Finite Group Databasebeta

← C17×#7649 #16865393

Subgroups — C17×#7649 #16865393

Order 4352.

468 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×C17×#124C211yesmaximal, characteristic
2C17×#1839C212yesmaximal, complemented
3C17×#1272C212yesmaximal, complemented
4C17×#1270C212yesmaximal
5C2×C17×#124C411yescharacteristic
6C2×C4×Q8×C17C411yescharacteristic
7C2×C4×D4×C17C411yescharacteristic
8C17×#391C2211yescharacteristic
9C17×#390C2211yescharacteristic
10C17×#388—22no—
11C17×#382C411yescharacteristic
12C4×C22⋊C4×C17—22no—
13C43×C17C2211yescharacteristic
14C17×#336—22no—
15C17×#336—22no—
16C17×#336C2214yes—
17C22×Q8×C17C2×C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×C4×C17—21no—
20C22×C8×C17—24no—
21C17×#124—22no—
22C17×#124—22no—
23C4×Q8×C17—42no—
24C4×D4×C17—42no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17—21no—
30C2×C22⋊C4×C17—22no—
31C2×C42×C17—21no—
32C2×C42×C17C2×C411yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34C2×C42×C17D412yes—
35C2×C42×C17D412yes—
36C2×C42×C17C2311yescharacteristic
37C17×#95—22no—
38C17×#95—22no—
39C24×C17—21no—
40C2×Q8×C17—21no—
41C2×Q8×C17—41no—
42C2×Q8×C17—41no—
43C2×Q8×C17M1612yes—
44C2×D4×C17—21no—
45C2×D4×C17—41no—
46C2×D4×C17—41no—
47C2×D4×C17M1612yes—
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17—44no—
52C22×C4×C17—42no—
53C22×C4×C17C22⋊C412yes—
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22×C411yescharacteristic
57C22×C4×C17C2×D412yes—
58C2×C8×C17—44no—
59C2×C8×C17—44no—
60C2×C8×C17—44no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—42no—
64C22⋊C4×C17—28no—
65C22⋊C4×C17—42no—
66C22⋊C4×C17—44no—
67C42×C17—22no—
68C42×C17—21no—
69C42×C17—21no—
70C42×C17—24no—
71C42×C17—42no—
72C42×C17—41no—
73C42×C17—42no—
74#7649C1711yesmaximal, characteristic, direct product factor, Sylow
75C23×C17—22no—
76C23×C17—41no—
77C23×C17—41no—
78C23×C17—42no—
79C23×C17—42no—
80C23×C17C2×C22⋊C411yescharacteristic
81Q8×C17—42no—
82Q8×C17—42no—
83D4×C17—42no—
84D4×C17—42no—
85C2×C4×C17—22no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—41no—
97C2×C4×C17—41no—
98C2×C4×C17—41no—
99C2×C4×C17—41no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—84no—
103C2×C4×C17#10012yes—
104C2×C4×C17C2×M1611yescharacteristic
105C2×C4×C17C2×M1611yescharacteristic
106C2×C4×C17#13514yes—
107C8×C17—84no—
108C4×#124C2×C1711yescharacteristic
109#1839C2×C1712yescomplemented
110#1272C2×C1712yescomplemented
111#1270C2×C1712yes—
112C22×C17—42no—
113C22×C17—44no—
114C22×C17—81no—
115C22×C17—82no—
116C22×C17—81no—
117C22×C17#40711yescharacteristic
118C22×C17#41011yescharacteristic
119C22×C17C2×#10011yescharacteristic
120C22×C17#41612yes—
121C22×C17#42112yes—
122C4×C17—21no—
123C4×C17—21no—
124C4×C17—24no—
125C4×C17—22no—
126C4×C17—41no—
127C4×C17—42no—
128C4×C17—42no—
129C4×C17—81no—
130C4×C17—81no—
131C2×#124C4×C1711yescharacteristic
132C2×C4×Q8C4×C1711yescharacteristic
133C2×C4×D4C4×C1711yescharacteristic
134#391C22×C1711yescharacteristic
135#390C22×C1711yescharacteristic
136#388—22no—
137#382C4×C1711yescharacteristic
138C4×C22⋊C4—22no—
139C43C22×C1711yescharacteristic
140#336—22no—
141#336—22no—
142#336C22×C1714yes—
143C2×C17—82no—
144C2×C17#146611yescharacteristic
145C2×C17#148812yes—
146C2×C17#152112yes—
147C2×C17#153311yescharacteristic
148C2×C17#154311yescharacteristic
149C22×Q8C2×C4×C1711yescharacteristic
150C22×D4C2×C4×C1711yescharacteristic
151C23×C4—21no—
152C22×C8—24no—
153#124—22no—
154#124—22no—
155C4×Q8—42no—
156C4×D4—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4C2×C4×C1711yescharacteristic
159C2×C4⋊C4C2×C4×C1711yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C42—21no—
164C2×C42C2×C4×C1711yescharacteristic
165C2×C42C2×C4×C1711yescharacteristic
166C2×C42D4×C1712yes—
167C2×C42D4×C1712yes—
168C2×C42C23×C1711yescharacteristic
169#95—22no—
170#95—22no—
171C17#764911yescharacteristic, direct product factor, Sylow
172C24—21no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8M16×C1712yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4M16×C1712yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4C22⋊C4×C1712yes—
187C22×C4C22⋊C4×C1711yescharacteristic
188C22×C4C22⋊C4×C1711yescharacteristic
189C22×C4C22×C4×C1711yescharacteristic
190C22×C4C2×D4×C1712yes—
191C2×C8—44no—
192C2×C8—44no—
193C2×C8—44no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C22⋊C4—28no—
198C22⋊C4—42no—
199C22⋊C4—44no—
200C42—21no—
201C42—21no—
202C42—24no—
203C42—22no—
204C42—42no—
205C42—41no—
206C42—42no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23C2×C22⋊C4×C1711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4C17×#10012yes—
236C2×C4C2×M16×C1711yescharacteristic
237C2×C4C2×M16×C1711yescharacteristic
238C2×C4C17×#13514yes—
239C8—84no—
240C22—42no—
241C22—44no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22C17×#40711yescharacteristic
246C22C17×#41011yescharacteristic
247C22C2×C17×#10011yescharacteristic
248C22C17×#41612yes—
249C22C17×#42112yes—
250C4—21no—
251C4—21no—
252C4—24no—
253C4—22no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—81no—
258C4—81no—
259C2—82no—
260C2C17×#146611yescharacteristic
261C2C17×#148812yes—
262C2C17×#152112yes—
263C2C17×#153311yescharacteristic
264C2C17×#154311yescharacteristic
subgroupG/H
center, socleC23×C17C2×C22⋊C4
commutatorC22×C4C22×C4×C17
Frattini, cosocleC2×C42C23×C17
prime radicalC22×D4×C17C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C17 ◁ C43×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#1839C2semidirect
3. C17×#1272C2semidirect
74. #7649C17direct (N × H)
109. #1839C2×C17semidirect
110. #1272C2×C17semidirect
171. C17#7649direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C17×#1242
C17×#18392
C17×#12722
C17×#12702
#764917