Finite Group Databasebeta

← C17×#12247 #16869991

Subgroups — C17×#12247 #16869991

Order 4352.

394 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2582C211yesmaximal, characteristic
2C17×#2060C211yesmaximal, characteristic
3C17×#1611C212yesmaximal
4C17×#1346C212yesmaximal
5C17×#1325C211yesmaximal, characteristic
6C17×#544C2211yescharacteristic
7C2×C4×Q8×C17—21no—
8C17×#483—22no—
9C17×#428—21no—
10C17×#398—21no—
11C17×#398C2212yes—
12C17×#391—22no—
13C17×#389—22no—
14C17×#384C2211yescharacteristic
15C17×#343C2211yescharacteristic
16C17×#339—21no—
17C17×#328—22no—
18C17×#324C2212yes—
19C22×Q8×C17—21no—
20C2×M16×C17—21no—
21C17×C4⋊Q8—41no—
22C17×#125—41no—
23C17×C22⋊Q8—22no—
24C4×Q8×C17—42no—
25C4×Q8×C17—42no—
26C17×#117—21no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17D412yes—
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C42×C17—22no—
35C2×C42×C17D411yescharacteristic
36C17×#107—41no—
37C17×#106—41no—
38C17×#106—42no—
39C17×#103—42no—
40C17×C22⋊C8—22no—
41C17×#95—22no—
42C17×#95—21no—
43C17×#95—22no—
44C17×#95D412yes—
45C2×Q8×C17—24no—
46C2×Q8×C17—22no—
47C2×Q8×C17—42no—
48C2×Q8×C17—42no—
49C22×C4×C17—21no—
50C22×C4×C17—22no—
51C22×C4×C17—21no—
52C22×C4×C17—22no—
53C22×C4×C17C2×D411yescharacteristic
54C22×C4×C17C2×D412yes—
55M16×C17—22no—
56M16×C17—41no—
57C2×C8×C17—41no—
58C2×C8×C17—42no—
59C4⋊C4×C17—22no—
60C4⋊C4×C17—21no—
61C4⋊C4×C17—21no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—42no—
64C4⋊C4×C17—41no—
65C4⋊C4×C17—42no—
66C4⋊C4×C17—44no—
67C4⋊C4×C17—42no—
68C4⋊C4×C17—42no—
69C4⋊C4×C17—42no—
70C4⋊C4×C17—81no—
71C22⋊C4×C17—41no—
72C42×C17—21no—
73C42×C17—41no—
74C42×C17—44no—
75C42×C17SD1612yes—
76#12247C1711yesmaximal, characteristic, direct product factor, Sylow
77C23×C17#12011yescharacteristic
78Q8×C17—48no—
79Q8×C17—82no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—24no—
86C2×C4×C17—22no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17—42no—
90C2×C4×C17—42no—
91C2×C4×C17—42no—
92C2×C4×C17—42no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17—42no—
96C2×C4×C17—41no—
97C2×C4×C17C2×SD1611yescharacteristic
98C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
99C8×C17—42no—
100C8×C17—82no—
101#2582C2×C1711yescharacteristic
102#2060C2×C1711yescharacteristic
103#1611C2×C1712yes—
104#1346C2×C1712yes—
105#1325C2×C1711yescharacteristic
106C22×C17—21no—
107C22×C17—21no—
108C22×C17#45011yescharacteristic
109C22×C17#45611yescharacteristic
110C22×C17B(4,𝔽2)11yescharacteristic
111C4×C17—24no—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—44no—
115C4×C17—42no—
116C4×C17—82no—
117C4×C17—81no—
118C4×C17—81no—
119C4×C17—81no—
120#544C22×C1711yescharacteristic
121C2×C4×Q8—21no—
122#483—22no—
123#428—21no—
124#398—21no—
125#398C22×C1712yes—
126#391—22no—
127#389—22no—
128#384C22×C1711yescharacteristic
129#343C22×C1711yescharacteristic
130#339—21no—
131#328—22no—
132#324C22×C1712yes—
133C2×C17—21no—
134C2×C17—21no—
135C2×C17#160411yescharacteristic
136C2×C17#219811yescharacteristic
137C2×C17#220511yescharacteristic
138C22×Q8—21no—
139C2×M16—21no—
140C4⋊Q8—41no—
141#125—41no—
142C22⋊Q8—22no—
143C4×Q8—42no—
144C4×Q8—42no—
145#117—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4D4×C1711yescharacteristic
151C2×C4⋊C4D4×C1712yes—
152C2×C4⋊C4C23×C1711yescharacteristic
153C2×C42—22no—
154C2×C42D4×C1711yescharacteristic
155#107—41no—
156#106—41no—
157#106—42no—
158#103—42no—
159C22⋊C8—22no—
160#95—22no—
161#95—21no—
162#95—22no—
163#95D4×C1712yes—
164C17#1224711yescharacteristic, direct product factor, Sylow
165C2×Q8—24no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4C2×D4×C1711yescharacteristic
174C22×C4C2×D4×C1712yes—
175M16—22no—
176M16—41no—
177C2×C8—41no—
178C2×C8—42no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—41no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C22⋊C4—41no—
192C42—21no—
193C42—44no—
194C42—41no—
195C42SD16×C1712yes—
196C23C17×#12011yescharacteristic
197Q8—48no—
198Q8—82no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4C2×SD16×C1711yescharacteristic
217C2×C4C17×AGL(1,ℤ/8)11yescharacteristic
218C8—42no—
219C8—82no—
220C22—21no—
221C22—21no—
222C22C17×#45011yescharacteristic
223C22C17×#45611yescharacteristic
224C22C17×B(4,𝔽2)11yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—42no—
230C4—81no—
231C4—81no—
232C4—81no—
233C4—82no—
234C2—21no—
235C2—21no—
236C2C17×#160411yescharacteristic
237C2C17×#219811yescharacteristic
238C2C17×#220511yescharacteristic
subgroupG/H
center, socleC22×C17B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C17
prime radicalC23×C17#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C17 ◁ C2×C4⋊C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
76. #12247C17direct (N × H)
164. C17#12247direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#25822
C17×#20602
C17×#16112
C17×#13462
C17×#13252
#1224717