Finite Group Databasebeta

← C17×#10213 #16867957

Subgroups — C17×#10213 #16867957

Order 4352.

498 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3253C211yesmaximal, characteristic
2C17×#3216C212yesmaximal, complemented
3C17×#1995C211yesmaximal, characteristic, complemented
4C17×#1505C212yesmaximal, complemented
5C17×#1307C211yesmaximal, characteristic, complemented
6C17×#533C2211yescharacteristic
7C17×#532C2212yes—
8C17×#497—22no—
9C17×#489—22no—
10C17×#476—22no—
11C17×#469—22no—
12C17×#462—22no—
13C17×#460—22no—
14C17×#459—22no—
15C17×#452—24no—
16C17×#432C2212yes—
17C17×#423C2211yescharacteristic
18C17×#376C2211yescharacteristic
19C17×#339—22no—
20C17×#327—24no—
21C2×#41×C17—22no—
22C2×Q16×C17—24no—
23C2×SD16×C17—24no—
24C2×D8×C17—22no—
25C2×M16×C17D412yes—
26C17×C4⋊Q8D412yes—
27C17×C4⋊Q8D412yes—
28C17×#127D412yes—
29C17×#125—21no—
30C17×#124—22no—
31C17×C22⋊Q8—22no—
32C17×#121—22no—
33C17×#121—22no—
34C4×Q8×C17—22no—
35C4×D4×C17—22no—
36C4×D4×C17—22no—
37C17×#117—21no—
38C2×C42×C17—21no—
39C2×C42×C17C2311yescharacteristic
40C17×#106—22no—
41C17×C4⋊C8—22no—
42C17×C4⋊C8—22no—
43C17×#103—24no—
44C17×#102—24no—
45C17×#102—24no—
46C17×C22⋊C8D414yes—
47C17×#97—22no—
48C17×#95—22no—
49#41×C17—84no—
50C2×Q8×C17—22no—
51C2×Q8×C17—22no—
52C2×D4×C17—22no—
53C2×D4×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17—42no—
56C22×C4×C17—41no—
57C22×C4×C17—42no—
58C22×C4×C17C2×D411yescharacteristic
59C22×C4×C17C2×D412yes—
60Q16×C17—84no—
61SD16×C17—84no—
62D8×C17—82no—
63M16×C17—42no—
64C2×C8×C17—22no—
65C2×C8×C17—24no—
66C4⋊C4×C17—22no—
67C4⋊C4×C17—22no—
68C4⋊C4×C17—22no—
69C4⋊C4×C17—22no—
70C4⋊C4×C17—42no—
71C22⋊C4×C17—41no—
72C22⋊C4×C17—42no—
73C22⋊C4×C17—42no—
74C42×C17—22no—
75C42×C17—41no—
76C42×C17—41no—
77C42×C17C2×D411yescharacteristic
78C42×C17C2×D412yes—
79C42×C17#4111yescharacteristic
80#10213C1711yesmaximal, characteristic, direct product factor, Sylow
81C23×C17—42no—
82C23×C17#12711yescharacteristic
83Q8×C17—42no—
84Q8×C17—42no—
85Q8×C17—44no—
86D4×C17—44no—
87D4×C17—42no—
88D4×C17—42no—
89D4×C17—82no—
90D4×C17—84no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—84no—
98C2×C4×C17—82no—
99C2×C4×C17#12012yes—
100C2×C4×C17#12011yescharacteristic
101C2×C4×C17#12111yescharacteristic
102C2×C4×C17#12112yes—
103C8×C17—44no—
104C8×C17—42no—
105#3253C2×C1711yescharacteristic
106#3216C2×C1712yescomplemented
107#1995C2×C1711yescharacteristic, complemented
108#1505C2×C1712yescomplemented
109#1307C2×C1711yescharacteristic, complemented
110C22×C17—21no—
111C22×C17—22no—
112C22×C17—82no—
113C22×C17—82no—
114C22×C17—82no—
115C22×C17#39211yescharacteristic
116C4×C17—21no—
117C4×C17—22no—
118C4×C17—22no—
119C4×C17—82no—
120C4×C17—82no—
121C4×C17—82no—
122C4×C17#45511yescharacteristic
123C4×C17#45611yescharacteristic
124#533C22×C1711yescharacteristic
125#532C22×C1712yes—
126#497—22no—
127#489—22no—
128#476—22no—
129#469—22no—
130#462—22no—
131#460—22no—
132#459—22no—
133#452—24no—
134#432C22×C1712yes—
135#423C22×C1711yescharacteristic
136#376C22×C1711yescharacteristic
137#339—22no—
138#327—24no—
139C2×C17—41no—
140C2×C17—162no—
141C2×C17#200711yescharacteristic
142C2×C17#200812yes—
143C2×#41—22no—
144C2×Q16—24no—
145C2×SD16—24no—
146C2×D8—22no—
147C2×M16D4×C1712yes—
148C4⋊Q8D4×C1712yes—
149C4⋊Q8D4×C1712yes—
150#127D4×C1712yes—
151#125—21no—
152#124—22no—
153C22⋊Q8—22no—
154#121—22no—
155#121—22no—
156C4×Q8—22no—
157C4×D4—22no—
158C4×D4—22no—
159#117—21no—
160C2×C42—21no—
161C2×C42C23×C1711yescharacteristic
162#106—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165#103—24no—
166#102—24no—
167#102—24no—
168C22⋊C8D4×C1714yes—
169#97—22no—
170#95—22no—
171C17#1021311yescharacteristic, direct product factor, Sylow
172#41—84no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4C2×D4×C1711yescharacteristic
182C22×C4C2×D4×C1712yes—
183Q16—84no—
184SD16—84no—
185D8—82no—
186M16—42no—
187C2×C8—24no—
188C2×C8—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—22no—
198C42—41no—
199C42—41no—
200C42C2×D4×C1711yescharacteristic
201C42C2×D4×C1712yes—
202C42#41×C1711yescharacteristic
203C23—42no—
204C23C17×#12711yescharacteristic
205Q8—42no—
206Q8—42no—
207Q8—44no—
208D4—44no—
209D4—42no—
210D4—42no—
211D4—82no—
212D4—84no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4C17×#12012yes—
222C2×C4C17×#12011yescharacteristic
223C2×C4C17×#12111yescharacteristic
224C2×C4C17×#12112yes—
225C8—44no—
226C8—42no—
227C22—21no—
228C22—22no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22C17×#39211yescharacteristic
233C4—21no—
234C4—22no—
235C4—22no—
236C4—82no—
237C4—82no—
238C4—82no—
239C4C17×#45511yescharacteristic
240C4C17×#45611yescharacteristic
241C2—41no—
242C2—162no—
243C2C17×#200711yescharacteristic
244C2C17×#200812yes—
subgroupG/H
center, socleC22×C17#392
commutator, Frattini, cosocleC2×C42C23×C17
prime radicalC17×#3253C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C2×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#3216C2semidirect
3. C17×#1995C2semidirect
4. C17×#1505C2semidirect
5. C17×#1307C2semidirect
80. #10213C17direct (N × H)
106. #3216C2×C17semidirect
107. #1995C2×C17semidirect
108. #1505C2×C17semidirect
109. #1307C2×C17semidirect
171. C17#10213direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#32532
C17×#32162
C17×#19952
C17×#15052
C17×#13072
#1021317