Finite Group Databasebeta

← C2×C17×#2984 #16917988

Subgroups — C2×C17×#2984 #16917988

Order 4352.

2022 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q8×C17C211yesmaximal, characteristic
2C17×#2984C2116yesmaximal, direct product factor
3C2×C17×#446C216yesmaximal, complemented
4C2×C8×Q8×C17C212yesmaximal, complemented
5C2×C17×#432C213yesmaximal
6C2×C17×#431C213yesmaximal
7C23×Q8×C17C411yescharacteristic
8C2×C4×Q8×C17C418yes—
9C2×C4×Q8×C17C413yes—
10C2×C4×Q8×C17C411yescharacteristic
11C2×C4×Q8×C17C2211yescharacteristic
12C2×C4×Q8×C17C2213yes—
13C2×C4×Q8×C17C2218yes—
14C22×C4⋊C4×C17C413yes—
15C22×C42×C17C2213yes—
16C17×#446C22148yescomplemented
17C8×Q8×C17C22116yescomplemented
18C17×#432C22124yes—
19C17×#431C22124yes—
20C2×C17×C4⋊C8C2216yes—
21C2×C17×C4⋊C8C2216yes—
22C2×C17×C22⋊C8C2211yescharacteristic
23C2×C17×C22⋊C8C2213yes—
24C2×C17×#97C2216yes—
25C2×C4×C8×C17C2216yes—
26C22×Q8×C17—28no—
27C22×Q8×C17C2×C418yes—
28C22×Q8×C17C2×C413yes—
29C22×Q8×C17C2×C411yescharacteristic
30C23×C4×C17C2×C413yes—
31C23×C4×C17C2×C413yes—
32C23×C4×C17C2311yescharacteristic
33C22×C8×C17C2312yes—
34C22×C8×C17C2316yes—
35C4×Q8×C17C2×C4112yes—
36C4×Q8×C17C2×C4124yes—
37C4×Q8×C17C2×C414yes—
38C4×Q8×C17C2×C418yes—
39C4×Q8×C17C2314yes—
40C4×Q8×C17C23112yes—
41C2×C4⋊C4×C17C2×C416yes—
42C2×C4⋊C4×C17C2×C4124yes—
43C2×C4⋊C4×C17C2×C413yes—
44C2×C4⋊C4×C17C2×C413yes—
45C2×C42×C17C2×C413yes—
46C2×C42×C17C2×C4112yes—
47C2×C42×C17C2×C413yes—
48C2×C42×C17C23112yes—
49C2×C42×C17C2313yes—
50C2×C42×C17C2313yes—
51C17×C4⋊C8C23124yes—
52C17×C4⋊C8C23124yes—
53C17×C22⋊C8C2314yes—
54C17×C22⋊C8C23112yes—
55C17×#97C23124yes—
56C4×C8×C17C23124yes—
57C24×C17C22×C411yescharacteristic
58C2×Q8×C17—232no—
59C2×Q8×C17—212no—
60C2×Q8×C17M1618yes—
61C2×Q8×C17C22×C4112yes—
62C2×Q8×C17C22×C414yes—
63C22×C4×C17—212no—
64C22×C4×C17—212no—
65C22×C4×C17—24no—
66C22×C4×C17C22×C413yes—
67C22×C4×C17C22×C4112yes—
68C22×C4×C17C22×C4112yes—
69C22×C4×C17C22×C413yes—
70C22×C4×C17C22×C413yes—
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C22×C412yes—
73C22×C4×C17C22×C413yes—
74C22×C4×C17C2412yes—
75C22×C4×C17C2411yescharacteristic
76C2×C8×C17—24no—
77C2×C8×C17—212no—
78C2×C8×C17C2414yes—
79C2×C8×C17C24112yes—
80C4⋊C4×C17C22×C4124yes—
81C4⋊C4×C17C22×C4112yes—
82C4⋊C4×C17C22×C4112yes—
83C42×C17C22×C4112yes—
84C42×C17C22×C4112yes—
85C42×C17C22×C416yes—
86C42×C17C22×C416yes—
87C42×C17C2416yes—
88C42×C17C2416yes—
89C2×#2984C1711yesmaximal, characteristic, direct product factor, Sylow
90C23×C17—22no—
91C23×C17—22no—
92C23×C17#13114yes—
