Finite Group Databasebeta

← C17×#9979 #16867723

Subgroups — C17×#9979 #16867723

Order 4352.

492 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3189C211yesmaximal, characteristic
2C17×#3095C212yesmaximal
3C17×#1818C211yesmaximal, characteristic
4C17×#1472C212yesmaximal
5C17×#1293C211yesmaximal, characteristic
6C2×C17×C22⋊Q8C2212yes—
7C2×C17×#117C2211yescharacteristic
8C17×#485—24no—
9C17×#484—22no—
10C17×#484—22no—
11C17×#483—22no—
12C17×#467—22no—
13C17×#451—22no—
14C17×#428—22no—
15C17×#419—22no—
16C17×#407C2212yes—
17C2×C17×C22⋊C8C2211yescharacteristic
18C2×C17×C22⋊C8C2211yescharacteristic
19C17×#336—22no—
20C17×#324—24no—
21C22×Q8×C17D412yes—
22C23×C4×C17C2311yescharacteristic
23C2×Q16×C17—22no—
24C2×M16×C17—22no—
25C22×C8×C17—21no—
26C22×C8×C17D412yes—
27C17×C22⋊Q8—22no—
28C17×C22⋊Q8—22no—
29C17×C22⋊Q8—22no—
30C17×C22⋊Q8—22no—
31C17×#117—22no—
32C17×#117—22no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17D412yes—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—22no—
39C2×C42×C17—21no—
40C17×#107—24no—
41C17×#107—22no—
42C17×#106—24no—
43C17×#103—24no—
44C17×#103—24no—
45C17×#103—24no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—22no—
48C17×C22⋊C8—22no—
49C17×C22⋊C8D414yes—
50C24×C17#4111yescharacteristic
51C2×Q8×C17—22no—
52C2×Q8×C17—22no—
53C2×Q8×C17—44no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17—42no—
59C22×C4×C17C2×D412yes—
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D412yes—
63Q16×C17—82no—
64M16×C17—44no—
65C2×C8×C17—22no—
66C2×C8×C17—24no—
67C2×C8×C17—42no—
68C2×C8×C17—42no—
69C2×C8×C17—41no—
70C4⋊C4×C17—22no—
71C4⋊C4×C17—22no—
72C4⋊C4×C17—22no—
73C4⋊C4×C17—22no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—42no—
76C22⋊C4×C17—42no—
77C22⋊C4×C17—42no—
78C22⋊C4×C17—42no—
79C42×C17—41no—
80C42×C17—41no—
81#9979C1711yesmaximal, characteristic, direct product factor, Sylow
82C23×C17—22no—
83C23×C17—41no—
84C23×C17—41no—
85C23×C17#12112yes—
86C23×C17#12111yescharacteristic
87Q8×C17—44no—
88Q8×C17—44no—
89C2×C4×C17—22no—
90C2×C4×C17—24no—
91C2×C4×C17—41no—
92C2×C4×C17—42no—
93C2×C4×C17—42no—
94C2×C4×C17—44no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—82no—
101C2×C4×C17#12012yes—
102C2×C4×C17#12011yescharacteristic
103C2×C4×C17#12711yescharacteristic
104C8×C17—44no—
105C8×C17—42no—
106C8×C17—81no—
107#3189C2×C1711yescharacteristic
108#3095C2×C1712yes—
109#1818C2×C1711yescharacteristic
110#1472C2×C1712yes—
111#1293C2×C1711yescharacteristic
112C22×C17—21no—
113C22×C17—21no—
114C22×C17—22no—
115C22×C17—22no—
116C22×C17—22no—
117C22×C17—82no—
118C22×C17#39211yescharacteristic
119C22×C17#47312yes—
120C4×C17—22no—
121C4×C17—21no—
122C4×C17—81no—
123C4×C17—82no—
124C4×C17—81no—
125C4×C17—82no—
126C4×C17#45812yes—
127C2×C22⋊Q8C22×C1712yes—
128C2×#117C22×C1711yescharacteristic
129#485—24no—
130#484—22no—
131#484—22no—
132#483—22no—
133#467—22no—
134#451—22no—
135#428—22no—
136#419—22no—
137#407C22×C1712yes—
138C2×C22⋊C8C22×C1711yescharacteristic
139C2×C22⋊C8C22×C1711yescharacteristic
140#336—22no—
141#324—24no—
142C2×C17—22no—
143C2×C17—42no—
144C2×C17#201711yescharacteristic
145C2×C17#202011yescharacteristic
146C2×C17#202211yescharacteristic
147C22×Q8D4×C1712yes—
148C23×C4C23×C1711yescharacteristic
149C2×Q16—22no—
150C2×M16—22no—
151C22×C8—21no—
152C22×C8D4×C1712yes—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157#117—22no—
158#117—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4D4×C1711yescharacteristic
161C2×C4⋊C4D4×C1711yescharacteristic
162C2×C4⋊C4D4×C1712yes—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C42—21no—
166#107—24no—
167#107—22no—
168#106—24no—
169#103—24no—
170#103—24no—
171#103—24no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8D4×C1714yes—
176C17#997911yescharacteristic, direct product factor, Sylow
177C24#41×C1711yescharacteristic
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—44no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4C2×D4×C1711yescharacteristic
187C22×C4C2×D4×C1712yes—
188C22×C4C2×D4×C1712yes—
189C22×C4C2×D4×C1711yescharacteristic
190Q16—82no—
191M16—44no—
192C2×C8—22no—
193C2×C8—24no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—41no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C42—41no—
207C42—41no—
208C23—22no—
209C23—41no—
210C23—41no—
211C23C17×#12112yes—
212C23C17×#12111yescharacteristic
213Q8—44no—
214Q8—44no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4C17×#12012yes—
228C2×C4C17×#12011yescharacteristic
229C2×C4C17×#12711yescharacteristic
230C8—44no—
231C8—42no—
232C8—81no—
233C22—21no—
234C22—21no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—82no—
239C22C17×#39211yescharacteristic
240C22C17×#47312yes—
241C4—22no—
242C4—21no—
243C4—81no—
244C4—82no—
245C4—81no—
246C4—82no—
247C4C17×#45812yes—
248C2—22no—
249C2—42no—
250C2C17×#201711yescharacteristic
251C2C17×#202011yescharacteristic
252C2C17×#202211yescharacteristic
subgroupG/H
center, socleC22×C17#392
commutator, Frattini, cosocleC23×C4C23×C17
prime radicalC24×C17#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
81. #9979C17direct (N × H)
176. C17#9979direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#31892
C17×#30952
C17×#18182
C17×#14722
C17×#12932
#997917