Finite Group Databasebeta

← C17×#7760 #16865504

Subgroups — C17×#7760 #16865504

Order 4352.

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

normal subgroups only

This group has 424 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×#2809C211yesmaximal, characteristic
2C17×#2058C212yesmaximal
3C17×#2016C211yesmaximal, characteristic
4C17×#1272C211yesmaximal, characteristic
5C17×#1271C212yesmaximal
6C22×Q16×C17—21no—
7C2×C17×C4⋊Q8C2211yescharacteristic
8C2×C17×#106—22no—
9C2×C17×#103—21no—
10C17×#400—21no—
11C17×#398—22no—
12C17×#391—21no—
13C17×#391C2211yescharacteristic
14C17×#389C2212yes—
15C17×#387—22no—
16C17×#340—22no—
17C17×#336C2211yescharacteristic
18C17×#336C2212yes—
19C22×Q8×C17—21no—
20C22×Q8×C17D411yescharacteristic
21C2×Q16×C17—42no—
22C2×Q16×C17—42no—
23C2×Q16×C17—42no—
24C22×C8×C17—21no—
25C22×C8×C17—22no—
26C17×C4⋊Q8—42no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D412yes—
32C2×C4⋊C4×C17D412yes—
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C42×C17—21no—
35C2×C42×C17C2311yescharacteristic
36C17×#106—44no—
37C17×#103—42no—
38C17×#95—21no—
39C17×#95—22no—
40C17×#95—22no—
41C17×#95—22no—
42C17×#95—22no—
43C2×Q8×C17—21no—
44C2×Q8×C17—22no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17—41no—
49C2×Q8×C17—41no—
50C2×Q8×C17—41no—
51C2×Q8×C17—41no—
52C2×Q8×C17SD1612yes—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—41no—
58C22×C4×C17C2×D411yescharacteristic
59C22×C4×C17C2×D412yes—
60Q16×C17—48no—
61C2×C8×C17—24no—
62C2×C8×C17—22no—
63C2×C8×C17—41no—
64C2×C8×C17—42no—
65C2×C8×C17—41no—
66C2×C8×C17—42no—
67C4⋊C4×C17—24no—
68C4⋊C4×C17—22no—
69C4⋊C4×C17—22no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—42no—
73C4⋊C4×C17—42no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—81no—
76C4⋊C4×C17—81no—
77C42×C17—41no—
78C42×C17—81no—
79#7760C1711yesmaximal, characteristic, direct product factor, Sylow
80C23×C17#12011yescharacteristic
81Q8×C17—42no—
82Q8×C17—42no—
83Q8×C17—42no—
84Q8×C17—81no—
85Q8×C17—81no—
86Q8×C17—81no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—41no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—42no—
103C2×C4×C17—81no—
104C2×C4×C17—81no—
105C2×C4×C17C2×SD1611yescharacteristic
106C2×C4×C17#13512yes—
107C2×C4×C17#13711yescharacteristic
108C2×C4×C17#13712yes—
109C8×C17—44no—
110C8×C17—42no—
111#2809C2×C1711yescharacteristic
112#2058C2×C1712yes—
113#2016C2×C1711yescharacteristic
114#1272C2×C1711yescharacteristic
115#1271C2×C1712yes—
116C22×C17#44811yescharacteristic
117C22×C17#45212yes—
118C22×C17#45311yescharacteristic
119C22×C17#45612yes—
120C22×C17B(4,𝔽2)11yescharacteristic
121C4×C17—21no—
122C4×C17—21no—
123C4×C17—22no—
124C4×C17—22no—
125C4×C17—42no—
126C4×C17—82no—
127C4×C17—82no—
128C4×C17—82no—
129C4×C17—81no—
130C22×Q16—21no—
131C2×C4⋊Q8C22×C1711yescharacteristic
132C2×#106—22no—
133C2×#103—21no—
134#400—21no—
135#398—22no—
136#391—21no—
137#391C22×C1711yescharacteristic
138#389C22×C1712yes—
139#387—22no—
140#340—22no—
141#336C22×C1712yes—
142#336C22×C1711yescharacteristic
143C2×C17#159611yescharacteristic
144C2×C17#160312yes—
145C2×C17#163411yescharacteristic
146C2×C17#165312yes—
147C2×C17#166011yescharacteristic
148C22×Q8—21no—
149C22×Q8D4×C1711yescharacteristic
150C2×Q16—42no—
151C2×Q16—42no—
152C2×Q16—42no—
153C22×C8—21no—
154C22×C8—22no—
155C4⋊Q8—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4D4×C1712yes—
161C2×C4⋊C4D4×C1712yes—
162C2×C4⋊C4D4×C1711yescharacteristic
163C2×C42—21no—
164C2×C42C23×C1711yescharacteristic
165#106—44no—
166#103—42no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172C17#776011yescharacteristic, direct product factor, Sylow
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8SD16×C1712yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4C2×D4×C1711yescharacteristic
189C22×C4C2×D4×C1712yes—
190Q16—48no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C42—41no—
208C42—81no—
209C23C17×#12011yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—81no—
214Q8—81no—
215Q8—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4C2×SD16×C1711yescharacteristic
235C2×C4C17×#13512yes—
236C2×C4C17×#13711yescharacteristic
237C2×C4C17×#13712yes—
238C8—42no—
239C8—44no—
240C22C17×#44811yescharacteristic
241C22C17×#45212yes—
242C22C17×#45311yescharacteristic
243C22C17×#45612yes—
244C22C17×B(4,𝔽2)11yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—82no—
251C4—82no—
252C4—82no—
253C4—81no—
254C2C17×#159611yescharacteristic
255C2C17×#160312yes—
256C2C17×#163411yescharacteristic
257C2C17×#165312yes—
258C2C17×#166011yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C17#120
commutator, Frattini, cosocleC2×C42C23×C17
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C17 ◁ C2×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
79. #7760C17direct (N × H)
172. C17#7760direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#28092
C17×#20582
C17×#20162
C17×#12722
C17×#12712
#776017