93C23×C17C23×C412yes—
94C23×C17C23×C411yescharacteristic
95Q8×C17—224no—
96Q8×C17C2×M16116yes—
97C2×C4×C17—22no—
98C2×C4×C17—26no—
99C2×C4×C17—224no—
100C2×C4×C17—26no—
101C2×C4×C17—26no—
102C2×C4×C17—24no—
103C2×C4×C17—224no—
104C2×C4×C17—24no—
105C2×C4×C17C2×M16112yes—
106C2×C4×C17C23×C412yes—
107C2×C4×C17C23×C416yes—
108C2×C4×C17C23×C416yes—
109C2×C4×C17C23×C416yes—
110C2×C4×C17C23×C411yescharacteristic
111C2×C4×C17C23×C416yes—
112C2×C4×C172−1+414yes—
113C2×C4×C17C2511yescharacteristic
114C8×C17—24no—
115C8×C17—212no—
116C22×C4×Q8C2×C1711yescharacteristic
117#2984C2×C17116yesdirect product factor
118C2×#446C2×C1716yescomplemented
119C2×C8×Q8C2×C1712yescomplemented
120C2×#432C2×C1713yes—
121C2×#431C2×C1713yes—
122C22×C17—22no—
123C22×C17—28no—
124C22×C17—22no—
125C22×C17#51912yes—
126C22×C17C22×M1612yes—
127C22×C17C2×#13114yes—
128C22×C17C2×#13112yes—
129C22×C17C24×C411yescharacteristic
130C4×C17—22no—
131C4×C17—26no—
132C4×C17—26no—
133C4×C17—26no—
134C4×C17#51912yes—
135C4×C17C22×M16112yes—
136C4×C17C2×2−1+412yes—
137C23×Q8C4×C1711yescharacteristic
138C2×C4×Q8C4×C1718yes—
139C2×C4×Q8C4×C1713yes—
140C2×C4×Q8C4×C1711yescharacteristic
141C2×C4×Q8C22×C1711yescharacteristic
142C2×C4×Q8C22×C1713yes—
143C2×C4×Q8C22×C1718yes—
144C22×C4⋊C4C4×C1713yes—
145C22×C42C22×C1713yes—
146#446C22×C17148yescomplemented
147C8×Q8C22×C17116yescomplemented
148#432C22×C17124yes—
149#431C22×C17124yes—
150C2×C4⋊C8C22×C1716yes—
151C2×C4⋊C8C22×C1716yes—
152C2×C22⋊C8C22×C1711yescharacteristic
153C2×C22⋊C8C22×C1713yes—
154C2×#97C22×C1716yes—
155C2×C4×C8C22×C1716yes—
156C2×C17—24no—
157C2×C17#298414yesdirect product factor
158C2×C17C2×#51911yescharacteristic
159C2×C17C23×M1611yescharacteristic
160C2×C17C22×#13111yescharacteristic
161C22×Q8—28no—
162C22×Q8C2×C4×C1718yes—
163C22×Q8C2×C4×C1713yes—
164C22×Q8C2×C4×C1711yescharacteristic
165C23×C4C2×C4×C1713yes—
166C23×C4C2×C4×C1713yes—
167C23×C4C23×C1711yescharacteristic
168C22×C8C23×C1712yes—
169C22×C8C23×C1716yes—
170C4×Q8C2×C4×C17112yes—
171C4×Q8C2×C4×C17124yes—
172C4×Q8C2×C4×C1714yes—
173C4×Q8C2×C4×C1718yes—
174C4×Q8C23×C1714yes—
175C4×Q8C23×C17112yes—
176C2×C4⋊C4C2×C4×C1716yes—
177C2×C4⋊C4C2×C4×C17124yes—
178C2×C4⋊C4C2×C4×C1713yes—
179C2×C4⋊C4C2×C4×C1713yes—
180C2×C42C2×C4×C1713yes—
181C2×C42C2×C4×C17112yes—
182C2×C42C2×C4×C1713yes—
183C2×C42C23×C1713yes—
184C2×C42C23×C1713yes—
185C2×C42C23×C17112yes—
186C4⋊C8C23×C17124yes—
187C4⋊C8C23×C17124yes—
188C22⋊C8C23×C1714yes—
189C22⋊C8C23×C17112yes—
190#97C23×C17124yes—
191C4×C8C23×C17124yes—
