmaximal, characteristic
2C17×#2023C212yesmaximal, complemented
3C17×#2013C211yesmaximal, characteristic, complemented
4C17×#1270C212yesmaximal, complemented
5C17×#1270C211yesmaximal, characteristic, complemented
6C22×SD16×C17—21no—
7C2×C17×C4⋊Q8C2211yescharacteristic, complemented
8C2×C17×#120—21no—
9C2×C17×#120—22no—
10C2×C17×#103—21no—
11C2×C17×#102—22no—
12C17×#394—22no—
13C17×#394—21no—
14C17×#390C2211yescharacteristic
15C17×#390C2212yes—
16C17×#340—22no—
17C17×#336C2211yescharacteristic, complemented
18C17×#336C2212yescomplemented
19C25×C17—21no—
20C22×Q8×C17—21no—
21C22×D4×C17—22no—
22C22×D4×C17—21no—
23C22×D4×C17D411yescharacteristic
24C22×D4×C17D412yes—
25C2×SD16×C17—42no—
26C2×SD16×C17—42no—
27C2×SD16×C17—42no—
28C22×C8×C17—21no—
29C22×C8×C17—22no—
30C17×C4⋊Q8—42no—
31C17×#120—42no—
32C17×#120—44no—
33C17×#120—42no—
34C17×#120—44no—
35C2×C4⋊C4×C17—22no—
36C2×C4⋊C4×C17D412yes—
37C2×C4⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—22no—
40C2×C22⋊C4×C17—22no—
41C2×C22⋊C4×C17—22no—
42C2×C22⋊C4×C17—22no—
43C2×C42×C17C2311yescharacteristic
44C17×#103—42no—
45C17×#102—44no—
46C17×#95—21no—
47C24×C17—21no—
48C24×C17—22no—
49C24×C17—42no—
50C24×C17—44no—
51C24×C17—81no—
52C24×C17—82no—
53C24×C17—81no—
54C2×Q8×C17—22no—
55C2×Q8×C17—41no—
56C2×Q8×C17—41no—
57C2×Q8×C17—41no—
58C2×Q8×C17—41no—
59C2×Q8×C17—41no—
60C2×D4×C17—21no—
61C2×D4×C17—22no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C2×D4×C17—42no—
65C2×D4×C17—42no—
66C2×D4×C17—44no—
67C2×D4×C17—44no—
68C2×D4×C17—44no—
69C2×D4×C17—81no—
70C2×D4×C17—82no—
71C2×D4×C17D812yes—
72C2×D4×C17SD1614yescomplemented
73C22×C4×C17—22no—
74C22×C4×C17—21no—
75C22×C4×C17—41no—
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D412yes—
78SD16×C17—48no—
79C2×C8×C17—22no—
80C2×C8×C17—41no—
81C2×C8×C17—41no—
82C2×C8×C17—42no—
83C2×C8×C17—42no—
84C2×C8×C17—42no—
85C4⋊C4×C17—24no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—42no—
88C4⋊C4×C17—42no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—44no—
91C22⋊C4×C17—42no—
92C22⋊C4×C17—42no—
93C22⋊C4×C17—42no—
94C22⋊C4×C17—42no—
95C22⋊C4×C17—42no—
96C22⋊C4×C17—42no—
97C42×C17—41no—
98#7750C1711yesmaximal, characteristic, direct product factor, Sylow
99C23×C17—22no—
100C23×C17—24no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—42no—
104C23×C17—41no—
105C23×C17—42no—
106C23×C17—42no—
107C23×C17—42no—
108C23×C17—42no—
109C23×C17—41no—
110C23×C17—42no—
111C23×C17—84no—
112C23×C17—88no—
113C23×C17—84no—
114C23×C17—88no—
115C23×C17—84no—
116C23×C17#12011yescharacteristic
117Q8×C17—42no—
118Q8×C17—42no—
119Q8×C17—81no—
120Q8×C17—81no—
121D4×C17—42no—
122D4×C17—44no—
123D4×C17—81no—
124D4×C17—82no—
125D4×C17—84no—
126D4×C17—88no—
127C2×C4×C17—21no—
128C2×C4×C17—22no—
129C2×C4×C17—22no—
130C2×C4×C17—21no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17—41no—
135C2×C4×C17—42no—
136C2×C4×C17—42no—
137C2×C4×C17—81no—
138C2×C4×C17—81no—
139C2×C4×C17C2×D811yescharacteristic
140C2×C4×C17C2×SD1612yes—
141C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C17AGL(1,ℤ/8)12yes—
143C8×C17—42no—
144C8×C17—82no—
145#2679C2×C1711yescharacteristic
146#2023C2×C1712yescomplemented
147#2013C2×C1711yescharacteristic, complemented
148#1270C2×C1712yescomplemented
149#1270C2×C1711yescharacteristic, complemented
150C22×C17—42no—
151C22×C17—44no—
152C22×C17—44no—
153C22×C17—44no—
154C22×C17—44no—
155C22×C17—816no—
156C22×C17—82no—
157C22×C17—82no—
158C22×C17—82no—
159C22×C17—81no—
160C22×C17—82no—
161C22×C17—81no—
162C22×C17—82no—
163C22×C17#44711yescharacteristic
164C22×C17#45012yes—
165C22×C17#45311yescharacteristic
166C22×C17#45512yes—
167C22×C17B(4,𝔽2)11yescharacteristic
