Finite Group Databasebeta

← C17×#9153 #16866897

Subgroups — C17×#9153 #16866897

Order 4352.

586 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 150 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×#2855C211yesmaximal, characteristic
2C2×C17×#383C211yesmaximal, characteristic
3C17×#1817C211yesmaximal, characteristic
4C2×C17×#328C212yesmaximal
5C17×#1294C212yesmaximal
6C2×C17×C22⋊Q8C2211yescharacteristic
7C22×C42×C17C2211yescharacteristic
8C2×C17×C22⋊C8C2212yes—
9C17×#391C412yescomplemented
10C17×#389C412yescomplemented
11C17×#387—22no—
12C17×#385—21no—
13C17×#383—24no—
14C17×#381—21no—
15C17×#380C2211yescharacteristic
16C2×C17×#95—21no—
17C2×C17×#95—21no—
18C2×C17×#95C2212yes—
19C17×#336—22no—
20C17×#328—24no—
21C22×Q8×C17C2×C411yescharacteristic
22C23×C4×C17—22no—
23C23×C4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C22×C8×C17—22no—
27C17×C22⋊Q8—22no—
28C17×C22⋊Q8—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17C2×C412yes—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—21no—
35C2×C42×C17—22no—
36C2×C42×C17—22no—
37C2×C42×C17C2×C412yes—
38C2×C42×C17D412yes—
39C17×C22⋊C8—22no—
40C17×C22⋊C8—22no—
41C17×#95—24no—
42C17×#95—24no—
43C17×#95—22no—
44C17×#95—22no—
45C17×#95—22no—
46C17×#95—22no—
47C17×#95—21no—
48C17×#95D414yes—
49C24×C17C2×D411yescharacteristic
50C2×Q8×C17—22no—
51C2×Q8×C17—42no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17—21no—
59C22×C4×C17—22no—
60C22×C4×C17—21no—
61C22×C4×C17—24no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17—44no—
65C22×C4×C17—44no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17C22⋊C412yes—
69C22×C4×C17C22⋊C412yes—
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D412yes—
73C22×C4×C17#4112yes—
74C2×C8×C17—44no—
75C2×C8×C17—42no—
76C4⋊C4×C17—22no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—44no—
79C4⋊C4×C17—42no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—42no—
83C42×C17—24no—
84C42×C17—24no—
85#9153C1711yesmaximal, characteristic, direct product factor, Sylow
86C23×C17—22no—
87C23×C17—22no—
88C23×C17C2×C22⋊C411yescharacteristic
89C23×C17#12011yescharacteristic
90C23×C17#12311yescharacteristic
91C23×C17#13512yes—
92C23×C17#13712yes—
93Q8×C17—82no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—44no—
109C2×C4×C17—44no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—44no—
118C2×C4×C17—84no—
119C2×C4×C17—84no—
120C2×C4×C17C4×D412yes—
121C2×C4×C17#12112yes—
122C8×C17—82no—
123#2855C2×C1711yescharacteristic
124C2×#383C2×C1711yescharacteristic
125#1817C2×C1711yescharacteristic
126C2×#328C2×C1712yes—
127#1294C2×C1712yes—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17—24no—
133C22×C17—24no—
134C22×C17#38611yescharacteristic
135C22×C17#41012yes—
136C22×C17#41611yescharacteristic
137C22×C17#41911yescharacteristic
138C22×C17#42112yes—
139C22×C17#45212yes—
140C22×C17#48512yes—
141C4×C17—22no—
142C4×C17—22no—
143C4×C17—42no—
144C4×C17—42no—
145C4×C17—84no—
146C4×C17—81no—
147C4×C17—81no—
148C4×C17—82no—
149C2×C22⋊Q8C22×C1711yescharacteristic
150C22×C42C22×C1711yescharacteristic
151C2×C22⋊C8C22×C1712yes—
152#391C4×C1712yescomplemented
153#389C4×C1712yescomplemented
154#387—22no—
155#385—21no—
156#383—24no—
157#381—21no—
158#380C22×C1711yescharacteristic
159C2×#95—21no—
160C2×#95—21no—
161C2×#95C22×C1712yes—
162#336—22no—
163#328—24no—
164C2×C17—22no—
165C2×C17—22no—
166C2×C17#151512yes—
167C2×C17#152112yes—
168C2×C17#187311yescharacteristic
169C2×C17#189711yescharacteristic
170C2×C17#189911yescharacteristic
171C22×Q8C2×C4×C1711yescharacteristic
172C23×C4—22no—
173C23×C4D4×C1711yescharacteristic
174C23×C4D4×C1711yescharacteristic
175C23×C4C23×C1711yescharacteristic
176C22×C8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C2×C4×C1711yescharacteristic
182C2×C4⋊C4C2×C4×C1712yes—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42C2×C4×C1712yes—
188C2×C42D4×C1712yes—
189C22⋊C8—22no—
190C22⋊C8—22no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—21no—
198#95D4×C1714yes—
199C17#915311yescharacteristic, direct product factor, Sylow
200C24C2×D4×C1711yescharacteristic
201C2×Q8—22no—
202C2×Q8—42no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4C22⋊C4×C1712yes—
220C22×C4C22⋊C4×C1712yes—
221C22×C4C22×C4×C1711yescharacteristic
222C22×C4C2×D4×C1711yescharacteristic
223C22×C4C2×D4×C1712yes—
224C22×C4#41×C1712yes—
225C2×C8—44no—
226C2×C8—42no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C22⋊C4—41no—
232C22⋊C4—42no—
233C22⋊C4—41no—
234C42—24no—
235C42—24no—
236C23—22no—
237C23—22no—
238C23C2×C22⋊C4×C1711yescharacteristic
239C23C17×#12011yescharacteristic
240C23C17×#12311yescharacteristic
241C23C17×#13512yes—
242C23C17×#13712yes—
243Q8—82no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4C4×D4×C1712yes—
271C2×C4C17×#12112yes—
272C8—82no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22C17×#38611yescharacteristic
280C22C17×#41012yes—
281C22C17×#41611yescharacteristic
282C22C17×#41911yescharacteristic
283C22C17×#42112yes—
284C22C17×#45212yes—
285C22C17×#48512yes—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—84no—
291C4—82no—
292C4—81no—
293C4—81no—
294C2—22no—
295C2—22no—
296C2C17×#151512yes—
297C2C17×#152112yes—
298C2C17×#187311yescharacteristic
299C2C17×#189711yescharacteristic
300C2C17×#189911yescharacteristic
subgroupG/H
center, socleC23×C17C2×C22⋊C4
commutatorC22×C4C22×C4×C17
Frattini, cosocleC23×C4C23×C17
prime radicalC24×C17C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C17 ◁ C22×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
9. C17×#391C4semidirect
10. C17×#389C4semidirect
85. #9153C17direct (N × H)
152. #391C4×C17semidirect
153. #389C4×C17semidirect
199. C17#9153direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#28552
C2×C17×#3832
C17×#18172
C2×C17×#3282
C17×#12942
#915317