Finite Group Databasebeta

← C17×#10216 #16867960

Subgroups — C17×#10216 #16867960

Order 4352.

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

normal subgroups only

This group has 468 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×#3255C211yesmaximal, characteristic, complemented
2C17×#3235C212yesmaximal
3C17×#1963C211yesmaximal, characteristic
4C17×#1506C212yesmaximal, complemented
5C17×#1307C211yesmaximal, characteristic, complemented
6C17×#533C2212yes—
7C17×#532C2211yescharacteristic
8C17×#501—22no—
9C17×#490—22no—
10C17×#482—24no—
11C17×#476—22no—
12C17×#475—22no—
13C17×#474—22no—
14C17×#470—22no—
15C17×#462—22no—
16C17×#432C2212yes—
17C17×#423C2211yescharacteristic
18C2×C17×#104—22no—
19C17×#376C2211yescharacteristic
20C17×#328—24no—
21C2×#41×C17—21no—
22C2×Q16×C17—22no—
23C2×M16×C17D412yes—
24C17×C4⋊Q8D411yescharacteristic
25C17×C4⋊Q8D412yes—
26C17×C4⋊Q8D412yes—
27C17×#127D411yescharacteristic
28C17×#125—22no—
29C17×#124—21no—
30C17×C22⋊Q8—22no—
31C17×C22⋊Q8—22no—
32C17×#121—22no—
33C4×Q8×C17—22no—
34C4×Q8×C17—22no—
35C4×D4×C17—22no—
36C17×#117—22no—
37C2×C42×C17—21no—
38C2×C42×C17C2311yescharacteristic
39C17×#107—24no—
40C17×#106—24no—
41C17×#106—22no—
42C17×C4⋊C8—22no—
43C17×C4⋊C8—22no—
44C17×#104—44no—
45C17×#103—24no—
46C17×#102—24no—
47C17×#102—24no—
48C17×C22⋊C8D414yes—
49C17×#97—22no—
50C17×#95—22no—
51#41×C17—42no—
52#41×C17—42no—
53C2×Q8×C17—22no—
54C2×Q8×C17—21no—
55C2×Q8×C17—22no—
56C2×D4×C17—22no—
57C2×D4×C17—21no—
58C2×D4×C17—41no—
59C22×C4×C17—41no—
60C22×C4×C17—41no—
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D412yes—
63Q16×C17—82no—
64M16×C17—42no—
65C2×C8×C17—22no—
66C2×C8×C17—24no—
67C4⋊C4×C17—22no—
68C4⋊C4×C17—22no—
69C4⋊C4×C17—22no—
70C4⋊C4×C17—22no—
71C4⋊C4×C17—22no—
72C4⋊C4×C17—42no—
73C4⋊C4×C17—42no—
74C22⋊C4×C17—42no—
75C22⋊C4×C17—42no—
76C42×C17—22no—
77C42×C17—41no—
78C42×C17—42no—
79C42×C17C2×D411yescharacteristic
80C42×C17C2×D412yes—
81C42×C17#4111yescharacteristic
82#10216C1711yesmaximal, characteristic, direct product factor, Sylow
83C23×C17—41no—
84C23×C17#12711yescharacteristic
85Q8×C17—42no—
86Q8×C17—42no—
87Q8×C17—42no—
88Q8×C17—82no—
89D4×C17—44no—
90D4×C17—42no—
91D4×C17—82no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—41no—
99C2×C4×C17—82no—
100C2×C4×C17—81no—
101C2×C4×C17—81no—
102C2×C4×C17#12012yes—
103C2×C4×C17#12011yescharacteristic
104C2×C4×C17#12112yes—
105C2×C4×C17#12111yescharacteristic
106C8×C17—44no—
107C8×C17—42no—
108#3255C2×C1711yescharacteristic, complemented
109#3235C2×C1712yes—
110#1963C2×C1711yescharacteristic
111#1506C2×C1712yescomplemented
112#1307C2×C1711yescharacteristic, complemented
113C22×C17—21no—
114C22×C17—22no—
115C22×C17—81no—
116C22×C17—82no—
117C22×C17#39211yescharacteristic
118C4×C17—21no—
119C4×C17—22no—
120C4×C17—22no—
121C4×C17—82no—
122C4×C17—81no—
123C4×C17—82no—
124C4×C17—82no—
125C4×C17#45311yescharacteristic
126C4×C17#45411yescharacteristic
127#533C22×C1712yes—
128#532C22×C1711yescharacteristic
129#501—22no—
130#490—22no—
131#482—24no—
132#476—22no—
133#475—22no—
134#474—22no—
135#470—22no—
136#462—22no—
137#432C22×C1712yes—
138#423C22×C1711yescharacteristic
139C2×#104—22no—
140#376C22×C1711yescharacteristic
141#328—24no—
142C2×C17—41no—
143C2×C17—161no—
144C2×C17#200611yescharacteristic
145C2×C17#200912yes—
146C2×#41—21no—
147C2×Q16—22no—
148C2×M16D4×C1712yes—
149C4⋊Q8D4×C1711yescharacteristic
150C4⋊Q8D4×C1712yes—
151C4⋊Q8D4×C1712yes—
152#127D4×C1711yescharacteristic
153#125—22no—
154#124—21no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157#121—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×D4—22no—
161#117—22no—
162C2×C42—21no—
163C2×C42C23×C1711yescharacteristic
164#107—24no—
165#106—22no—
166#106—24no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169#104—44no—
170#103—24no—
171#102—24no—
172#102—24no—
173C22⋊C8D4×C1714yes—
174#97—22no—
175#95—22no—
176C17#1021611yescharacteristic, direct product factor, Sylow
177#41—42no—
178#41—42no—
179C2×Q8—22no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C2×D4×C1711yescharacteristic
188C22×C4C2×D4×C1712yes—
189Q16—82no—
190M16—42no—
191C2×C8—24no—
192C2×C8—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—22no—
203C42—41no—
204C42—42no—
205C42C2×D4×C1711yescharacteristic
206C42C2×D4×C1712yes—
207C42#41×C1711yescharacteristic
208C23—41no—
209C23C17×#12711yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—82no—
214D4—42no—
215D4—44no—
216D4—82no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4C17×#12012yes—
228C2×C4C17×#12011yescharacteristic
229C2×C4C17×#12112yes—
230C2×C4C17×#12111yescharacteristic
231C8—44no—
232C8—42no—
233C22—21no—
234C22—22no—
235C22—81no—
236C22—82no—
237C22C17×#39211yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—81no—
243C4—82no—
244C4—82no—
245C4C17×#45311yescharacteristic
246C4C17×#45411yescharacteristic
247C2—41no—
248C2—161no—
249C2C17×#200611yescharacteristic
250C2C17×#200912yes—
subgroupG/H
center, socleC22×C17#392
commutator, Frattini, cosocleC2×C42C23×C17
prime radicalC17×#532C22
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
1. C17×#3255C2semidirect
4. C17×#1506C2semidirect
5. C17×#1307C2semidirect
82. #10216C17direct (N × H)
108. #3255C2×C17semidirect
111. #1506C2×C17semidirect
112. #1307C2×C17semidirect
176. C17#10216direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#32552
C17×#32352
C17×#19632
C17×#15062
C17×#13072
#1021617