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—22no—
171C4×C17—22no—
172C4×C17—82no—
173C4×C17—81no—
174C4×C17—82no—
175C22×SD16—21no—
176C2×C4⋊Q8C22×C1711yescharacteristic, complemented
177C2×#120—21no—
178C2×#120—22no—
179C2×#103—21no—
180C2×#102—22no—
181#394—22no—
182#394—21no—
183#390C22×C1711yescharacteristic
184#390C22×C1712yes—
185#340—22no—
186#336C22×C1711yescharacteristic, complemented
187#336C22×C1712yescomplemented
188C2×C17—82no—
189C2×C17—84no—
190C2×C17#159511yescharacteristic
191C2×C17#160112yes—
192C2×C17#162911yescharacteristic
193C2×C17#163912yes—
194C2×C17#165911yescharacteristic
195C25—21no—
196C22×Q8—21no—
197C22×D4—22no—
198C22×D4—21no—
199C22×D4D4×C1711yescharacteristic
200C22×D4D4×C1712yes—
201C2×SD16—42no—
202C2×SD16—42no—
203C2×SD16—42no—
204C22×C8—21no—
205C22×C8—22no—
206C4⋊Q8—42no—
207#120—42no—
208#120—44no—
209#120—42no—
210#120—44no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4D4×C1712yes—
213C2×C4⋊C4D4×C1711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C42C23×C1711yescharacteristic
220#103—42no—
221#102—44no—
222#95—21no—
223C17#775011yescharacteristic, direct product factor, Sylow
224C24—21no—
225C24—22no—
226C24—42no—
227C24—44no—
228C24—81no—
229C24—82no—
230C24—81no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×D4—21no—
238C2×D4—22no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—81no—
247C2×D4—82no—
248C2×D4D8×C1712yes—
249C2×D4SD16×C1714yescomplemented
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4C2×D4×C1711yescharacteristic
254C22×C4C2×D4×C1712yes—
255SD16—48no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—24no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C42—41no—
275C23—22no—
276C23—24no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—41no—
286C23—42no—
287C23—84no—
288C23—88no—
289C23—84no—
290C23—88no—
291C23—84no—
292C23C17×#12011yescharacteristic
293Q8—42no—
294Q8—42no—
295Q8—81no—
296Q8—81no—
297D4—42no—
298D4—44no—
299D4—81no—
300D4—82no—
301D4—84no—
302D4—88no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4C2×D8×C1711yescharacteristic
316C2×C4C2×SD16×C1712yes—
317C2×C4C17×AGL(1,ℤ/8)11yescharacteristic
318C2×C4C17×AGL(1,ℤ/8)12yes—
319C8—42no—
320C8—82no—
321C22—42no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—816no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—82no—
332C22—81no—
333C22—82no—
334C22C17×#44711yescharacteristic
335C22C17×#45012yes—
336C22C17×#45311yescharacteristic
337C22C17×#45512yes—
338C22C17×B(4,𝔽2)11yescharacteristic
339C4—21no—
340C4—21no—
341C4—22no—
342C4—22no—
343C4—82no—
344C4—81no—
345C4—82no—
346C2—82no—
347C2—84no—
348C2C17×#159511yescharacteristic
349C2C17×#160112yes—
350C2C17×#162911yescharacteristic
351C2C17×#163912yes—
352C2C17×#165911yescharacteristic
subgroupG/H
center, socleC23×C17#120
commutator, Frattini, cosocleC2×C42C23×C17
prime radicalC17×#2679C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C17 ◁ C2×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#2023C2semidirect
3. C17×#2013C2semidirect
4. C17×#1270C2semidirect
5. C17×#1270C2semidirect
7. C2×C17×C4⋊Q8C22semidirect
17. C17×#336C22semidirect
18. C17×#336C22semidirect
72. C2×D4×C17SD16semidirect
98. #7750C17direct (N × H)
146. #2023C2×C17semidirect
147. #2013C2×C17semidirect
148. #1270C2×C17semidirect
149. #1270C2×C17semidirect
176. C2×C4⋊Q8C22×C17semidirect
186. #336C22×C17semidirect
187. #336C22×C17semidirect
223. C17#7750direct (N × H)
249. C2×D4SD16×C17semidirect

Maximal subgroups

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

subgroupindex
C17×#26792
C17×#20232
C17×#20132
C17×#12702
#775017