192C17C2×#298411yescharacteristic, direct product factor, Sylow
193C24C22×C4×C1711yescharacteristic
194C2×Q8—232no—
195C2×Q8—212no—
196C2×Q8M16×C1718yes—
197C2×Q8C22×C4×C17112yes—
198C2×Q8C22×C4×C1714yes—
199C22×C4—212no—
200C22×C4—212no—
201C22×C4—24no—
202C22×C4C22×C4×C1711yescharacteristic
203C22×C4C22×C4×C1712yes—
204C22×C4C22×C4×C17112yes—
205C22×C4C22×C4×C1713yes—
206C22×C4C22×C4×C1713yes—
207C22×C4C22×C4×C1713yes—
208C22×C4C22×C4×C17112yes—
209C22×C4C22×C4×C1713yes—
210C22×C4C24×C1712yes—
211C22×C4C24×C1711yescharacteristic
212C2×C8—24no—
213C2×C8—212no—
214C2×C8C24×C1714yes—
215C2×C8C24×C17112yes—
216C4⋊C4C22×C4×C17124yes—
217C4⋊C4C22×C4×C17112yes—
218C4⋊C4C22×C4×C17112yes—
219C42C22×C4×C17112yes—
220C42C22×C4×C17112yes—
221C42C22×C4×C1716yes—
222C42C22×C4×C1716yes—
223C42C24×C1716yes—
224C42C24×C1716yes—
225C23—22no—
226C23—22no—
227C23C17×#13114yes—
228C23C23×C4×C1712yes—
229C23C23×C4×C1711yescharacteristic
230Q8—224no—
231Q8C2×M16×C17116yes—
232C2×C4—22no—
233C2×C4—26no—
234C2×C4—224no—
235C2×C4—26no—
236C2×C4—26no—
237C2×C4—24no—
238C2×C4—224no—
239C2×C4—24no—
240C2×C4C2×M16×C17112yes—
241C2×C4C23×C4×C1712yes—
242C2×C4C23×C4×C1716yes—
243C2×C4C23×C4×C1716yes—
244C2×C4C23×C4×C1716yes—
245C2×C4C23×C4×C1711yescharacteristic
246C2×C4C23×C4×C1716yes—
247C2×C4C17×2−1+414yes—
248C2×C4C25×C1711yescharacteristic
249C8—24no—
250C8—212no—
251C22—28no—
252C22—22no—
253C22—22no—
254C22C17×#51912yes—
255C22C22×M16×C1712yes—
256C22C2×C17×#13114yes—
257C22C2×C17×#13112yes—
258C22C24×C4×C1711yescharacteristic
259C4—26no—
260C4—26no—
261C4—26no—
262C4—22no—
263C4C17×#51912yes—
264C4C22×M16×C17112yes—
265C4C2×C17×2−1+412yes—
266C2—24no—
267C2C17×#298414yesdirect product factor
268C2C2×C17×#51911yescharacteristic
269C2C23×M16×C1711yescharacteristic
270C2C22×C17×#13111yescharacteristic
subgroupG/H
centerC22×C4×C17C24
commutatorC22C24×C4×C17
Frattini, cosocleC2×C4C25×C17
socleC23×C17C23×C4
prime radicalC24×C17C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#2984C2direct (N × H)
3. C2×C17×#446C2semidirect
4. C2×C8×Q8×C17C2semidirect
16. C17×#446C22semidirect
17. C8×Q8×C17C22semidirect
89. C2×#2984C17direct (N × H)
117. #2984C2×C17direct (N × H)
118. C2×#446C2×C17semidirect
119. C2×C8×Q8C2×C17semidirect
146. #446C22×C17semidirect
147. C8×Q8C22×C17semidirect
157. C2×C17#2984direct (N × H)
192. C17C2×#2984direct (N × H)
267. C2C17×#2984direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Q8×C172
C17×#29842
C2×C17×#4462
C2×C8×Q8×C172
C2×C17×#4322
C2×C17×#4312
C2×#298417