Finite Group Databasebeta

← C17×#9206 #16866950

Subgroups — C17×#9206 #16866950

Order 4352.

578 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2618C211yesmaximal, characteristic
2C17×#2313C211yesmaximal, characteristic, complemented
3C17×#1519C212yesmaximal
4C2×C17×#324C211yesmaximal, characteristic, complemented
5C17×#1289C212yesmaximal, complemented
6C2×C17×#123—21no—
7C2×C17×#121C2211yescharacteristic
8C22×C4⋊C4×C17C2211yescharacteristic
9C2×C17×C22⋊C8C2212yes—
10C17×#398C412yescomplemented
11C17×#396—22no—
12C17×#394C412yescomplemented
13C17×#387—24no—
14C17×#386—21no—
15C17×#381—21no—
16C17×#380C2211yescharacteristic
17C17×#380C2212yes—
18C4×C22⋊C4×C17—21no—
19C17×#327—24no—
20C17×#324—22no—
21C22×D4×C17—21no—
22C22×D4×C17C2×C411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C22×C8×C17—22no—
27C17×#123—44no—
28C17×#121—22no—
29C17×#121—22no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17—22no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17C2×C412yes—
37C2×C4⋊C4×C17D412yes—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—22no—
40C2×C22⋊C4×C17—22no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—22no—
43C2×C42×C17—22no—
44C17×C22⋊C8—22no—
45C17×C22⋊C8—22no—
46C17×#95—22no—
47C17×#95—24no—
48C17×#95—22no—
49C17×#95C2×C412yes—
50C17×#95D414yes—
51C24×C17—21no—
52C24×C17C2×D411yescharacteristic
53C2×D4×C17—22no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17—42no—
57C2×D4×C17—42no—
58C2×D4×C17—81no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—22no—
63C22×C4×C17—22no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—22no—
67C22×C4×C17—24no—
68C22×C4×C17—81no—
69C22×C4×C17—81no—
70C22×C4×C17C22⋊C412yes—
71C22×C4×C17C22⋊C412yes—
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D412yes—
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17#4112yes—
76C2×C8×C17—42no—
77C2×C8×C17—82no—
78C4⋊C4×C17—24no—
79C4⋊C4×C17—24no—
80C4⋊C4×C17—22no—
81C4⋊C4×C17—44no—
82C4⋊C4×C17—82no—
83C4⋊C4×C17—81no—
84C22⋊C4×C17—44no—
85C22⋊C4×C17—42no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—42no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—82no—
90C42×C17—82no—
91#9206C1711yesmaximal, characteristic, direct product factor, Sylow
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17C2×C22⋊C411yescharacteristic
100C23×C17#12011yescharacteristic
101C23×C17#12311yescharacteristic
102C23×C17#13512yes—
103C23×C17AGL(1,ℤ/8)12yes—
104D4×C17—82no—
105D4×C17—82no—
106D4×C17—82no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—41no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—41no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—44no—
120C2×C4×C17—42no—
121C2×C4×C17—44no—
122C2×C4×C17—42no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17—41no—
127C2×C4×C17—44no—
128C2×C4×C17—81no—
129C2×C4×C17C4×D412yes—
130C2×C4×C17#12112yes—
131C8×C17—82no—
132#2618C2×C1711yescharacteristic
133#2313C2×C1711yescharacteristic, complemented
134#1519C2×C1712yes—
135C2×#324C2×C1711yescharacteristic, complemented
136#1289C2×C1712yescomplemented
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—82no—
146C22×C17—82no—
147C22×C17—81no—
148C22×C17—81no—
149C22×C17—81no—
150C22×C17#38611yescharacteristic
151C22×C17#41611yescharacteristic
152C22×C17#41811yescharacteristic
153C22×C17#44912yes—
154C22×C17#48212yes—
155C4×C17—22no—
156C4×C17—22no—
157C4×C17—42no—
158C4×C17—42no—
159C4×C17—82no—
160C4×C17—84no—
161C4×C17—81no—
162C2×#123—21no—
163C2×#121C22×C1711yescharacteristic
164C22×C4⋊C4C22×C1711yescharacteristic
165C2×C22⋊C8C22×C1712yes—
166#398C4×C1712yescomplemented
167#396—22no—
168#394C4×C1712yescomplemented
169#387—24no—
170#386—21no—
171#381—21no—
172#380C22×C1711yescharacteristic
173#380C22×C1712yes—
174C4×C22⋊C4—21no—
175#327—24no—
176#324—22no—
177C2×C17—21no—
178C2×C17—22no—
179C2×C17—21no—
180C2×C17—22no—
181C2×C17—161no—
182C2×C17#187511yescharacteristic
183C2×C17#191211yescharacteristic
184C2×C17#191311yescharacteristic
185C22×D4—21no—
186C22×D4C2×C4×C1711yescharacteristic
187C23×C4D4×C1711yescharacteristic
188C23×C4D4×C1711yescharacteristic
189C23×C4C23×C1711yescharacteristic
190C22×C8—22no—
191#123—44no—
192#121—22no—
193#121—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×C4×C1711yescharacteristic
200C2×C4⋊C4C2×C4×C1712yes—
201C2×C4⋊C4D4×C1712yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C42—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210#95—22no—
211#95—24no—
212#95—22no—
213#95C2×C4×C1712yes—
214#95D4×C1714yes—
215C17#920611yescharacteristic, direct product factor, Sylow
216C24—21no—
217C24C2×D4×C1711yescharacteristic
218C2×D4—22no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—41no—
223C2×D4—81no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4C22⋊C4×C1712yes—
236C22×C4C22⋊C4×C1712yes—
237C22×C4C22×C4×C1711yescharacteristic
238C22×C4C2×D4×C1712yes—
239C22×C4C2×D4×C1711yescharacteristic
240C22×C4#41×C1712yes—
241C2×C8—42no—
242C2×C8—82no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—22no—
246C4⋊C4—44no—
247C4⋊C4—82no—
248C4⋊C4—81no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—82no—
255C42—82no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊C4×C1711yescharacteristic
264C23C17×#12011yescharacteristic
265C23C17×#12311yescharacteristic
266C23C17×#13512yes—
267C23C17×AGL(1,ℤ/8)12yes—
268D4—82no—
269D4—82no—
270D4—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—81no—
293C2×C4C4×D4×C1712yes—
294C2×C4C17×#12112yes—
295C8—82no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—82no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22C17×#38611yescharacteristic
310C22C17×#41611yescharacteristic
311C22C17×#41811yescharacteristic
312C22C17×#44912yes—
313C22C17×#48212yes—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—82no—
319C4—84no—
320C4—81no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2—22no—
325C2—161no—
326C2C17×#187511yescharacteristic
327C2C17×#191211yescharacteristic
328C2C17×#191311yescharacteristic
subgroupG/H
center, socleC22×C17#386
commutatorC22×C4C22×C4×C17
Frattini, cosocleC23×C4C23×C17
prime radicalC2×C17×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#2313C2semidirect
4. C2×C17×#324C2semidirect
5. C17×#1289C2semidirect
10. C17×#398C4semidirect
12. C17×#394C4semidirect
91. #9206C17direct (N × H)
133. #2313C2×C17semidirect
135. C2×#324C2×C17semidirect
136. #1289C2×C17semidirect
166. #398C4×C17semidirect
168. #394C4×C17semidirect
215. C17#9206direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#26182
C17×#23132
C17×#15192
C2×C17×#3242
C17×#12892
#920617